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HALL'S MATHEMATICAL SERIES COMPLETE /KRITAMETIC BY WERNER SCMOOLBOOKCOMPAMY SiDUCATiOM.AU PUBUSHESS NEW YQRK; CMiCASO - BOSTO?! _ i iiii[i i iii i i i ii i i i i i iiwiw i i i i i ii iii < i ii i "iiiii iiiiii<iiiiiiiiiiiiiii'iii i iii it iiiiiii n ii ^^^^^'-^ 5<!C DE No. 3 IN i AMERICAN ^ ,. ..u, A. F. GUNN, L^i^nx A^'t, 204 PINE S^-^-^TT, SAN FRAN.._...^.. Digitized by tine Internet Archive in 2008 witii funding from IVIicrosoft Corporation http://www.arcliive.org/details/completearitlimetOOIiallricli HALL'S MATHEMATICAL SERIES COMPLETE ARITHMETIC ORAL AND WRITTEN BY FRANK H. HALL AUTHOR OF "THE WERNER ARITHMETICS," "THE ARITHMETIC READERS,' ETC WERNER SCHOOL BOOK COMPANY NEW YORK CHICAGO BOSTON HALL'S MATHEMATICAL SERIES THE WERNER ARITHMETICS A Three-Book Course for Graded Schools Book I. For third and fourth grades, cloth, 256 pages, 400. Book II. For fifth and sixth grades, cloth, 288 pages, 40c. Book III. For seventh and eighth grades, cloth, 288 pages, 50c. TEACHERS' HAND BOOK giving oral work preparatory for Book I, suggestions to teach- ers who are using the Werner Arithmetics, answers to proDiems in Books II and III, and a large amount of supplementary seat- work. Cloth, 131 pages, 25c. THE HALL ARITHMETICS A Two- Book Course for Graded or Ungraded Schools Hall's Elementary Arithmetic, cloth, 248 pages, - 35c. Hall's Complete Arithmetic, cloth, 448 pages, - 60c. IW ■~'», Copyright, 1899 By WERNER SCHOOL BOOK COMPANY Typography by R. R. Donnelly Sons Co.. Chicago ^ PREFATORY NOTE. In Part I. of this book (pp. 11-149), classification is made subordinate to gradation. Every problem is selected, not with reference to the place it occupies in a scientific classifica- tion of mathematical topics, but rather with reference to the supposed thought-power of the pupil. But systematic arrange- ment of the matter presented is not ignored. Seven topics are treated. These appear on the first ten pages of the book, and each topic is re-presented in each ten-page group. Com- pare pages 15, 25, 35, 45, etc.; pages 17, 27, 37, 47, etc. In Part II. (pp. 151-369), as in Part I., each page is a unit of the greater ten-page unit. The first six pages of every ten-page group are devoted to some general topic. Upon the seventh and eighth pages the algebraic phase of this topic appears; upon the ninth, elementary work in geometry, and upon the tenth, miscellaneous problems. This arrangement makes the book convenient for reference and review, and, it is believed, will greatly aid the pupil in properly correlating his own mathematical knowledge. In Part III. (pp. 371-442), the matter is arranged under four general heads, viz.: Denominate Numbers, Short Meth- ods in Multiplication and Division, Practical Approximations, and Miscellaneous Problems. With the denominate number tables, many practical problems in measurements are pre- sented. The "short methods" are, for pedagogical reasons, placed near the close of the book. The miscellaneous prob- lems include many typical sets of "examination questions" supplied to the author for this use. Altogether, Part III. provides for a complete and thorough application of the prin- ciples presented in Parts I. and II. F. H. H. Jacksonville, IllixNois, Marc h, 1899. CONTENTS — PART I. "The Foundation," Suggestions to Teachers, Simple Numbers, - Common Fractions, Decimals, Denominate Numbers, Measurements, Ratio and Proportion, Percentage, Percentage, Review Problems, Miscellaneous Problems, - 5-9 10 11, 21, 31, 41, 51, 61, etc. 12, 22, 32, 42, 52, 62, etc. 13, 2.^, 33, 43, 53, 63, etc. 14, 24, 34, 44, 54, 64, etc. 15, 25, 35, 45, 55, 65, etc. 16, 26, 36, 46, 56, 66, etc. 17, 27, 37, 47, 57, 67, etc. 18, 28, 38, 48, 58, 68, etc. 19, 29, 39, 49, 59, 69, etc. 20, 30, 40, 50, 60, 70, etc THE FOUNDATION. To THE Pupil. — Read each problem and (a) tell its meaning, (b; solve it, and (c) tell the suggested number story. Do this until you can easily give the meaning of all problems similar to these, solve them, and tell the suggested number stories without reference to the notes that follow. 8^ + 2^ = * ^- .^ = # 6^X2 = ^ f + }=(t^i. f-i \ = « ) f bu. X 7 = (3) 6f + 5J=V) 6i- 5i = (0 12^X|=(8) 2 bu. 3 pk. 5bu. 1 pk. $|XJ=(13) 2 bu. 2 pk. Ibu. 3 pk. 12^X2J=06) (") (.2) 12jx3 = (i») 12^X21. = (22) 6^ -- 2<Z! = * 6^ -^ 2 = * 4-J = W \ bu. ^ 4 = (5) l-i = (») 6| «yd.-2 = (,o) 7^^2i=(.4 ) 5J tons -5- 2 = (15) 2 ft.)46 ft. (, 1 2)46 ft. 2jft.)15ft. (20) (18) 2} bu.)13i bu. 3)27 1 ft. (23) (21) 1. I plus J, means, \ and \, I can change thirds and fourths to twelfths; | — twelfths; \ = twelfths; twelfths and twelfths = twelfths. Story— Harry played ball | of an hour, and ^^hide and seek'' J of an hour; in all he played twelfths of an hour. (See First Book, pages 125, 135, 145, 147, 149.) 2. I minus ^, means, f less J. I can change fourths and thirds to twelfths. I = twelfths; y= twelfths; twelfths less * See Elementary Arithmetic, pp. 89 and 153. 5 g THE FOUNDATION. ■-. — twelfths = twelfths. Story —William had a piece of wire I of a fcor. long; he cut from it apiece \ of a foot long; what re- mained was twelfths of a foot long. 3. I of a bushel multiplied by 7, means, take 7 times three fourth- bushels; 7 times 3 fourth-bushels = fourth-bushels, or and bushels. Story— A farmer fed to his horses | hu. of oats each day for a week ; in all he fed and bushels. 4. 4 divided by J, means, find how many times 1 third is contained in 4. I can change 4 to thirds. 4 is 12 thirds. 1 third is contained in 12 thirds times. Story — The teacher divided 4 oranges among some boys, giving to each ^ of an orange ; there were boys. 5. i of a bushel divided by 4, means, find 1 fourth of J of a bushel. 1 fourth of ^ of a bushel = of a bushel. Story — A boy divided \ of a bushel of oats equally among 4 ponies ; each pony received of a bushel. 6. 6| plus 5|, means, 6| and 5|. 6| and 5| = and . Story — In one jar there ivere 6^ lb- of butter ; in another there were 5^ lb. ; in both there were and • pounds. 7. 6| minus 5^, means, 6| less 5J. 6| less b^ - - — and -. Story — A grocer had 6^ lb. cheese, from ivhich he sold 5^ lb. ; there remained and pounds. 8. 12^ multiplied by |, means, take | of 12^. 3 fourths of 12^ = cents. Story — At 12f a pound, I of a pound of meat costs cents. 9. I divided by ^, means, find how many times 1 sixth is contained in 2 thirds. I can change thirds to sixths; 2 thirds = sixths; 1 sixth is contained in 4 sixths times. Story — Mrs. Smith divided ^ of a pie am,ong some boys, giving to each J of a pie ; there were boys. 10. 6| yd. divided by 2, means, find 1 half of 6| yards. 1 half of 6| yards = and yards. Story— A salesman cut 6'| yd. of ribbon into 2 equal pieces ; there were and yards in each piece. 11. 2 bu, 3 pk. plus 2 bu. 2 pk., means, 2 bu. 3 pk. and 2 bu. 2 pk. 2 bu. 3 pk. and 2 bu. 2 pk. = bu. pk. Story —A dealer put 2 bu. 3 pk. of oats into one bag and 2 bu. 2 pk. into another bag; in both there were bushels peck. 12. 5 bu. 1 pk. minus 1 bu. 3 pk., means, 5 bu. 1 pk. less 1 bu. 3 pk. 5 bu. 1 pk. less 1 bu. 3 pk. = bu. pk. Story — Mr. Bean had 5 bu. 1 pk. of oats, from which he fed 1 bu. 3 pk. ; there remained bushels pecks. 13. $1 multiplied by J, means, take 1 half of 3 fifths of a dollar. 1 half of 3 fifths of a dollar = of a dollar. Story— At | of a dollar a yard, 1 half of a yard of ribbon is worth of a dollar. THE FOUNDATION. 7 14. 7^ divided by 2|, means, find how many times 2 J is contained in 7|. I can change 2| and 7| to halves; 2| = halves; 7| = halves. halves are contained in halves times. Story — A farmer put 7J bushels of oats into hags, putting 2^ bushels in each bag ; there were bags. 15. 5f tons divided by 2, means, find 1 half of 5| tons. 1 half of 5f tons = and tons. Story — 5| tons of coal ivere divided equally between two families ; each family received and tons. 16. 12"? multiplied by 2|, means, take 2 times 12<? plus 3 fourths of 12^ (2| times 12?). 2 times 12«* plus 3 fourths of 12? = cents. Story — At 12^ a pound 2% lb. of cheese cost cents. 17. 46 ft. divided by 2 ft., means, find how many times 2 ft. are contained in 46 ft. 2 ft. are contained in 46 ft. times. Story — A mechanic cut 46 ft. of moulding into pieces, each piece being 2 ft. long ; there were pieces. 18. 46 ft. divided by 2, means, find 1 half of 46 ft. 1 half of 46 ft. = ft. Story— Henry divided 46 ft. of wire into 2 equal parts ; each part was feet long. 19. 12^ multiplied by 3, nieans, take 3 times 12\ (3 times \, plus 3 times 12). 3 times 12 ^ - . Story — One side of a triangle having equal sides is 12\ ft. ; the perimeter of the triangle is and feet. 20. 15 feet divided by 2| feet, means, find how many times 2| feet are contained in 15 feet. I can change 15 ft. and 1\ ft. to half-feet; 15 ft. = half- feet; 2| feet = half- feet; half- feet are contained in half-feet times. Story — A mechanic had 15 feet of moulding which he cut into pieces, each piece being 2^ ft. long ; there were pieces. 21. 27 1 feet divided by 3, means, find 1 third of 27^ feet. 1 third of 27^ feet = and feet. Story —The perimeter of a triangle having equal sides is 27^ feet; each side is and feet. 22. 12| multiplied by 2|, means, take 2 times 12^, plus J of 12^. 2 times 12^, plus i of 12 J = and . Story— At $12\ per ton, 2^ tons of hay cost and dollars. . 23. 13| bu. divided by 2^ bu., means, find how many times 2^ bu. are contained in 13| bu. I can change 2^ and 13^ to fourths. 2^ = fourths. 13^ = fourths. fourths are contained in fourths times. Story — A farmer had 131 bu. of oats which he put into bags, putting 2\ bu. in each bag. There ivere bags. THE FOUNDATION. .6 + .2 = * .6 - .2 = * .6X2=:* .24 + .05 = (i) . .64 - .05 =: (2) .12 x 9 == (3) .24 + .5 = (e) .64 - .5 = (7) 20 x -2 == (s) 176.4 276.4 20x3.2 = (13) 148.75 148.23 (11) (12) .1 X.l = (i6) .3x.2 = (i9) .6^.2 = * .6^2 = * .08 ^ .02 = (4) $.08 ^ 2 = (5) 8 ^ .4 = (9) , $6.36 -^ 3 = (1 0) 4 -^ .05 = (1 4) 5.6 tons ^ 4 = (1 5) 2.5 -^ .05 = (1 1) 5)$24.5 (1 s) $.5 )$24.5 (2o) 5)$8.2 (21) 1. .24 plus .05, means, 24 hundredths and 5 hundredths. .24 and .05 = hundredths. Story— William had $.24; he earned 4-05; he then had . 2. .64 minus .05, means, 64 hundredths less 5 hundredths. .64 less .05 = hundredths. Story—Martha had $.64; she spent $.05; she then had . 3. .12 multiplied by 9, means, take 9 times 12 hundredths. 9 times .12 = hundredths, or and hundredths. 4. .08 divided by .02, means, find how many times 2 hundredths are contained in 8 hundredths. .02 are contained in .08 times. Story — I paid $.08 for oranges at $.02 each ; I bought oranges. 5. $.08 divided by 2, means, find 1 half of 8 hundredths of a dollar. One half of $.08 = , Story — I paid $.08 for 2 lemons ; one lemon cost . 6. .24 plus .5, means, 24 hundredths and 5 tenths. 5 tenths =: 50 hundredths. .24 and .50 = hundredths. Story — John had $.24; Alfred had $.5; together they had . 7. .64 minus .5, means, 64 hundredths less 5 tenths. 5 tenths =; 50 hundredths. .64 less .50 = hundredths. Story — Sarah had $.64 ; Mary had $.5 ; Sarah had $ more than Mary. * See Elementary Arithmetic, pp. 153 and 155. THE FOUNDATION. 9 8. 20 multiplied by .2, means, find 2 tenths of 20. One tenth of 20 = ; 2 tenths of 20 = . Story— At $20 an acre, .2 of an acre of land would cost dollars. 9. 8 divided by .4, means, find how many times 4 tenths are con- tained in 8. 8 = 80 tenths. 4 tenths are contained in 80 tenths times. Story — I paid $8 for potatoes at $A {4 dimes) a bushel; I bought bushels. 10. $6.36 divided by 3, means, find 1 third of $6.36. One third of $6.36 = . Story— I paid $6.36 for 3 barrels of apples; 1 barrel cost . 11. The sum of 176.4 and 148.75 is . 12. The difference of 276.4 and 148.23 is . 13. 20 multiplied 3.2, means, take 3 times 20, plus 2 tenths of 20. 3 times 20 = . 2 tenths of 20 = . 3 times 20, plus 2 tenths of 20 = . Story — At $20 an acre, 3.2 acres of land are worth dollars. 14. 4 divided by .05, means, find how many times 5 hundredths are contained in 4. 4 = 400 hundredths. 5 hundredths are con- tained in 400 hundredths times. Story — I paid $4 for tablets at $.05 {5f) each; I bought tablets. 15. 5.6 tons divided by 4, means, find 1 fourth of 5.6 tons. One fourth of 5.6 = and tons. Story — A farmer sold 4 loads of hay, the entire weight of the hay was 5.6 tons ; the loads averaged and tons. 16. .1 multiplied by .1, means, find 1 tenth of 1 tenth. One tenth of 1 tenth = . 17. 2.5 divided by .05, means, find how many times 5 hundredths are contained in 2.5. 2.5 = 250 hundredths. 5 hundredths are con- tained in 2.5 (250 hundredths) times. Story — I paid $2.5 for pencils at $.05 (5^) each; I bought pencils. 18. $24.5 divided by 5, means, find 1 fifth of $24.5. One fifth of $24.5 = . Story — I paid $24.5 for 5 tons of coal; 1 ton cost 19. .3 multiplied by .2, means, find 2 tenths of 3 tenths. One tenth of 1 tenth = . One tenth of 3 tenths = -. Two tenths of 3 tenths = . 20. $24.5 divided by $.5, means, find how many times 5 tenths of a dollar are contained in $24.5 (245 tenths dollars). 5 tenths are con- tained in 24.5 (245 tenths) times. Stoi^—I paid $24.5 for apples at $.5 (5 dimes) a bushel; I bought bushels. 21. $8.2 divided by 5, means, find 1 fifth of $8.2. One fifth of $8.2 ($8.20) = . Story— I paid $8.2 for 5 yards of cloth; 1 yard cost . SUGGESTIONS TO TEACHERS. ORDER OF PROCEDURE IN PART I. Step 1. — Prepare the pupil by means of oral instruction for the work of a given page. This preparation may be, in part, the slow reading to the pupil of the figure problems'^ upon the page, the teacher hesitating at each blank for the pupil to supply the word. In this preparatory work, no book should be in the hands of the pupil. New words should be first presented through the voice of the teacher in the expression of thought. They may then be written upon the blackboard by the teacher, erased, and the pupil exercised in both oral and written reproduction. Step 2. — The book should be put into the hands of the pupil, and he should read, first silently, then orally, the figure problems upon any page for which proper preparation has been made. Step 3. — The pupil may attempt the letter problems* at his desk without further assistance. If he is unable to solve these, review the figure problems and give others similar to them. In some in- stances it may be well to have the figure problems solved upon the blackboard as preparation for the letter problems. 2. Wherever the word ''story" follows a problem, as on pp. 12 and 13, require the pupil to make a statement showing how the problem might originate in business or other experience; thus, the story for problem (f), page 12, might be, A man divided 4| acres of land into lots each containing ^ of an acre. There were 95 lots. 3. The separatrix, as an aid in " pointing off " in multiplication and division of decimals, is mentioned in a note on page 133. Its use is illustrated on page 143. Its occasional use in blackboard work earlier in the course may be helpful; but a too early reliance upon formal rules is to be avoided if the purpose of the work is the development of the thought-power of the pupil. * The figure problems are those designated by figures ; the letter problans, those designated by letters. 10 ARITHMETIC, PART I. SIMPLE NUMBERS. 1. Three is an exact divisor of 6, of 9, of , of . of , and of . Four is an exact divisor of . 2. Eighteen is exactly divisible by 2, by , by , and by . 28 is exactly divisible by . 3. Two is an exact divisor of 4, 6, , , , etc. 4. A number that is exactly divisible by 2 is called an even number; 8, 10, , , , are even numbers. 5. Two is not an exact divisor of 1, 3, 5, , , etc. 6. A number that is not exactly divisible by 2 is called an odd number; 9, 11, , , , , are odd num- bers. 7. 37 is an number. 24 is an number. 8. Tell which of the following are odd numbers and which are even numbers: 375, 256, 320, 197, 281, 378, 584, 252, 323, 569. (a) Copy the numbers given in problem 8 and find their sum.* (b) Find the sum of all the odd numbers from 1 to 15 inclusive. (c) Find the sum of all the even numbers from 2 to 16 inclusive. * Problems designated by letters are for the slate. 11 12 COMPLETE ARITHMETIC. FRACTIONS. 1. When two or more fractions have denominators that are alike, they are said to have a common denominator. ^1^, ^, ^5^, and y^g- have a . f and f do not have Reduce to equivalent fractions having a common denominator. 2. h h i' (Change to 12ths.) i= i= ^ = 3. h h h (Change to ) ^ = i= ^ = 4. }, |. (Change to ) | = \ = 5. ^, 4- (Change to ) | = | = (a) f and f (b) f and f (c) | and |. I can change eighths and thirds to ths. f = ^T- i = YT' twenty-fourths and twenty-fourths = twenty-fourths. (d) Find the sum of 576|- and 239i. 7. From 4 subtract 1 I can change sevenths and thirds to sts. i =YT' i = YT- twenty-firsts less twenty-firsts = twenty-firsts. (e) Find the difference of 572f and 368i 8. Divide |- by -^^. This means, find how many times ■gig- is contained in |-. I can change |- to twentietlis. 3. = .^^, 1 twentieth is contained in twentieths times. Story* (f) Find the quotient of 4|- divided by ^q. (Change 4| to twentieths.) Story. ^ (g)147f + 38|. (h)374f-18H. (i) 27ibu. -^ 2^bu4 * See Note (9) page 6. t See paragraph 2, page 10. I Tell the meaning. Change 27J and 2\ to lialves. Story. See note (14) page 7. PART I. 13 DECIMALS — THOUSANDTHS. 1. .125 and .006 are thousandths. 2. .125 and .06 are thousandths. 3. .125 and .6 are thousandths. 4. .275 less .006 are thousandths. 5. .275 less .06 are thousandths. 6. 2.4 and 3.104 are and thousandths. (a) Add 27.006, 14.8, and 25.06. 7. 36.025 less 1.008 equals and thousandths. (b) From 8.175 subtract 3.023. 8. 3 times 2.005 is and thousandths. (c) Multiply 26.008 by 4. 9. Divide .035 by .007. This means 4* . (d) Find the quotient of .375 divided by .005. Story.^ 10. Divide .035 by 7. This means 5* — -. (e) Find the quotient of 35.049 acres divided by 7. Story. 11. At $.05 each, for $1.25 I can buy pencils. 1.25 -^.05 = 12. At $.05 each, for $2 I can buy pencils. 2 - .05 = (f) (g) (h) (i) (J) Add. Subtract. Multiply. Divide. Divide. 46.27 54.75 7.054:i: $.00 5)$.385 § 5)$.385 § 25.308 8.326 5 ♦These figures refer to notes on page 8. t See page 10, paragraph 2. X Write the decimal point in the product immediately after writing the tenths' figure of the product. §" Point off" in division of decimals by thinking what the problems mean. Problem (i) means, find how many times 5 thousandths arc contained in 385 thousandths. 5 thousandths are contained in 383 thousandths 77 times. Problem (j) means, ^wd 1 fifth of 385 thousandihs. J of 385 thousandths is 77 thousandths, or .077. 14 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. 2000 lb. = 1 ton. 1. 500 lb. are of a ton. 2. 1600 lb. are of a ton. 3. 1000 lb. are of a ton. 4. 200 lb. are of a ton. 6. I weigh pounds. 6. My father's horse weighs pounds. 7. A keg of nails weighs 100 pounds. 8. One ton of nails is kegs. 9. A load of hay weighs from to pounds.* 10. A load of coal weighs from to pounds.* 11. When hay is $12 a ton, 1000 lb. cost . 12. When coal is $8 a ton, 1500 lb. cost ■ . 13. When meal is $16 a ton, 500 lb. cost . 14. When hay is $6 a ton, 200 lb. cost — . 15. When hay is $7 a ton, 200 lb. cost . 16. When hay is $5 a ton, 200 lb. cost . 17. When hay is $8 a ton, 200 lb. cost . 18. When hay is $6 a ton, 2200 lb. cost . 19. Wlien hay is $8 a ton, 2200 lb. cost . 20. When hay is $5 a ton, 2200 lb. cost . 21. 2500 lb. are and tons. 22. 3000 lb. are and tons. 23. 4500 lb. are and tons. 24. 5000 lb. are and tons. 25. When hay is $12 a ton, 2500 lb. cost . 26. When hay is $10 a ton, 3000 lb. cost . 27. When hay is $8 a ton, 4500 lb. cost . ♦When you see a load of hay or a load of coal, courteously ask the man in charge of it to tell you how much it weighs. PART 1. MEASUREMENTS. 15 1-inch cube. 1. A 2-inch square contains square inches. 2. A 2 -inch cube contains cubic inches. 3. A 3-inch square contains square inches. 4. A 3 -inch cube contains cubic inches. 5. A 2 -foot square contains 6. A 2 -foot cube contains - 7. A 3-foot square contains 8. A 3-foot cube contains - - square feet, cubic feet. - square feet, cubic feet. 9. The area of a 2 -inch square is 10. The soUd content of a 2 -inch cube is 11. The area of a 3-inch square is 12. The solid content of a 3-inch cube is inches. inches. 13. The area of a 2-foot square is feet. 14. The soHd content of a 2-foot cube is . 15. Find the area of — Find the sohd content of — * (1) A 4-inch square. (a) A 4-inch cube. (2) A 5 -inch square. (3) A 6 -inch square. (4) A 12-inch square. (b) A 5 -inch cube. (c) A 6 -inch cube. (d) A 1 2 -inch cube. ♦Problems designated by letters are for the slate. 16 COMPLETE ARITHMETIC. RATIO AND PROPOKTION. 1. One fourth of 20 is . 20 is ^ of — 2. One half of 11 is . 11 is i of — 3. One third of 10 is . 10 is i of — 4. One fifth of 45 is . 45 is i of — 5. Two thirds of 60 are . 60 is -| of — 6. Two thirds of 36 are . 36 is | of — 7. Two thirds of 15 are . 15 is | of — 8. Two thirds of 21 are . 21 is | of — 9. 10 is of 15. 15 is 1 of 10, or— times 10. 10. 15 is of 20. 20 is of 15, or — times 15. 11. 20 is of 25. 25 is of 20, or — times 20. 12. 25 is of 30. 30 is of 25, or — times 25. 13. Three fourths of 16 are 2 thirds of — . 14. Two thirds of 15 are 1 half of . 15. Three fourths of 24 are 2 thirds of 16. Two thirds of 12 are 1 third of — 17. Eight is of 12. A man can earn as much in 8 days as he can earn in 12 days. If he can earn $21 in 12 days, in 8 days he can earn dollars. 18. Fifteen is and times 6. A man can earn times as much in 15 days as he can earn in 6 days. If he can earn $10 in 6 days, in 15 days he can earn dollars. (a) If a man can earn $104 in 6 weeks, how much can he earn in 15 weeks. *It may be well, after the pupil has solved " mentally " such problems as Nos. 5 and 6, to require him to solve them with a pencil— this, as a preparation for similar problems in which larger numbers are used. t Allow the pupil to say, " 15 is 3 halves of 10," but remind him that the expres- sion "3 halves of 10," means, 5 times 1 haJfoflO. PART I. 17 PERCENTAGE* 50 per cent = .50 = ^. 25 per cent = .25 = ^. 33iper cent = .33L = -J. 20 per cent = .20 = f 1. 50 per cent of 10 = 10 is 50% of • . 2. 25 per cent of 12 = 12 is 25% of . 3. 33i per cent of 12 == 12 is 33^% of . 4. 20 per cent of 10 = 10 is 20% of . 5. 50 per cent of 8 = 8 is 50% of . 6. 25 per cent of 8 = 8 is 25% of . 7. 20 per cent of 15 = 15 is 20% of . 8. 33| per cent of 15 = 15 is 33^% of . (5)t (5)t 9. 5 is per cent of 10. 5 is % of 20. 10. 5 is per cent of 15. 5 is % of 25. 11. 4 is per cent of 8. 4 is % of 12. 12. 4 is per cent of 16. 4 is % of 20. 13. 3 is per cent of 6. 3 is % of 15. 14. Henry had 25 cents; he spent 20 per cent of his money; he spent cents. 15. Peter spent 10 cents; this was 25 per cent of all he earned; he earned cents. 16. Eoscoe earned 60 cents and spent 30 cents; he spent per cent of what he earned. *To THE Teacher. — Frequently review this page, and give many similar oral problems as a preparation for page 27. fThese figures designate the percentage cases to which the problems below them belong. 18 COMPLETE ARITHMETIC. PERCENTAGE. (1) 1. 50% of 18 is . 50% of 19 is - (a) Find 50 per cent of 724 ; (b) of 725. 2. 25% of 16 is . 25% of 17 is - (c) Find 25 per cent of 896 ; (d) of 897. 3. 33-1% of 18 is . 33i% of 19 is (e) Find 331- per cent of 726 ; (f) of 727. 4. 20% of 25 is . 20% of 26 is - (g) Find 20 per cent of 875 ; (h) of 877. (2) 5. 7 is 50% of . 7^ is 50% of . (i) 376 is 50% of what? (j) 376 1- is 50% of what? 6. 9 is 25% of . 9^- is 25% of . (k) 524 is 25% of what ? (1) 524i- is 25% of what ? 7. 8 is 33i% of . 8^ is 33^% of . (m) 652 is 33a% of what ? (n) 652 1- is 33^% of what ? (S) 11 is % of 33. 11 is % of 55. 30 is % of 60. 8. 11 is - % of 22. 9. 11 is - — % of 44. 10. 15 is - % of 30. (i) 11. One per cent (1%) of a number is 1 hundredth of the number ; 2 % is 2 hundredths ; 3 % is 3 hundredths. 12. One per cent of 500 is . 2% of 500 is . 13. One per cent of 600 is . 3% of 600 is . PART 1. 19 REVIEW. 1. Tell which of the following are odd numbers and which are even numbers: 14, 17, 19, 20, 24, 27. 2. The fractions y^y, -^-g, and \\ have a denomi- nator. 3. Change | and |- to equivalent fractions having a com- mon denominator. I can change thirds and fifths to . 2 _ 4 _ 3 - y - 4. Add -| and i. I can change 7ths and 3ds to . I = -g-j-. i = -g^. twenty-firsts and twenty- firsts = twenty-firsts. 5. Divide |- by ^-^. This means, find how many times ^ is contained in f . I can change 5ths to . | = ^-g-. One twentieth is contained in twentieths times. 6. Multiply 6.05 by 3. Three times . (a) Find the product of 34.02 multipHed by 4. 7. Divide $6.28 by 2. This means, 10* . (b) Find the quotient of $475.65 divided by 5. 8. When hay is $12 a ton, 1500 lb. cost dollars. 9. When coal is $7 a ton, 3000 lb. cost dollars. 10. A 2 -inch square is how many times as large as a 1-inch square ? 11. A 2 inch cube is how many times as large as a 1-inch cube ? (c) How many square inches in an oblong 9 inches by 23 inches ? 12. Three fourths of 20 are 1 half of . (d) Three fourths of 84 are 1 half of what number ? ♦See note 10, pag^ft 20 COMPLETE ARITHMETIC. MISCELLANEOUS. 1. Mr. A owned 15^ acres of land; he sold 6^ acres; ha then had acres. (a) Mr. B owned 546^ acres of land; he sold 228^ acres. How many acres had he remaining ? 2. A bushel of oats weighs 32 lb.; 2 bushels of oats weigh pounds; 3 bushels weigh pounds. (b) Sixty bushels of oats weigh how many pounds less than one ton ? 3. Two pounds of coffee @ $.23 a pound cost . (c) Find the cost of 84 pounds of coffee @ $.23 a pound.* 4. From a piece of cloth containing 12 yards there were sold 5|- yards and 2^ yards; there were left yards. (d) From a piece of cloth containing 55 yards, there were sold 24|- yd. and 17|- yd. How many yards were left? 5. A boarding-school uses one gallon of milk each day ; at 5^ a quart the milk for one week costs . (e) A boarding-school uses 4 gal. of milk each day; at 5^ a quart how much will the milk for the month of October cost ? 6. John bought 20 oranges at 3^ each and sold them at 6f each ; he gained cents. (f) John's father bought 25 bushels of potatoes at 32^ a bushel and sold them at 45^ a bushel. How much did he gain? *In multiplication of decimal fractions, require the pupil to locate the decimal point in the product immediaMy after he has written the figure to the right of the decimal point in any whole or pjirtial product. Thus, in the ^-23 multiplication of $.23 by 84 he writes (and thus locates) the decimal point of the product immediately after writing the figure 9 in the first partial pro- ^ duct. This will be quite clear to the pupil if he understands that he first — '- — finds 4 times .23; then 80 times .23; then the sum of these two partial 519'32 products. PART 1. 21 SIMPLE NUMBERS. 1. Seventeen is an integral number. 2. Five eighths is a fractional number. 3. Seventeen and five eighths is a mixed number. 4. 25 is an number. 5|- is a number 5. |-J- is a number. 3.5 is a ■ number. 6. .8 is a number. 18 is an number. 7. A number is exactly divisible by 2 if the right-hand figure is 0, 2, 4, 6, or 8. Tell which of the following are exactly divisible by 2 : 241, 136, 274, 393, 247, 826. 8. A number is exactly divisible by 5 if its right-hand figure is or 5. Tell which of the following are exactly divisible by 5 : 184, 275, 320, 145. 9. The following are exactly divisible by 2 : 16, 38, 54, 68, , , , . 10. The following are exactly divisible by 5 : 85, 140, 175, 180, , , , . 11. The following are exactly divisible by 10: 30, 40, 80, 120, , , , . 12. There are twice as many 5's as there are lO's in a number. 90 is tens. 90 is fives. (a) Divide $2150 by 50.* (b) Divide $215.00 by 50. (c) Divide $2295 by 51. (d) Divide $290.70 by 61. (e) Divide $2058 by 49. (f) Divide $156.80 by 49. (g) Divide $3276 by 52. (h) Divide $322.40 by 52. (i) Divide $2496 by 48. (j) Divide $254.40 by 48. * This means, find 1 fiftieth of §2150. 1 fiftieth of $2150 is dollars. Story- If 50 acres of land cost $2150, 1 acre cost . t In solving such problems as (b), (d), etc., require the pupil to write the deci- mal point in the quotient, immediately after writing the units' figure of the quotient If the pupil thinks what the problem means, he will easily " point off " correctly. 22 COMPLETE ARITHxAIETIC. COMMON FRACTIONS. Reduce to equivalent fractions having a common denominator : 1. I and ^. (Change to .) | = -i- = (a) I and f . (b) f and |. (c) f and -J. 2. Add I and f (Change to ths.)* (d) Find the sum of 248 1 and 467f 3. From |- subtract |. (Change to ths.)-|- (e) Find the difference of 837} and 284|. 4. Divide 6 by |. This means, find how many times ^ are contained in 6. I can change 6 to thirds. 6 = -g . 2 thirds are contained in thirds times. Story.-^ (f) Find the quotient of 48 divided by |. Story.^ 5. Divide 8|- feet by 2. This means, find 1 half of 8|- feet 1 half of 8^ feet ^ Story. (g) Find the quotient of 86^ miles divided by 2. Story. \^ (h) 3^ miles)154 miles. This means, find how many times 3\ miles are contained in 154 miles. (Change 3^ and 154 to halves.) Story — A canal-boat moved at the rate of 3^ miles an hour; to move 154 miles would require hours. (i) 3)1654- miles. This means, find 1 third of 1651 niiles. Story — A train moved 165^ miles in 3 hours ; it moved at the rate of miles an hour. * See page 12, problem 6. t See page 12, problem 7. t See note (4), page 6. g 48 feet of ribbon was cut into pieces 5 of a foot long ; there were pieces. I A train moved 86J miles in 2 hoiirs ; this was at the rate of miles an hour. PART I. 23 DECIMAL FRACTIONS. 1. One thousandth of a dollar is mill. 2. Four thousandths of a dollar are 4 . 3. Four hundredths of a dollar are 4 . 4. Four tenths of a dollar are 4 . 5. .1 of $60 = .1 of $4 = .1 of $64 =* 6. .1 of $.1 =. .1 of $.5 = .1 of $.7 = 7. .1 of $6 = .1 of $8.5 = .1 of $5.3 = 8. .1 of $.01 = .1 of $.03 = .1 of $.07 = 9. .1 of $6.42 = .1 of $7.56 = .1 of $8.42 := 10. .1 of $875 = .1 of $74.2 = .1 of $5.35 = (a) Multiply $374 by .3. This means. , find 3 tenths of 3 Operation. $374t .3 Explanation. One tenth of $374 = Three tenths of $374 = $112.2 NUMBER STORY. If one acre of land is worth $374, 1 tenth of an acre of land is worth . 3 tenths of an acre of land are worth . (b) (0) (d) (e) (f) Multiply. Multiply. Multiply. Divide. Divide 346 23.5 2.73 $.05)$.85$ 5)$.85§ .3 .3 3 " * By means of problems 5 to 10 and other similar exercises, lead the pupil to see that he can obtain 1 tenth of an integral number by "pointing off " one figure on the right, and of a mixed decimal by removing the point one place to the left. tTh(j pupil should understand that he multiplies S37.4 (not $374) by 3, and should be taught to write the decimal point in the product immediately after wnting the tenths' figure of the product. See page 10, paragraph 3. t This mesins, find how many times 5 hundredths are contained in 85 hundrcd'hs. I This means, ^nrf t fifth of 85 hundredths. 24 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. A bushel of oats weighs 32 lb. A bushel of wheat weighs 60 lb. A bu. of corn (shelled) weighs 56 lb. A bu. of corn (not shelled) weighs 70 lb.* 1. Two bushels of oats weigh lb.; 5 bu. weigh (a) How much more than 1 ton do 75 bu. of oats weigh 2. Two bu. of wheat weigh ; 3 bu. weigh . (b) Seventy-five bushels of wheat weigh how much more than two tons ? (c) How much less than three tons ? 3. Two bushels of shelled corn weigh ; 3 bu. weigh (d) Forty-five bushels of shelled corn weigh how much more than 1^ tons ? (e) How much less than 1^ tons ? 4. Two bushels of corn (not shelled) weigh ; 3 bushels weigh ; 4 bushels weigh . (f) How much more than three fourths of a ton do 25 bushels of corn (unshelled) weigh ? (g) How much less than one ton ? 5. One hundred pounds of oats are bushels and pounds, or and 32nds bushels. (h) Five hundred pounds of oats are how many bushels ? (i) How many bushels in one and one half tons of wheat ? (j) How many bushels in two tons of unshelled corn ? (k) (1) (m) (n) 32 lb.)768 lb. 32 lb.)769 lb. 32 lb.)770 lb. 32 lb.)772 lb. * Explain to the pupil that the expression " a bushel of corn (not shelled)," meau3, the amount 0/ unshelled com required to maHe I Ifushel 0/ shelled com. PART I. MEASUREMENTS. 25 ^ .^y / ^ i 1 1 1; I 1. A 3-foot square (or its equivalent) is called a square yard. It contains square feet. 2. A 3-foot cube (or its equivalent) is called a cubic yard. It contains cubic feet. 3. The area of a 4-foot square is square feet, or and square yards. or or 4. The solid content of a 4-foot cube is cubic feet, (a) How many cubic yards in a 4-foot cube ? 5. The area of a 5 -foot square is square feet. and square yards. 6. The solid content of a 5 -foot cube is cubic feet. (b) How many cubic yards in a 5 -foot cube ? The area of a - square yards. 6-foot square is square feet. cubic feet. 8. The solid content of a 6-foot cube is - (c) How many cubic yards in a 6-foot cube ? 9. Find the area of — Find the solid content of — (1) A 7-foot square. (d) A 7-foot cube. (2) An 8-foot square. (e) An 8-foot cube. (3) A 9-foot square. (f) A 9-foot cube, 26 COMPLETE ARITHMETIC. KATIO AND PROPORTION. 1. One fifth of 25 is . 25 is | of . 2. One fourth of 21 is . 21 is | of -. 3. One third of 22 is . 22 is | of . 4. One half of 45 is . 45 is i of . 5. Three fourths of 24 are . 24 is | of . 6. Three fourths of 36 are . 36 is |- of — — . (b) 96 is f of what number? (d) 96 is f of what number? (f) 96 is 1^ of what number? of 18. 18 is of 12. of 24. 24 is of 18. of 30. 30 is of 24. 10. Two thirds of 18 are 1 half of . 11. Three fourths of 24 are 2 thirds of . 12. Two thirds of 30 are 1 half of . (g) One third of 132 is 1 half of what number? (h) Two thirds of 252 are 1 half of what number? (i) Three fourths of 96 are 2 thirds of what number? (j) Two thirds of 96 are 3 fourths of what number? 13. Nine is of 12. A man can earn — as much in 9 days as he can earn in 12 days. If he (a) Findf of 96. (c) Find 1 of 96. (e) Find^ of 96. 7. 12 is - 8. 18 is - 9. 24 is - can earn $40 in 12 days, in 9 days he can earn dollars, (k) If a man can earn $896 in 12 months, how many dol- lars can he earn in 9 months ? (1) If Mr. Conrad's horses consume 726 bushels of oats in a year, how many bushels will be required to feed them 8 months ? PART I. 27 PERCENTAGE. 16| per cent = .161 = 14f per cent = .14| = 12^per cent = .121- = {1) {2) 1. 16f per cent of 12 = 12 is 16 1% of — 2. 14f per cent of 21 = 21 is 14f % of — 3. 12|- per cent of 16 = 16 is 12|-% of — 4. 16f per cent of 18 = 18 is 16|% of — 5. 14| per cent of 28 = 28 is 14f % of — 6. 12|- per cent of 24 = 24 is 12|-% of — 7. 50 per cent of 12 = 12 is 50% of . 8. 25 per cent of 16 = 16 is 25% of . 9. 20 per cent of 25 = 25 is 20% of . 10. 331- per cent of 18 = 18 is 33^% of - — . (5) (5) 11. 2 is per cent of 6. 2 is % of 4. 12. 2 is per cent of 8. 2 is % of 14. 13. 2 is per cent of 12. 2 is % of 10. 14. 2 is per cent of 16. 1 is % of 2. 15. There were 40 pears on a tree; 25% of them fell off; pears remained on the tree. 16. William sold 7 melons; these were 12|-% of all the melons h3 raised ; he raised melons. 17. Mary had 30 little chickens; a hawk killed five of them ; the hawk killed . per cent of her chickens. 18. Twelve and one half per cent of $48 is dollars (a) Find 12i- per cent of $992. (b) Find 16| per cent of $852. J8 COMPLETE ARITHMETIC. PERCENTAGE. (1) 1. 16|% of 24= 16f% of 25 = {Si) Find 16| per cent of 342 ; (b) of 343. 2. 12|-% of 24 = 121-% of 25 = (c) Find 12|- per cent of 976 ; (d) of 978. 3. 142% of 28 = 14f % of 30 = (e) Find 14| per cent of 994; (f) of 996. (2) 4. 5 is 121-% of . 5i is 12J% of . (g) 246 is 12|-% of what ? (h) 246^ is 12^-% of what? 5. 5 is 14f % of . 5J is 142% of . (i) 351 is 14f% of what? (j) 356| is 14f % of what? 6. 5 is 16f % of . 5^ is 16f % of . (k) 239 is 16f % of what? (1) 241^ is 16|% of what? (3) 7. 12 is % of 24. 12 is % of 36. 8. 12 is % of 48. 12 is % of 60. , 9. 12 is % of 72. 12 is % of 84. 10. 12 is % of 96. 15 is % of 45. (1) 11. One per cent of $300 is . 2% of $300 = 12. One per cent of $320 is . 2% of $320 = 13. One per cent of $325 is . 2% of $325 = 14. One per cent of $342 is . 2% of $342 = (m) Find 3% of $342. (n) Find 4% of $342. (o) Find 5% of $342. (p) Find 6% of $342. (q) Find 3% of $536. (r) Find 4% of $536. PART I. 29 REVIEW. 1. Two is an exact divisor of 28, 274, , . 2. Five is an exact divisor of 75, 230, , . 3. Ten is an exact divisor of 80, , , . 4. Twenty is an integral number. |- and .5 are numbers. 5\ and 3.2 are numbers. 5. The fractions -^^, -^q, , , have a common denominator. | and do not have a common denonii- liator. 6. The sum of .135 and .6 is . .135 - .06 = 7. Divide .045 by .009. (This means 4* .) (a) Find the quotient of .875 divided by .005. 8. Divide .045 by 9. (This means 5* .) (b) Find the quotient of 54.063 divided by 9. 9. When hay is $8 a ton, 2500 lb. cost . 10. Five bushels of wheat weigh pounds. (c) Sixty-five bushels of wheat weigh how much more than 65 bushels of oats ? 11. The area of a 10-inch square is ■ square inches. (d) How many square feet in a 15-foot square? (e) How many square yards in a 1 5-foot square ? 12. The solid content of a 10-inch cube is cu. in. (f) How many cubic feet in a 15-foot cube? (g) How many cubic yards in a 15-foot cube? 13. Three fourths of 60 are . 60 is f of . (h) Find I of 132. (i) 132 is f of what number? 14. Two thirds of 27 are 1 half of . (j) Two thirds of 132 are 1 half of what number? * These figures refer to notes on page 8. 30 COMPLETE ARITHMETIC. MISCELLANEOUS PROBLEMS. 1. Byron bought 6 melons for 54^ ; he sold them at 15^ each ; on each melon he gained cents. ' (a) A merchant bought 6 barrels of apples for $13.50, and sold them at $2.75 a barrel. How much did he gain on each barrel ? 2. James carried 4 dozen eggs to market ; he sold them for 12^ a dozen ; the dealer paid James for the eggs with sugar at 5^ a lb. ; James should receive and ^ pounds of sugar. (b) A farmer took four cords of wood to market ; he sold it for $6.75 a cord and took its value in coal at $6 a ton. How many tons of coal should he receive ? 3. At 5^ a quart, 1 gallon of milk is worth cents. 2|- gallons are worth cents. (c) At 5^ a quart, how much are 45|^ gallons of milk worth ? 4. Two and 1 half feet are inches. 4 weeks are days. (d) Forty-six and 1 half feet are how many inches ? (e) Fifty-two weeks are how many days ?* (f ) Seventy-five gallons are how many quarts ? 5. William had 30^- ; he spent | of his money for ink and pencils and the remainder for paper ; the paper cost cents. (g) William's father had $585 ; he spent f of his money for a carriage and liarness and the remainder for horses. How much did the horses cost ? * The pupil should see that in the solution of this problem he may take fifty-two 7's or seven 52's. If he adopts the latter method his thought may be— (Me day in each week would make 52 days, and seven days in each week would make 7 times 52 days; or he may think that fifty-two 7's are equal to seven 52's, and that, as a matter of convenience, he finds seven 52's. PART I. 31 SIMPLE NUMBERS. 1. A number that has no exact integral divisors except itself and 1, is called a prime number. 1, 2, 3, 5, 7, 11, , , , , , are prime numbers. 2. Integral numbers that are not prime are said to be composite. 4, 6, 8, 9. , , , , are composite numbers. 3. 24 is a number. 23 is a - 29 is a number. 26 is a - 28 is a number. 27 is a - 31 is a number. 37 is a - number, number, number, number. 4. If an integral number is expressed by two or more figures, and the right-hand figure is 5, the number is . 5. Every even number, except 2, is . 6. Some odd numbers are , and some are . 9 is . 11 is . 13 is . 21 is . (a) Write all the prime numbers from 1 to 53 inclusive, and find their sum. (b) (c) (d) (e) (f) Add. Subtract. Multiply. Divide. Divide. 346 5423 754J S52)$7436* 52)$7436t 275 1896| 26 142 879 (g) (h) (i) (J) 27 74624 846 53 bu.)2438 bu. 53)2438 bu. 624 1827 35|- * This means, find liow many times $52 are contained in $7436. $52 are contained in $7436, 143 times. Story— A man paid $71,36 for land at $52 an acre; there were lUS acres. t This means, find 1 fifty-second of $7436. 1 fifty-second of $7436 is $143. Story— A man paid $7it36 for 52 acres of land; one acre cost $1US. 32 COMPLETE ARITHMETIC. COMMON FRACTIONS. Reduce to their lowest terms : 112_ 12— 5— 8_ 8_ io__ ■*-Tcr— 3li — 3J— TiT— T8— TIT — (a)T'A= (b)TVtr= (c)tVV = Reduce to whole or mixed numbers : 2. j/= -v-= -v-= -V-= -V-= ¥= -V- = (d)i|i= (e)H^= (f)^«A = Reduce to equivalent fractions having a common denominator : 3. i, i, ^. (Change to .)* i = i = i = (§) i> f' I- W h h f- w i> h T- 4. Add ^, ^, and J. (Change to ths.) (j) Find the sum of 75|, 86|, and 47f 5. From -|- subtract i. (Change to ths.) (k) Find the difference of 946| and SSSf 6. Divide 7 by |. This means . Story.^ (1) Find the quotient of 36 divided by J. Story. % (m)2^ gallons)! 2 6 gallons. This means, ^nd how many '■ times 2\ gallons are qontained in 126 gal. (Change 2^ and 126 to fourths.) Story. ^ (n) 4)276^ gallons. This means, find 1 fourth of 2161 gal- Ions. Story — In 4 days Mr. Smith sold 276 jt gallons of milk ; this was at the rate of gallons per day. * The pupil is expected to find by trial that he can change halves, thirds, and eighths, to twenty-fourths. t See note (4) page 6, and problem 4, page 22. t Mr. Brown put 36 bushels of peaches into J-bushel baskets; there were — baskets. § Put 126 gal. of milk into 2i-gal. cans. PART 1. 33 DECIMAL FRACTIONS. 1. One tenth of $2.45 is . .2 of $2.45 = (a) Multiply $2.45 by 2.3. This means, find 2 times $2.45 plus 3 tenths of $2.45. Operation. Explanation. ^2.45 One tenth of $2.45 is . 2,3 Three tenths of |2.45 are -. Two times |2.45 are . $.735 + 14.90 = $5,635. $.735* $4.90 $5,635 NUMBER STORY. If 1 ton of coal is worth $2.45, 1 tenth of a ton of coal is worth 3 tenths of a ton of coal are worth . 2 tons of coal are worth . 2.3 tons of coal are worth . (b) Multiply $3.65 by 2.4.* (.1 of $3.65 is $.365.) (c) Multiply $52.8 by 3.2. (.1 of $52.8 is $5.28.) 2. I bought 7f yards of print at 6^ a yd. and gave the salesman half a dollar; I should receive in change . (d) I bought 5.3 tons of coal at $4.20 a ton and gave the salesman 3 10-dollar bills. How much change should I receive ? (e) (f) (g) (h) (i) Add. Subtract. Multiply. Divide. Divide. 64f 78 3.75 $. 5)$47.5t 5 )$47.5j: 28.37 24.42 2.6 * Write the decimal point in each partial product immediately after making tfie figure that represents the tenths in that product. t This means, ^?id how many times 5 tenths are contained in U75 tenths. % This means, find 1 fifth offU7.5. 34 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. 1. One ton is pounds. 2. One tenth of a ton is — 3. Three tenths of a ton are 4. Nine tenths of a ton are pounds. — pounds. — pounds. pounds. 6. Three hundredths of a ton are pounds. 7. Seven hundredths of a ton are pounds. 5. One hundredth of a ton is 8. One thousandth of a ton is — 9. Two thousandths of a ton are 10. Six thousandths of a ton are ■ 11. 2400 lb. = 1 and - 12. 2800 lb. = 1 and - 13. 3200 lb. = 1 and - 14. 2460 lb. = 1 and - 15. 2468 lb. = 1 and - pounds. — pounds. - pounds. tenths (1.2) tons, tenths ( ) tons, tenths ( ) tons. tons. tons. (a) Change 4870 lb. to tons, (c) Change 5260 lb. to tons, (e) Change 6480 lb. to tons. 16. 2 tons are lb. 17. 2.3 tons are lb. 18. .03 of a ton are lb. 19. 2.04 tons are lb. 20. 2.34 tons are lb. (b) 4980 lb. (d) 5750 lb. (f) 7260 lb. •3 of a ton are 2.4 tons are — lb. lb. .04 of a ton are 2.03 tons are lb. 3.23 tons are lb. lb (g) Change 3.52 tons to lb. (h) 4.37 tons, (i) Change 5.18 tons to lb. (j) 3.72 tons, (k) Change 1.48 tons to lb. (1) 2.324 tons, (m) At ^ a cent a pound, find the cost of 3.45 tons of scrap iron. PART I. 35 MEASUREMENTS. A / An \ An right angle / acute angle \ obttise angle 1. Wlien two lines meet at a point, they are said to form an angle. 2. When a jackknife is half way open, the handle and blade form a right angle. If it is less than half way open, they form an angle. If it is more than half way open, but not fully open, they form an angle. 3. A square D has 4 right angles. 4. An oblong en has right angles. 5. Rectangular means right-angled. 6. A rectangular figure, or rectangle, may be a square, or it may be an ohlong. 7. A rectangle 6 inches by 6 inches is a . 8. A rectangle 4 in. by 6 in. is an . 9. A rectangular surface 4 feet by 6 feet is an ; its area is square feet and its perimeter is feet. 10. A rectangular surface 8 feet by 8 feet is a ; its area is square feet ; its perimeter is ft. 11. Which has the greater area, a 5-in. square, or an oblong 4 in. by 6 in.? Compare their perimeters. Find the area and the perimeter of each of the following rectan- gular surfaces : (a) 23 feet by 6 feet. (b) 25 inches by 7 inches. (c) 15 yards by 8 yards. (d) 32 feet by 23 feet, (e) 30 inches by 20 inches. (f) 21 yards by 24 yards. 36 COMPLETE ARITHMETIC. RATIO AND PROPORTION. 1. One sixth of 12 is . 12 is -J of 2. One fifth of 12 is . 12 is i of 3. One fourth of 11 is . 11 is ^ of (a) Find 1 third of 275. (b) 275 is 1 third of what? (c) Find 1 half of 377. (d) 377 is 1 half of what ? 4. Two fifths of 30 are . 30 is f of . 5. Three fifths of 30 are . 30 is f of . (e) Find f of 90. (f) 90 is | of what number? (g) Find f of 90. (h) 90 is f of what number? 6. 14 is of 21. 21 is of 14. 7. 21 is . of 28. 28 is • of 21. 8. 28 is of 35. 35 is of 28. 9. Two fifths of 30 are 1 half of . 10. Three fifths of 30 are 2 thirds of . (i) Two fifths of 90 are 1 half of what number? (j) Three fifths of 90 are 2 thirds of what number? 11. Twelve is of 8, or and ■ times 8. A man can earn and times as much in 12 days as he can earn in 8 days. If he can earn $20 in 8 days, in 12 days he can earn . (k) If a man can earn $73 in 8 months, how many dollars can he earn in 12 months ? 12. If 8 lb. of sugar are worth 50^, 12 lb. are worth cents. (1) If 8 tons of coal are worth $34.20, how much are 12 tons worth ? (m) If 12 barrels of apples are worth $27.60, how much are 8 barrels worth ? PART I. 37 PERCENTAGE. 11-J- per cent = 10 per cent = (1) 1. ll^per cent of 27 = 2. 10 per cent of 20 = 3. 11| percent of 18 = 4. 10 per cent of 30 = 5. 50 per cent of 3 = 6. 25 per cent of 9 = 7. 33i per cent of 7 = 8. 20 percent of 11 = 9. 16f per cent of 19 = 10. 141 per cent of 15 = (5) 11. G is per cent of 24. 12. 6 is per cent of 12. 13. 3 is per cent of 24. 14. 3 is per cent of 18. .10 = 27 20 18 30 3 9 7 11 19 15 .1 ■7- 1 {2) \\\% of . 10% of . 11|% of . 10% of . 50% of . 25% of . 331-% of . 20% of . 16f % of . 142% of -^ — . (5) is % of 18. is % of 30. is % of 21. is % of 27. 15. Helen bought a piece of flannel that was 40 inches long; by washing it shrank 10% in length; after washing it was inches long. 16. Before washing, a piece of flannel was 40 inches in length; after washing it was 35 inches long; it shrank by washing per cent* 17. A dealer had 25 bu. of apples; he lost 20% of them by decay ; there remained bushels. (a) A dealer had 2375 bu. of apples; he lost 20% of them by decay. How many bushels remained ? * It shrank what part of its original length? 'SS COMPLETE ARITHMETIC. PERCENTAGE. 1. lli%of36= lli%of38 = (a) Find 11^ per cent of 1044; (b)of 1047. 2.10^0 of 50= 10% of 51= 10% of 52 = (c) Find 10% of 870 ; (d) of 874. 3. 7 is 11-1-% of . 7| is 11^% of . (e) 371 is lli% of what? (f) 371i is lli-% of what ? 4. 8 is 10% of . 81- is 10% of . (g) 89 is 10% of what ? (h) 89.2 is 10% of what ? (3) 12|- is % of 50. 6 is % of 54. 3J is % of 10. 5. 12 is ■ % of 24. 6. 6 is - — % of 60. 7. 3|- is - % of 7. 8. 36 is ■ % of 72. (1) 9. One per cent of $700 is 10. One per cent of $730 is 11. One per cent of $732 is (i) Find 3 % of $732. (k) Find 5 % of $732. (m) Find 7 % of $320. 72 is % of 144. 2% of $700 = 2% of $730 = 2% of $732 = (j) Find 4% of $732 (1) Find 6% of $732. (n) Find 7% of $326. 12. Two dollars are 1 hundredth of $200. 13. Two dollars are 1% of . $4 are 2% of 14. Six dollars are 3% of . $8 are 4% of 15. Five dollars are 1% of . $10 are 2% of PART I. 39 REVIEW. 1. All integral numbers are either prime or composite. 17 is . 31 is . 95 is . 242 is . 370 IS 2. Keduce each of the following improper fractions to a whole number or to a mixed number : -y-, ^^-, -y-, ^^-, \K (a) ^±. (b) ^K (c) ^5 6. (d) ^K 3. Eeduce each of the following to its lowest terms : 8 9 12 15 11 12 10 ^0" ^T ITT ¥6- 3T T¥ TUir (e)TVV (OtVV (g)Hf WtW 4. Eeduce the following to equivalent fractions having a common denominator: i>if (Change to .) ^= i= ^ = (i) h I h (3) h h h W I' h h 5. If ^ of a bushel of potatoes costs 20^, 2|- bushels will cost . (1) If ^ of a ton of straw costs $2.25, how much will 3.2 tons cost? 6. Change the following to tons : 4200 lb. (m) 5750 lb. (n) 7320 lb. (o) 3150 lb. 7. Find the area and the perimeter of each of the fol- lowing rectangular surfaces : 7 feet by 5 feet. (p) 27 feet by 12 feet. (q) 15 inches by 15 inches. 8. If 2 gallons of molasses are worth 6 Of/, 3 gallons are worth cents. (r) If 2 loads of brick are worth $12.60, how much are 3 loads worth ? 40 COMPLETE ARITHMETIC. MISCELLANEOUS PROBLEMS. 1. In a school there are 35 pupils ; ^ of the pupils are boys ; of the pupils are girls ; there are boys and girls. (a) In a school there are 392 pupils ; | of the pupils are boys. How many girls are in the school ? 2. If 3 melons cost 36^, at the same rate 5 melons will cost cents. (b) If 3 acres of land cost $525, how much will 5 acres cost at the same rate ? 3. Harry exchanged 5 lb. of butter at 20^ a pound for coffee at 25^ a pound; he should receive pounds of coffee. (c) Harry's father exchanged 6 cords of wood at $5.20 a cord for cedar posts at 20^ each. How many posts should he receive ? 4. 3 + 2 + 6+4 + 5 + 8+1 + 7 + 9 + 4 + 7 + 3 + 2 = (d) 275 + 361 + 554 + 732 + 598 + 236 + 347 + 256 = 5. If Mark saves $4 a month, in 1 year he will save dollars. (e) If Mark's father saves $21.50 a month, how much will he save in 1 year ? 6. A common brick is 8 inches long, 4 in. wide, and 2 in. thick ; it has two faces each of which is 4 in. by 8 in., two faces each of which is « in. by in., and two faces each of which is in. by in. (f) Find the sum of the areas of all the faces of a common brick. (g) Change 674 inches to feet and inches. PART I. 41 SIMPLE NUMBERS. 1. Any exact integral divisor of a number (except the number itself and 1) is called a factor of the number. The factors of 6 are and . The factors of 10 are and . 2. A factor that is itself a prime number is called a prime factor. A factor that is itself a composite number is called a composite factor. 2 and 3 are factors of 24. 4 and 6 are factors of 24. 3. Every composite number may be resolved into prime factors. The prime factors of 12 are 2, 2, and 3. The prime factors of 18 are 2, 3, and 3. The prime factors of 15 are • and . The prime factors of 14 are and . The prime factors of 30 are , , and . 4. From the above it will be seen that a number is equal to the product of its prime factors. 3 and 7 are the prime factors of ; 2, 2, and 7, of . 2)30 The prime factors of 2)50 The prime factors of 3)15 30 are 2, 3, and 5. 5)25 50 are 2, 5, and 5. ■ 5 2 X 3 X 5 = 30. 5 2 X 5 X 5 = 50. (a) What are the prime factors of 40 ? (b) Of 60 ? (c) Of 65 ? (d) Of 72 ? (e) Of 86 ? (f) Of 85 ? Multiply. Divide. Divide. (g) 724 by 28. (h) 748 lb. by 32 lb. (i) 1254 lb. by 9. (j) 846 by 23. (k) 834 lb. by 32 lb. (1) 1046 lb. by 9 (m) 637 by 25. (n) 928 lb. by 32 lb. (o) 1134 lb. by 9. (p) 926 by 27. (q) 796 lb. by 32 lb. (r) 1 341 lb. by 9. 39 _ 42 COMPLETE ARITHMETIC. COMMON FRACTIONS. Reduce to improper fractions : 1. 7i = y. 8| = ^. 9| = ^. 7|=,. (a) 27^= (b)48f= (c)75f = Reduce to whole or mixed numbers : Ol7_ 35— 39— 38— 45-- ^' --6- - S- - "¥" - ~3 " - ~V - (d)2 8A= (e)^^^ (^)H^ = Reduce to equivalent fractions having a common denominator : 3. i, i, i. (Change to .) ^= ^ = i = (g) h h T- W -6' T' I- © h h h 4. Add i, ^, and ^. (Change to .) (j) Find the sum of 86|, 95|, and 87f 5. From | subtract ^. (Change to .) (k) Find the difference of 873f and 249|. 6. Divide |- by -i. This unesins, find how many times ^ is contaiiud in |. I can change fourths and thirds to ths. I = y^. ^ = -YY- twelfths are contained in twelfths times. Story — James can husk a row of corn in i of an hour ; in f of an hour he can husk and rows. (l)i-^i. (m)l-i. (n)f-f (o) 5-1- acres)288 acres. This means, find how many times 5^ acres are contained in 288 acres. (p) 5 )6 2 5 1^ acres. This means, find 1 fifth of 625\ acres. Story. PART I. 43 DECIMAL FRACTIONS. 1. One hundredth of $500 is . .02 of $500 = 2. One hundredth of $540 is . .02 of $540 - 3. One hundredth of $542 is -. .02 of $542 -^ 4. .01 of $600= .01 of $60= .01 of 5. .01 of $.1 = .01 of $.5 = .01 of $.7 = 6. .01 of $6.4 = .01 of $7.5 = .01 of $3.2 = 7. .01 of $24.2 = .01 of $37.1 = .01 of $53.1 = (a) Multiply $374 by .03. This means, ^Tif? 3 hundredths of $374. Operation. Explanation. $374 * One hundredth of $374 is .03 Three hundredths of ^374 are $11.22 NUMBER STORY. If one acre of land is worth f 374, 1 hundredth of an acre of land is worth . 3 hundredths of an acre of land are worth (b) Multiply $347 by .03. (c) $537 x .04. (d) Multiply $24.6 by .03. (e) $39.4 x .04. (f) At $875 an acre, how much will .04 of an acre of land cost ? (g) (h) (i) (j) (k) Add. Subtract. Multiply. Divide. Divide. 286.3 146f 356 $.0 4)$5.76t 4)$5.76| 184f 78.2 .05 *The pupil should understand that he multiplies $3.74 (not §371) by 3, and should be taught to write the decimal point in the product immediately after writing the tenths' figure of the product. fThis means, ^nd how many times U hundredths are contained in 576 hundredths. X This means, ^?id 1 fourth of $5.76. 44 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. 1. 4000 lb. are 2. 4200 lb. are 3. 4400 lb. are 4. 4600 lb. are tons, tons, tons, tons. (a) At $5.20 a ton, how much will 4600 lb. of coal cost? Operation. 4600 lb. = 2.3 tons. $5.20 2.3 $1,560 $10.40 $11,960 Explanation. One ton costs ^5.20. 1 tenth of a ton costs $.52. 3 tenths of a ton cost 2 tons cost . 2.3 tons cost . Find the cost : (b) 4800 lb. @ $5.40 per ton. (d) 6200 lb. @ $5.35 per ton. (f) 5600 lb. @ $5.70 per ton. 5. Three yd. are feet. 6. Three hr. are minutes. 7. Three min. are seconds. 8. Three bu. are quarts. 9. Three lb. are ounces. (c) 4200 lb. @ $5.60. (e) 6400 lb. @ $5.25. (g) 5400 lb. @ $5.80. 3 yd. are inches. 3 yr. are months. 3 wk. are days. 3 gal. are quarts. 3 pk. are pints. (h) Change 148 yd. to feet, (i) Change 15 yd. to inches. (j) Change 24 hr. to min. (1) Change 35 yr. to mo. (n) Change 37 lb. to oz. (p) Change 43 bu. to qts. (k) Change 24 hr. to seconds, (m) Change 52 wk. to days, (o) Change 49 gal. to quarts, (q) Change 56 pk. to pints. (r) How many square inches in a 5 -foot square PART I. MEASUREMENTS — RECTANGULAR SOLID 46 1. A rectangular solid has faces. Each face is a rectangle. 2. Some or all of the faces of a rectangular solid may be squares. 3. If each face of a rectangular solid is a square, the solid is called a . 4. If some of the faces of a rectangular soHd are oblongs, the sohd is not a cube. 5. The area of a rectangular surface 3 inches by 4 inches is square inches. 6. The solid content of a rectangular solid 3 in. by 4 in. by 2 in. is cubic inches. 7. The area of a rectangular surface 3 inches by 5 inches is square inches. 8. The solid content of a rectangular solid 3 in. by 5 in. by 2 in. is cubic inches. 9. The area of a rectangular surface 2 inches by 5 inches is square inches. 10. The solid content of a rectangular solid 2 in. by 5 in. by 3 in. is cubic inches. 46 COMPLETE ARITHMETIC. RATIO AND PROPORTION. 1. One seventh of 28 is . 28 is i of . 2. One fifth of 28 is . 28 is ^ of . 3. One third of 28 is . 28 is i of . (a) Find 1 fourth of 387. (b) 387 is 1 fourth of what ? (c) Find 1 fifth of 724. (d) 724 is 1 fifth of what ? 4. Five sixths of 60 are . 60 is | of . 5. Four fifths of 40 are . 40 is | of . (e) Find | of 420. (f) 420 is | of what number ? (g) Find 4 of 920. (h) 920 is ^ of what number ? 6. 16 is ot 24. 24 is of 16. 7. 24 is of 32. 32 is of 24. 8. 32 is of 40. 40 is of 32. 9. 40 is of 48. 48 is of 40. 10. Five sixths of 30 are 1 third of . 11. One sixth of 48 is 2 thirds of . (i) Five sixths of 366 are 1 half of what number? (j) One sixth of 852 is 2 thirds of what number? 12. Twelve is of 9, or and times 9. A man can earn and times as much in 12 days as he can earn in 9 days. If he can earn $24 in 9 days, in 12 days he can earn dollars. (k) If a man can earn $840 in 9 months, how many dol- lars can he earn in 12 months. 13. If 9 lb. of nails are worth 33 cents, 12 lb. are worth cents. (1) If 9 cords of wood are worth $42.75, how much are 12 cords worth ? PART I. 47 PERCENTAGE. 66| per cent = .66| = |. 75 per cent = .75 =: -|. 1. 66f per cent of 12 = 2. 75 per cent of 12 = 3. 66f per cent of 24 = 4. 75 per cent of 24 = 5. 50 per cent of 5 = 6. 25 per cent of 13 = 7. 33i per cent of 10 = 8. 20 per cent of 16 = 9. 16f per cent of 25 = 10. 14f per cent of 22 = 11. 12-1- per cent of 33 = 12. 11| per cent of 19 = 13. 10 per cent of 21 = (5) 12 is 66f % of 12 is 75% of - 24 is 66|% of 24 is 75% of - 5 is 50% of - 13 10 16 25 22 33 19 21 14. 4 is 15. 4 is 16. 4 is (3) — per cent of 24. — per cent of 28. — per cent of 32. — per cent of 36. per cent of 40. 17. 4 is 18. 4 is 19. 9 is per cent of 12. s 25% of - s 33^% of s 20% of - s 16f % of s 14f % of s 121-% of s lli% of s 10% of 7 is 7 is 7 is 7 is 8 is 12 is — % of 35. — % of 28. — % of 21. — % of 14. -% of 12. % of 18. 20. Twenty-five per cent of 80 sheep are sheep. 21. Twenty-one sheep are 25% of sheep. 22. Twenty-five sheep are per cent of 75 sheep. 23. Forty sheep are per cent of 60 sheep. 48 COMPLETE AKITHMETIC. PEKCENTAGE. 1. 75 per cent of 36 = 66f per cent of 36 = (a) Find 75% of 796. (b) Find 66f % of 822. (c) Find 3% of $375. This means, ^tic? 3 hundredths of 375. Operation. Explanation. $375 One per cent of |375 = $3.75. Q3 Three per cent of $375 = $11.25. $11.25 NUMBER STORY. Mr. A collected money for Mr. B. It was agreed that Mr. A should keep S% of all he might collect to pay him for his trouble. He collected $375 ; he should keep $11.25 and "pay over "the re- mainder to Mr. B. (d) How much should Mr. A " pay over " to Mr. B ? (e) Find 7% of $465. (f) Find 9% of $324. (g) Find 3% of $422. (h) Find 7% of $538. 2. 36 is 75% of . 36 is 66|% of . (i) 453 is 75% of what? (j) 562 is 66|% of what? (k) Forty-eight dollars are 3 % of what ? This means, $48 are 3 hundredths of how many dollars ? Operation. Explanation. $48 -f- 3 = $16. C)ne hundredth of the unknown number is $16 X 100 = $1600 ^^^' ^^^ hundredths (the whole) are NUMBER STORY. A lawyer collected some money for 3% of the amount collected ; his share (commission) was $48 ; the amount collected was $1600. (1) How much should the lawyer pay over to the man for w^hom he collected the money ? PART 1. 49 REVIEW. 1. The prime factors of 45 are , , and . (a) What are the prime factors of 100 ? (b) Of 125 ? 2. Two, 3, and 5 are the prime factors of . (c) Of what number are 3, 3,2, and 7 the prime factors ? 3. Keduce ^f to its lowest terms. ^ = (d) Reduce ^^ and ^-^ to their lowest terms. 4. Reduce 8|- to an improper fraction. 8|- = (e) Reduce 57| and 72|- to improper fractions. 5. Reduce -\^- to a mixed number. -\8- = (f) Reduce ^^ and ^^ to mixed numbers. 6. Reduce f and f to equivalent fractions having a common denominator. |- = |- = (g) Reduce y^ and ^ to equivalent fractions having a common denominator. 7. Multiply 60 by .7. This means, find 7 tenths of 60. One tenth of 60 = ; 7 tenths of 60 = . (h) Multiply $537 by .07. This means, ^ti^ 7 hundredths of $537. (See page 43.) 8. At $1 per ton, 4240 lb. of coal cost . (i) Find the cost of 4240 lb. of coal at $7 per ton. 9. The volume of a rectangular solid 5 inches by 4 inches by 2 inches is cubic inches. (j) Find the volume of a rectangular soUd 9 inches by 7 inches by 7 inches. 10. If 9 lb. of tea are worth $6, 12 lb. are worth . (k) If 9 acres of land are worth $346.50, how much are 12 acres worth at the same rate ? (12 is 1^ times 9.) 50 COMPLETE ARITHMETIC. MISCELLANEOUS PROBLEMS. 1. From June, 1881, to June, 1899, it is years. (a) How many years from Aug., 1492, to Aug., 1897 ? 2. A man sold a horse for one hundred twenty dollars ; this was seventeen dollars and twenty cents more than the horse cost him ; the horse cost him . (b) A man sold a farm for fourteen thousand seven hun-- dred fifty dollars ; this was eight hundred seventy-five dollars more than he paid for the farm. How much did the farm cost him ? 3. From 5460 lb. of coal there were sold 2 tons. pounds remained. (c) From 18940 lb. of coal there were sold 8|- tons. How many pounds were left ? 4. From Mendota to Galesburg it is 80 miles ; a train going 30 miles an hour, that leaves Mendota at 8:30, should arrive at Galesburg at . (d) From Chicago to Denver it is about 1000 miles. If a train leaves Chicago for Denver at 9 o'clock Monday morning and goes at the rate of 30 miles an hour, when will it arrive at Denver ? 5. If butter is 25^ a pound and coffee is 30^ a pound 3 lb. butter will pay for lb. coffee. (e) Fifteen and one half pounds of butter at 24^ a pound, will pay for how many pounds of coffee at 32^ a pound? 6. If f of a yard of lace is worth 15^, 2^ yards are worth cents. (f) If I of a yard of cloth costs $1.05, how much will 27^- yards cost ? PART I. 51 SIMPLE NUMBERS.* 1. Four, 6, 8, 10, 12, etc., are multiples of 2. 2. Six, 9, 12, 15, 18, etc., are multiples of 3. 3. Twelve is a multiple of 2. 12 is also a multiple of 3, and of 4, and of 6. 4. Twelve is a common multiple of 2, 3, , and . 5. Fifteen is a common multiple of and . 6. Common multiples of 4 and 6 are, 12, 24, , etc. The least common multiple of 4 and 6 is 12. 7. Common multiples of 6 and 8 are, 24, 48, , eta The least common multiple of 6 and 8 is . 8. Common multiples of 8 and 12 are, , , etc. The least common multiple of 8 and 12 is . 9. Common multiples of 6 and 9 are, , , etc. The least common multiple of 6 and 9 is . 10. The prime factors of 18 are , , and . 11. The prime factors of 70 are , , and . (a) What are the prime factors of 140. (b) Of 160 ? (c) Of 135 ? (d) Of 175 ? (e) Of 250 ? (f) Of 225 ? Multiply. Divide. Divide, (g) 635 by 53. (h) 944 lb. by 56 Ib.f- (i) 1536 lb. by 12.t (j) 728 by 54. (k) 846 lb. by 56 lb. (1) 1445 lb. by 12. (m) 834 by 52. (n) 739 lb. by 56 lb. (o) 1374 lb. by 12. (p) 947 by 51. (q) 873 lb. by 56 lb. (r) 1653 lb. by 12. (s) 836 by 55. (t) 965 lb. by 56 lb. (u) 1738 lb. by 12. * Do much oral work in preparation for this page. By using these terms, make the pupil as familiar with multiple, common multiple, and least common multiple, as he is with house, schoolhouse, and stone schoolhouse. t Tell the meaning. Tell a number story. 52 ^ COMPLETE ARITHMETIC. COMMON FRACTIONS. L. c. m, is the abbreviation for least common multiple. 1. Add 1 and \^. (a) 1+1- L. c. m. of 8 and 12 is . (b) f + i i = -ST- TJ = Ti' (c) l+f- iT + li = ^T=l¥¥- (d) l + f 2. Fromf subtract \. (e) l-i- L. c. m. of 9 and 6 is . (f) 1-^- i T¥- "B - T8-- (g)TV-l- 'ff-A-TT. (h)f-|. 3. Divide |J by f (i) l^i- L. c. m. of 12 and 8 is . (i) f^-J- ■H = ^T- i = IT- (k)f-i lf-A = (1) 4^f 4. Multiply 12 by 2f . (m) 48x2t. This means . (n) 252 X 3f 23 times 12 - . Story. (0) 175 X 2|. 5. Multiply 12 by |. (P) 96 xf. TVii" moTn" (q) 84 X |. 1 of 12 = . Story. (r) 95 X f. (s) 6^ dollars)375 dollars. Thi&mQd^ns, findhowmany times $6\ are contained in $375. (Change 6^ and 375 to fourths.) Story, (t) 6)756 dollars. This means, find 1 sixth of $156. Story. PART I. 53 DECIMAL FRACTIONS. 1. One tenth of $432 is . .2 of $432 = 2. One hundredth of $432 is . .02 of $432 := (a) Multiply $432 by .25. This means, find 2 tenths of $432, plus 5 hundredths of $432. Operation. Explanation. $432 One hundredth of $432 is . .25 5 hundredths of $432 are . One tenth of $432 is . 2 tenths of $432 are . $21.60 + $86.4 = $21.60* $86.4 t $108.00 NUMBER STORY. If 1 acre of land is worth $432, 1 hundredth of an acre is worth 5 hundredths of an acre are worth 1 tenth of an acre is worth . 2 tenths of an acre are worth 25 hundredths of an acre are worth (b) Multiply $325 by .23. (c) $482 x .32. (d) Multiply $278 by .43. (e) $356 x .36. (f) Multiply $536 by .07. (g) $351 x .7. (h) Multiply $249 by 2.6. (i) $426 x .45. (J) (k) (1) (m) (n) Add. Subtract. Multiply. Divide. Divide. 8.75 56-J- 675 $.05)$6. 5)$6. 7.324 12.9 .36 ~" * The pupil should understand that he multiplies $4.32 (not $432) by 5, and should be taught to write the decimal point in the partial product immediately after writing the tenths' figure, 6, of the partial product. t The pupil should understand that he multiplies $43.2 (not $432) by 2, and should be taught to write the decimal point in the partial product immediately after writing the tenths' figure, 4, of the partial product. 54 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. 1. Sixteen and one half feet are 1 rod. 2. Three hitindred twenty rods are 1 mile. 3. One rod is and yards.* 4. Two rods are feet. 4 rods are feet. 5. Six rods are feet. 10 rods are feet. 6. One mile is rods. J mile is rods. 7. \ mile is rods. -i- mile is rods. 8. The telegraph poles along the line of a railroad are usually ten rods apart ; they are feet apart. From the first telegraph pole to the third it is rods ; from the first to the fifth it is rods. (a) How far is it from the first telegraph pole to the thirty-third ? 9. The roads in the country are usually 4 rods wide. A 4-rod road is feet wide. 10. A 100-foot street is rods and foot wide. 11. From the schoolhouse to , it is of a mile, or rods. (b) Change 5 mi. to rods. (c) Change 40 rd. to feet. (d) Change 28 rd. to yards, (e) Change 7 mi. to rods. REVIEW. 12. When hay is $12 a ton, 3000 lb. cost . 13. When oats are 30^ a bushel, 96 lb. cost . 14. When wheat is 80^ a bushel, 180 lb. cost . (f) When com (not shelled) is 45^' a bushel, how much will 3640 lb. cost ? (g) When coal is $5.40 a ton, how much will 2600 lb. cost ? * Discover by actual measurement the number of yards in a rod. PART 1. MEASUREMENTS. 55 1. A pile of vjood 8 feet long, 4 feet wide, and 4 feet high (or its equivale7it) is called a cord. (a) How many cubic feet in a rectangular solid 8 feet by 4 feet by 4 feet ? 2. A pile of wood 16 feet long, 4 feet wide, and 4 feet high contains cords. (b) How many cubic feet in 2 cords ? 3. A pile of wood 8 feet long, 4 feet wide, and 6 feet high contains cords. • (c) How many cubic feet in 1|- cords ? (d) In f of a cord ? (e) How many cubic feet in 5 ^- cords ? (f) In 6 J cords ? REVIEW. 4. A 1-inch square = 5. A 1-inch cube = — 6. A 2 -inch square = 7. A 2-inch cube = — - of a 2 -inch square, of a 2 -inch cube. - of a 3-inch square, of a 3 -inch cube. (g) Find the area of a surface 12 ft. by 24 ft.* (h) Find the volume of a solid 12 ft. by 8 ft. by 4 ft.f *Take care that the pupil understands that he does not (cannot) multiply 24 ft. by 12 ft., but 24 square feet by 12, or 12 square feet by 24. fin this problem the pupil multiplies 12 cubic feet by 8, and the product thus obtained by 4. 56 COMPLETE ARITHMETIC. RATIO AXD PROPORTION. 1. One eighth of 24 is . 24 is | of . 2. One seventh of 24 is . 24 is ^ of . 3. One fifth of 24 is . 24 is i of . (a) Find l of 49.26. (b) 49.26 is J of what? (c) Find i of $9.31. (d) $9.31 is i of what? 4. Four sevenths of 56 are . 56 is f of 5. Three sevenths of 42 are . 42 is f of (e) Find -f of 875. (f) 876 is f of what number? (g) Find I of 924. (h) 927 is f of what number? 6. 18 is of 27. 27 is of 18. 7. 27 is of 36. 36 is of 27. 8. 45 is of 54. 54 is of 45. 9. Three sevenths of 28 are 2 thirds of . 10. Six sevenths of 28 are 3 fourths of . (i) Three sevenths of 364 are 1 half of what number? (j) One seventh of 434 is 2 thirds of what number ? 11. Twenty is of 25. If 25 bags of salt are worth $15, 20 bags of salt are worth dollars. (k) If 25 acres of land are worth $640, how much are 20 acres worth at the same rate ? 12. Twenty-five is of 20, or and times 20. If 20 bushels of apples are worth $12, 25 bushels of apples are worth dollars. (1) If 20 barrels of salt are worth $22.40, how much are 25 barrels of salt worth at the same rate ? 13. If 12 qt. of nuts are worth 40^-, 9 qt. of nuts are worth cents. PART I. 57 PERCENTAGE. 40 per cent = .40 = |. 60 per cent = .60 = -|. {1) (2) 1. 40 per cent of 50 = 50 is 40% of 2. 60 per cent of 15 = 15 is 60% of — 3. 40 per cent of 45 = 16 is 40% of — 4. 60 per cent of 45 = 18 is 60% of — 5. 75 per cent of 28 = 18 is 75% of — 6. 66f per cent of 21 = 18 is 66f % of - 7. 40 per cent of 60 = 20 is 40% of — 8. 60 per cent of 60 = 24 is 60% of — d. 75 per cent of 32 = 27 is 75% of — 10. 66f per cent of 36 = 22 is 66|% of . (5) {3) 11. 8 is per cent of 32. 8 is % of 24. 12. 8 is per cent of 40. 8 is % of 16. 13. 8 is per cent of 64. 8 is % of 56. 14. 8 is per cent of 48. 8 is % of 72. 15. 8 is • per cent of 80. 8 is % of 20. 16. 9 is per cent of 15. 18 is % of 27. 17. 18 is per cent of 24. 4-1- is % of 13. 18. Eussel earned 60^ ; lie spent 10% of his money for a tablet and 20% of it for a book; the tablet cost cents and the book cost cents. 19. Ten per cent of the sheep in a certain flock were black ; there were 8 black sheep ; in all there were sheep. 68 COMPLETE ARITHMETIC. PERCENTAGE. (1) 1. 40 per cent of 55 = 60 per cent of 55 = (a) Find 40% of 575. (b) Find 60% of 365. (c) Find 75% of 676. (d) Find 66f % of 591. (e) Find 3% of $254. (f) Find 7% of $254. (g) A lawyer's commission for collecting money was 7% ; he collected $635. How much of the money should he keep and how much should he " pay over " to the man for whom he collected the money ? (^) 2. 18 is 40% of . 18 is 60% of . (h) 346 is 40% of what ? (i) 345 is 60% of what ? (j) 534 is 75% of what ? (k) 534 is 66|% of what? (1) $36 is 3% of what? (m)$84 is 7% of what? (n) A lawyer's commission for collecting money was 7% ; his commission amounted to $63. How much did he collect and how much should he " pay over " to the man for whom he collected the money ? (5) 3. 18 is per cent of 45. 33 is % of 55. 4. 33 is per cent of 44. 18 is % of 27. (o) Eighteen dollars are what per cent of $600 ? This mesiiis, find how many hundredths of $600, $18 ao^e. Operation. Explanation. $600 -^ 100 =r $6. One per cent of $600 is . $18 -^ $6 = 3 times. ^^^ ^''^ ''^ "'^'^^ P^'* ''^"^ "!* ^^^^ ^ / o ^/.^^ as t^6 are contained times in $18. $18 are 3% of $600. 6. $84 are % of $1200. $72 are % of $800. PART I. 59 REVIEW. 1. The prime factors of 63 are , — -, and . (a) What are the prime factors of 215 ? (b) Of 470 ? 2. Three, 3, 3,. and 2 are the prime factors of . (c) Of what number are 3, 5,2, and 23 the prime factors ? 3. Add I and |. The 1. c. m. of 8 and 3 is . (d) Add I, -A, and |. The 1. c. m. of 9, 12, and 3 is (e) Divide | by -^. The 1. c. m. of 9 and 12 is ■ 4. Multiply $324 by |. This means . (f) Multiply $324 by .25. 5. One half of a rod is and feet. (g) Twenty-six and one half rods are how many feet ? 6. Three eighths of a mile are rods. (h) How many rods in 7| miles ? (i) In 4|- miles ? 7. Two rods are — — yards. 3 rd. are yards. (j) How many yards in 42 rods ? (k) In 64 rods ? 8. When wheat is 70^ a bushel, 240 lb. cost . (1) When wheat is 90^ a bushel, how much will one ton of wheat cost ? 9. A pile of wood 32 feet long, 4 feet wide, and 4 feet high contains cords. (m) How many cu. ft. in a pile of wood 32 ft. by 4 ft. by 2 f t. ? 10. If 12 lb. of sugar are worth 45^, at the same rate 8 lb. are worth cents. (n) If 12 boxes of soap are worth $22.50, how much are 8 boxes worth at the same rate ? 60 COMPLETE ARITHMETIC. MISCELLANEOUS PROBLEMS. 1. The perimeter of a rectangular surface 7 feet by 9 feet is feet. (a) How many rods of fence are required to enclose a rectangular field 38 rods by 54 rods ? 2. If the sun rises at 7 o'clock and sets at 4:30, from sunrise to sunset it is hours and minutes. (b) If the sun rises at 7:14 and sets at 4:28, how long is it from sunrise to sunset ? 3. A man bought 16 dozen eggs at 10^ a dozen; he lost 25 per cent of them by decay; he sold the remainder at 12^ a dozen ; he lost cents. (c) A man bought 360 dozen eggs at 9^ a dozen ; he lost 25% of them by decay; he sold the remainder at 11^ a dozen. How much money did he lose ? 4. From midnight Monday to midnight Wednesday it is hours. (d) How many hours in a week ? (e) In the month of January ? 5. At 45^ each, 3 copies of "Eobinson Crusoe" wiU cost dollar and cents; for $1.80 I can buy copies. (f) How much will 7 copies of " Robinson Crusoe " cost ? (g) How many copies of " Robinson Crusoe " can I buy for $10.35 ? (h) For $4.95 ? (i) Add four thousand three hundred twenty-four and twenty-five thousandths, and seven hundred forty-six and thirty-four hundredths, and seventy-five and eight tenths. (j) From forty-five hundred subtract forty-five hundredths. PART I. 61 SIMPLE NUMBERS. 1. Ten times 60 are . 10 times 40 are . 2. Ten times 46 are . 10 times 25 are . 3. Ten times 300 are . 10 times 325 are . 4. Ten times 2 times a numher are times the number. (a) Multiply 347 by 20. Operation. Explanation. 347 Two times 347 are . 20 Ten times 694 are . fiQAO ^^^ times 2 times (20 times) 347 are . (b) Multiply 564 by 30. (c) Multiply 468 by 40. (d) Multiply 735 by 50. (e) Multiply 642 by 60. 5. One hundred times 2 times a number are times the number. (f) Multiply 86 by 200. (g) Multiply 94 by 300. (h) Multiply 75 by 400. (i) Multiply 48 by 700. REVIEW. 6. The prime factors of 48 are , , , , and . (j) What are the prime factors of 290 ? (k) Of 430 ? Multiply. Divide. Divide. (1) 347 by 70. (m) 8540 lb. by 70 lb.* (n) 1480 lb. by 20.t (o) 575 by 70. (p) 9470 lb. by 70 lb. (q) 1350 lb. by 20. * This means, find how many times 70 lb. are contained in 8540 lb. 70 lb. are contained in 8540 lb. times. Story^In 85U0 lb. of ear corn there are btishels. tThis means, find 1 twentieth of 1480 lb. 1 twentieth of 1480 lb. is lb. Story —Iwish to put U80 lb. of wheat into 20 sacks; I must put lb. in each sack. 62 COMPLETE ARITHMETIC. COMMON FKACTIONS. 1. Add 3V and -^ * (a) S75^-{-256^\. 2. From 7| subtract 2ft (b) 3467^ - 12824. 3. Divide I by 3V. (c) 3| -^ ^. « (9) 4. Divide 8 by |. (d) 28 -i- 1. (4) 5. Divide 3f by If (e) 47^ -^ 1|. (h) 6. Divide 21^ ft. by 3. (f) 87| ft. -^ 3. (21) 7. Divide 1| ft. by 3. (g) 5|- ft. ^ 3. (15) 8. Divide 13|- ft by 3. (h) 745f ft. ^ 3. (21) 9. Multiply 20^ by 2f.t (i) 96 x 2f. (I6) 10. Multiply 20^' by |. (j) 124 x J. (8) 11. Multiply $f by 6. (k) 64f x 6. (19) 12. Multiply $1- by J. (1) $i x 26f (13) 13. Multiply $1^ by i. (m) $27|- x |. (13) 14. Multiply $61^ by 2f § (n) $86|- x 2 J. (22) 15. At $4|^ per ton, 2 tons of coal cost dollars; ^ a ton costs ; 2i tons cost . 16. At $8^ per ton, 2|- tons of hay cost . * The pupil is expected to find by trial that the 1. e. m. of 10 and 12 is . fTake 1 from the 7, change it to fifths and add it to the J. t In problems 9 to 13 inclusive, think of the number to be multiplied as the price per yard; this will suggest the following: Problem 9, 21 yd, of ribbon at 20^ a yard. Problem 10, J yd. of ribbon at 20^ a yard. Problem 11, 6 yd. of ribbon at $1 a yard. Problem 12, J yd. of ribbon at $J a yard. Problem 13, i yd. of ribbon at 8U a yard. g This means, 2 times 86^ plus J of 86i. 2 times S6i are dollars, i of 86i is dollars. $13 + $31 = dollars. Stoi-y— At $6i a ton, 2i tons of coal will cost . i See note, page 6. PART I. 63 DECIMAL FRACTIONS. (a) Multiply $546 by 3.24. This means, find 3 times $546, plus 2 tenths of $546, plus 4 hundredths of $546. Operation. Explanation. $546 One hundredth of |546 is . 3.24 ^ hundredths of $546 are . One tenth of $546 is . ^^^•^^ 2 tenths of $546 are . $109.2 3 times $546 are . $1638 $21.84 + $109.2 + $1638 = $1769.04 NUMBER STORY. K one acre of land is worth $546, 1 hundredth of an acre is worth . 4 hundredths of an acre are worth . 1 tenth of an acre is worth . 2 tenths of an acre are worth . 3 acres are worth . 3.24 acres are worth . (b) Multiply $437 by 2.36 ; (c) $375 by 5.27. (d) Multiply $352 by 6.21 ; (e) $284 by 7.32. (f) Multiply four hundred seventy-three dollars by four and thirty-five hundredths. (g) When oil meal is $28 a ton, how much will 4680 lb. cost ? (4680 lb. = tons.) (h) (i) (j) (k) (1) Add. Subtract. Multiply. Divide. Divide. .886 8.744 8.2* $.5 )$31 5)$31. .075 .956 M_ * One hundredth of 8.2 is .082. The pupil should write the decimal jwint in the first partial product immediately after xvnting the tenths' figure oj (he partial product. 64 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. ]Sfn ^^^ CITY SCALE. Load of E^ Dec.gj, -1^9^ ^^^^ Clyde H. Hall rp^ Fay D. Winslow. . ' Gross Weight, ^J^ Ih. Tare, ^1^ Ih. Net Weight, ???? Ih. Clarence Marshall, Weiaher (a) Find the value of the load of hay, ticket No. 186, at $8.25 a ton. Find the value of the following : Commodity. Gross Weight. Tare. Price per Ton. (b) Hay. 5610 1870 $12.50 (c) Coal. 6380 2140 $6.50 (d) Bran. 3340 1560 $13.50 1. One hundred feet are rods and foot. 2. Two hundred feet are rods and feet. 3. Three hundred feet are rods and feet. 4. Four hundred feet are rods and feet. 5. Five hundred feet are rods and feet. (e) Change 8 miles to rods, (f) Change 30 rods to feet, (g) Change 36 rods to yards, (h) Change 2160 lb. oats to bu. (i) Change 5240 lb. to tons, (j) Change 274 ft. to inches. PART I. 65 MEASUREMENTS. 1. One hundred sixty sqiiare rods are one acre. A piece of land 1 rod wide and 160 rods long is one r-. 2. A piece of land i of a mile long and 1 rod wide is 3. Land 2 rd. wide and rd. long is one acre. 4. Land 4 rd. wide and rd. long is one acre. 5. Land 8 rd. wide and rd. long is one acra. 6. Land 10 rd. wide and • rd. long is one acre. 7. Land 5 rd. wide and rd. long is one acre. 8. Land 4 rods by 4 rods is th of an acre. 9. Land 4 rods by 10 rods is - — of an acre. 10. Land 8 rods by 10 rods is of an acre. 11. Land 1 rod wide and 1 mile long is — — acres. 12. Land 2 rods wide and 1 mile long is acres. 13. Land 66 ft. wide and 1 mile long is acres. 14. Land 5 rods by 16 rods is square rods. (a) How many square rods in a rectangular piece of land 28 rd. by 36 rd.? REVIEW. (b) Find the area of a rectangular surface 18 ft. by 26 ft.* (c) Find the volume of a rectangular solid 15 ft. by 8 ft. by 23 ft.f 15. How many cords in a pile of wood 4 ft. wide, 4 ft. high, and 36 ft. long? (d) How many cords in a pile of wood 4 ft. high, 4 ft. wide, and 276 ft. long? (e) How many cords in a pile of wood 4 ft. wide, 4 ft. high, and as long as your schoolroom ? * See foot-note (*), page 55. f See foot-note (f), page 55. COMPLETE ARITHMETIC. RATIO AND PROPORTION. 1. One ninth of 36 is . 36 is ^ of . 2. One eighth of 36 is . 36 is i of . 3. One seventh of 36 is . 36 is -i- of . (a) Find 1 eighth of 353.6. (b) 353.6 is i of what? (c) Find. 1 fifth of 1685. (d) 1685 is i of what? 4. Five eighths of 80 are . 80 is f of . 5. Three fifths of 60 are . 60 is f of . (e) Find |- of 680. (f) 680 is | of what number ? (g) Find I of 435. (h) 435 is | of what number ? 6. 22 is of 33. 33 is of 22. 7. 33 is of 55. 55 is of 33. 8. 55 is of 99. 99 is of 55. 9. 44 is — _ of 77. 77 is of 44. 10. Three eighths of 32 are 2 thirds of . ' 11. Five eighths of 32 are 2 thirds of . (i) Three eighths of 528 are 2 thirds of what number? (j) Five eighths of 528 are 2 thirds of what number? 12. Twenty is of 16, or and — times 16. A man can earn and times as much in 20 days as he can earn in 16 days. If he can earn $44 in 16 days, in 20 days he can earn dollars. (k) If a man can earn $740 in 16 weeks, how much can he earn in 20 weeks ? 13. If 20 bushels of apples are worth $12, 15 bushels are worth dollars. (1) If 20 gal. of milk are worth $3.24, how much are 15 gal. worth ? PART I. 67 PEBCENTAGE. 80 per cent = .80 = f 831^ per cent = .83^-: |. W (^) 1. 80 per cent of 60 = 60 is 80% of . 2. 83i per cent of 60 - 30 is 83^% of , 3. 80 per cent of 40 = 40 is 80% of . 4. 83|- per cent of 36 ■= 35 is 83^% of . 5. 75 per cent of 60 = 60 is 75% of . 6. 66f per cent of 18 = 32 is 66f % of . 7. 40 per cent of 35 = 18 is 40% of . 8. 60 per cent of 35 = 27 is 60% of . (3) (3) 9. 9 is per cent of 54. 9 is % of 45. 10. 9 is per cent of 63. 9 is % of 36. 11. 9 is per cent of 27. 9 is % of 72. ' 12. 9 is per cent of 18. 9 is % of 90. 13. 9 is per cent of 81. 14 is % of 35. 14. 21 is per cent of 35. 14 is % of 21. 15. 21 is * per cent of 28. 35 is % of 42. 16. 28 is per cent of 35. 12 ^ is % of 25. 17. Mr. Dow had 80 bushels of apples; he lost 25% of tham by decay ; he lost bushels and had bushels left. 18. Fourth-of-July night Willie had 10^; this was 12|^ per cent of what his father gave him to spend ; his father gave him cents. 19. Sarah had 45 chicks; a hawk killed 18 of them; the hawk killed per cent of her chickens. 68 COMPLETE ARITHMETIC. PERCENTAGE. 1. 80 per cent of 55 = 83^ per cent of 48 = (a) Find 80% of 435. (b) Find 83-i-% of 492. (c) Find 40% of 435. (d) Find 75% of 492. (e) Find 7% of $435. (f) Find 9% of $492. (g) A lawyer's commission for collecting money was 9 %;* he collected $834. How much of this money should he keep ? and how much should he " pay over " to the man for whom he collected the money ? (^) 2. 20 is 80% of . 20 is 83^ % of . (h) 280 is 80% of what ? (i) 280 is 83^% of what ? (j) 276 is 60% of what? (k) 276 is 75% of what? (1) 224 is 7% of what ? (m) 531 is 9% of what ? (n) A lawyer's commission for collecting money was 9 % ;* his commission amounted to $54. How much did he col- lect ? and how much should he " pay over " to the man for whom he collected the money ? (5) 3. Fifty-six dollars are what per cent of $800 ? This means, $56 are how many hundredths o/ $800 ? 4. $36 is % of $400. $55 is % of $500. 5. $2.40 is % of $80. $3.50 is % of $50. 6. A lawyer collected $900 ; he retained as his commis- sion $63 of this sum, and paid the remainder, $837, to the man for whom he collected the money ; the lawyer's com- mission for collecting was %. * % of what ? 95t of the amount collected. PART I. 69 REVIEW. 1. Ten times 6 times a number are times the num- ber. 10 times 6 times 8 are . 8 x 60 = 2. One hundred times 5 times a number are times the number. 9 multipHed by 500 = (a) Multiply 78 by 80. (b) Multiply 96 by 700. 3. Divide 12^ hj 2i. (Change to halves.) Story. (c) Divide 345| by 2f. (Change to .) Stori/. (d) Divide 345.6 by 2.4. This means, . Story. (e) Multiply $338 by 2.4. (f) Multiply $338 by 2.43. (g) When land is $375 an acre, how much will 3.35 acres cost ? Find the value of the following : Commodity. Gross Weight. Tare. Price per Ton. (h) Straw. 4360 1b. 1780 1b. $4.25 (i) Coal. 6240 lb. 1830 lb. $6.75 4. One acre is square rods. ^ acre = 5. One mile is rods. ^ mile = 6. A piece of land ^ of a mile long and 2 rods wide is acres. 7. A piece of land 160 rods long and as wide as the schoolroom is about acres. (j) How many square rods in a rectangular piece of land 47 rods by 6 rods ? Is this more or less than 1 acre ? Is it more or less than 2 acres ? 8. One cord is cubic feet. ^ cord = (k) How many cubic feet of wood in a pile 7 feet by 5 feet by 6 feet ? Is this more or less than 1 cord ? Is it more or less than 2 cords ? 70 COMPLETE ARITHMETIC. MISCELLANEOUS PROBLEMS. 1. If a man can save $5 a month, in 3 years he can save dollars. (a) If a man can save $2 7 in a month, how much can he save in 9 years ? 2. Byron bought 2 doz. oranges for 40^; he sold them at 3^ each ; he gained cents. (b) A merchant bought 35 barrels of apples for $85 ; he sold them at $2.75 a barrel. How much did he gain ? 3. Henry had 25 chickens; a hawk caught 20% of them; he sold the remainder at 22^ each; he received dollars and cents. (c) Henry's father had 75 bushels of apples ; he lost 20% of them by decay; he sold the remainder at 85^ a bushel. How much did he receive for them ? 4. If coffee costs $f a pound, for 1 dollar I can buy pounds. 1^1 = (d) If tea costs $| a pound, how many pounds can I buy for $57? (Change $57 to fifth-dollars.) 6. The sum of two numbers is 34 ; one of the numbers is 12 ; the other number is . (e) The sum of two numbers is 346.2 ; one of the num- bers is 75.36. What is the other number? 6. In a piece of slate 1 foot square and 1 inch thick there are cubic inches. (f) How many cubic inches in a piece of slate 2 feet square and 2 inches thick ? (g) If |- of a ton of coal is worth $4.20, how much is 1 ton worth ? PAKT I. 71 SIMPLE NUMBERS. 1. One tenth of 6 is . 1 tenth of 40 is - 2. One tenth of 46 is . 1 tenth of 25 is - 3. One tenth of 300 is . 1 tenth of 325 is 4. One half of 1 tenth of a number is one th of the number. One third of 1 tenth of a number = (a) Divide 472 by 20. Operation. Explanation. 20)47 "^2 C)ne tenth of 472 is . — TT-T- One half of 1 tenth (^) of 472 is . Zo.o (b) Divide 741 by 30. . (c) Divide 548 by 40. (d) Divide 735 by 50. (e) Divide 960 by 60. 5. One third of 1 hundredth of a number is one th of the number. One fourth of 1 hundredth of a number = (f) Divide 972 by 300. (g) Divide 972 by 400. (h) Divide 895 by 500. (i) Divide 976 by. 800. REVIEW. 6. Common multiples of 15 and 6 are , , etc. The least common multiple of 15 and 6 is -^ — . 7. The prime factors of 63 are , , and . (j) What are the prime factors of 124 ? (k) Of 178 ? 8. 3, 3, and 5 are the prime factors of . Forty-five is exactly divisible by 3 ; by 5 ; by ; by . (1) ■ (m) 128 cu. feet.)1536 cu. ft. 12)1584 cu. ft. (n) (o) 160 sq. rd.)2400 sq. rd. 15)2445 sq. rd. 72 COMPLETE ARITHMETIC. COMMON FRACTIONS. 1. Add y3_ and J. The 1. c. m. of 15 and 6 is - (a) Find the sum of 456^-^, 341f, 245|, and 564. 2. From 9^?^ subtract 4|. 1^-^ =30. i= to- (b) Find the difference of 4275^2-^. and 1328|. 3. Multiply I by 9. This means 3* . (c) Find the product of 453-|- multiplied by 9. 4. Multiply 2f by -i. This means 13* (d) Find the product of 45 8 J multiplied by i 5. Multiply 17 by |. This means 8* - (e) Find the product of 741 multiplied by |. 6. Multiply 17 by 2f. This means 16* , (f) Find the product of 741 multiplied by 2|. 7. Multiply 16i by 2^. This means - — 22* — (g) Find the product of 732^- multiplied by 2i. 8. Divide 8 by f . (4)* Change 8 to ths. (h) Find the quotient of 97 divided by f. Stori/. 9. Divide ^ by -J. (9)* Change to ^ths. (i) Find the quotient of 3^1 divided by J. Story. 10. Divide 7| by 2^. (14)* Change to ths. (j) Find the quotient of 55^ divided by 2f. Sto7y. 11. Divide Slf (|) by 4.- This means 15* — (k) Find the quotient of 7f divided by 8. Story. a) (^) (^) (^) (P) Add. Subtract. Multiply. Divide. Divide. 375ff 4351 346| 3| ft. )232 ft. 3)365 1 ft. 2465 182|^ 12 * These figures refer to notes on pages 6 and 7. See also foot-notes, page i PART I. 73 DECIMAL FRACTIONS. 1. One tenth of S6 is . .1 of $6.25 is . 2. One hundredth of $6 is . .01 of $6.25 is $.0625. 3. Eead each of the following in two ways: $.2436,* $.0532, $.6403, $.0042, $.0002, $.6042, $.8002. (a) Multiply $6.25 by 4.23. This means, find 4 times $6.25 + 2 tenths of $6.25 + 3 hundredths of $6.25. Operation. Explanation. $6.25 One hundredth of $6.25 is . 4.23 3 hundredths of $6.25 are . <m 1 oycj. One tenth of $6.25 is . $1250 " tenths of $6.25 are . 4 times $6.25 are . $25.00 $26.4375 $.1875 +$1.250 + $25 = NUMBER STORY. If one ton of coal is worth $6.25, 1 hundredth of a ton is worth . 3 hundredths of a ton are worth . 1 tenth of a ton is worth . 2 tenths of a ton are worth . 4 tons are worth . 4.23 tons are worth . (b) Multiply $7.35 by 3.46 ; (c) $4.45 by 5.24. * (1) 24^, 3 m., and 6 tenths of a mill. (2) 2436 ten-thousandths of a dollar. f To THE Teacher. — Read the foot-note on page 133 ; also, the first part of page 143. If the pupil finds difficulty in " pointing off," teach him to use a separatrix in the multiplicand as suggested on page 133. While multiplying 6.25 by 3 hundredths, it may appear "^06.25 on the slate thus : ^9;^ Do not at this stage of the work allow the pupil to " point oft" " by counting the decimal places in the mul- tiplicand and multiplier. Rather, lead him to think the meaning of the problem. See foot-notes, page 53. 74 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. 1. 5280 feet are 1 mile. - (a) Change 32*0 rd. to feet. 2. Six hundred feet are — 3. 4. 5. Seven hundred feet are - Eight hundred feet are - Two miles are rods. — rods are 1 mile. (b) Change 160 rd. to feet. rods and feet. — rods and feet. — rods and feet. 1^ miles are rods. (c) Change 6 mi. to rods, (e) Change 2 mi. to feet. (g) Change 2^ mi. to feet. Find the value of the following : Commodity. Gross Weight, (i) Corn meal. 4380 lb. (j) Oat straw. 3460 lb. (k) Old iron. 3240 lb. (1) Paper-rags. 3120 lb. (m) Ear corn. 3690 lb. (n) Oats. 2410 lb. 6. 120 inches are feet. (o) Change 428 in. to ft. 7. 72 feet are yards. (q) Change 853 ft. to yd. 8. 33 feet are rods. (s) Change 627 ft. to rd.f 9. 11 yards are rods. (d) Change 30 rd. to feet, (f) Change 75 rd. to yards, (h) Change 2|- mi. to rods. Tare. Price. 1540 lb. $15.25 per ton. 1620 lb. $4.50 per ton. 1380 1b. ^^ per lb.* 1260 1b. 1^ per lb. 1230 1b. 35^perbu. 1250 1b. 28^perbu. 126 inches are feet. (p) Change 580 in. to ft. 73 feet are yards. (r) Change 725 ft. to yd. 66 feet are rods. (t) Change 594 ft. to rods. 22 yards are rods. (u) Change 759 yd. to rd. (v) Change 638 yd. to rd. *1860 lbs at 1^ a pound would be worth 818.60 ; at J a cent a jwund the value is I of S18.60, or . tTo divide 627 ft. by 16 J ft., change both numbers to halves. See page 7, npte (20). PART I. 75 MEASUREMENTS. 1. A piece of land 10 rods by 16 rods contains square rods. It is acre. 2. A piece of land 20 rods by 16 rods contains square rods. It is • acres. (a) Change 640 sq. rd. to acres, (b) Change 1280 sq. rd. to acres, (c) How many acres in a piece of land 40 rods by 64 rods ? 3. Eighty square rods are of an acre. 4. Forty square rods are of an acre. 5. One hundred twenty square rods are of an acre. 6. A piece of land 20 rd. by 26 rd. contains square rods. (d) How many acres in a piece of land 20 rd. by 26 rd. ? Find the number of acres in each of the following : (e) 20 rods by 28 rods. (f) 20 rods by 30 rods, (g) 12 rods by 60 rods. (h) 24 rods by 50 rods, (i) 36 rods by 50 rods. (j) 48 rods by 60 rods. 7. A pile of wood 8 ft. by 4 ft. by 4 ft. contains cubic feet. It is cord. 8. Sixty-four cubic feet are of a cord. 9. Thirty-two cubic feet are of a cord. (k) How many cubic feet in a pile of wood 12 ft. by 8 ft by 4 ft. ? (1) How many cords in a pile 1 2 ft. by 8 ft. by 4 ft. ? Find the number of cords in each of the following : (m) 6 feet by 4 feet by 8 feet, (n) 6 ft. by 8 ft. by 8 ft. (o) 6 feet by 12 feet by 4 feet, (p) 6 ft. by 12 ft. by 8 ft 76 COMPLETE ARITHMETIC. RATIO AND PROPORTION, 1. One fourth of 3.6 is * 3.6 is ^ of . 2. One fifth of 3.5 is . 3.5 is i of . (a) Find 1 fourth of 78.4. (b) 78.4 is ^ of what ? (c) Find 1 fifth of 97.5. (d) 97.5 is ^ of what? 3. Three fourths of 3.6 are .f 3.6 is I- of — 4. Two thirds of 2.4 are . 2.4 is | of . (e) Find f of 28.8. (f) 28.8 is | of what number? (g) Find I of 37.2. (h) 37.2 is | of what number ? 5. 1.4 is of 2.1.$ 2.1 is of 1.4. 6. 2.1 is of 2.8. 2.8 is of 2.1. 7. Two thirds of 2.1 are 1 half of . 8. One half of 2.4 is 2 thirds of . (i) Two thirds of 53.7 are 1 half of what number? (j) One half of 65.6 is 2 thirds of what number? 9. If a man can earn $12 in 2.1 days, in 2.8 days he can earn dollars. (k) If a man can earn $75.45 in 2.1 months, how much can he earn in 2.8 months ? 10. If 2.8 tons of coal are worth $12, 2.1 tons are worth dollars. (1) If 2.8 tons of coal are worth $9.60, how much are 2.1 tons worth ? 11. If a pile of wood 4 ft. wide, 4 ft. high, and 8 ft. long is worth $5, a pile 4 ft. wide, 4 ft. high, and 32 ft. long is worth dollars. * One fourth of 36 tenths is tenths. t Three fourths of 36 tenths are tenths, or 2 and tenths. i Fourteen tenths are of 21 tenths. PART I. 77 w 1. 121% of 48 = 2. 25 % of 48 = 3. 37i% of 48 -^ 4. 50 % of 48 = 5. 62^% of 40 = 6. 871-% of 56 = (5) 7. 3 is % of 24. 8. 21 is % of 24. 9. 4 is % of 32. 10. 12 is % of 32. 11. 5 is % of 40. PERCENTAGE. 37^ per cent = .37^ = |. 621 per cent = .62-i- = |. 87|- per cent = .87^ = |. {2) 48 is 121 % of 48 is 25 % of 48 is 37|-% of 48 is 50 % of 40 is 62^% of 56 is 87^% of (5) — % of 24. — % of 24. — % of 32. — % of 32. — % of 40. 9 IS 15 is 20 is 28 is 15 is 12. Mr. A owed Mr. B $72. Mr. A had suffered many losses and it was hard for him to pay this amount. Mr. B kindly discounted the bill 12|^% and Mr. A paid it. He paid dollars. 13. Harry paid 87^% of his money for books and had S2 left ; before he bought the books he had dollars ; the books cost dollars. 14. Mr. Hill set 40 trees in his new orchard ; 8 of them died the first year ; per cent of the trees were dead ; per cent were alive. 15. Twenty per cent of 50 is per cent of 30. 16. Forty per cent of 50 is per cent of 80. 78 . COMPLETE ARITHMETIC. PERCENTAGE. {!) 1. 37^ per cent of 64 = 62i per cent of 64 = (a) Find 37^% of 192. (b) Find 62^% of 192. (c) Find 87^% of 192. (d) Find 75% of 192. (e) Find 5% of $192. (f) Find 7% of $192. (g) A lawyer's commission for collecting money was 8%;* he collected $875. How much of this money should he keep ? and how much should he " pay over " to the man for whom he collected the money ? (^) 2. 15 is 37^ per cent of . 35 is 62-i-% of . (h) 165 is 37J% of what? (i) 165 is 62|-% of what? (j) 161 is 87|-% of what? (k) 168 is 75% of what ? (1) 135 is 3% of what? (m) 168 is 7% of what? (n) A lawyer's commission for collecting money was 8%;* his commission amounted to $34.80. How much did he collect ? and how much did he " pay- over " to the man for whom he collected the money ? f (3) 3. 18 is per cent of 48. 30 is per cent of 48. 4. Forty-five dollars is what per cent of $500 ? This means, $45 are hovj many hundredths o/ $500 ? J 5. A lawyer collected $500 ; he retained as his commis- sion $55 of this sum, and paid the remainder, $445, to the man for whom he collected the money ; the lawyer's com- mission for collecting was %. * 8^ of what ? %i of the amount collected. t $34.80 is 8 hundredths of what ? X One hundredth of $500 is $5. See page 58, problem (o). PART I. 79 REVIEW. 1. One fourth of 1 tenth of a number is one th of the number. 1 fourth of 1 tenth of 120 is . (a) Divide 836 by 40. (b) Divide 75.6 by 40. 2. One fifth of 1 hundredth of a number is one th of the number. 1 fifth of 1 hundredth of 3500 is . (c) Divide 950 by 500. (d) Divide 246.5 by 500. (e) Divide 3660 by 600. (f) Divide 5740 by 700. 3. Multiply 19 by 3|. (This means .)* (g) Multiply 276 by 5|. (h) Multiply 276 by 5.75. Compare the answers to problems (g) and (h). 4. The sum of 86 hundredths and 5 tenths is . (i) Add 735 ten-thousandths and 642 thousandths. 5. One mile is feet. 1 mile is rods. (j) Change 5i mi. to feet, (k) Change 8|- mi. to rods. 6. A piece of land 4 rods by 80 rods is acres. (1) How many acres in a piece of land 18 rd. by 60 rd. 7. A pile of wood 4 ft. wide, 4 ft. high, and 40 ft. long contains cords. (m) How many cords in a pile of wood 4 ft. wide, 4 ft. high, and 432 ft. long ? Perform the operations indicated and tell number stories : (n) 48 cu. ft. X 6 X 8 = (o) 2304 cu. ft. ^ 128 cu. ft.=t (p) 96 sq. rods x 25 = (q) 2400 sq. rd. -^160 sq. rd.=$ (r) 960 rods x 4 = (s) 3840 rods ^ 320 rods = (t) 2350 lb. X 7 = (u) 16450 lb. -^ 2000 lb. = * See page 7, note 16. t Number story for (n) and (o). In a pile of wood 48 ft. long, 6 ft. wide, and 8 ft. high there are cords. I Number story for (p) and (q). A rectangular piece of land 96 rods by 25 rods contains acre& 80 COMPLETE ARITHMETIC. MISCELLANEOUS PKOBLEMS. 1. If I put 5. oz. of candy in each bag, 3 lb. of candy will fill bags, with ounces over. (a) If I put 2 bushels of oats in each bag, how many bags will one ton of oats fill ? 2. Henry was born on January 5, 1885; on Jan. 5, 1896, he \vas years old. (b) Queen Victoria was born on May 24, 1819. How old was she on the 24th day of May, 1897 ? 3. A dealer bought 5 gal. of milk at 14^ a gal. ; he sold it at 5^ a quart, but lost 2 quarts in over-measurement and waste ; he gained cents. (c) A grocer bought 3 bbl. sugar, each containing 340 lb. ; he paid 4^^* a lb. If he sells it at the rate of 20 lb. for a dollar, but loses 40 lb. by over-weights and waste, how much does he gain ? 4. Edward bought 2 lb. candy at 18^ a pound, 2 lb. walnuts at 20^ a pound, and 1 qt. of pop-corn for 10^; for all he paid cents. Find the amount of each of the following bills: (d) (e) 3 lb. Steak @ 15^ 5 bu. Apples @ 35^ ^ bu. Potatoes @ 40^ |- bu. Beans @ $1.50 18 lb. Sugar @ 5^ 7 gals. Milk @ 15c. 5 cans Corn @ 8^ 2 cd. Wood @ $4.50 7 heads Cabbage @ 8(f 1| tons Hay @ $8.50 5. If the divisor is 6 and the quotient is 4 J, the dividend is . (f) If the divisor is 128 and the quotient 16 J, what is the dividend ? PART I. 81 SIMPLE NUMBERS. 1. Alfred rode his bicycle for 3 consecutive hours ; the first hour he rode 1 miles ; the second hour, 8 miles ; the third hour, 6 miles ; altogether he rode miles ; his average speed was (24 -r- 3) miles an hour. 2. A farmer bought 20 sheep; for 10 of them he paid $35, and for the other 10, $25 ; for all he paid dollars; the average cost per head was dollars. (a) If 8 cows cost $204, what is the average cost per head? (b) Five boys were examined in spelling; the mark? received were 90, 95, 85, 80, and 75. What was the avei - age standing of the boys ? (c) The temperature at noon of each day for 1 week was as follows : Sunday, 94 ; Monday, 91 ; Tuesday, 96 ; Wednes- day, 95; Thursday, 90; Friday, 89; Saturday, 89. What was the average temperature at noon for the week ? (d) Six beef cattle weighed as follows : 1560 lb., 1430 lb., 1640 lb., 1350 lb., 1420 lb., and 1660 lb. What was the average weight per head ? REVIEW. 3. The prime factors of 70 are ■, , and . (e) What are the prime factors of 195 ? (f) Of 185 ? 4. 2, 3, and 7 are the prime factors of . Forty-two is exactly divisible by 2 ; by 3 ; by ; by ; by , and by . (g) Multiply 864 by 50 (h) Divide 7650 by 50. (i) Multiply 875 by 700. (j) Divide 8750 by 700. (k) Multiply 736 by 400. (1) Divide 9384 by 400. 82 COMPLETE ARITHMETIC. COMMON FRACTIONS. 1. Add -f^ and J. The 1. c. m. of 16 and 6 is — (a) Find the sum of 376y\, 254|-, 321|, and 342. 2. From 1-^^ subtract 3|. l^-g r. ^^. | = ^^. (b) Find the difference of 6274^ig and 842|. 3. Multiply -^^ by 8. This means . (c) Find the product of 357y^g multiplied by 8. 4. Multiply If by ^. This means . (d) Find the product of 573f multiplied by \. 5. Multiply 19 by f . This means . (e) Find the product of 643 multiplied by f . 6. Multiply 19 by 2f. This means (f) Find the product of 643 multiplied by 4|. 7. Multiply 15|- by 2|. This means — (g) Find the product of 345 -|- multiplied by 4i. 8. Divide 8 by |. (Change 8 to ths.) (h) Find the quotient of 86 divided by f. Story. 9. Divide y'g by \ . (Change to —ths.) (i) Find the quotient of '^^^ divided by -\. Story. 10. Divide 8| by 2|. (Change to ths.) (j) Find the quotient of 57| divided by 2|. Story. 11. Divide $1|- (|) by 5. This means . (k) Find the quotient of 8| divided by 9. (1) Find the quotient of 45 3| divided by 4. (m) (n) (0) (p) (q) Add. Subtract. Multiply. Divide. Divide. 436|^ 573^ 245 3| in. )246 in. 3)246 1 in. 148| 245|^ 24f PART I. 83 DECIMAL FRACTIONS. Division. Case I. 1. Divide .45 by .09. This means 4* . (a) Find the quotient of 46.75 divided by .25.f Stor/j. 2. Divide 9 by .6. This means 9* . (b) Find the quotient of 874 divided by .5.| Storij. 3. Divide 6 by .05. This means 14* . (c) Find the quotient of 96 divided by .25.§ Stori/. 4. Divide 4.5 by .09. This means . (d) Find the quotient of 46.5 divided by .25. |j Stori/. (e) Find the quotient of 57.5 divided by 2.5.^ Story. (f) Find the quotient of 68.5 divided by .25.** Story. (g) Find the quotient of 76 divided by .2 5. ft Story. Division. Case II. 5. Divide $.63 by 7. This means 5* . (h) Find the quotient of $87.15 divided by 7. Story. (i) Find the quotient of $375.50 divided by 25. Story. (1) Tell the meaning of each of the following, (2) solve, and (3) tell a suggested number story : (j) Divide $256 by $8. (k) Divide $256 by 8. (1) Divide $24.36 by $.04. (m) Divide $24.36 by 4. (n) Divide $53.6 by $.8 (o) Divide $53.6 by 8. * These figures refer to notes on pages 8 and 9. t Find how many times 25 hundredths are contained in 4675 hundredths. $ Find how m»ny times 5 tenths are contained in 8740 (874.0) tenths. § Find how many times 25 hundredths are contained in 9600 (96.00) hundredths. II Find how many times 25 hundredths are contained in 4650 (46.50) hundredths. Tf Find how many times 25 tenths are contained in 575 (57.5) tenths. ** Find how many times 25 hundredths are contained in 6850 (68.50) hundredths, tt Find how many times 25 hundredths are contained in 7600 (76.00) hundredths 84 COMPLETE AKITHMETIC. DENOMINATE NUMBERS. 1. 24 sheets of paper are 1 qvAix. 2. 20 quires of paper are 1 ream. 3. One ream of paper is sheets. 4. One half of a ream is quires, or sheets. 5. One fourth of a ream is quires, or sheets. 6. George bought a ream of paper for $2 and sold it at the rate of 2 sheets for a cent ; he gained cents. 7. Arthur bought a ream of paper for $2.25 and sold it at 15^ a quire ; he gained cents. REVIEW. 8. Ten thousand feet are nearly miles. 9. Fifteen thousand feet are nearly miles. (a) Twenty thousand feet are how many feet less than four miles ? (b) Five miles are how many feet more than 25000 feet? 10. One thousamd feet are nearly one th of a mile. (c) Two fifths of a mile are how many feet more than 2000 feet ? (d) f of a mile, than 3000 ft. ? 11. Nine hundred feet are rods and feet. 12. One thousand feet are rods and feet. 13. One hundred feet are yards and foot. 14. Two hundred feet are yards and feet. 15. Three hundred feet are yards. (e) Change 3 mi. to ft. (f) Change 3 mi. to rd. (g) Change 3 A. to sq. rd. (h) Change 3 cd. to cu. ft. (i) Change 3|- mi. to ft. (j) Change 3|- mi. to rd. (k) Change 3|- A. to sq. rd. (1) Change 3|- cd. to cu. ft. (m) Change 3^ rd. to ft. (n) Change 3| yd. to in. PARI I. 85 MEASUREMENTS. 1. In one cubic yard there are cubic feet. (a) To dig a cellar 15 feet long, 12 ft. wide, and 6 feet deep would require the removal of how many cubic feet of earth ? (b) How many cubic yards ? (c) How many square feet in the floor of the cellar de- scribed in problem (a) ? (d) How many sq. yards ? 2. To make an excavation 2 yards long, 2 yards wide, and 2 yards deep would require the removal of cubic yards. (e) How many cubic feet in the excavation described in problem 2 ? 3. The area of one of the faces of a 2 -yard cube is square yards ; of all the faces, square yards. (f) The area of all the faces of a 2 -yard cube is how many square feet ? REVIEW. 4. A piece of lam) 20 rods by 8 rods contains . (g) How many acres in a piece of land 20 rd. by 46 rd. ? 5. A shed 16 feet long, 10 feet wide, and 6 feet high contains cubic feet. (h) How many cords of wood will the shed described in problem 5 contain ? Find the number of acres in each of the following : (i) 80 rods by 24|- rods. (j) 40 rods by 73 rods. (k) 20 rods by 52 rods. (1) 30 rods by 84 rods. Find the number of cords in each of the following : (m) 8 ft. by 12 ft. by 8 ft. (n) 16 ft. by 8 ft. by 8 ft. (o) 16 ft. by 4 ft. by 8 ft. (p) 20 ft. by 8 ft. by 8 ft. 86 COMPLETE ARITHMETIC. RATIO AND PROPORTION. 1. One fourth of .36 is . .36 is i of . 2. One fifth of .45 is . .45 is i of . 3. One eighth of .32 is . .32 is J of . (a) Find 1 fourth of 9.86.* (b) 9.78 is ^ of what ? ■ (c) Find 1 fifth of 8.24. (d) 8.24 is ^ of what ? 4. Five eighths of .32 are . .35 is |- of . 5. Three fifths of .45 are . .45 is | of . (e) Find | of 8.40. (f) 8.40 is | of what number ? (g) Find I of 9.45. (h) 9.45 is f of what number? 6. .32 is of .40. .40 is of .32. 7. .24 is of .40. .40 is of .24. 8. .24 is of .56. .56 is of .24. 9. .40 is — of .64. .64 is of .40. 10. Three fourths of .32 are 1 hal^ of hundredths. (i) Three fourths of 9.36 are 1 half of what number ? (j) One half of 9.36 is 4 fifths of what number ? 11. If .32 of an acre of land is worth $20, .40 of an acre is worth dollars. (k) If .32 of an acre of land is worth $24.56, how much is .40 of an acre worth at the same rate ? 12. If .40 of a ton of coal is worth $4.00, .32 of a ton is worth . (1) If .40 of a ton of coal is worth $3.25, how much is .32 of a ton worth ? * Think as suggested by the following : J of 9 is 2 with 1 remainder ; 1 = 10 tenths ; 10 tenths + 8 tenths are 18 tenths ; J of 18 tenths is 4 tenths with 2 tenths remainder ; 2 tenths = 20 hundredths ; 20 hundredths + 6 hun- 4)9^ dredths = 26 hundredths ; i of 26 hundredths = G hundredths with 2 ^•'*^ hundredths remainder ; 2 hundredths = 20 thousandths ; i of 20 thousandths = } thousandths. PART I. 87 PERCENTAGE. 30 per cent = .30 = .3 70 per cent = .70 = .7 90 per cent = .90 = .9 = -j%. 3 10"- 7 r"o- (1) 1. 30% of 60 = 2. 40% of 60 = 3. 50% of 60 = 4. 60% of 60 = 5. 70% of 140 = 6. 80% of 160 = 7. 90% of 180 = (3) 8. 4 is 9. 28 is 10. 5 is 11. 15 is 12. 6 is % of 40. % of 40. % of 50. % of 50. % of 60. (2) 60 is 30% of — 60 is 40% of — 60 is 50% of — 60 is 60% of — 140 is 70% of . 160 is 80% of — 180 is 90% of — (5) 12 is % of 40. 36 is % of 40. 45 is % of 50. 35 is % of 50. 42 is % of 60. 13. In a certain orchard there were 120 trees; 10% of them were pear trees and 30% of them were cherry trees; there were pear trees and cherry trees. 14. In another orchard there were 24 pear trees ; these were 30% of all the trees in the orchard; there were trees in the orchard. 15. There were 60 trees in an orchard ; 42 of them were apple trees and the remainder pear trees ; there were pear trees ; • per cent of the trees in the orchard were apple trees and per cent of them were pear trees. 16. Twenty-five % of the trees in an orchard died and were removed ; there remained 36 trees ; trees died. 88 COMPLETE ARITHMETIC. PERCENTAGE. 1. 30 per cent of 42 = (^V of 42 is 4.2.) 2. 90 per cent of 51 = 90 per cent of 31 = (a) Find 30% of 275. (b) Find 70% of 362. (c) Find 90% of 436. (d) Find 30% of 475. (e) Find 3% of 536. (f) Find 7% of -824. (g) A lawyer's commission for collecting money was 10%; he collected $954. How much of the money should he keep ? and how much should he " pay over " to the man for whom he collected the money ? (^) 3. 3.2 is 10 per cent of . 4.6 is 10% of . 4. 9.6 is 30 per cent of . 14.7 is 70% of . (h) 372.3 is 30% of what? (i) 562.1 is 70% of what? (j) 291.6 is 90% of what? (k) 752.4 is 30% of what? (1) 279 is 3% of what? (m) 539 is 7% of what? (n) A lawyer's commission for collecting money was 10%; his commission amounted to $76.20. How much did he collect ? and how much should he " pay over " to the man for whom he collected the money ? (5) 5. 5.1 is per cent of 51. 8.3 is % of 83. 6. 84 is per cent of 840. 8.40 is % of 840. 7. 7.54 is . per cent of 75.4. 7.54 is % of 754. 8. A lawyer collected $754; he retained as his commis- sion $75.40 of this sum and paid the remainder to the man for whom he collected the money ; the lawyer's commission for collecting was %. PART I. 89 KEVIEW. 1. Six sheep cost $33 ; the average cost per head was . (a) In a certain schoolroom, on Monday there were 35 pupils ; Tuesday, 38 ; Wednesday, 36 ; Thursday, 37, and Friday, 39. What was the average daily attendance ? 2. Divide j% by |. (Change to ths.) Story. (b) Divide $323| by $1|. (Change to ths.) Story. (c) Divide $323.75 by $1.25. This means . Compare the answers to problems (b) and (c). Why are they alike? 3. In ^ of a ream of paper there are quires. 4. At 3 sheets for 2 cents, 1 quire of paper costs . (d) At 3 sheets for 2 cents, how much will one ream of paper cost ? 5. In 2 square yards there are square feet. 6. In 2 cubic yards there are cubic feet. (e) In 18 cubic yards there are how many cu. ft. (f) In 48 square yards there are how many square feet ? (g) How many square feet of blackboard in this room ? * (h) How many sq. yd. of blackboard in this room ? * (i) How many cubic feet of earth must be removed to make an excavation 6 ft. by 9 ft. by 15 ft. ? (j) How many cubic yards ? (k) 326 + 432 + 28 + 175 + 326 + 47 + 63 + 82 + 96 = (1) 274 + 386 + 27 + 237 + 528 + 35 + 72 + 51 + 46 = (m) 438 + 275 + 43 + 324 + 136 + 32 + 85 + 37 + 91 = (n) 243 + 139 + 18 + 914 + 326 + 12 + 72 + 36 + 75 = * Require an approximate answer only. 90 COMPLETE ARITHMETIC. MISCELLANEOUS REVIEW. 1. Two boys start from two cities that are 67 miles apart and ride toward each other; one rides 8 miles an hour; the other rides 12 miles an hour; in one hour they ride miles ; in 3 hours they are • miles apart. (a) Two trains start from two cities that are 1000 miles apart and move toward each other; one goes 25 miles an hour; the other goes 30 miles an hour. At the end of 12 hours how far are the trains apart ? 2. A man bought land for $80 and sold it for $88 ; he gained $2 an acre ; there were acres. (b) A man bought land for $1850, and sold it for $2220, th(}reby gaining $5 an acre. How many acres did he buy ? (c) How much did he pay per acre for it ? 3. A boy earned some money, spent f of it, and had $6 left ; he spent dollars. (d) A man earned some money, spent | of it, and had $25.20 left. How much did he spend ? 4. The product of two numbers is 28 ; one of the num- bers is 7 ; the other number is . (e) The product of two numbers is 240 ; one of the num- bers is 15. What is the other number ? 5. Fifty per cent of my money is $12, and John has 3 times as much as I have ; John has dollars. (f) Fifty per cent of Mr. A's money is $36.45 and Mr. B has 3 times as much as Mr. A. Mr. B has . 6. I have $24; 50% of my money equals 33-i^ per cent of Bernie's money ; Bernie has dollars. (g) Mr. C has $72.30; 50% of Mr. C's money equals 33^% of Mr. D's money. How much money has Mr. D ? PART I. SIMPLE NUMBERS. Niimher of poitnds of milk received. '91 PATRONS. M. 236 149 375 97 201 385 87 230 155 435 T. 224 167 384 84 196 397 146 231 160 424 W. 259 134 363 93 205 388 175 229 145 412 T. 263 158 356 86 208 394 82 233 170 456 F. 248 175 325 92 199 396 185 228 180 428 S. 227 136 364 87 215 399 96 229 175 435 S. 23G 145 356 88 218 401 133 235 194 457 am't. L. R. Orr Levi Smith Henry Judd Peter Johnson Wm. Jones E.C.Ford {^) (b) (0) (d) (e) (f) (g) i^) (i) (J) Otto Fisk Fay Winslow Albert Davis Jno. Harris Total (k) (1) (m) (n) (o) (P) (q) (!•) (s) The above is a statement of the milk received at a small creamery for one week, (a) to (j) Find the amount delivered by each patron, (k) to (q) Find the amount received at the creamery each day. (r) Find the sum of (k) to (q) inclusive, (s) Find the sum of (a) to (j) inclusive. (t) Find tlie average amount received daily at the creamery. EEVIEW. etc. 1. Common multiples of 15 and 10 are , - The least common multiple of 15 and 10 is . 2. The prime factors of 75 are , , and . (;u) What are the prime factors of 2750 ? (v) Of 280 ; 3. 3, 3, and 7 are the prime factors of . Sixty-three is exactly divisible by 3 ; by ; by ; and by . (w) 27 cu. ft.)1296 cii. ft. 27)1296 cu. ft. 92 • COMPLETE AKITHMETIC. COMMON FRACTIONS. 1. Add yig and -^-^. The 1. c. m. of 15 and 10 is (a) Find the sum of 5S4.j\, 257^V 225i and 324i. 2. From 8^1^ subtract 5^\. 1-L = 37. tV = 3 o- (b) Find the difference of 7433?^ and ISl^V 3. Multiply -^\ by 8. This means . (c) Find the product of 235yV multiplied by 8. 4. Multiply If by i. This means . (d) Find the product of 724f multiplied by J. Story, 6. Multiply 16 by |. This means . (e) Find the product of 721 multiplied by |. 6. Multiply 16 by 2|. This means . (f) Find the product of 721 multiplied by 7|. 7. Multiply 18|- by 2i. This means (g) Find the product of 432 1- multiplied by 4i 8. Divide 7 by |. (Change 7 to ths.) (h) Find the quotient of 94 divided by |. Story. 9. Divide -^\ by yV (Change to ths.) (i) Find the qiiotieiit of 4^i divided by J^. Story. 10. Divide 6f by 2 J. (Change to ths.) (j) Find the quotient of 74f divided by 3^. 11. Divide 1| (V) by 3. This means . (k) Find the quotient of 7| divided by 3. (1) (m) (n) (0) (p) Add. Subtract. Multiply. Divide. Divide. 375|| 436J 753| 3^ y d.)512 yd. 5)636f yd. 286yV 182y3_- 24 " PART I. 93 DECIMAL FRACTIONS. Division. Case I. 1. Divide .48 by .12. This means . (a) Find the quotient of 54.24 divided by .12. Story. 2. Divide 6 by .4. This means . (b) Find the quotient of 736 divided by .4. Story. 3. Divide 3 by .04. This means . (c) Find the quotient of 78 divided by .12. Story. 4. Divide 4.8 by .12. This means . (d) Find the quotient of 40.8 divided by .12. Story. (e) Find the quotient of 43.2 divided by 1.2. Story. (f) Find the quotient of 51.6 divided by .12. Story (g) Find the quotient of 75 divided by .12. Story. Division. Case II. 5. Divide $.56 by 8. This means . (h) Find the quotient of $98.40 divided by 8. Story. (i) Find the quotient of $436.80 divided by 12. Story. (1) Tell the meaning of each of the following, (2) solve, and (3) tell a suggested number story. (j) Divide $924 by $7. (k) Divide $924 by 7. (1) Divide $29.34 by $.06. (m) Divide $29.34 by 6. (n) Divide $55.3 by $.7. (o) Divide $55.3 by 7. (p) Multiply $265 by .3. (q) Multiply $265 by 2.3. (r) Multiply $265 by .03. (s) Multiply $265 by .23. (t) Multiply $265 by 4.23. (u)At $.12 a dozen, how many dozen eggs can I buy for $63.84? 94 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. (a) Copy, complete, and receipt the following Bill : Waukegan, III., July 30, 1896. Mr. Kichard Yates, Bought of C. F. Brown & Co. July 1 a 1 i( 5 (( 8 <c 8 « 19 15 lb. Sugar, 5 lb. Raisins, 2 1- lb. Cheese, 1 sack Flour, ^ bu. Potatoes, 5 lb. Coffee, $.05 .07 .16 $.60 .35 Eeceived Payment, (a) 25 REVIEW. 1. 500 sheets of paper are 1 ream and sheets. 2. 500 feet are rods and feet. 3. 500 rods are 1 mile and rods. 4. 500 sq. rd. are 3 acres and sq. rd. 5. 500 inches are feet and inches. 6. 500 feet are yards and feet. 7. 500 pounds are of a ton. 8. When hay is $8 a ton, 5000 lb. cost dollars. (b) Load of straw, gross weight, 3380 lb.; tare, 1580 lb. Find the value at $5.50 a ton. (c) Multiply 40 sq. rods by 24 and divide the product by 160 square rods. Stor?/. (d) Multiply 24 cubic feet by 8 and the product thus ob- tained by 6. Divide the last product by 128 cu. ft. Story. PART I. 95 MEASUREMENTS. 3^ in. .s This diagram represents a piece of land. -h|n It is drawn on a scale of | inch to the rod. CM 1. The land represented by the above diagram is rods long. It is rods wide. (a) Find the perimeter of the piece of land represented by the diagram, (b) Find its area in square rods, (c) Find its area in acres, (d) How much is the land worth at S40 an acre ? 2. Draw carefully upon your slate or paper, on a scale of J in. to the rod, a diagram of a rectangular piece of land that is 30 rods by 36 rods. (e) Find the perimeter of the land represented by the diagram you haVe drawn, (f) Find its area in square rods, (g) Find its area in acres. (h) How much is the land worth at $60 an acre ? (i) How much will it cost to build a fence around it at 50^ a rod? (j) Find its area in square yards. 96 COMPLETE ARITHMETIC. RATIO AND PROPORTION. 1. One fifth of 2 (2.0) is * ^ of 3 is . 2. One fourth of 6 (6.0) is . i- of 4 is . 3. One eighth of 4 (4.0) is . ^ of 2 is . (a) Find 1 fourth of 70. (b) Find \ of 79. (c) Find 1 fifth of 83. (d) Find -i- of 66. 4. 1 is of 27. 2 is of 27. 5. 3 is of 27, or — .f 4 is of 27. 6. 5 is of 27. ■ 6 is of 27, or — . 7. 7 is of 27. 8 is of 27. 8. 9 is of 27, or — . 10 is — — of 27. 9. 11 is of 27. 12 is of 27, or - -. 10. 13 is of 27. 14 is of 27. 11. 15 is of 27, or — . 16 is — — of 27. 12. 17 is of 27. 18 is of 27, or — . 13. 1.8 is of 2.7. 2.7 is of 1.8. 14. .18 is of .27. .27 is of .18. 15. .19 is of .27. .27 is ■ of .19. 16. 1.9 is of 2.7. 2.7 is of 1.9. (e) 120 is what part of 140 ? % (f) 140 of 120 ? (g) 175 is what part of 185 ? (h) 185 of 175 ? (i) 120 is what part of 128 ? (j) 128 of 120 ? (k) 120 is what part of 160 ? (1) 160 of 120 ? (m) 24 is what part of 144 ? (n) 48 of 144 ? (o) 96 is what part of 144 ? (p) 100 of 144 ? (q) 120 is what part of 150 ? (r) 125 of 150 ? * J of 20 tenths. 1 3 is three twenty-sevenths of 27, or ^ of 27. X In each such problem as (e) , the number that follows the word of always becomes the denominator. 120 is \l% = (} = $ of 140. PART I. 97 PEKCENTAGE. 1. One % of $435 =• 2. One % of S832 = 3. One % of $38.4 = 4. One % of $56.30 = 5. One % of $8324 = {1) (a) Find 7% of $435. (b) Find 9% of $832. (c) Find 8% of $38.4. (d) Find 6% of $56.30, (e) Find 5% of $8324. {2) 6. 63 is 7% (7 hundredths) of . (f) 364 is 7% of what? (g) 635 is 5% of wha/. ? 7. 72 is 9% (9 hundredths) of . (h) 657 is 9% of what ? (i) 584 is 8% of whnt ? 8. 54 is 6% (6 hundredths) of . (j) 456 is 6% of what ? (k) 504 is 7% ot v/hat ? (-5) is % of $200.* is what % of $200 ? is % of $300. 9. $12 (1) $54 11. $21 (n) $84 13. $36 (p) $96 15. $40 (r) $95 is what % of $500 ? is what % of $300 ? is % of $400. is what % of $400 ? is % of $500. 10. $13 is -^.-% of $200 (m) $45 is what % of $200 ' 12. $22 is ^ % of $300 (o) $85 is what % of $300 ' 14. $38 is % of $400 (q) $98 is what % of $400 ' 16. $42 is % of $500, (s) $96 is what % of $500 ? *Lead the pupil to observe that he has already learned to solve " Case 3 " prob- lems in percentage in two ways. Sometimes he has found what part one number is of another and has changed the fraction thus obtained to hundredths. See prob- lems 11 to 17, page 57. Sometimes he has found one per cent of the base and has used this number as a divisor to obtain the required result. See page 58, problem (o); iiage 68, problems 3 to 5 ; page 78, problems 4 to 6. The " Case 3 " problems on this page and the greater number of those that follow in Part I., can be most readily solved by the second method. 12 is what 5^ of 200? One % of 200 is 2; therefore 12 is 6 per cent qf200; 13 is 6i per cent; 6I4. is 27 per cent, and U5 is 22\ per cevi. 98 COMPLETE ARITHMETIC. PERCENTAGE. 1. Three % of $620 = (a) Find 9% of $620. 2. Eighteen is 3% of . (b) $75 is 3% of what? 3. Eighteen is % of 600. (c) 138 is what % of 600? 4. Eight % of $220 - (d) Find 8% of $738. 5. Forty-eight is 8% of . (e) $76 is 4% of what? 6. Thirty-six is % of 600. (f) 174 is what % of 600? (g) A farmer had 825 sheep; he sold 8% of them. How many sheep did he sell ? (h) How many sheep did he have left? (i) A dealer bought some sheep and immediately sold 72 of them ; these were 8 % of all he purchased. How many sheep did he purchase ? (j) How many sheep did he have left? (k) Mr. B had 400 sheep; he sold 48 of them. What % of his sheep did he sell ? (1) A man had $485 ; he spent 7% of his money. How much money did he spend ? (m) How much money did he have left? (n) Peter Gray paid $35.50 for a suit of clothes ; this was % of all he earned in a year. How much did he earn in a year? (o) Judge Branson had $600 in the bank; he drew out $198. What % of his deposit did he draw out ? 7. A lawyer collected $800 ; he retained as his commis- sion $56 of this sum and paid the remainder, dollars, to the man for whom he collected the money ; the lawyer's commission for collecting was %. (p) A lawyer collected $850; he retained as his commis- sion for collecting $51 of this sum. What per cent did the lawyer charge for making the collection ? PART I. ,99 REVIEW. (a) A creamery received milk as follows : Monday, 8250 lb. ; Tuesday, 7640 lb. ; Wednesday, 8375 lb. ; Thursday, 7230 lb.; Friday, 6953 lb.; Saturday, 8424 lb. ; Sunday, 7651 lb. Find the number of pounds received during the week, (b) Find the average daily receipts. 1. Add ^^ and y^^. (Change to ths.) (c) Find the sum of 375||, 246|, 82l3i_, and 234. (d) Find the product of 372 multiplied by 6|. (e) Find the product of 372 multiplied by 6.75. Compare the answers to problems (d) and (e). (f) Make a bill containing the following items, and find the amount : Aug. 2, 24 lb. sugar @ 5^; Aug. 5, ^ lb. tea @ 70^; Aug. 10, 2 gal. syrup @ 35^ ; Aug. 13, 5 bu. po- tatoes @ 55^; Aug. 18, 5 cans peaches @ 18^; Aug. 18, 5 gal. kerosene @ 15^. 2. Draw a diagram of a piece of land 40 rods by 36 rods, scale, ^ in. to a rod. The diagram is inches long and inches wide. (g) How many square rods in the piece of land described in problem 2 ? (h) How many acres ? (i) The perimeter of the field is how many rods ? 3. 3 is of 25. 3 twenty-fifths are — hundredths. 4. 4 is of 25. 4 twenty-fifths are — hundredths. (j) 85 is what part of 95 ? (k) 95 of 85 ? (1) 95 is what part of 105 ? (m) 105 of 95 ? (n) 105 is what part of 115 ? (o) 115 of 105 ? 100. COMPLETE ARITHMETIC. MISCELLANEOUS PROBLEMS. 1. Horace spent |- of his money and had 24^ left: he spent — — • cents. Before spending any money he had cents. (a) Mr. Green spent | of his money and had $56.48 left. How much money did he spend ? (b) Before spending any money how much did he have ? 2. At 2|^^ an ounce, |- of a pound of cinnamon costs cents. (c) Find the cost of 5^ pounds of cloves at 1^^ per ounce. 3. A farmer mixed 10 bushels of oats with 15 bushels of corn ; together there were bushels of grain. What part of the mixture was oats ? What part was corn ? (d) A dealer mixed 250 bushels of oats with 350 bushels of corn. What part of the mixture was oats ? (e) What part was corn ? 4. Eddie, Willie, and Fay divided a melon among them- selves. Eddie had 1 fourth of the melon. Will had ^ of what remained after Eddie had taken his share. What part of the melon did Will have ? What part of the melon was left for Fay ? (f) An estate was divided as follows : The widow received 1 third of it, and each of five children received 1 fiftli of what remained after the widow had received her share. What part of the whole estate did each child receive ? 5. A had 3 sevenths of an estate' and B had 4 sevenths of it. A's share was equal to what part of B's share ? (g) If the estate mentioned in problem 5 was wor^b $9660, what was the value of A's share ? (h) Of B's share ? PART I, 1*01 SIMLPE NUMBERS.'- Time-booh^ showing hours worked for one week* Name. Daily Wages M. T. W. T. F. S. Hours Days Am't earn'd John Miller T.Benjamin D. Gordon N. Hanson A. Gillett Wm. Price R.L. Wing J. Carlson S. Spencer S. L. Judd $2.00 2.50 3.00 2.00 2.40 3.50 2.60 2.60 3.00 2.20 10 10 12 9 10 10 10 10 11 10 10 9 12 10 10 12 9 8 10 12 8 10 12 12 10 12 9 12 10 9 8 10 11 10 12 10 10 10 12 10 12 10 10 10 12 12 10 10 15 12 9 10 11 12 12 12 15 (a) (b) (c) (d) (i) (i) (k) (1) (m) (n) (o) (P) (q) u (t) (u) (V) (w) (X) (y) (z) (aa) (bb) (cc) (dd) Total hours (ee) (ff) (gg) (hh) (ii) ij) (kk) (11) (mm) The above is from the time-book of a small manufactory. (a to j) Find the number of hours worked by each employe. ' (k to t) Find the number of days (10 hours) worked by each employe. (u to dd) Find the amount earned by each employe. (ee to jj) Find the number of hours of work done each day. (kk) Find the sum of (a) to (j) inclusive ; of (ee) to (jj) inclusive. (11) Find the sum of (k) to (t) inclusive. (mm) Find the sum of (u) to (dd) inclusive. -, etc. KEVIEW. 1. Common multiples of 15 and 20 are , The least common multiple of 15 and 20 is . *Copy this time-book and write numbers in place of the letters ; that is, instead of (a), write the number of hours worked by John Miller ; in place of (k), write the nxunber of days worked by John Miller, etc. Call 8 hours a day's work. 102 COMPLETE ARITHMETIC. •COMMON FRACTIONS. 1. Add I, yV and f The 1. c. m. of 8, 10, and 5 ^a) Eind the sum of 896|, 507^3_^ and 344|. 2. From 9|- subtract 3.2. 1^ = ^ .2=^. (b) Find the difference of 1754f and 592.7. 3. Multiply 2y\- by 5. This means IS (c) Find the product of 757y\- multiplied by 5. 4. Multiply 1|- by i This means . (d) Find the product of 536|- multiplied by i ^^^ry. 5. Multiply 21 by |. This means . (e) Find the product of 681 multiplied by |. 6. Multiply 21 by 2f. This means . (f) Find the product of 681 multiplied by 8|. 7. Multiply 11 1- by 2 J. This means — (g) Find the product of 273|- multiplied by 4^. 8. Divide 7^ by |. (Change 7} to ths.) (h) Find the quotient of 97|- divided by |. Story. 9. Divide | by y3_. (Change to ths.) (i) Find the quotient of 7-| divided by -f^. Story. 10. Divide 7f by 2i (Change to ths.) (j) Find the quotient of 86|- divided by S^. 11. Divide 2| (Y) by 3. This means , (k) Find the quotient of 8| divided by 9. 12. Divide 21f by 5. This means (1) Find the quotient of 62 IJ divided by 5. (m) (n) (o) (p) (q) Add. Subtract. Multiply. Divide. Divide. 478|- 539|- 738| 3J ft.)720i- ft. 3)745 f ft. 896f 182.3 28 PART I. 103 DECIMAL FRACTIONS. Case I. This means . (b) 47.275-1- .025= Story. This means . (d) 375.6 -i- .05 = Story.'' This means . (f) 256.4 -^.004= Story. (h). 35 ^.004= Story. (j) .35 ^ .025 = Story. Case II. This means . Division. 1. Divide .072 by .012. (a) 65*088 -f- .012 = 2. Divide 6 by .05. (c) 468 ^ .05 = 3. Divide 8 by .004. (e) 75 -^ .004 = (g) 38.72 ^ .004 = (i) 73-^.025 = Division. 4. Divide $.066 by 8. • (k) $24,368 -4- 8 = (1) $35,056 ^ 8 = Story. (1) Tell the meaning of each of the following, (2) solve, and (3) tell a suggested number story. (m) $336.6 -J- $6 = (n) $336.6 -4- 6 = (o) $336.6 -^ $.6 = (p) $336,624 -^ 6 = (q) $356 X .7 = (r) $356 x 2.7 = ' (s) At $.25 a dozen, how many lemons can I buy for $36.75 ? (t) I paid $36.75 for 25 barrels of apples. How much did the apples cost per barrel ? (u) If one acre of land is worth $36.75, how much are 5.6 acres worth ? (v) If 2 chairs cost $37.24, how much will 9 chairs cost at the same rate ? * Pupils will most readily give a number story in such work as this, by letting the figures represent units of money, thus : At $.05 each, for $375.6 I could buy 7512 tablets. When the pupils are well grounded tn this work, they may be encouraged to seek variety in their number stories. 104 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. 1. Twelve dozen are one gross. 2. At 48^ a gross, 1 dozen buttons cost cents. 3. At 5^ a dozen, 1 gross of buttons cost cents. 4. At 50^ a gross, 6 dozen buttons cost cents. 5. At 36^ a gross, 3 dozen buttons cost cents. 6. George bought pens at 50^ a gross and sold them at the rate of 2 for 1 cent ; on 1 gross he gained . (a) Henry bought pens at 60^ a gross and sold them at the rate of 3 pens for 2 cents. How much did he gain on one gross ? (b) How much on 7 gross ? REVIEW. (c) Make a receipted bill of the following goods sold by yourself to Mr. Frank H. Armstrong: Aug. 2, 15 lb. shingle nails @ 4^; Aug. 10, 6 lb. fence staples @ 5^'; Aug. 20, 130 lb. No. 9 fence wire @ 2|^^; Aug. 25, 1 keg 20d. nails @ $3.25.* 7. One fourth of a ream of paper is sheets. 8. One fourth of a ton of coal is pounds. 9. One fourth of a mile rods. 10. When coal is $6 per ton, 7000 lb. cost dollars. (d) Load of bran; gross weight, 2850 lb.; tare, 1050 lb. Find the value at $7.50 a ton. (e) Multiply 36 square feet by 24, and divide the product thus obtained by 9 square feet. Story. (f) Multiply 12 cubic feet by 15, and the product thus obtained by 6. Divide the last product by 27 cubic feet. Story. * Ask the pupil to bring to school a shingle nail, a fence staple, a piece of No. 9 wire, and a JOd. nail ; also to learu the current price of the articles named. PART I. 105 MEASUREMENTS. 1. Draw carefully upon your slate or paper, on a scale of ^ inch to the rod, a diagram of a rectangular piece of land that is 30 rods by 40 rods. (a) Find the perimeter of the land represented by the diagram you have drawn. (b) Find its area in square rods. (c) Find its area in acres. (d) How much is the land worth at $72.50 an acre ? (e) How much will it cost to build a fence around it at 45^ a rod ? (f) A strip 4 rods wide across one side of the field con- tains how many square rods ? (g) A strip 8 rods wide across one side of the field con- tains how many acres ? 2. Land 5 rods by 40 rods contains . 3. Land 7 rods by 40 rods contains . 4. Land 9 rods by 40 rods contains . Find the number of acres in each of the following : (h) 32 rods by 5 rods. (i) 32 rods by 26 rods. (j) 32 rods by 22 rods. (k) 32 rods by 34 rods. Find the number of cords in each of the following : (1) 12 ft. by 8 ft. by 4 ft. (m) 16 ft. by 6 ft. by 4 ft. (n) 20 ft. by 4 ft. by 4 ft. (o) 21 ft. by 8 ft. by 4 ft. Find the number of square yards in each of the following : (p) 30 feet by 40 feet. (q) 25 feet by 33 feet, (r) 18 feet by 21 feet. (s) 35 feet by 40 feet. Find the number of cubic yards in each of the following : (t) 12 ft. by 6 ft. by 9 ft. (u) 15 ft. by 12 ft. by 6 ft. (v) 15 ft. by 9 ft. by 12 ft. (w) 21 ft. by 6 ft. by 18 ft 106 COMPLETE ARITHMETIC. RATIO AND PROPORTION r. One inch is of a yard. 2. Two inches are of a yard. 3. Three inches are of a yard. 4. Four inches are of a yard. 5. One foot and 2 inches are of a yard. 6. One foot and 8 inches are of a yard. 7. One square rod is of an acre. 8. Two square rods are of an acre. (a) Twelve square rods are what part of an acre ? (b) Twenty-eight square rods are what part of an acre ? (c) Thirty-six square rods are what part of an acre ? 9. 100 square rods are of one acre. 1 acre is of 100 square rods, or and times 100 square rods. 10. Two thirds of 1 ft. 6 in. are 3 fourths of inches. (d) Two thirds of 6 ft. 9 in. are 3 fourths of how many in.? (e) Two thirds of 96 sq. rd. are 1 half of how many sq. rd.? 11. If 36 sq. rd. of land is worth S60, 48 sq. rd. is worth dollars. (f) If 36 sq. rd. of land is worth $75.48, how much is 48 sq. rd. worth at the same rate ? 12. If 2 ft. 3 in. of silver wire is worth 24^', 1 yard is worth cents. (g) If 2 ft. 3 in. of platinum wire is worth $1.65, how much is 1 yd. worth at the same rate ? 13. If 4 sq. rd. of land is worth $5, 1 acre is worth dollars. (ii) If 4 sq. rd. of land is worth $7.25, how much is 1 acre worth at the same rate ? PAKT I. 107 PERCENTAGE. 1. One % of 640 = 2. H% of 640 = 3. One % of 420 = 4. 2^% of 420 = 5. One % of $972 = (a) Find 7|-% of $972. 6. One % of $876 = (b) Find 9^% of $876. 7. One % of $46.80 = (c) Find 8^% of $46.80. (2) 8. 84 is 7% (7 hundredths) of . (d) 595 is 7% of what? (e) 876 is 6% of what ? 9. 63 is 9% (9 hundredths) of . (f) 747 is 9% of what ? (g) 737 is 11% of what ? 10. 72 is 12% (12 hundredths) of . (h) 572 is 11% of what? (i) 560 is 10% of what?* (5) 11. 18 is % of 300. 12. 19 is % of 300. (j) 87 is what % of 300 ? (k) 88 is what % of 300? 13. 24 is % of 400. 14. 26 is % of 400. (1) 196 is what % of 400 ? (m) 380 is what % of 400? 15. 42 is what % of 600 ? 16. 46 is % of 600. (n) 384 is what % of 600 ? (o) 576 is what % of 600? 17. Two % of 430 = (p) Find 9% of 430. 18. Twenty is 4% of . (q) 96 is 4% of what? 19. Thirty is % of 600. (r) 276 is what % of 600 ? 20. Twenty-five per cent of 24 is what % of 21. Seventy-five per cent of 24 is what % of 200? 300? *The pupil should see that he may solve this problem as he solved problem (h), or he may solve it by making use of the fact that lO^i = 1 tenth. 108 COMPLETE ARITHMETIC. PERCENTAGE. 1. A nursery-man set out 200 trees; 8% of them died; - trees were dead and trees were alive. (a) A nursery-man set out 650 trees; 8% of them died. How many trees were dead ? (b) How many lived ? 2. A farmer sold 2 1 bushels of oats ; these were 7 % of all he raised ; he raised bushels and had bushels left. (c) A farmer sold 98 bushels of oats; these were 7% of all he raised. How many bushels did he raise ? (d) How many bushels had he left ? 3. A grocer sorted 200 bushels of apples; 12 bushels were " specked " and the remainder were sound ; % of the apples had begun to decay and % of them were sound. (e) A grocer sorted 400 bushels of apples; 56 bushels were " specked " and the remainder were sound. What % of the apples had begun to decay ? (f) What % were sound ? 4. Mr. Briggs earns $900 a year; he saves 8% of his wages ; he saves dollars and spends • dollars. (g) Mr. Jones earns $1250 a year; he saves 8% of his wages. How much money does he save ? 5. Mr. Piper paid $18 for a suit of clothes ; this was 12% of his month's salary; his salary was dollars. (h) Mr. Brean paid $162 for a horse, buggy, and harness; this was 12% of his year's salary. How much was his yearly salary ? 6. George earned $300 ; he spent $15 ; he spent % of what he earned. (i) Henry earned $350 ; he spent $63. What per cent of what he earned did he spend ? PART 1. 109 REVIEW. A manufacturer employed 20 men. The daily wages of these men were as follows : Five received $3 00 each; five received $2.50 each; five received $2.00 each; and five re- ceived $1.50 each. (a) Find the whole amount paid to these- men each day. (b) Find the average daily wages of the men. (c) Find the amount paid per week to the men if every man worked every day except Sunday. (d) Find the amount paid for the month of January, if the month began on Monday and every man was present every day except the Sundays of the month. (e) Divide 336 by 4f (Change to ths.) Story, (f) Divide 336 by 4.8. (336.0 ^ 4.8 =. ?) Story. Compare the answers to problems (e) and (f). (g) Bought a ream of paper for $2.00 and sold it at 18^ per quire. How much was my gain ? (li) Bought a box (1 gross) of pens for 75^ and sold them at the rate of 4 pens for 3 cents. How much did I gain ? (i) If 10 bushels of potatoes cost $3.50, how much will 15 bu. cost at the same rate ? 1. Draw a diagram of a piece of land 32 rods by 25 rods, scale ^ in. to a rod. The diagram is inches long and and inches wide. (j) How many square rods in the piece of land described in problem 1 ? (k) How many acres ? (1) The perimeter of the field is how many rods ? (m) How much will it cost to build a fence around it at 35^ a rod ? (n) How much will it cost to plow it at $1.25 an acre ? 2. 7 is of 25. 7 twenty-fifths are — hundredths. 110 COMPLETE ARITHMETIC. MISCELLANEOUS PROBLEMS. 1. A woman exchanged 3|- lb. of butter at 20^' a pound for sugar at 5^ a pound. She received pounds of sugar. (a) A man exchanged 9 cords of wood @ $4.50 a cord for coal at $6.75 a ton. How many tons of coal did he receive ? (b) How many pounds of coal did he receive ? (c) How many loads of 3000 lbs. each did he receive ? 2. The side of a square measures 5 ft. 8 in.; the peri- meter of the square is • feet • inches. (d) The side of a square measures 47 ft. 10 in. What is the measurement of the perimeter of the square ? 3. An estate was divided among a widow and four chil- dren ; the widow received 1 third of the estate ; each child received 1 fourth of what remainded after the widow had received her share ; each child received — of the estate. (e) The estate mentioned in problem 3 was worth $7140. How much did the widow receive ? (f) How much did all the cliildren together receive ? (g) What each child received was equal to what part of what the widow received ? 4. The width of an oblong is 8. inches ; its length is twice its width ; its perimeter is inches ; its area is square inches. (h) The width of a piece of land in the shape of an oblong is 32 rods ; its length is twice its width. How many rods of fence required to enclose it ? (i) How much will the fence cost at 35^ a rod ? (j) How many square rods in the piece ? (k) How many acres in the piece ? (1) What is it worth at $27.50 an acre? 5. At $1.40 a yard, 5|- yd. of carpet cost . (m) At $1.28 a yard, how much will 27^ yd. of carpet cost ? PART I. Ill SIMPLE NUMBERS * 1. The sum of all the odd numbers from 1 to 7 inclusive is ; of all the even numbers from 2 to 8 inclusive, . (a) Add all the odd numbers from 1 to 37 inclusive. (b) Add all the even numbers from 2 to 38 inclusive. 2. 5+4 + 6 + 7 + 3+8 + 6 + 5+4 + 3 + 5 + 6 + 3 + 7-|.5-|-9_50= . (c) From the sum of 9864, 792, and 8756, subtract 4598. (d) Multiply 8754 by 27. (e) 9328 x 46 = (f) Multiply 5964 by 38. (h) Multiply 3875 by 45. (j) Multiply 874 by 126. (1) Multiply 556 by 208. (n) Multiply 734 by 160 (p) Divide 7618 by 52. (r) Divide 9351 by 54. (t) Divide 9636 by 55. (v) Divide 7532 by 56. (x) Divide 10336 by 57. (z) Divide 3960 by 27. (bb) Divide 6888 by 27. (dd) Divide 6848 by 128. (ff) Divide 5776 by 128. (hh) Divide 2772 by 144. (jj) Divide 4860 by 144. (11) Divide 6440 by 160. (nn) Divide 11800 by 160. (oo) 5952 -^ 160 = *TotheTeacheb.— If pupils are inaccurate in their work in simple numbers, give them daily practice in exercises similar to those api)earing upon this page. Lead the pupils to feel that nothing short of perfect accuracy is highly commend- able. A " 90 ji pai)er ' ' iu aiithmetic is a failure. A " 90 jJ accountant "is vjorthless. (g) 3896 X 52 = (i) 2543 X 69 = (k) 963 X 245 = (m)831x 306 = (0) 673 X 250 = (q) 12551 -.-49 = (B) 11292 -.-48 = (u) 9545^ -46 = (w) 10635 -.-45 = (y) 10505 -.-44 = (aa) 8757 -f -27 = (cc) 5649-. -27 = (ee) 4768-. -128 = (gg) 9696-. -128 = (ii) 3960 -. -144 = (kk) 7932 -. -144 = (mm) 114280 -.-160 = 112 COMPLETE ARITHMETIC. COMMON FRACTIONS. 1. Add i |, and ^. The 1. c. m. of 9, 4, and 6 is — -. (a) Add 346f, 375, 486|, 296, and 855|. 2. From 12| subtract 31^. 15| - 2^ = (b) From 2381 subtract 154i-. (c) 596| - 148|- = 3. Multiply 3| by 4. 4. Multiply 7f by 8. (d) Multiply 537| by 7. (e) Multiply 375f by 9. 5. Multiply 12 by 2f 6. Multiply 11 by 3^. (f) Multiply 874 by 2i. (g) Multiply 954 by 3J. 7. Multiply 11^ by 4i. This means ■ . (h) Multiply 3731- ^y 4i. (i) Multiply 525|- by 4f •8. Divide 8^ by |. (Change 8^ to ths.) (j) Divide 48i by |. (k) Divide 17| by |. 9. Divide 6f by If 6 f = ^. 1^ =^. (1) Divide 75f by If (m) Divide 82f by If 10. Divide 1-J (-J) by 3. 11. Divide 19 J by 3.* (n) Divide 577J by 3. (o) Divide 625-J by 3. 12. James rode his wheel at the rate of 9| mi. an hour: In 4 hours he rode and miles. In ^ hour he rode and miles. In 4^ hours he rode and miles. A locomotive moved at the rate of 49| miles an hour : (p) How far did it move in 4 hours ? (q) In 5 hours ? (r) How far did it move in ^ of an hour? (s) In ^ of an hr.? (t) How far did it move in 4^ hours ? (u) In 5} hours ? 13. If I of a yard of print is sufficient for a child's apron, 8 J yards are sufficient for aprons. 8^ ^ | = (v) 79|- yards divided by 2 J yards. * Do not change 19J to sixths. Say, rather, 1 third of 191 is 6, with a remainder of 11; 1 third of IJ (5) is i\. PART 1, 113 DECIMAL FKACTIONS/ 2 _ ■3 - 1 _ T- i = i = 2 _ f- 4 tenths. ^ = tenths. -1- = tenths. |- = — hundredths. -I = hundredths. ^ hundredths. -^ ■ tenths. ^ = • tenths. |- = • tenths. ^ = • tenths. I = tenths. |- = 6 hundredths. ^ hundredths, f hundredths. ^ hundredths, f hundredths. f ^ = — tenths, f = — hundredths. |- -J = — tenths. -J = — hundredths. -J hundredths. | hundredths. 4" hundredths. ^ hundredths, -i- 3 _ 6 ■6 ~ 1 _ T — i= 1 — -w — 5 _ • tenths. ■ tenths. ^ = • tenths. ^ =. ■ tenths. ^ = — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. — thousandths. (1) Tell the meaning of each of the following, (2) solve, and (3) tell a suggested number story. (a) Multiply $345 by .7. (c) Multiply $345 by .08. (e) Multiply $345 by 3.28. (g) Multiply $345 by .025. (i) Divide $648 by $9. (k) Divide $648 by $.9. (m) Find the cost of 7.2 tons of coal at $4.75 a ton. (n) I paid $38.25 for coal at $4.50 per ton. How many tons did I buy ? (0) I paid $663 for 156 tons of coal. What was the price per ton ? * The pupil will probably need a pencil in the solution of some of the problems in the first fourteen lines. After he is able to fill the blanks, require him to write them as suggested in the following . i = .5; J = .50; i = .500; J = .3J; i = 33i; i = .338J, etc. (b) Multiply $345 by 5.7. (d) Multiply $345 by .28. (f) Multiply $345 by .005. (h) Multiply $345 by .325. (j) Divide $648 by 9. (1) Divide $648 by $.09. 114 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. 100 lb. = 1 hundredweight (cwt.). 1. At 60<f per cwt., 1 ton of bran costs dollars. 2. At 40^ per cwt., |- ton of feed costs dollars. 3. At 60^ per cwt., ^ ton of corn meal costs . (a) At $2.50 per cwt., how much will f of a ton of flour cost ? (b) ^V of a ton ? (c) | of a ton ? 4. At $2.40 per cwt., 50 lb. of flour are worth . 5. At 50^ per cwt, 1050 lb. cost ; 850 lb. cost ; 1250 lb. cost ; 950 lb. cost ; 1150 lb. cost . (d) At $2.70 per cwt, how much will 1450 lb. cost ? (e) 1950 lb. ? (f) 1350 lb. ? 6. At $2.40 per cwt, 25 lb. of flour are worth . 7. At 60^ per cwt, 25 lb. are worth cents ; 75 lb. are worth cents; 175 lb. are worth . (g) At $2.60 per cwt, how much wiU 1225 lb. cost ? (h) 1275 lb.? (i) 1525 lb.? 8. At $2.50 per cwt, 20 lb. of flour are worth cents. 9. At 60^ per cwt, 20 lb. are worth cents; 40 lb. are worth cents ; 60 lb. are worth cents ; 80 lb. are worth cents. (j) At $2.50 per cwt, how much will 1220 lb. cost; (k) 1240 lb.? (1) 1260 lb.? 10. At $2.30 per cwt, 10 lb. of flour are worth . 11. At 80^ per cwt, 10 lb. of corn are worth cents; 20 lb. are worth cents ; 30 lb. are worth cents ; 40 lb. are worth cents ; 50 lb. are worth cents ; 60 lb. are worth cents ; 70 lb. are worth cents. (m) At $2.50 per cwt., how much will 1210 lb. cost? PAET I. 115 MEASUEEMENTS. 1. Draw carefully upon your slate or paper, on a scale of j- in. to the foot, a diagram of the floor of a room that is 17 ft. long and 14 ft. wide. (a) How many square feet in the floor ? (b) How many feet of flooring will be required for the floor if, on account of the waste and the loss in matching, you must buy 1 fourth more feet of flooring than there are square feet to be covered ? (c) Making no allowance for doors, how many feet in length are the mop-boards of the room ? (d) If this floor is to be covered with a carpet that is 1 yard wide and the strips are to "run lengthwise" of the room, how many feet long will each strip be ? (e) How many strips must be purchased ? (f ) How much of one strip must be turned under or cut off? " (g) How many feet (in length) of carpet must be pur- chased if there is no waste in matching the strips ? (h) How many yards must be purchased ? (i) How much will the carpet cost at 60^ a yard ? (j) If the figure of the carpet is such that there is waste in matching, more carpet must be purchased than would otherwise be required. How much more if the waste on each strip (except the first) is 1 foot ? (k) How much, including the waste, will the carpet cost at 60^ a yard ? (1) If the same floor is to be covered with carpeting that is 1 yard wide, costing 40^ a yard, the strips to run length- wise, and the waste in matching being 6 in. on each strip except the first, how much will the carpet cost ? 116 COMPLETE AKITHMETIC. RATIO AND PROPORTION. 1. Six inches are of a rod. 2. Twelve inches are of a rod. 3. One foot and 6 inches are of a rod. 4. Two feet are of a rod. 6. Two feet and six inches are of a rod. 6. Three feet and six inches are of a rod. 7. Four feet and six inches are of a rod. 8. One rod is of a mile. 2 rods are . 9. Three rods are of a mile. 4 rods are . (a) Twelve rods are what part of a mile ? (b) 10 rods ? (c) 20 rods ? (d) 30 ? (e) 40 rods ? . (f) 60 rods ? 10. 200 rods are of a mile. 1 mile is of 200 rods, or and times 200 rods. 11. 12 feet are of a rod. 1 rod is of 12 feet, or and times 12 feet. 12. Two thirds of 7^ feet (-i/) are 1 half of feet. 13. Two thirds of 4J ft. (|) are 1 half of feet. (g) Two thirds of 25|- ft. are 1 half of how many feet ? (h) Two thirds of 37^ rd. are 1 half of how many rods ? 14. If a piece of rubber hose 5J feet long is worth 60^, a piece 16^ feet is worth . (i) If it is worth $1.65 to make 5^ ft. of concrete walk, how much is it worth to make IQ^ ft. of similar walk ? 15. Twenty-five rods are of 37^ rods. 37|- rods are of 25 rods, or and times 25 rods. (j) If it costs $75.30 to make 25 rods of road, how much will it cost to make 37^ rods at the same rate ? PART I. 117 PERCENTAGE. {!) 1. 50% more than 60 is * 50% less than 60 =t 2. 25 % more than 60 is . 25 % less than 60 = 3. 20% more than 60 is . 20% less than 60 = (a) 50% more than $846 = (b) 50% less than $846 = (c) 25% more than $576 = (d) 25% less than $576 = (e) 2 % more than $475 = (f) 2 % less than $475 = {2) 4. 60 is 50% more than 4 60 is 50% less than .§ 5. 60 is 2 5 % more than . 60 is 2 5 % less than . 6. 60 is 20% more than . 60 is 20% less than . (g) $420 is 50% more than how many dollars ? (h)$420 is 50% less than how many dollars ? (i) $420 is 25% more than how many dollars ? (j) $420 is 25% less than how many dollars ? (k) $42 is 2 % more than how many dollars ? (1) $420 is 20% less than how many dollars ? (5) 7. 50 is % more than 40. || 30 is % less than 40.^ 8. 60 is % more than 50. 35 is % less than 70. 9. 30 is — -% more than 20. 40 is % less than 50. (m)$660 is how many per cent more than $440 ? (n) $630 is how many per cent less than $840 ? * 50^ of 60 more than 60. 50^ = J. J of 60 = 30. 60 + 30 = 90. t 50/» of 60 less than 60. 50^ = J. ^ of 60 = 30. 60 - 30 = 30. :j: Think of x as standing for the number sought {the base). Then 60 = x and 1 half of X, or f of x. Since 60 = i of x, 1 half of x equals 20, and the whole of a; = 2 times 20, or 40. g 60 = re less 1 half of x, or i of x, II 50 is 10 more than 40, or Jg o/AO more than hO. \% = \ — 'i^ff,. II 30 is 10 less than 40, or l% of UO less than W. i8 = i = 255«. 118 COMPLETE ARITHMETIC. PERCENTAGE. 1. Ira earns 80 cents a day; Ernest earns 25% more than Ira; Ernest earns a day. Arthur earns 25% less than Ira ; Arthur earns a day. (a) Mr. James earns $860 a year; Mr. Brown earns 25% more than Mr. James. How much does Mr. Brown earn ? (b) Mr. White earns 2 5 % less than Mr. James-. How much does Mr. White earn ? 2. Peter earns $60 a month ; this is 25% more than Paul earns ; Paul earns a month.* (c) Mr. Harris earns $1200 a year ; this is 25% more than Mr. Williams earns. How much does Mr. Williams earn ? 3. Eichard earns $60 a month; this is 25% less than Harry earns ; Harry earns a month, f (d) Mr. Smith earns $840 a year; this is 25% less than Mr. Eice earns. How much does Mr. Eice earn ? 4. George earns $40 a month ; Joseph earns $50 a month ; George earns per cent less than Joseph ; Joseph earns per cent more than George. J (e) Mr. Dow earns $1000 a year; Mr. Wheeler earns $1250 a year. Mr. Wheeler earns how many per cent more than Mr. Dow ? (f) Mr. Dow earns how many per cent less than Mr. Wheeler ? 15 is % more than 10. 10 is % less than 15. 20 is % more than 15. 15 is % less than 20. 25 is % more than 20. 20 is % less than 25. * Think of x as standing for what Paul earns. Then $60 = a; and 1 fourth of x, or 5 fourths of X. If $60 is J of x, then x is dollars. t Let X stand for what Harry earns. Then $60 = a; less 1 fourth of x, or i of a;. If 860 is i of X, then x is dollars. X In the first part of this problem Joseph's money is the base, and in the second paxt George's money is the base. Joseph earns $10 more than $40, or I of $40 more than $40. George earns $10 less than $50, or J of $50 less than $50. PART I. 119 REVIEW. (a) Find the sum of all the odd numbers from 59 to 73 inclusive. (b) Find the sum of all the even numbers from 64 to 86 inclusive. (c) Divide 12 by f. (d) Multiply 12 by |. Compare the answers to problems (c) and (d). 1. -|^ = hundredths. ^ = thousandths. (e) Change | to hundredths. (f) Change f to lOOOths. (g) Change f to hundredths. (h) Change | to lOOOths. (i) Change -J to hundredths. (j) Change |- to lOOOths. 2. At $3 per cwt., 500 lb. of pork cost dollars; 650 lb. cost ; 510 lb. cost ; 520 lb. cost . (k) At S3.60 per cwt., what is the value of 500 lb. of pork? (1) Of 550 lb.? (m) Of 510 lb.? (n) Of 520 lb.? 3. At 60^ a gross, 1 dozen pens cost cents ; 3 dozen cost cents; 7 dozen cost cents. (o) At $5.28 a gross, what is the value of 1 dozen fruit- cans ? (p) Of 8 dozen ? (q)At $7.20 a gross, what is the cost of 3 dozen fruit- cans ? (r) Of 3 cans ? 4. A floor 15 feet by 17 feet is to be covered with carpet that is 1 yard wide. If the strips are laid lengthwise of the room, and there is no waste in matching the figure, strips will be required, each of which is feet long. (s) How many yards of carpet will be required for the room described in problem 4 ? (t) How much will it cost at 90^ a yard? 120 COMPLETE ARITHMETIC. MISCELLANEOUS PROBLEMS. 1. The state of Illinois is about 380 miles long and 225 miles wide at its widest part. A map of Illinois drawn upon a scale of 10 miles to the inch must be inches long and and inches wide. 2. A certain county of Illinois appears upon the map described in problem 1 as a rectangular figure 1.8 inches by 3 inches. The county is miles wide and miles long. (a) How many square miles in the county described in problem 2 ? Tell the amount of change that should be given in each of the following instances : 3. Bought 4|- lb. meat @ 12^ ; gave the salesman $1. 4. Bought 3^ lb. cheese @ 16^-; gave the salesman $1. 5. Bought 2|- lb. cheese @ 16^; gave the salesman 50^. 6. Bought 2|- dozen eggs @ 14^ ; gave the salesman $2. (b) Bo't 2.3 tons coal @ $6.50 ; gave the salesman S20. (c) Bo't 1.4 tons hay @ $9.50 ; gave the salesman $15. (d) Bo't 3^ cords wood @ $4.50 ; gave the salesman $16. 7. Byron bought 8 melons for 50^; he sold one half of them at 10^ each and the other half at 5^ each; he gained cents. (e) A drover bought 36 sheep for $125 ; he sold one half of them at $4 each and the other half at $4.50 each. How much did he gain ? 8. A house rents for $10 per month ; the rent for one year and six months is dollars. (f) A house rents for $32 per month. How much is the rent for 2 years and 4 months ? PAKT I. SIMPLE NUMBERS * TICKETS OF ADMISSION SOLD AT A COUNTY FAIR. 121 Price Wed. Thur. Fri. Sat. Total. Children's Tickets Adults' Tickets 15^ 25?* 50^ 1645 2243 143 123 1154 1754 174 175 3561 3871 186 162 1424 2124 75 137 (e) Single Carriage Tickets. . Double Carriage Tickets . (a) (b) (c) (d) (a) to (d) Find the number of tickets sold each day. (e) Find the number of children's tickets sold. (f) Find the number of adults' tickets sold. (g) Find the number of single carriage tickets sold, (h) Find the number of double carriage tickets sold. (i) Find the sum of (a) to (d) inclusive, and (e) to (h) in- clusive. (j) Find the receipts for tickets sold Wednesday; (k) Thursday; (1) Friday; (m) Saturday; (n) total receipts for tickets sold. (o) Find the receipts for all the children's ticket sold dur- ing the week ; (p) adults' tickets ; (q) single carriage tickets ; (r) double carriage tickets. (s) Find the sum of (o), (p), (q), and (r), and compare this sum with the answer to problem (n). - days.f - days4 - days, (t) How many days from June 28th to Dec. 25th ? From April 20 th to Dec. 10 th ? 1. From June 28th to July 3rd it is 2. From June 28th to Aug. 3rd it is 3. From June 28th to Sept. 3rd it is (ui * Insist upon accuracy : " 90 per cent of accuracy ' t To June 30th is 2 days ; to July 3rd, 3 more days. t Two and 81 and 3. is failure. 122 COMPLETE ARITHMETIC. COMMON FRACTIONS. Reduce the following fractions to their lowest terms : 124— 18— 21— 24_ 15— 18 — ^' 3^ — 2 T — J-Q — T¥ — 3" 5" — TS - (a) 111= .(b)Hf= (c)AV= (d)/A = Reduce the following improper fractions to whole or mixed numbers : 2. V- = = -V- = l* = u = (e)^ = (0 l^A = (g) H^ = = (h) H^ = Reduce the following mixed numbers to improper : fractions : 3. 7f = 9f = 8| = 12i = iif = 101 = (i) 564 = (J) 74f = (k) 86| = (1) 73f = Reduce the following fractions to equivalent fractions having a common denominator : 4. -|, ^, and ^. The 1. c. m. of 8, 4, and 6 is . (m) I- and -J. (n) ^V and f . (o) ^% and 3%. (p) Change 47 to a fraction whose denominator is 6. 6. The sum of two numbers is 78 ; one of the numbers is 32 ; the other number is . (q) The sum of two fractions is f ; one of the fractions is |. What is the other fraction ? 7. If it takes f of a yard of cloth to make one vest, from three yards vests can be made. (r) How many vests, each containing f of a yard, can be made from 36 yards? (s) From 48 yards? (t) From GO yards ? PART I. 123 DECIMAL FRACTIONS. Change the following common fractions to decimal fractions : Il_ 1— 3— 2— 3— 4_ ^' ^ — T- T- T- 6^- J - (a)i = * (b)|= (c)-J= (d)|= (e)^ = (f) f = (g) f = (h) f = (i) 4 = (J) f = (k)i= (1)1= (mH= (n)|= (0)1 = Change the following decimal fractions to common fractions, and reduce to their lowest terms : 2. .8 = .5 = A= .25 r= .75 = .121- ^ | .3^ ^ (p) .125 = (q) .375 = (r) .625 = (s) .875 = (t).175= (u).225= (v).325= (w).675 . (1) Tell the meaning of each of the following, (2) solve, and (3) tell a suggested number story : J (x) Multiply $475 by .9. (y) Multiply $475 by 2.9. (z) Multiply $534 by .07. (aa) Multiply $534 by .37. (bb) Multiply $534 by 2.37. (cc) Multiply $534 by .003. (dd) Multiply $534 by .043. (ee) Multiply $534 by .243. (ff) Divide $724 by $8. (gg) Divide $724 by 8. (hh) Divide $724 by $.8. (ii) Divide $724 by $.08. (jj) At $12.50 per ton, find the cost of 3.7 tons of hay. (kk) At $9.75 per ton, find the cost of 3.4 tons of hay. * Change these to thousandths. There are 1(XX) thousandths in a whole (a unit) ; in J of a unit there are J of 1000 thousandths. This answer may be written, .333J or 333+. t Observe that you can divide both the numerator and denominator of 12J hun- dredths by 12i. % Do not omit the number story, or business problem. The number story for problem (x) might be : Mr. Hoyt bought .9 of an acre of land at $lt75 an acre; the land cost him dollars. 124 COMPLETE ARITHMETIC. DENOMINATE NUMBEKS. Find the cost : * 1. 3000 lb. hay at $6 per ton. 2. 500 lb. hay @ S6. 3. 1500 lath at $3 per M. 4. 250 lath @ S3. 5. 1250 lb. pork at $4 per cwt. 6. 125 lb. pork @ $4. 7. 2500 lb. coal at $5 per ton. 8. 400 lb. coal @ $5. 9. 650 lb. beef @ $5 per cwt. 10. 25 lb. beef at $5. 11. 2500 brick @ $8 per M. 12. 250 brick @ $8. Find the value : 13. 1 brick at $8 per M. 14. 100 brick @ $8. 15. 1 lb. hay at $10 per ton. 16. 100 lb. hay @ $10. 17. 1 lb. of beef @ $4 per cwt. 18. 10 lb. beef @ $4. (a) 2150 brick at $8 per M. (b) 625 brick @ $8. (c) 2150 lb. pork at $4 per cwt. (d) 625 lb. pork @ $4. (e) 2150 lb. hay at $10 per ton. (f) 650 lb. hay @ $10. (g) 3240 lath at $3 per M. (h) 750 lath @ $3. (i) 3240 lb. coal @ $5.50 per ton. (j) 760 lb. coal @ $5.50. (k) 3240 lb. beef @ $6.50 per cwt. (1) 86 lb. beef @ $6.50. 19. At 60^ a gross, 2 doz. buttons cost cents. (m) At $1.60 per gross, 75 doz. buttons cost how much ? (n) Make a receipted bill of the following goods sold by yourself to Christopher Columbus : Sept. 1, 1230 lb. pork @ $4.60 per cwt, and 1080 lb. beef @ $6.75 per cwt. 20. At $2.40 per ream, 1 quire of paper costs cents ; 12 sheets cost cents; 6 sheets cost cents. (o) Bought paper @ $2.40 per ream and sold it at 15^ a quire. What was the gain on 3 reams ? (t)) At 7^ a quire, how much will 1440 sheets of paper cost ? ♦ Pork and beef are ustially bought and sold by the hundredweight (cwt.), hay and coal by the ton, and lath and brick by the thousand (M.). PART I. 125 MEASUREMENTS. 1. Draw carefully upon your slate or paper, on a scale of ^ in. to the foot, a diagram of a rectangular grass plot 50 ft. by 40 ft. Kepresent a gravel walk 5 feet wide just outside the grass plot and extending entirely around it. The dia- gram of the grass plot is and inches long and inches wide. The width of the walk as shown in the diagram is of an inch. The entire length of the figure including the diagram of the grass plot and walk is and inches. (a) How many square feet in the grass plot described in problem 1 ? (b) How many square yards ? (c) The perimeter of the grass plot described in problem 1 is how many feet ? (d) How many yards ? (e) The perimeter of the outside of the gravel walk de- scribed in problem 1, is how many feet ? (f) How many yards ? (g) The outside perimeter of the gravel walk described in problem 1 is how much greater than the inside perimeter ? (h) How many square feet in the walk ? (i) How many square yards in the walk ? (j) How much did it cost to make the walk at 54^ per square yard ? 2. A floor 14 ft. by 17 ft. is to be covered with a carpet that is one yard wide. If the breadths are 1 7 feet long, breadths will be required. If the breadths are 14 ft. long, breadths will be required. (k) If there is no waste in matching, how many yards of carpet will be required for the room described in problem 2 if 17-foot strips are used? (1) How many yards if 14-foot strips are used ? 126 COMPLETE ARITHMETIC. RATIO AND PROPORTION. 1. One cwt. is of a ton. 2 cwt. = 2. Three cwt. are of a ton. 4 cwt. = 3. Five cwt. are of a ton. 6 cwt. = 4. Seven cwt. are of a ton. 8 cwt. = 5. Fifteen cwt. are of a ton. One ton is of 15 cwt., or and times 15 cwt. 6. Sixteen cwt. are of a ton. One ton is of 16 cwt., or and times 16 cwt. 7. Eight cwt., are of a ton. One ton is of 8 cwt., or and times 8 cwt. 8. Twelve cwt. of hay is of a ton of hay. (a) If one ton of hay is worth $12.75, how much is 12 cwt. of hay worth at the same rate ? 9. Eighteen cwt. of coal is of a ton of coal. (b) If one ton of coal is worth $6.50, how much is 18 cwt. of coal worth at the same rate ? 10. Six cwt. of flour is of a ton of flour. One ton of flour is of 6 cwt. of flour, or and times cwt. (c) If 6 cwt. of flour is worth $12.30, how much is 1 ton of flour worth at the same rate ? 11. Forty pounds are of 1 cwt. (d) If 1 cwt. of sugar is worth $5.25, how much is 40 lb. of sugar worth ? 12. Eighty pounds are of 1 cwt. 1 cwt. is of 80 lb., or times 80 lb. (e) If 80 lb. of beef is worth $5.60, how much is 1 cwt. of beef worth at the same rate ? PART I. 127 PERCENTAGE. 1. 40% more than 25 is * 40% less than 25 is . 2. 16f % more than 30 is . 16f % less than 30 is 3. 33i% more than 36 is . 33i% less than 36 is (a) 40% more than $570 = (b) 40% less than $570 = (c) 16|% more than $834 = (d) 16f % less than $834 = (e) 33-1-% more than $726 = (f) 33^% less than $726 = (2) 4. 49 is 40% more than .f 36 is 40% less than — 4 5. 42 is 16|% more than . 40 is 16f % less than 6. 48 is 33i-% more than . 30 is 33^% less than (g) $245 is 40 per cent more than how many dollars ? (h) $831 is 40 per cent less than how many dollars ? (i) $658 is 16| per cent more than how many dollars ? (j) $645 is 16f per cent less than how many dollars ? (3) 7. 70 is % more than 50.§ 50 is — % less than 60 || 8. 40 is % more than 30. 30 is % less than 50. (k) $450 is how many per cent less than $750 ? (1) $400 is how many per cent less than $480 ? (m)$480 is how many per cent more than $360 ? *40^of\vhat? iO%=l. § of 25 =10. 25 + 10 = 35. t Think of x as standing for the number sought (the base). Then 49 = z and 2 fifths of z, or I of X. Since 49 is g of a;, 1 fifth of x is 7, and the whole of a; = 5 times 7, or 85. $ 36 = a; less 2 fifths of x, or f of x. §70 is 20 more than 50, or §g of 50 more than 50. ig '= §. i = 40$^. i 50 is 10 less than 60, or Jg of 60 less than 60. |g = J, or 161 %. 128 COMPLETE ARITHMETIC. PEKCENTAGE. 1. Hiram had $50 ; Samuel had $40 ; Hiram had % more than Samuel; Samuel had % less than Hiram. (a) Mr. Cooper has 880 bushels of corn; Mr. Judd has 660 bu. Mr. Judd has how many per cent less than Mr. Cooper ? (b) Mr. Cooper has how many per cent more than Mr. Judd? 2. In 1896 a farmer raised 60 bu. of oats to the acre; this was a 50% better yield than he had in 1895; in 1895 he raised bushels to the acre.* (c) In 1896 a farmer sold 726 bushels of wheat; this was 50% more than he sold in 1895. How many bushels did he sell in 1895? 3. William earns $30 a month; Benjamin earns 40% more than Wilham ; Benjamin earns dollars a month. Joseph earns 50% more than Benjamin; Joseph earns dollars a month. (d) Isaac earns $27.50 a month; Ralph earns 40% more than Isaac. How much does Ralph earn per month ? (e) Clinton earns 50% more than Ralph. How much does CHnton earn per month ? 4. Good cheese is worth 18^ a pound; this is 50% higher than it was 1 year ago ; a year ago it was worth cents. (f) A certain brand of flour is worth $5.55 per barrel. If this is 50% more than it was worth a year ago, what was it then worth ? 18 is % more than 12. 12 is % less than 18. 24 is % more than 18. 18 is % less than 24. * Think of x as standing for the yield of 1895. Then 60 bu. = a; and 1 half of x, or I of X. Since 60 bu. = § of a;, a; = bushels. PART I. 129 REVIEW. 1. From April 25 th to May 5 th it is days. (a) How many days from April 25th to August 5 th ? 2. From Feb. 20th of a leap-year to March 10th, it is days. (b) How many days from Feb. 20th of a leap-y^ar to July 4th? (c) Divide 64 by f . (d) Multiply 64 by |. Compare the answers to problems (c) and (d). 3. ^ = hundredths. ^ = thousandths. (e) Change | to hundredths, (f) Change -f to lOOOths. (g) Change -^ to hundredths, (h) Change f to lOOOths. Change the following decimal fractions to common fractions and reduce them to their lowest terms : 4. .37|-=* .621- = .87^= .16|= .33-1- = (i) .15 = (j) .85 = (k) .65 = (l) .75 = (m) .55 = Find the cost : 5. 5500 brick at $8 per M. 6. 250 brick @ $8. (n) 7600 brick at $6 per M. (o) 520 brick @ $6. (p) There is a 4-foot gravel. walk around a rectangular grass plot that is 30 ft. by 40 ft. How many feet in the perimeter of the grass plot ? (q) How many feet in the per- imeter of the outside of the walk ? (r) How many square feet in the grass plot? (s) How many square feet in the gravel walk? (t) If one ton of coal is worth $7.50, how much is 14 cwt. of coal worth ? (u) If one hundredweight of corned beef is worth $4.50, 90 lb. of beef is worth how much? * Divide the numerator and the denominator of -^^^ by 12^. 130 COMPLETE ARITHMETIC. MISCELLANEOUS PEOBLEMS. 1. I am thinking of the surface of a box that is 6 in. long, 4 in. wide, and 3 in. high. The surface of the bottom is square inches. The surface of one side is square inches. The surface of one end is square inches. The surface of the entire box — top, bottom, sides, and ends — is square inches. (a) Find the entire surface of a box that is 15 in. long, 12 inches wide, and 8 inches high. (b) Make a receipted bill of the following goods sold by yourself to your teacher : March 6, 150 lb. sugar @ 4^^; 7 lb. coffee @ 35^; March 18, 4 lb. cheese @ 15^-; 1 vinegar barrel, $1.00; March 25, 6 gal. kerosene @ 12^; 3 gal. syrup @ 45^. (c) A farmer sold 15 head of cattle weighing 19650 lb. at $5.40 cwt. How much did he receive for them ? (d) What was the average weight per head ? (e) If there is 1 half a ream in a package of paper, how many sheets in 6 such packages ? (f) A merchant had $275 in the bank Monday morning. His deposits for the week were as follows : Monday, $86 ; Tuesday, $55 ; Wednesday, $72 ; Thursday, $64 ; Friday, $83 ; Saturday, $124. He drew from the bank during the week $354.24. How much remained on deposit ? 2. I gave 3 doz. eggs, worth 15^ a dozen, in part pay- ment for 20 lb. sugar at 4^ a pound. There remains unpaid cents. (g) I gave 2250 lb. hay at $10 per ton in part payment for 2 tons of coal @ $6.50 per ton. How much remains unpaid ? PART I. 131 SIMPLE NUMBERS. 1. The sum of two numbers is 38 ; ona of the numbers is 12 ; the other number is . (a) The sum of two numbers is 12346; one of the num- bers is 4734. What is the other number ? 2. The difference of two numbers is 17; the less num- ber is 45 ; the greater number is . (b) The difference of two numbers is 547 ; the less ' number is 3476. What is the greater number? 3. The difference of two numbers is 14; the greater number is 45 ; the less number is . (c) The difference of two numbers is 607; the greater number is 4045. What is the less number ? 4. In a problem the multiplier is 6 and 1 n^^JJiplicand. ^ ■'^ 6 multiplier, the product is $42 ; the multiplicand is . "142" product. (d) In a problem the multiplier is fifteen and the product is nine hundred forty-five. What is the multipHcand ? 5. In a problem the multiplicand is ^25 multiplicand. $25 and the product is $125; the multi- ^i-r?r- '"^^P-^®^- ^ $125 product, puer is . (e) In a problem the multiplicand is two hundred thirty- five dollars and the product is five thousand four hundred five dollars. What is the multiplier ? 6. In a problem the divisor is $12 and the quo- $12) ? tient is 8 ; the dividend is . 8 (f) In a problem the divisor is $146 and the quotient is 27. What is the dividend ? 7. In a problem the quotient is 12 and the $9) divisor is $9; the dividend is . 12 132 COMPLETE ARITHMETIC. COMMON FRACTIONS. 1. Three fourths of 36 feet are — — feet, f of 37 = (a) Three fourths of 984 feet are how many feet ? (b) Three fourths of 985 = (c) Three fourths 986 = 2. 36 feet are three fourths of feet. 37 ft. are three fourths of and feet. 38 ft. are J of . (d) 576 ft. are three fourths of how many feet ? (e) 577 is three fourths of what? (f) 578 is f of what? 3. Paul spent ^ of the money his father gave him for a book and ^ of it for a knife, and had 12^ left. Before he spent any money he had cents.* (g) A man spent ^ of his month's wages for fuel and J- of it for groceries, and had $17.25 left. How much was his wages ? (h) How much did he spend for fuel ? (i) How much did he spend for groceries ? 4. Harris paid $4 for chickens at f of a dollar each. He bought chickens. 4 ^ |- means, find how many times 2 fi/ths are contained in 4 (20 fifths). (j) A man paid $375 for wheat at $f a bushel. How many bushels did he buy ? (k) A man paid $368 for apples at $lf a barrel. How many barrels did he buy ? 5. Ellis sold 2|- cords of wood at $4|- a cord ; he should receive for the wood and dollars. (2^ times 4|- means, 2 times 4^ and 1 half of 4|-.) (1) A man sold 12^ acres of land at $56|^ an acre. How much should he receive for the land ? * Think of x as standing for the money his father gave him. Then he spent 1 half of X and 1 third of x, or 5 sixths of x, and had of x remaining. PART I. 133 DECIMAL FRACTIONS. 1. At S60 an acre, 3.2 acres of land are worth dollars. 3.2 times 60 means, 3 times 60, plus .2 of 60. (a) At 185 an acre, how much are 6.7 acres of land worth ? (b) 6.2 acres at $75 an acre ? 2. At $70 an acre, 3.02 acres of land are worth . 3.02 times 70 means, 3 times 70, phcs 2 hundredths of 70. (c) At $65 an acre, how much are 5.03 acres of land worth? (d) 8.06 acres at $95 an acre? 3. At $40 an acre, 2.24 acres of land are worth dollars. 2.24 times 40 means, 2 times 40, plus 2 tenths of 40, plus 4 hundredths of 40. (e) At $45 an acre, how much are 4.35 acres of land worth ? (f) 7.26 acres at $64 an acre ? Find the cost : (g) 5.34 acres @ $265. ^265 Price per acre. .-IN no A ^ ciAat:^ 5.34 Number of acres, (h) 7.34 acres @ $465. (i) 6.23 acres @ $52.5. $10.60 Value of .04 of an acre. r\ n r\o ^ (3>ooc; $79.5 Value of .3 of an acre. 0) 7.03 acres @ $325. ^^3^5^ Value of 5 acres. (k) 3.27 acres @ $43.5. (1) 5.37 acres @ $54.6. $1415.10 Value of 5.34 acres. (m)1.56 acres @ $276. note.— Whlle the pupil is multiplying by (Xi) 24.3 acres &> S342 ^'^ separatrix may stand between the 2 and ^ ^ ' ' 6 of the multiplicand; thus, $2'' 65. This will (o) 32.6 acres @ $41.6. help him to remember that he is really multi- plying $2.65, the value of 1 hundredth of an acre, by 4. When he is ready to multiply by 3, the separatrix should be erased and written thus: $26^5. This will help him to re- member that he is really multiplying $26.5, the value of 1 tenth of an acre, by 3. After a time he can simply imagine the separatrix in its place. Require the pupil to write the decimal point in each partial product when, in the process of multiplica- tion, he reaches the place where it belongs. The pupil may now be taught that when he has solved a problem in multiplication of decimals, if he has " pointed off" correctly, the decimal places in the product will he equal to those in the multiplicand and mu>liplier counted together. 134 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. 1. From March 26th to April 2d, it is days. If March 26th is Monday, April 2d is . 2. From April 20th to May 5th, it is days, or weeks and day. If April 20th is Friday, May 5 th is 3. From April 20th to May 12th it is days, or weeks and day. If April 20th is Friday, May 12th is (a) How many days from April 20th to June 9 th ? (b) If April 20th is Friday, what day of the week is June 9th ? (c) June 12th ? 4. In a year in which there is a Feb. 29 th, there are days, or weeks and days. 5. In a year in which there is no Feb. 29th, there are days, or weeks and day. 6. If the first day of February of a common year is Mon- day, the first day of February of the next year is . 7. If the first day of February of a leap-year is Monday, the first day of February of the next year is . 8. If the tenth of February of a leap-year is Saturday, the tenth day of February of the next year is . 9. If the 17th day of April of a leap-year is Wednesday^ the 17th day of April of the next year is . Find how many weeks and days : (d) Apr. 10 to July 4.* (e) May 5 to July 10. (f) May 18 to Aug. 5. (g) June 15 to Oct. 4. (h) July 12 to Sept. 1. (i) Aug. 1 to Sept. 25. ♦ Think as suggested in the following ; April 10 to Apr. 30, 20 days; Apr. 30 to May 31, 31 days , May 31 to June 30, 30 days ; June 30 to July 4, 4 days. 20 days + 31 days + 30 days + 4 days = ? PAKT I. 135 MEASUREMENTS. The following diagram of a house and lot is drawn on a scale of 24 feet to an inch. Norlh side of Lot (a) How many feet long is the lot, not including the walk ? (b) How many feet wide is the lot ? (c) How many feet from the sidewalk to the house ? (d) How far from the house to the back of the lot ? (e) How far from the house to the north side of the lot ? (f) How many feet long is the house ? (g) How many feet wide is the front of the house ? (h) How many feet wide is the rear of the house ? (i) How far from the south side of the lot to the house ? (j) How wide is the sidewalk ? (k) How much will the sidewalk cost at 12^ per square foot ? * (1) How much is the lot worth at $25 a foot front ? f * There are 306 square feet in the walk. In finding the cost the pupil may think that at 1^ a foot it would cost $3.06, and at 120 a foot, 12 times $3.06 ; or he may think that if 1 foot costs 12(?, 306 feet would cost 306 times 12^. t The expression "■foot front " stands for a strip 1 foot wide and as long as the lot is "deep." 136 COMPLETE ARITHMETIC. RATIO AND PEOPORTION. 1. One ton is and times 800 lb. If 800 lb. of hay is worth $3.00, 1 ton is worth dollars. (a) If 800 lb. of flour is worth $18.50, how much is 1 ton worth at the same rate ? 2. One mile is times 80 rd. If it costs $30 to build 80 rd. of fence, to build one mile of similar fence will cost dollars. (b) If it costs $375 to make 80 rd. of road, how much will it cost to make 1 mile of road at the same rate ? 3. One hour is and times 24 minutes. Harry rode 5 miles in 24 minutes ; this was at the rate of an hour. (c) A locomotive moved 19 miles in 24 minutes. This was equal to what rate per hour ? 4. One minute is times 20 seconds. A pump rod made 18 strokes in 20 seconds; this was at the rate of strokes a minute. (d) A certain pulley revolves 106 times in 20 seconds. What is its rate of revolution per minute ? (e) How many times does it revolve in one hour ? (f) Find how many times your pulse beats in 20 seconds. This is at what rate per minute ? Per hour ? 5. One pound is and times 12 ounces. If 12 oz. cheese is worth 15^, 1 lb. is worth . (g) If 12 ounces of onion seed is worth 72^, how much is 1 pound worth ? 6. If 50 sq. rods of land is worth $20, at the same rate 1 acre is worth dollars. (h) If 50 square rods of land is worth $345, how much is 1 acre worth at the same rate ? PART I. 137 PERCENTAGE. w 1. 12^% more than 56 is . 12^% less than 56 = 2. 10% more than 60 is . 10% less than 60 = 3. 30% more than 50 is . 30% less than 50 = (a) 121-% more than $972 = (b) 12^-% less than $972 = (c) 10% more than $845 = (d) 10% less than $845 = (e) 30% more than $650 = (f) 30% less than $650 = (^) 4. 45 is 12|% more than .* 49 is 12|-% less than -t 5. 77 is 10% more than . 54 is 10% less than 6. 39 is 30% more than . 28 is 30% less than (g) $747 is 12^ per cent more than how many dollars ? (h) $868 is 12|- per cent less than how many dollars ? (i) $572 is 10 per cent more than how many dollars ? (j) $729 is 10 per cent less than how many dollars ? (k) $299 is 30 per cent more than how many dollars ? (5) 7. 88 is % more than 80. 56 is % less than 64. 8. 63 is % more than 56. 35 is % less than 50. (1) $350 is how many per cent less than $500 ? (m) $450 is how many per cent more than $400 ? * Think of x as standing for the number sought. Then 45 = a; and 1 eighth of x, or I of X. Since 45 is 9 eighths of x, 1 eighth of a; is 5, and the whole of a; is 8 times 5 = 40. 1 49 = X less 1 eighth of x, or | of x. Since 49 is § of x, 1 eighth of x Is 7, and the whole of X is 8 times 7 = 56. 138 COMPLETE AKITHMETIC. PERCENTAGE. 1. Five years ago the population of a certain city was 6000; it has increased 33-|- % ; its present population is . (a) Eight years ago the population of a certain city was 9750; it has increased 33i-%. What is its present popula- tion? •2. A man sold a horse for $100 which was 25 % more than he gave for it. He gave dollars for the horse. (b) A man sold a farm for $6825 which was 25% more than he gave for it. How much did he give for the farm ? 3. Ahce has 50^; Jane has 60^; Jane has % more than Alice; Alice has % less than Jane.* (c) Mr. Lyon has 600 acres of land; Mr. Whitney has 400 acres. Mr. Lyon has how many per cent more than Mr. Whitney ? (d) Mr. Whitney has how many per cent less than Mr. Lyon? 4. By selling a horse for $60, the owner would lose 25%; the horse cost dollars. (e) By selling a farm for $4320, the owner would lose 25%. How much did the farm cost him? 5. By selling a horse for $60, the owner would gain 25%; the horse cost dollars. (f) By selling a farm for $5300, the owner would gain 25%. How much did the farm cost him? 6. 25% of the vinegar in a cask leaked out and 36 gal. remained; before the leakage there were gallons. (g) 25% of the water in a tank leaked out and 465 gallons remained. How many gallons in the tank before the leakage? *lu oue paxt of this problem 50 is the Dase; in the other part, 60 is the base. PART I. 139 REVIEW. 1. The sum of three numbers is 27 ; two of the numbers are 8 and 1 ; the other number is . (a) The sum of three numbers is 2756 ; two of the num- bers are 784 and 975. What is the other number? 2. Arthur rode on his bicycle three consecutive hours ; the first hour he rode 12 miles; the second hour, 10 miles, and the third hour 8 miles ; his average speed per hour was miles. (b) The attendance at a certain school for one week was as follows : Monday, 35 ; Tuesday, 38 ; Wednesday, 37 ; Thursday, 36 ; Friday, 34. What was the average daily attendance ? 3. At $74 per ton, 2 J tons of coal cost .* (c) At $348 1- an acre, how much will 5|- acres cost ? (d) At $348.50 an acre, how much will 6.5 acres cost ? (e) Compare the answers to (c) and (d). How much is their difference ? 4. The first day of January, 1897, was Friday. Tell the day of the week of the first day of January of each of the following years : 1898, ; 1899, ; 1900, ; 1901;t ; 1902, ; 1903, ; 1904, ; 1905, ; 1906, ; 1907, ; 1908, ; 1909, ; 1910, . (f) Upon what day of the week will the first day of Janu- ary, 1925, occur? 5. A plot of a certain garden is drawn on a scale of 20 feet to an inch. A line 3| inches long represents feet. (g) A certain map is drawn on a scale of 25 mi. to an inch. A line 15^ in. long represents how many miles ? * 2 times 7J and 1 half of 7i. t Remember that the year 1900 is not a leap year. 140 COMPLETE ARITHMETIC. MISCELLANEOUS PROBLEMS. 1. In a pane of glass 9 in. by 12 in. there are sq. inches. (a) How many square incnes in 36 panes of glass each 9 in. by 12 in.? (b) How many square feet ? 2. Mr. Black received $30 per month as rent for a house. In one year he received dollars. (c) At $35 per month, how much is the rent of a house for 2 years and 6 months ? 3. I paid $2.00 for coffee at 25^ a pound; I purchased pounds. (d) Paid $34.75 for coffee at 25^ a pound. How many pounds were purchased ? 4. In a floor 12 ft. by 12 ft. there are square feet; there are square yards. (e) In a lot 24 feet by 96 feet, there are how many square feet ? (f ) How many square yards ? 5. If a train moves at the rate of 20 miles an hour, to move 110 miles will require — — hours. (g) If a train moves at the rate of 35 miles an hour, how long will it take to go 1000 miles? 6. A boy bought 10 chickens for 25^ each, and 10 for 35^ each; the average price paid was • cents. (h) A man bought 10 horses at $135 each and 10 at $124.50 each. Wliat was the average price? 7. A man sold a horse at |- of what it cost him, thereby losing $10; the horse cost him dollars; he sold it for dollars. (i) A man sold a farm for f of what it cost him, thereby losing $1275. How much did the farm cost him? (j) For how much did he sell it ? PART I. .141 SIMPLE NUMBERS. Review page 11. 1. Name five odd numbers; five even numbers. 2. Name three exact divisors of 36; of 48; of 75. Review page 21. 3. Name five integral numbers ; five fractional numbers ; five mixed numbers. 4. ^f and .7 are numbers. 4^ and 7.2 are . Review page 31. 5. Name five prime numbers ; five composite numbers. 6. Which of the following are prime and which are com posite? 2, 22, 5, 37, 45, 49, 53, 72, 87. Review page 41. .7. What are the prime factors of 36 ? Of 35 ? Of 34 ? Of 33? 8. Of what number are 2, 2, 3, and 5 the prime factors* Review page 51. 9. Name three common multiples of 4 and 6. 10. What is the least common multiple of 4 and 6 ? Review page 61. (a) Multiply 746 by 20. (b) Multiply 394 by 80. (c) Multiply 547 by 300. (d) Multiply 834 by 70G. Review page 71. (e) Divide 891 by 30. (f) Divide 1265 by 50. (g) Divide 728 by 40. (h) Divide 2478 by 70. Review page 81. (i) A farmer bought 30 sheep ; for 5 of them he paid $6 per head ; for 1 he paid $5 per head ; for the remainder he paid $70. How much did the 30 sheep cost him ? (j) What was the average price per head ? 142 COMPLETE ARITHMETIC. COMMON FRACTIONS. Review page 12. 1. Name three fractions that have a common denom- inator ; three that do not have a common denominator. 2. Change the following to equivalent fractions having a common denominator: | and f. Review page 32. 3. Tell the terms of each of the following : 4, |, y^ 4. Keduce each of the following to its lowest terms : ^^, 18 15 45 27 /o\240 /K\ 5 75 TY' YO^ ¥T' 3" 3* K"^) SWO' V"/ '^JZ- 5. Eeduce each of the following to a whole or mixed Tinmbpr- il 25 24 32 97 /p\335 /A\ 4 76 Review page 42. 6. Eeduce each of the following to an improper fraction : 9f,7|,llf,5f (e)28f. (f) 47|. (g) 94f (h) 86^- Review page 52. (i) 7 1 5 _ (J) tV + tV = (k)A + A (1) i - i = (m)TV-A = (n) tV - tV (o) |x 1 = (p) A- X f = (<1) 4 X i « \i^ i = (s)tV^tV = (t) T^TT^A Review pages 62 and 72. (u) Find the product of 794^ multiplied by 6^.* (v) Find the quotient of 835 1- bu. divided by 2^ bu.* (w)Find the quotient of 654^ bu. divided by 9.* * Solve, and tell a suggested number story. '6.40 3.7 PART I. 143 DECIMAL FRACTIONS. Review pages 13, 23, 33, 43, 53, 63, 73, and 133. Observe again the fact that when a problem in multiplication of decimals has been solved ac- 4.480 curately, the number of decimal places in the 19.20 product is equal to the number of decimal places 23 680 ^^ ^^^ multiplicand and multiplier counted to- gether. This fact should be used as a test of ^73.42 the accuracy of the work rather than as a rule 3,56 for "pointing off." AAr.rn Observe that when the decimal point in the Q<^7in ^^^^ partial product has been located, the re- 99rk 9A mainder of the "pointing off" may be done '- mechanically by placing the point in each of the 261.3-752 other partial products and in the complete product, directly under the one in the first par- -00 5)38.455 -* iial product. 7691. Review pages 83, 93, and 103. .05)38. 4 7-51 ^^^ abstract work in division of decimals may rjnQ r be regarded as belonging to Case I. ; that is, the pupil may consider that he is to find how many 5^)38 4- 75t times the divisor is contained in the dividend. Before beginning to divide, place a separa- 76.95 trix {^) in the dividend immediately after that figure in the dividend that is of the same denom- 5 )oo. 47o§ ination as the right hand figure of the divisor. 7.695 When in the process of division this separatrix is reached, the decimal point must he written in .5)78.0" II the quotient. 156. *Find how many times 5 thousandths are contained in 38455 thofusandths. fFind how many times 5 hundredths are contained in .05)78.00 ^ 3847 hundredths. ~TTTT~ J Find how many times 5 tenths are contained in 384 15 dU. tenths. § Find how many times 5 units are contained in 38 units. j! Find how many times 5 tenths are contained in 780 tenths. If Find how many times 5 hundredths are contained in 7800 hundredths. 144 COMPLETE ARITHMETIC. DENOMINATE NUMBERS. Review page 14. 1. One half a ton is lb. 1 tenth of a ton = lb. 1 hundredth of a ton = lb. 1 thousandth of a ton = lb. Review page 24. 2. A bushel of wheat weighs pounds. (a) 72 bu. of wheat weigh how much more than 2 tons 1 Review page 34. (b) Change 3.26 tons to pounds. (c) 4.7 tons are how many pounds ? (d) Change 3264 lb. to tons, (e) 5624 lb. to tons. Review page 44. (f) Find the cost of 7360 lb. coal at $7.25 per ton. (g) Find the cost of 5360 lb. hay at $9.50 per ton. Review page 54. (h) Change 28 rods to feet, (i) 7 miles to rods, (j) Change 506 yd. to rods, (k) 2880 rods to miles. Review page 64. (1) The gross weight of a load of bran was 2850 lb. ; tare 1275 lb. Find the cost at $8 per ton ? (m)The gross weight of a load of oats was 2970 lb.; tare 1050 lb. How many bushels ? (n) Find the cost at 25^ a bushel. Review page 74. (o) A mountain is 11000 ft. high. How many feet more than 2 miles high is it ? (p) A mountain is 5 mi. high. How many feet high is it ? PART I. 145 MEASUREMENTS. Review pages 15 and 25. 1. Which is the larger, a five foot square or an oblong 4 feet by 6 feet ? (a) Which is the larger, a 25 ft. square or an oblong 26 ft. by 24 ft.? (b) Find the area of a 15 ft. square. (c) Find the solid content of a 1 5 ft. cube. Review page 35. 2. Every rectangular figure has sides and right angles. If the sides are equal, the figure is a . If two of the sides are longer than the other two, the figure is an . 3. Draw a 4-sided figure that is neither a square nor an oblong. Is the figure you have drawn rectangular ? 4. All the angles of a square are angles. 5. All the angles of an oblong are angles. 6. Angles that are not right angles are either angles or angles. Review page 45. 7. Every rectangular solid has faces. These faces may be either squares or . If they are all squares the sohd is a . 8. Cut from a potato or a turnip a solid with six faces, some of which .are not rectangular. Observe the acute angles and the obtuse angles. Review pages 55, 65, 75, and 85. (d) In 2^ cords there are how many cubic feet? (e) In 2 1- acres there are how many square rods ? 146 COMPLETE ARITHMETIC. RATIO AND PROPORTION. Review pages 16, 26, 36, 46, 56, 66, 76, 86, and 96. 1. A 1-ft. square equals what part of a 2 -ft. square ? 2. A 1-ft. cube equals what part of a 2-ft. cube? 3. A 2-ft. square equals what part of a 3-ft. square ? 4. A 2-ft. cube equals what part of a 3-ft. cube ? 5. A 2-yd. square equals how many times a 1-yd. square ? 6. A 2-yd. cube equals how many times a 1-yd. cube ? 7. A 3-yd. sq're equals what part of a 4-yd. square ? (a) A 10-rod square equals what part of a 12-rd. sq're ? (b) A 12-rd. sq're equals what part of a 16-rd. sq're ? (c) A 12-rd. sq're equals what part of a 24-rd. sq're ? (d) A 12-rd. sq're equals what part of a 36-rd. sq're ? 8. A 3-ft. cube equals what part of a 4-ft. cube ? (e) A 3-ft. cube equals what part of a 5 -ft. cube ? (f) A 3-ft. cube equals what part of a 6 -ft. cube ? (g) A 3-ft. cube equals what part of a 9 -ft. cube ? 9. A l-ft. square equals what part of a 1-ft. square ? (h) A ^-ft. cube equals what part of a 1-ft. cube ? 10. A 1-ft. sq're equals how many times a ^-ft. sq're ? (i) A 1-ft. cube equals how many times a i-ft. cube ? 11. The surface of a 1-foot cube equals what part of the surface of a 2 -foot cube ? (j) The surface of a 2-ft. cube equals what part of the surface of a 3-ft. cube ? (k) The surface of a 1-ft. cube equals what part of the surface of a 1-yd. cube ? (1) If a 1-inch cube of silver is worth $3.60, how much is a 3 -inch cube of the same metal worth ? PART I. 147 PERCENTAGE. Review pages 17 and 18. 1. 25% of 24 is . . (a) Find 25% of $3479. 2. 24 is 25% of . (b) S3479 is 25% of what? 3. 8 is % of 24. (c) 75 is what % of 375 ? Review pages 27 and 28. 4. 121-% of 72 is . (d) Find 12J% of $650. 5. 72 is 12% of . (e) $650 is 12i-% of what? 6. 12 is % of 72. (f) 35 is what % of 245 ? Review pages 37 and 38. 7. 10% of 45 is . (g) Find 10% of $725. 8. 45 is 10% of . (h) $725 is 10% of what ? 9. 5 is % of 45. (i) 55 is what % of 440 ? Review pages 47 and 48. 10. 66|% of 48 is . (j) Find 66f of $756. 11. 48 is 66f% of . (k) $756 is 66|% of what? 12. 36 is % of 48. (1) $450 is what % of $600 ? Review pages 57 and 58. 13. 60% of 75 is . (m)Find 60% of $810. 14. 75 is 60% of . (n) $810 is 60% of what? 15. 20 is % of 40. (o) $84 is what % of $210 ? Review pages 67 and 68. 16. 80% of 60 is . (p) Find 80% of $640. 17. 60 is 80% of . (q) $640 is 80% of what? 18. 50 is % of 60. (r) $550 is what % of $660 ? Review pages 77, 78, 87, and 88. 19. 37|^% of 24 is . (s) Find 37^% of $576. 20. 24 is 37^% of . (t) $576 is 37^% of what ? 21. 24 is % of 80. (u) $675 is what % of $750 ? 148 COMPLETE ARITHMETIC. PERCENTAGE. Review pages 97 and 98. 1. One % of 357 = (a) Find 13% of 357. 2. 39 is 3% of . (b) 264 is 8% of what ? 3. 15 is % of 300.* (c) 60 is what % of 750 ?t Review pages 107 and 108. 4. One % of 736 = (d) Find 8 J% of 736. 5. 108 is 9% of . (e) 375 is 5% of what ? 6. 57 is % of 300. (f) 41.5 is what % of 830? Review pages 117 and 118. 7. 25% more than 80 = 25% less than 80 = 8. 40 is 25 % more than 4 45 is 25 % less than . 9. 75 is % more than 60.§ 60 is % less than 75. 10. AUce has $40 ; Jane has $50 ; Mary has $60. (g) Jane has what per cent more than Ahce ? (h) Mary has what per cent more than Jane ? (i) Mary has what per cent more than Alice ? (j) Jane has what per cent less than Mary ? (k) Alice has what per cent less than Jane ? (1) Alice has what per cent less than Mary ? (m) Alice's money equals what % of Jane's money ? (n) Alice's money equals what % of Mary's money ? (o) Jane's money equals what % of AUce's money ? (p) Jane's money equals what % of Mary's money ? (q) Mary's money equals what % of Alice's money ? (r) Mary's money equals what % of Jane's money ? * First find 1% of 300. t First find l5i of 750. $ Let a; = the number sought, the base ; then 40 = a; and 1 fourth of x, or I of z. I 75 is how many more than 60 ? 15 is what ji of 60 ? PART 1. 149 Review pages 113 and 123. 1' i= hundredths, (a) -J = thousandths. (b) Change .275 to a common fraction and reduce it to its lowest terms. (c) .375. (d) .425. (e) .575. (f) .625. Review pages 133 and 143. (g) Find the cost of 6.28 acres of land at $2.75 an A.* (h) Find the cost of 3.46 tons of coal @ $6.75 a ton. i) Divide 6.25 by 5. (6.^25 -^ 5 units.) j) Divide 6.25 by .5 (6.2^5 ^ 5 tenths.) k) Divide 6.25 by .05. (6.25V 5 hundredths.) 1) Divide 36 by 5. (36' divided by 5 units.) m) Divide 36 by .5. (36.0' -^ 5 tenths.) n) Divide 36 by .05. (36.00' ^ 5 hundredths.) o) Divide 36 by .005. (36.000' -^ 5 thousandths.) p) Divide 57.26 by 7. (57.'26 divided by 7 units.) q) Divide 57.26 by .7. (57.2'6 h- 7 tenths.) r) Divide 57.26 by .07. (57.26' -^ 7 hundredths.) s) Divide 57.26 by .007. (57.260' -^ 7 thousandths.) t) Divide 67.5 by 25. (67.'5 divided by 25 units.) u) Divide 67.5 by 2.5. (67.5' -^ 25 tenths.) v) Divide 67.5 by .25. (67.50' -4- 25 hundredths.) w) Divide 67.5 by .025. (67.500' -^ 25 thousandths.) x) Divide 6.75 by 25. (25 units in 6 units = 0., etc.) y) Divide 6.75 by 2.5. (6.7'5 -^ 25 tenths.) * Reqxiire the pupil to put the work on the blackboard and to explain by telling (1) the cost of 1 hundredth of an acre; (2) of 8 hundredths; (3) of 1 tenth; (4) of 2 tenths; (5) of 1 acre; (6) of 6 acres; (7) of 6.28 acres. How many decimal places in the product? How many in the multiplicand ? How many in the multiplier ? CONTENTS— PART II. Pages Notation, - . . . . . . I5i_i58 Addition, . 161-168 Subtraction, ---._.. 171-178 Multiplication, - - - - . . 181-188 Division, ----... I9i_i98 Properties of Numbers, - - . _ 201-206 Divisibility of Numbers, - - - _ . 211-216 Fractions, - - - 221-228,231-238,241-248,251-256 Percentage, 261-266,271-276 Discounting Bills, ----- 281 Discounts from List Price, - - - - 282 Selling on Commission, - - - - 283 Taxes, ----... 284 Insurance, --.... 285 Interest, - - - - . ... 291-296 Promissory Notes, - - - . . 301-306 Stocks and Bonds, - - - . _ 311-316 Ratio and Proportion, - . . . 321-328, a31-338 Powers and Eoots, ----- 341-348, 351-358 Metric System, ------ 361-368 Algebra, - - - 157, 158; 167, 168; 177, 178, etc. Geometry, - - - . 159, 169, 179, 189, 199, 209, etc. Miscellaneous Problems, - 160, 170, 180, 190, 200, 210, etc. 150 PART II. NOTATION. 1. The expression of numbers by symbols is called notation. 2. In mathematics two sets of symbols are employed to represent numbers ; namely, ten characters — 1, 2, 3, 4, 5, 6, 7, 8, 9, — called figures ; and the letters a, h, c, d, . . . X, y, z. Note. — The figures from 1 to 9 are called digits. The term significant figures is sometimes applied to the digits. The tenth character (0) is called a cipher, zero, or naught. THE ARABIC NOTATION. 3. The method of representing numbers by figures and places is called the Arabic Notation. It is the principle of 'position in writing numbers that gives to the system its great value. s ,' IS I III u c3 +3 02 ^ ^ 02 oaJo ?5.SS (Da)a>.-ga)a<x> ^T^^^oj'^rS z^ ^^ ^ ^ ^-^ ^ ^ ^^ Ph-S ^2 OOO^OOO 1 1 S ^ 1^ § 1 .^ .^ .^ I .-§ .^ .-§ S*r^"^'-i^^lrf d G ^ -^ iz: c a 2 4 3 8.596 2438.596 151 152 COMPLETE ARITHMETIC. 4. A figure standing alone or in the first place represents primary units, or units of the first order; a figure standing in the second place represents units of the second order; a figure standing in the third place represents units of the third order ; a figure standing in the first decimal place rep- resents units of the first decimal order, etc. 5. The following are the names of the units of eight orders : Fourth decimal order . . . ten-thousandths. Third decimal order . . . thousandths. Second decimal order . . . hundredths'. First decimal order .... tenths. DECIMAL POINT. First order primary units. Second order tens. Third order hundreds. Fourth order thousands. 6. In a row of figures representing a number (342.65), the figure on the right represents the lowest order given ; the figure on the left, the highest order given. In general, any figure represents an order of units higher than the figure on its right (if there be one), and lower than the figure on its left (if there be one). 7. Ten units of any order equal one unit of the next higher order; thus, ten hundredths equal one tenth; ten tenths equal one primary unit, etc. 8. The naught, or zero, is used to mark vacant places; thus, the figures 205 represent 2 hundred, no tens, and 5 primary units. PART II. 153 Note 1. — Observe that a figure always stands for units. If it occupies the first place, it stands for primary units ; if it occupies the second place, it stands for tens (that is, units of tens); the third place, for hundreds ; the first decimal place, for tenths ; the second decimal place, for hundredths, etc. Thus, a figure 5 always stands for five — jive primary units, jive thousand, jive hundredths, jive tenths, according to the place it occupies. Note 2. — In reading integral numbers, the primary unit should be, and usually is, most prominent in consciousness. Thus, the number 275 is made up of 2 hundreds, 7 tens, and 5 primary units ; but 2 hundreds equal two hundred (200) primary units, and seven tens equal seventy (70) primary units ; these (200 + 70 + 5) we almost unconsciously combine in our thought, and that which is present in consciousness is 275 primary units. So in the number 125,246, there are units of six orders, which we reduce in thought to primary units, and say, one hundred twenty-five thousand two hundred forty-six primary units. Note 3. — In reading decimals, too, the primary unit should be prominent in consciousness. Thus, .256 is made up of 2 tenths, 5 hundredths, and 6 thousandths ; but 2 tenths equal 200 thousandths, and 5 hundredths equal 50 thousandths ; these (200 + 50 + 6) we combine in our thought, and that which should be present in con- sciousness is 256 thousandths of a primary unit. 9. Exercise. Write in figures : 1. Two hundred fifty-four thousand one hundred. 2. One hundred seventy-five and two hundred six thou- sandths. 3. Eighty-four and three hundred five thousandths. 4. Three hundred seven and eighty-seven hundredths. 5. Seven thousand four hundred twenty-four. 6. Twenty-four thousand six hundred fifty-one. 7. One hundred thirty-five thousand two hundred, (a) Fiifid the sum of the seven members. 154 COMPLETE ARITHMETIC. 10. Exercise. Read in two ways as suggested in the following : 324.61. (1) 3 hundreds, 2 tens, 4 primary units, 6 tenths, 1 hundredth. (2) Three hundred twenty-four and sixty-one hundredths. Use the word and in place of the decimal point only. 1. 2746.2. 5. 2651.4. 2. 546.85. 6. 80.062. 3. 24.006. 7. 2085.7. 4. 1.6285. 8. 120.08. 11. Exercise. Observe that any number may be read by giving the name of the units denoted by the right-hand figure to the entire number ; thus, 146 is 146 primary units ; 21.8 is 218 tenths ; 3.25 is 325 hun- dredths. 1. 27 = 2 tens + 7 primary units = 27 primary units. 2. 2.7 = 2 primary units + 7 tenths = tenths. 3. .27 = 2 tenths + 7 hundredths = hundredths. 4. .027 = 2 hundredths + 7 thousandths = thou- sandths. 5. .436 = 4 tenths + 3 hundredths + 6 thousandths = thousandths. 6. 5.247 = 5 primary units + 2 tenths + hundredths -}- 7 thousandths = 5247 ths. 7. 3.24 = hundredths. 8. 5.206 = thousandths. 9. 25.13 = hundredths. 10. 14.157 = thousandths. 11. 275.4 = tenths. Note. — Exercise 11 and Exercise 12 are important as a prepar- ation for the clear understanding of division of decimals. PART II. 155 12. Exercise. Observe that any part of a number may be read by giving the name of the units denoted by the last figure of the part to the entire part; thus, 24.65 is 246 tenths and 5 hundredths; 14.275 is 1427 hundredths and 5 thousandths. In a similar manner read each of the following : 1. 2.75 = tenths and hnndredths. 2. 32.46 = • tenths and hundredths. 3. 1.425 = hundredths and ■ thousandths. 4. 24.596 = tenths and thousandths. 5. 321.45 = tenths and hundredths. 6. 14.627 = hundredths and thousandths. 7. 2.6548 = hundredths and ten-thousandths. 13. Exercise. Observe that in reading a mixed decimal in the usual way, we divide it into two parts and give the name of the units denoted by the last figure of each part to each part ; thus, 2346.158 is read 2346 (primary units) and 158 thousandths. Read the following in the usual manner. Do not use the word and in reading the numbers in the second column : 1. 200.006. .206. 6. 800 and 24. 824. 2. 400.0005. .0405. 7 9000 and 6. 9006. 3. 500.025. .525. 8. 2400 and 8. 2408. 4. 200 and 40. 240. 9. 17000 and 4. 17004. 5. 700 and 35. 735. 10. 46500 and 40. 46540. 14. Exercise. Write in figures : 1. Two hundred and eight thousandths. 2. Two hundred eight thousandths. 3. Six hundred and twelve thousandths. 4. Six hundred twelve thousandths. 156 ^ COMPLETE ARITHMETIC. 15. Reference Table. 1 o 'A I a o 1 xh 1 "3 a a .2 1 .2 o 9 O H 5 157,896,275,832,456,297,143,215,367,291,326,415. 16. Note the number of decimal places in each of the following expressions : 1. .4 = 4 tenths. (1 decimal place.) 2. .27 = 27 hundredths. (2 decimal places.) 3. .346 = 346 thousandths. (3 decimal places.) 4. .2758 = 2758 ten-thousandths. 5. .07286 = 7286 hundred thousandths. 6. .000896 = 896 millionths. (6 decimal places.) 7. .000,468,275 = billionths. (9 decimal places.) 8. .000,000,000,462 = trillionths. 9. .000,000,000,000,527 = quadrillionths. 10. In any number of thousandths there are decimal places. 11. In any number of millionths there are decimal places. 12. In any number of billionths there are decimal places. 13. In any number of hundredths there are decimal places. 14. In any number of ten-thousandths there are decimal places. 15. In any number of hundred thousandths there are decimal places. PAKT II. 167 Algebra— Notation. 17. Letters are used to represent numbers ; thus, the let- ter a, h, or c may represent a number to which any value may be given. 18. Known numbers, or those that may be known with- out solving a problem, when not expressed by figures, are usually represented by the first letters of the alphabet ; as, a, h, c, d. Illustrations. (a) To find the perimeter of a square when its side is given. Let a = one side.* Then 4 a = the perimeter. Hence the rule : To find the perimeter of a square, multiply the number denoting the length of its side by 4. (b) To find the perimeter of an oblong when its length and breadth are given. Let a = the length. Let b - the breadth. Then 2a + 26, or (a + 6) X 2 = the perimeter. Hence the rule : To find the perimeter of an oblong, multiply the sum of the numbers denoting its length and breadth by 2. 19. Unknown numbers, or those which are to be found by the solution of a problem, are usually represented by the last letters of the alphabet ; as x, y, z. Illustration. (a) There are two numbers whose sum is 48, and the second is three times the first. What are the numbers ? Let X = the first number. Then 3 a; = the second number, and a; + 3 £c = 48. 4 aj = 48. X = \2. 3 a; = 36. * That is, the number of units in one side. The letter stands for the number. 158 COMPLETE ARITHMETIC. 20. The sign of multiplication is usually omitted between two letters representing numbers, and between figures and letters ; thus, a xi, is usually written ab ; b x 4, is written 4: b. 6 ab, means, 6 times a times b, or 6 x a x b. 21. Exercise. Find the numerical value of each of the following expressions, if a = 8, b = 5, and c = 2 : 1. <x + & + c = 5. 2ab = 2. a^b — c= 6. Sabc = 3. 2a + b + c= 7. 2ab + 5c = 4:. a-\-b-2c= S. ab-\-bG = (a) Find the sum of the eight results. 22. Exercise. Find the numerical value of each of the following expressions if a = 20,b = o, and c = 2 : 1. 3(a-\-b) = * b. a-^b = 2. 2(a-b)=: 6. (a + &)^c = t 3. 4(a + 6 + c)=: 7. {a-\-b)^3c = 4. 2(a-{-b-c)= 8. (a + 2b) ^2c = (b) Find the sum of the eight results. 23. Exercise. 1. If 2x = 20, 2. If bx = 40, 3. If 6x = 72, a; = ? i» = ? x= 1 4. U2x-\-Sx = 60, 5. If 3^ + 4^ = 56, ' x='i. x = 'i * This means, 3 times the sum of a and 6. t This means, the sum of a and &, divided by c. PART II. 159 Geometry. 24. A geometrical line has length, but neither breadth nor thickness. XoTE. — Lines drawn upon paper or upon the blackboard are not geometrical lines, since they have breadth and thickness. They represent geometrical lines. 25. A straight line is the shortest distance from one point to another point. 26. A curved line changes its direction at every point. 27. A broken line is not straight, but is made up of straight lines. 1. The line AB is a . 2. The line CD is a . 3. The line ^i^is a . 4. The line FG is a . 5. The Hne JEK is a . 6. The perimeter of a square is a - equal lines. line made up of 7. The perimeter of a regular pentagon is a made up of equal lines. hne 8. The circumference of a circle is a line, every point in which is equally distant from a point called the center of the circle. 9. Imagine a straight Hne drawn upon the surface of a stovepipe. Can you draw a straight line upon the surface of a sphere ? 160 COMPLETE ARITHMETIC. 28. Miscellaneous Review. 1. If a equals one side* of a regular pentagon, the peri- meter of the pentagon is . 2. li b equals the perimeter of a square, the side of the square equals h -i . 3. If a equals a straight line connecting two points and b equals a curved line connecting the same points, then a is than b. 4. Find the difference between two hundred seven thou- sandths, and two hundred and seven thousandths. 5. How many zeros in 1 million expressed by figures ? 1 billion? 1 trillion? 6. How many decimal places in any number of million- ths? billionths? trillionths? 7. How many decimal places in 25 thousandths ? in 275 thousandths ? in 4346 thousandths ? 8. A figure in the second integral place represents units how many times as great as those represented by a figure in the second decimal place ? 9. li a = 6, b = 2, and ^ = 8, what is the numerical value of the following ? 12 a -\- Sb — 6 d. 10. John had a certain amount of money and James had 5 times as much; together they had 354 dollars. How many dollars had each ? Let X = the number of dollars John had. Then 6 x = the number of dollars James had, and X -\- 5 X = 354 dollars. 6 a; = 354 dollars. X = 'i 5x = ? *The expression " a equals one side " means that a equals the number of units in one side. Remember that in this kind of notation the letters employed stand Jot numbers. ADDITION. 29. Addition (in arithmetic) is the process of combining two or more numbers into one number. XoTE 1. — The word number, as here used, stands for measured magnitude, or number of things. 30. The sum is the number obtained by adding. 31. The addends are the numbers to be added. 32. The sign, +, which is read plus, indicates that the numbers between which it is placed are to be added ; thus, 6 + 4, means that 4 is to be added to 6. 33. The sign, =, which is read equal or equals, indicates that that which is on the left of the sign, equals that which is on the right of the sign ; thus, 3+4 = 7. 5+4 + 2 = 6 + 5. 34. Principles. 1. Only like numbers can he added. 2. The denmnination of the sum is the same as that of the addends. 35. Primary Facts of Addition. There are forty-five primary facts of addition. See Elementary Book, pp. 32 and 82. The nine which many pupils fail to mem- orize perfectly are given below. (Note i, p. 443.) 7 8 9 8 9 8 9 9 9 6 5 4 6 5 7 6 7 8 13. 13 13 14 14 161 15 15 16 17 162 COMPLETE ARITHMETIC, 36. Examples of Addition. 1. 2. 3. 3754 37.426 $24,305 2862 1.48 $6,752 1457 375.062 $375.08 8073 413.968 $406,137 4. 5. 6. 275 acres. 43 gal. 2 qt. 1 pt. 5a-\-2b 146 acres. 24 gal. 1 qt. 1 pt. 27a + 36 27 acres. 63 gal. 3 qt. 1 pt. 4:6a -{-U 448 acres. 131 gal. 3 qt. 1 pt. 78a + 9b 37. Observe that in written problems in addition the figures that stand for units of the same order are usually written in the same column. 1. In example 2, what figures represent units of the second decimal order ? Of the third decimal order ? 2. In example 5, what figures represent units of the first integral order ? 38. Observe that in written problems in addition of denominate numbers, the figures that stand for units of the same denomination and order are usually written in the same column. 1. In example 5, what figures represent units of gallons ? Of quarts ? 2. In example 5, what figures represent tens of gallons ? 3. In example 4, what figures represent hundreds of acres ? PART II. 163 Addition— Simple Numbers. 39. Find the sum of 275, 436, and 821. Operation. Explanation. ^rj^ The sum of the units of the first order is 12; this is ^o/? equal to one unit of the second order and 2 units of the Q^iy^ first order. "Write the 2 units of the first order, and add the 1 unit of the second order to the other units of 1532 the second order. The sum of the units of the second order is 13 ; this is equal to 1 unit of the third order and 3 units of the second order. Write the 3 units of the second order, and add the 1 unit of the third order to the other units of the third order. The sum of the units of the third order is 15; this is equal to 1 unit of the fourth order and 5 units of the third order, each of whicli is written in its place. The sum of 275, 436, and 821 is 1532. 40. Problems. 1. Add 3465, 4268, 3279, 6534, 5731. 2. Add 5732, 6721, 3466, 4269, 6535. 3. Add 2768, 5329, 4685, 3752, 8467. 4. Add 4671, 5315, 6248, 1533, 7232. 5. Add 375, 506, 258, 327, 580, 648, 846. 6. Add 436, 307, 449, 498, 736, 274, 888. 7. Add 625, 494, 742, 673, 574, 654, 638. 8. Add 564, 693, 684, 502, 376, 726, 877. (a) Find the sum of the eight sums. To THE Teacher — Impress upon the pupil the fact that in arithmetic nothing short of accuracy is commendable. One figure wrong in one problem in ten is failure. The young man or the young woman who cannot solve ten problems like those on this page without an error, is worthless as an accountant. 164 COMPLETE ARITHMETIC. Addition— Decimals. 41. Find the sum of 4.327, 8.29 and .836. Operation. Explanation. Ac>nn The sum of the units of the third decimal order is q\q 13 ; this is equal to 1 unit of the second decimal order \oa ^i^d 3 units of the third decimal order. Write the 3 ! units of the third decimal order and add the 1 unit of 13.453 the second decimal order to the other units of that order. The sum of the units of the second decimal order is 15 ; this is equal to 1 unit of the first decimal order and 5 units of the second decimal order. Write the 5 units of the second decimal order and add the 1 unit of the first decimal order to the other units of that order. The sum of the units of the first decimal order is 14; this is equal to 1 unit of the first integral order and 4 units of the first decimal order. Write the 4 units of the first decimal order and add the 1 unit of the first integral order to the other units of that order. The sum of the units of the first integral order is 13; this is equal to 1 unit of the second integral order and 3 units of the first integral order, each of which is written in its place. The sum of 4.327, 8.29, and .836 is 13.453. 42. Pkoblems. 1. Add 474.36, 21.37, 38.007, and 487. 2. Add 78.63, 61.993, .725, and 724.64. 3. Add .7, .84, .375, .0275, and .25326. 4. Add 85.997, 47.9994, 72, and 53.93. 5. Find the sum of seven hundred ninety-eight and nine hundred ninety-four thousandths, and seven hundred ninety- four thousandths. (a) Find the sum of the five sums. To THE Pupil. — Can you solve these five problems and find the sum of the five sums on first trial without an error ? PART II. 165 Addition— United States Money. 43. Find the sum of the money represented in the follow- ing columns : $32445 28.47 375.28 6.94 175.89 27.56 475.39 802.21 354.48 916.37 144.50 75.34 Explanation. Sums of the units of each order 81 70 82 56 39 246.25 $3962.01 Second decimal order (cents) Fhst decimal order (dimes) . First integral order .... Second integral order . Third integral order . . . Observe that the 8 of the first sum is included in the 70 of the second sum; that the 7 of the second sum is included in the 82 of the third sum; that the 8 of the third sum is included in the 56 of the fourth sum, and that the 5 of the fourth sum is included in the 39 of the fifth sum. Hence the sum of the five sums is rep- resented by the figures 8962.01. 44. Problems.* 1. 2. 3. $256.35' $275 $725 145.24 146 854 321.75 281 719 286.44 675 697 308.92 284 716 244.31 552 448 986.24 496 504 275.46 628 715 383.27 682 603 (a) Find the sum of the four sums. 4. $743.65 854.76 678.25 713.56 843.18 769.45 724.54 616.73 847.90 * To THE Pupil —Remember that nothing short of absolute accuracy is of any value in such work as this. 166 COMPLETE ARITHMETIC. Addition— Denominate Numbers. 45. Find the sum of 7 bu. 2 pk. 5 qt., 3 bu. 3 pk. 3 qt., 6 bu. 1 pk. 7 qt., and 9 bu. 3 pk. 5 qt. Explanation. The sum of the number of quarts is 20 ; this is equal to 2 pecks and 4 quarts. Write the 4 quarts and add the 2 pecks to the pecks given in the second colump. The sum of the number of pecks is 11 ; this is equal to 2 bushels and 3 pecks. Write the 3 pecks, and add the 2 bushels to the bushels given in the third column. The sum of the number of bushels is 27, which is wTitten in its Operation. 7 bu. 2 pk. 5 qt. 3 bu. 3 pk. 3 qt. 6 bu. 1 pk. 7 qt. 9 bu. 3 pk. 5 qt. 27 bu. 3 pk. 4 qt. place. The sum is 27 bu. 3 pk. 4 It. 46. Problems. 1. Add. 2. Add. 6 bu. 2 pk. 6 qt. 5 bu. pk. 7 qt. 4 bu. 2 pk. 2 qt. 4 bu. 1 pk. 6 qt. 5 bu. 2 pk. 2 qt. 3 bu. pk. 1 qt. 6 bu. 3 pk. 7 qt. 5 bu. pk. 5 qt. 4 bu. 3 pk. 3 qt. 1 bu. 1 pk. 2 qt. 8 bu. 2 pk. 6 qt. 2 bu. 3 pk. 3 qt. 7 bu. pk. 5 qt. 4 bu. 2 pk. 4 qt. 5 bu. 1 pk. 4 qt. 2 bu. pk. 6 qt. 8 bu. pk. 6 qt. 6 bu. pk. 5 qt. 7 bu. 3 pk. 2 qt. 4 bu. 1 pk. qt. 6 bu. 3 pk. 1 qt. 7 bu. pk. 2 qt. 6 bu. 1 pk. 7 qt. 3 bu. 2 pk. 2 qt. 7 bu. 2 pk. qt. 3 bu. 1 pk. 2 qt. 6 bu. 1 pk. 6 qt. 3 bu. 2 pk. 1 qt. (a) Find the sum of the two sums. PART II. 167 Algebraic Addition. 47. A coefficient is a number that indicates how many times a literal quantity* is to be taken ; thus, in the expres- sion 4:ab, 4 is the coefficient of ab.-f When no coefficient is expressed, it is understood that 1 is the coefficient; thus, in the expression 4:a-\-b, the coefficient of 6 is 1. 48. The terms of an algebraic expression are the parts that are separated by the sign + or -. There are three terms in the following : a& + 3c + 4ahc. There are only two terms in the following: 8a x 46+ 5a -^ 6h. 49. Positive terms are usually preceded by the plus sign. 60. Negative terms are preceded by the minus sign. If no sign is expressed, the term is understood to be positive. 51. When the literal part of two or more terms is the same, the terms are said to be similar. 52. Problems. Unite the terms in each of the following algebraic expressions into one equivalent term : 1. bx -\- Zx -]- 2x = 5. 4a& + 2ab + 3a5 = 2. 4:X-\-5x-Sx= 6. 2ah + 5ah - 4:ab = 3. 6a ~2a + 4a= 7. 3bc - 6hc + 6hc = 4. Sh + 4:h-2h= 8. hx + 2hx -f- Sbx = 53. Language Exercise. Referring to the problems given above, use the following words in complete sentences: Coefficient, terms, positive, negative, similar, literal. * The word quantity in algebra means number. The expression literal quantity means number expressed by letters. fThe term coefficient is sometimes applied to the literal part of an expression; thus, in the expression abc, ab is the coefficient of c. Usually, however, the term coefficient has reference to the numerical coefficient. 168 COMPLETE ARITHMETIC. Algebraic Addition. 54. Kegarding the following positive numbers as represent- ing gains and the negative numbers as representing losses, find the total gain (or loss) in each case ; that is, find the algebraic sum of the numbers in each group : 1. 2. 3. 4. 5. 6. 70 85 45 8a 96 2ah 25 -35 -65 3a ~_3& - 6«& 95 ~50 ^ Note. — The positive sums of Nos. 1, 2, 4, and 5 indicate actual gain ; the negative sums of Nos. 3 and 6 indicate actual loss. 55. Regarding each of the following positive numbers as representing a rise and each of the negative numbers as rep- resenting a fall of the mercury in a thermometer, find the total rise (or fall) in each case ; that is, find the' algebraic sum of the numbers in each group : 1. 2. 3. 4. 5. 6. 16 12 8 8a ih 5c 10 -4 4 2a ^u 3c 5 6 -16 3a 4& -12c 31 14 -4 Note. — The positive sums of Nos. 1, 2, 4, and .5 indicate actual rise ; the negative sums of Nos. 3 and 6 indicate actual fall. 56. Problems. Find the sum : 1. 2. 3. 12 + 4-6 3a + 2&+ c 2a6 + Qhc 4-h6-3 2a- & + 3c 3a6+ he 5-2 + 8 a + 5& - 6c -ah + Uc PART II. Geometry. 169 57. A circle is a plane figure bounded by a curved line, every point in which is equally distant from a point within, called the center. 58. The line that bounds a circle is a circumference. 59. A straight line passing through the center of a circle and ending in the circumference is a diameter. 60. A straight line from the center of a circle to the cir- cumference is a radius. ^ 61. Any part of a circumference is an arc. 62. For the purpose of measurement, every circumference is considered as divided into 360 equal parts, called degrees. 63. Two lines meeting at a point form an angle. The point in which the two lines meet is the vertex of the angle. 64. Every angle' may be regarded as having its vertex at the center of a circle, and the angle is measured by the part of the arc intercepted; thus, the angle hlk is measured by the arc mn. 65. The angle efg is an angle of 90 degrees, called also a right angle. The angle abc is an angle of 90 degrees. 66. All the angles about a point together equal four right angles. 170 COMPLETE ARITHMETIC. 67. fliscellaneous Review. 1. The angle ahd is an angle of about — degrees. The angle dhc is an angle of — degrees. The angle abd + the angle dhc ■ angle ahc. The angle ahc is an angle of — degrees. 2. The angle mln is an angle of about — degrees. The angle mln + the angle nls = the angle mis. The angle mis is an angle of about — . 3. Copy the following figures and add by column and by line. Prove by comparing the sum of the sums of the columns with the sum of the sums of the lines. That pupil who can solve this prob- lem without an error, on first trial, has taken an important step toward making himself valuable as an accountant. Mon. Tues. Wed. Thur. Frid. Sat. Total. A 2.70 2.95 2.80 3.00 2.65 2.45 B 3.43 3.12 3.26 3.62 3.28 3.39 C 3.00 2.90 3.15 3.20 2.95 3.05 D 2.00 1.76 2.22 1.93 1.98 ■1.87 E 4.15 4.25 4.15 4.35 4.45 4.25 F 3.00 3.30 3.12 3.18 3.24 3.15 G 5.10 4.90 4.95 5.05 5.15 4.95 H 2.10 2.12 2.20 2.04 2.06 2.25 I 3.50 3.60 3.40 3.30 3.50 3.60 K 3.05 2.90 3.15 ^.95 3.15 2.00 Total . . 1 SUBTEACTTOK 68. Subtraction (in arithmetic) is the process of taking one number from (out of) another. ^OTE 1. — The word number, as here used, stands for measured magnitude, or number of things. 69. The minuend is the number from which another number is taken. 70. The subtrahend is the number taken from another number. 71. The difference is the number obtained by subtracting. 72. The sign — , which is read mimts, indicates that the number that follows the sign is to be taken from (out of) the number that precedes it ; thus, 8-3, means, that 3 is to be taken from (out of) 8. 73. Principles. 1. Only like numhcrs can he subtracted. 2. The denomination of the difference is the same as that of the minuend and the subtrahend. 74. Primary Facts of Subtraction. There are eighty-one primary facts of subtraction which should be learned while learning the facts of addition. (Note 2, p. 443.) 171 172 COMPLETE ARITHMETIC. 75. Examples of Subtraction. 1. 2. 3. 2687 57.38 $675.46 1298 28.146 $282.75 1389 29.234 $392.71 4. 5.. 6. 576 pounds 25 bu. 3 pk. 5 qt. 25a+ 6h 288 pounds 12 bu. 1 pk. 7 qt. 10 a + 2 b 288 pounds 13 bu. 1 pk. 6 qt. na-\-4b 76. Observe that in written problems in subtraction the subtrahend is usually placed under the minuend and the difference under the subtrahend; and that, as in addition, the units of the same order are written in the same column. 1. In example 2, what figures represent units of the third decimal order? of the second integral order? of the first decimal order ? of the first integral order ? 2. In example 5, what figures represent units of the first integral order ? of the second integral order ? 77. Observe that in subtraction of denominate numbers the figures that stand for units of the same denomination and order are usually written in the same column. 1. In example 4, what figures represent tens of pounds ? hundreds of pounds ? 2. In example 5, what figures represent bushels and units of the first order ? 78. Observe that in both addition and subtraction the decimal points, if there are any, usually appear in column. PART II. 173 Subtraction— Simple Numbers. 79. Find the difference of 8274 and 5638. Operation. Explanation. 8274 Eight is greater than 4. In the minuend, take one 5638 unit of the second order from the 7 units of the second ()c.oa order. This unit of the second order, combined with the 4 units of the first, makes 14 units of the first order. Eight units of the first order from 14 units of the first order leave 6 units of the first order. Three units of the second order from 6 (7—1) units of the second order leave 3 units of the second order. Six is greater than 2. In the minuend take one unit of the fourth order from the 8 units of the fourth order. This unit of the fourth order, combined with the 2 units of the third order, makes 12 units of the third order. Six units of the third order from 12 units of the third order leave 6 units of the third order. Five units of the fourth order from 7 (8 — 1) units of the fourth order leave 2 units of the fourth order. The difference of 8274 and 5638 is 2636. 80. Problems. 1. From 35642 subtract 12456. 2. From 875^44 subtract 64358. 3. From 90070 subtract 13256. 4. From 8164 subtract 3275. 5. From the sum of 8539, 2647, 3984, 1461, 7353, 6016, and 2364, subtract 22364. 6. From the sum of 1352, 3425, 2640, 3724, 6575, 7360, and 6276, subtract 21352. 7. From 6 thousand 7 hundred 25, subtract 1 thousand 8 hundred 36. 8. From seven thousand four hundred sixty-five, subtract two thousand three hundred fifty-four. (a) Find the sum of the eight differences. 174 , COMPLETE ARITHMETIC. Subtraction— Decimals. 81. Find the difference of 28.36 and 15.432. Operation. " Explanation. 28.36 One unit of the second decimal order (1 from 6) 15.432 equals 10 units of the third decimal order. Two from 12 9*^8 ^^ leaves 8. Three from 5 (6—1) leaves 2. Four is greater than 3. One unit of the first integral order (1 from 8) equals 10 units of the first decimal order. 10 + 3 m 13. Four from 13 leaves 9. Five from 7 (8 — 1) leaves 2. One from 2 leaves 1. The difference of 28.36 and 15.432 is 12.928. 82. Problems. 1. From 100 take .3456. 6. 100 - 44.764. 2. From 100 take 5.246. 7. 250 - 159.63. 3. From 100 take 44.236. 8. 250 - 36.75. 4. From 100 take .6544. 9. 250 - 140.37. 5. From 100 take 4.754. 10. 250 - 163.25. (a) Find the sum of the ten differences. 83. Miscellaneous. 1. The sum of two numbers is 3.7464; one of the num- bers is 1.3521. What is the other number? 2. The difference of two numbers is 2.3254; the less number is 7.6746. Wliat is the greater number? 3. The difference of two numbers is 2.3943; the greater number is 10. What is the less number? 4. From 10 subtract 7.6744. (b) Find the sum of the four results. To THE Pupil. — Work with care. Make no errors. The sum of the results should be correct on first trial. PART II. 175 Subtraction— United States Money. 84. Find the difference of $27.25 and $14.51. Explanation. One cent from 5 cents = 4 cents. Five dimes from 12 dimes = 7 dimes. Four dollars from 6 dollars (7 — 1) = 2 dollars. One ten-dollars from 2 ten-dollars = 1 ten-dollars. The difference of $27.25 and .*$14.51 is $12.74. Operation. $27.25 $14.51 $12.74 85. Problems — Addition and Subtraction.* Find the sum that each depositor has to his credit : A. B. Deposit Check $30.50 Deposit $254.20 Deposit 38.60 Check $12.50 Check 10.80 Check 3.60 Check 5.40 Balance . C. Deposit $745.80 Check $87.50 Check 89.20 Check 96.40 Check 94.60 Deposit 45.80 Balance Check Deposit Check Check Balance 21.75 18.34 6.24 D. $175.30 54.20 E. Deposit, $1000. F. Deposit, $864. Deposit $824.70 Check $69.50 Check 78.25 Check 81.66 Deposit Check 93.76 Balance Checks, $275, $324, $400. Checks, $375, $146, $279. 61.40 (a) Find the amount of the six balances. * That bank clerk who makes one error a day in carrying out his balances, which he does not himself discover and correct, will not retain his position. 176 COMPLETE ARITHMETIC. Subtraction— Denominate Numbers. 86. Find the difference of 15 yd. 2 ft. 4 in. and 8 yd. 1 ft. 10 in. Operation. Explanation. 15 yd. 2 ft. 4 in. Ten inches are more than 4 in.; 1 ft. 8 yd. 1 ft. 10 in. (from the 2 ft.) equals 12 in.; 12 in. and 7 vd ft 6 in ^ ^^^' ^^'® ^^ "^•' ^^ ^^' ^^^^ 1^ i^* leave 6 in. One ft. from 1 ft. (2-1) leaves ft. Eight yd. from 15 yd. leave 7 yd. The difference of 15 yd. 2- ft. 4 in. and 8 yd. 1 ft. 10 in. is 7 yd. 6 in. 87. Problems. 1. From 12 yd. 1 ft. 8 in. subtract 5 yd. 2 ft. 3 in. 2. From 8 yd. 2 ft. 6 in. subtract 5 yd. 1 ft. 10 in. 3. From 9 yd. 1 ft. subtract 2 yd. 2 ft. 7 in. 4. From 6 yd. 2 ft. 5 in. subtract 4 yd. 8 in. 5. From 7 yd. subtract 5 yd. 1 ft. 1 in. 6. From 7 yd. 1 ft. 4 in. subtract 4 yd. 9 in. 7. From 11 yd. 6 in. subtract 4 yd. 1 ft. 2 in. 8. From 10 yd. 2 ft. subtract 7 yd. 5 in. (a) Find the sum of the eight differences. 88. Problems. 1. How many days from April 25 to May 1 ? 2. How many days from April 25 to May 10 ? 3. How many days from April 25 to May 20 ? 4. How many days from April 25 to June 1 ? 5. How many days from April 25 to June 10 ? 6. How many days from April 25 to June 30 ? 7. How many days from April 25 to July 5 ? (b) Find the sum of the seven answers. Tues. Wed. Thurs. Fri. Sat. 60 20 la -4& -12c -20 80 50 -30 2a 5 a -6& 2h - 6c - 6c PART II. 177 Algebraic Subtraction. 89. Eegarding the following minuends as representing A's gain (or loss), and the subtrahends as representing B's gain (or loss), subtract B's from A's.* Mon. A, 80 B, 30 50 Note. — The positive differences for Monday, Tuesday, Thursday, and Friday indicate that A's gain was greater (or his loss less) than B's. The negative differences for Wednesday and Saturday indicate that A's gain was less (or his loss greater) than B's. 90. Eegard the following minuends as representing dis- tances one boat sails from a given point, and the subtrahends as representing distances another boat sails from the same point. Distances sailed north are here represented by posi- tive numbers, and distances sailed south by negative num- bers. Find how far the first boat is from the second. 1. 2. 3. 4. 5. 6. 1st B., 75 85 15 8a -U -15c 2nd B., 25 -35 35 2a -9b - 5c 50 120 -20 Note. — The positive differences in Nos. 1, 2, 4, and 5 indicate that the first boat is north of the second boat. The negative differ- ences in Nos. 3 and 6 indicate that the first boat is south of the second boat. 91. Eule for algebraic subtraction : Conceive the sign (or signs) of the subtrahend to be changed (— to + and + to — ), then proceed as in addition. *Seepagel68, Art. 54. 178 COMPLETE ARITHMETIC. Algebraic Subtraction. 1. A gained $1200 and lost $250 ; B gained $500 and lost $350. How much more was A's wealth increased by the two transactions than B's ? $1200 -$250 12^-56 Ua-Sh $500 - $350 5a-7b 6a-5b $700 + $100 ^ ~ 2. C gained $1500 and $650; D gained $600 and lost $250. How much more was C's wealth increased by the two transactions than D's ? $1500 + $650 lb a + 13b IS a -\- 6 b $600 -$250 6a- 5b 4.a-Sb + 3. E gained $1300 and lost $450; F gained $400 and $250. How much more was E's wealth increased by the two transactions than F's ? $1300 -$450 IS a -9b 17 a -8b $400 + $250 4:a + 5b 4a + 6& $900 - $700 4. G gained $1200 and lost $500 ; H gained $900 and lost $100. How much more was G's wealth increased by the two transactions than H's ? $1200 -$500 12a -5b 16a-Sb $900 - $100 9a- b 4^-25 $300 - $400 5. Eeview the foregoing and observe that in every instance subtracting a positive number is equivalent to adding an equal negative number, and subtracting a negative number is equivalent to adding an equal positive number. B PART II. 179 Geometry. 92. Direction of Lines. A M N O P D C 1. When one straight line meets another straight hne in such a manner that two right angles are formed by the lines, the two lines are said to be perpendicular to each other. 2. Two lines side by side extending in the same direction are said to be parallel. 3. Of the lines given above : AC is to BD. BD is to AC. MN is to OP. 4. A line extending in the direction of the horizon is said to be horizonidX. A line on the floor of the room is hori- zontal; a line on the ceiling is horizontal; a line on the blackboard, every point in which is equally distant from the floor, is horizontal. For convenience, lines drawn upon paper, that are parallel with the top and bottom of the paper, may be regarded as representing horizontal lines. 5. A line suspending a piece of lead (plumbum) is called a plumb-line. A line in the direction of a plumb-line is said to be vertical. A vertical line is perpendicular to a hori- zontal line. Lines on the blackboard may or may not be vertical or horizontal. For convenience, lines drawn upon paper that are parallel with the sides of the paper may be regarded as vertical. 180 COMPLETE ARITHMETIC. 93. Miscellaneous Reviews. 1. An angle that is equal to one half of a right angle, is an angle of degrees. 2. The angle ADB is an angle of de- grees. 3. If from a right angle, an angle of 30 remaining angle is an D degrees be angle of — taken, the — degrees. 4. The angle FHG is an angle of grees. 5. During the month of November, 1897, there were con- sumed at the Illinois Institution for the Education of the Blind, 64 loads of coal. The weight of each load in pounds is given below. Find the total weight. 6100 8020 5490 5190 8380 6860 6800 7130 4850 6230 6560 7090 8010 6780 6690 7790 7080 6980 5780 6810 6620 6240 6980 8600 6450 6420 5990 9100 6570 6310 4740 6740 7950 6300 5520 5380 4750 6530 3630 7640 8840 6950 4930 5650 7290 6980 5150 5900 4960 4920 6420 6200 8330 5880 6770 6620 6300 7030 6220 7170 7080 5160 6020 9210 MULTIPLiCATIOK 94. Multiplication is the process of taking a number (of things) a number of times. The word number as first used in the above statement stands for measured magnitude. The second word nuinher does not stand for measured magnitude, but rather for pure number, representing simply the times the number (of things) is to be repeated. 95. The multiplicand is thie number (of things) taken or repeated.* 96. The multiplier is the number that shows how many times the multiplicand is to be repeated. 97. The product is the number (of things) obtained by multiplying. 98. The sign, x, which is read multiplied hi/, indicates that the number preceding the sign is a multiplicand, and the number following it a multiplier. For other uses of this sign, see notes 3, 4, and 5, page 443. 99. Principles. 1. The multiplier is always an abstract number. 2. The denomination of the product is always the same as that of the multiplicand. 100. Primary Facts of Multiplication. There are sixty-four primary facts of multiplication. See note 6, page 444. * " The multiplicand, however written, must always be understood to express measured quantity; it is always concrete."— Psychology of Number, McCleUan & Deivey, page 76. 181 182 COMPLETE ARITHMETIC. 101. Examples of Multiplication. 1. 2. 3. 4865 37.258 $375.42 3 3 3 14595 111.774 $1126.26 4. 5. 6. 364 tons 61 yd. 2 ft. 5 in. 8a - 6b 5 5 5 1820 tons 309 yd. ft. 1 in. 4:0 a - 30 b 102. Observe that in each of the above examples the multiplier is a pure number. 103. Observe that in each of the above examples the denomination of the product is the same as the denomina- tion of the multiplicand. 104. Multiplication and Addition Compared. Find the sum of each of the following groups of numbers and compare the result with the product in the corresponding problem in article 101 : 1. 2. 3. 4865 37.258 $375.42 4865 37.258 $375.42 4865 37.258 $375.42 4. 5. 6. 364 tons 61 yd. 2 ft. 5 in. 8a- 6b 364 tons 61 yd. 2 ft. 5 in. 8a - 6b 364 tons 61 yd. 2 ft. 5 in. 8a - 6b 364 tons 61 yd. 2 ft. 5 in. 8a - 6b 364 tons 61 yd. 2 ft. 5 in. 8a - 6b PART II. 183 Multiplication— Simple Numbers. 105. Find the product of 563 and 7. Operation. Explanation. 563 Seven times 3 units of the first order are 21 units of the 7 first order ; they are equal to 1 unit of the first order and oQ^-| 2 units of the second order. Write the 1 unit of the first order and add the 2 units of the second order to the next partial product. Seven times 6 units of the second order are 42 units of the second order ; 42 + 2 = 44 ; 44 units of the second order equal 4 units of the second order and 4 units of the third order. Write the 4 units of the second order and add the 4 units of the third order to the next partial product. Seven times 5 units of the third order are 35 units of the third order; 35 + 4 = 39 ; 39 units of the third order equal 9 units of the third order and three units of the fourth order. Write the 9 units of the third order and the 3 units of the fourth order. The product of 563 and 7 is 3941. 106. Find the product of 3426 and 57. Operation. Explanation. 3426 Seven times 3426 equals 23982. Fifty times 3426 57 equals 171300. Fifty times the number plus 7 times nn(\c,.') the number equals 57 times the number. Therefore 71 ^0 ^<idi"g 23982 and 171300 gives 57 times 3426. The product of 3426 and 57 is 195282. 195282 107. Problems. 1. Multiply 3241 by 27. 6. 6521 x 54 = 2. Multiply 6759 by 27. 7. 3572 x 74 - 3. Multiply 4328 by 36. 8. 6428 x 74 = 4 Multiply 5672 by 36. 9. 3521 x 29 = 5. Multiply 3479 by 54. 10. 6479 x 29 = (a) Find the sum of the ten products. 184 COMPLETE ARITHMETIC. Multiplication — Decimals. 108. Find the product of 728.37 and .6. Operation. Explanation. 72^8.37 To multiply by .6 means to take 6 times 1 tenth .6 of the number. One tenth of 728.37 is 72.837. Six .0-7 029 ti^es 72.837 equals 437.022. Note 1.— The separatrix is used to indicate the place of the decimal point in the number that is one tenth of the multiplicand. Note 2. — The decimal point should be written in the product when, in the process of multiplication, the place is reached where it belongs. Do not multiply all the figures and then attempt to deter- mine the place of the point. 109. Find the product of 746.2 and .25. Operation. Explanation. 7^46.2 To multiply by .25 means to take 25 times 1 hun- .25 dredth of the number. 37 32Q ■ One hundredth of 746.2 is 7.462. Five times 7.462 149 24 equals 37.310. Twenty times 7.462 equals 149.24. 37.310 + 149.24 = 186.550. 186.550 110. Observe that when a multiplication of decimals is complete, the number of decimal places in the product is equal to the number of decimal places in the multiplicand and multiplier. 111. Problems. 1. Multiply 324.6 by .7. 6. 324.6 x 54 = 2. Multiply 324.6 by .27. 7. 324.6 x .48 = 3. Multiply 324.6 by 2.7. 8. 324.6 x 6.3 = 4. Multiply 324.6 by 27. 9. 324.6 x 3.07 = 5. Multiply 324.6 by 5.4. 10. 324.6 x .08 = (a) Find the sum of the ten products. PART II. 185 Multiplication— United States Money. 112. Find the product of $34.50 and 53.4. Operation. $3^4.50 53.4 $13,800 $103.50 $1725.0 $1842.300 Explanation. To multiply by 53.4, means to take 53 times the multiplicand plus 4 tenths of the multiplicand. One tenth of ^34.50 is ^3.45. Four tenths of $34.50 is 113.80. Three times 134.50 is $103.50. Fifty times $34.50 is $1725. The sum of the partial products is $1842.30. Practical application of the foregoing. If one acre of land costs $34.50, how much will 53.4 acres cost ? One tenth of an acre costs $3.45. Four tenths of an acre cost $13.80. Three acres cost $103.50. Fifty acres cost $1725. 53.4 acres cost $1842.30. 113. Complete the following bill and find the amount: Inst, for the Blind, 1897. To Geo. E. Sybrant, Dr. Dec. 4 27 bbl. Apples @ $2.25 I 10 56 bush. Potatoes @ .52 ( 12 13 bush. Beans @ 1.75 < 16 18 34 bush. Turnips @ .35 50 bbl. Flour @ 4.90 ( 21 74 lb. Butter @ .19 ' 22 53 lb. Tea @ .42 i 24 37 bush. Onions @ .55 u 31 58 lb. Ham @ .14 To THE Pupil. — Remember that any inaccuracy in solving busi- ness problems makes the work valueless. Accuracy ranks next in importance to integrity in the selection of an accountant. 186 COMPLETE ARITHMETIC. Multiplication— Denominate Numbers. 114. Find the product of 3 tons 850 lb. and 8. Operation. Explanation. 3 tons 850 lb. Eight times 850 lb. equals 6800 lb. 8 6800 lb. equals 3 tons 800 lb. 27 tons 800 lb Write the 800 lb. and add the 3 tons to the next partial product. Eight times 3 tons equals 24 tons; 24 tons plus 3 tons equals 27 tons. 3 tons 800 lb. multiplied by 8 equals 27 tons 800 lb. 115. Problems. 1. If one side of a square garden measures 6 rd. 8 ft, what is the perimeter of the garden ? 2. The circumference of a certain bicycle track is 13,rd. 1 2 ft. How far does the rider travel who goes around it .12 times ? 3. The length of a rectangular field is 15 rd. 10 ft. and the width 9 rd. 8 ft. What is the perimeter of the field ? 4. There is a walk 5 feet wide around a rectangular grass plat 3 rd. 6 ft. by 2 rd. 10 ft. What is the outside perim- eter of the walk ? 5. How far does the person travel who walks once around the grass plat described in problem 4, if he keeps his track in the center of the walk ? (a) Find the sum of the five answers. 116. Problems. 1. If a train moves at the rate of a mile in 1 min. 25 sec, in how long a time will it move 325 miles ? 2. If the circumference of a wagon wheel is 15 feet 6 inches, how far will the wagon move while the wheel re- volves 1000 times? PART II. 187 Algebraic Multiplication. 117. Examples. No. 1. No. 2. 8 + 4-3 a + Sb-4:C 32 + 16 - 12 4a + 12?)- 16c No. 3. No. 4. 3ft -2b-^c 2ft + 3& - 5c f^ 2(^ 3ac^ - 2bd + cc/ 4a£^ + Qhd - lOcd 1. Observe that in the above examples we multiply each term of the multiplicand by the multiplier. 2. Prove example No. 1 by uniting the terms of the mul- tiplicand and comparing 4 times the number thus obtained with the number obtained by uniting the terms of the product. 3. Verify example No. 2 by letting a = b,b = ?>, and c = 2. 4. Verify example No. 3 by giving the following values to the letters : «, = 7, 5 = 4, c = 3, ^ = 5. 5. Verify example No. 4 by giving any values you may choose to each letter. 118. Problems. 1. Multiply 3a5 - 2bc + 5c by 2d. 2. Multiply 2ax + 4:bx - ?/ by 5. 3. Multiply Zbc + ab -be by Sd. 4. Multiply X — 1/ -\-zhj Sab. 6. Multiply ax-}-bx — ex by 2'i/.- 6. Verify each of the above problems by giving the fol- lowing values to the letters : a = S, b = 2, c = 4, d = 5, x=7, 2/ = 6, = 8. 188 COMPLETE ARITHMETIC. Algebraic Multiplication. 119. Exponent. 1. a X d, or aa which means a multipKed by a, is usually written a\ This is read a square or a second power. 2. b^ (to be read h cube or h third power) means h taken three times as a factor. It is h xh xb- 3. a* (to be read a fourth power, or simply a fourth) means that a is taken four times as a factor. It is a x cc x a X cc- 4. The small figure at the right of a letter tells the number of times the letter is to be used as a factor. The figure so used is called an exponent. When the exponent is 1, it is not usually expressed ; thus, a means a\ 120. Problems. On the supposition that a = 2, b = 3, and c = 4, find the numerical value of each of the following expressions : 1. a'-{-2ah-th' 6. 5a'h-2hc 2. Sa¥-}-5hc' 7. a'b' - c' 3. ^a'h' + ShV • 8. a'h' + c' 4. 2a'h f 2ay' 9. d'b'c' - ab\ 5. 3&V4-5a6 10. 2aTc'^ (a) Find the sum of the numerical values of the above. 121. No. 1. 4:ax -Y 2by -\- c a' Examples. No. 2. Wx -\-2by-c 2b' 4a'x + 2a'by + d'c 6b'x + 4% - 2&'c Verify each of the above examples by letting a = 2,b = S, c = 4:, X- 5, y = 6. PART II. Geometry. 122. Parallelograms. 189 1. Any side of any one of the above figures is parallel to the opposite side of the same figure. Hence the figures are called parallelograms. 2. Each of the above figures has four sides. Hence the figures are called quadrilaterals. 3. If all the sides of a figure are equal, the figure is said to be eqioilateral. 4. If all the angles of a parallelogram are right angles (angles of 90°) the figure is said to be rectangular. 5. Wliich of the above figures are equilateral ? 6. Which of the above figures are rectangular ? 7. Which of the above figures are not equilateral ? 8. Which of the above figures are not rectangular ? 9. Which of the above figures are parallelograms ? 10. Which of the above figures are quadrilaterals ? 11. Can you draw a quadrilateral that is not a parallelo- gram ? 12. Is any one of the above figures an equilateral rect- angular parallelogram ? 13. In a rhomboid or rhombus two of the angles are less than right angles and two of them are greater than right angles. Convince yourself by cutting a rhomboid from paper and comparing it with rectangular figures that two of the angles of a rhomboid are as much less than two right angles as the other two are greater than two right angles. 190 COMPLETE ARITHMETIC. 123. Miscellaneous Reviews. 1. If one of the angles of a rhombus is an angle of 80 degrees, what is the number of degrees in each of the other angles ? 2. Draw a rhomboid one of whose angles is an angle of 70 ; give the number of degrees in each of the other angles. 3. An oblong has four right angles. The angles of a rhomboid are together equal to how many right angles ? 4. If an oblong is a feet long and l feet wide, the number of square feet in the area is a&.* If the side of a square is a feet, the number of square feet in its area is . 5. If a rectangular solid is a feet long, h feet wide, and c feet thick, the number of cubic feet in its solid contents is abc. If the side of a cube is a feet, the number of cubic feet in its solid contents is . 6. If a man earns h dollars each week and spends c dollars, in one week he will save dollars ; in 7 weeks he will save dollars. 7. A framed picture, on the inside of the frame, is 18 in. by 22 in. ; the frame is 4 inches wide. How many inches in the outside perimeter of the frame ? 8. Think of two fields: one is 9 rd. by 16 rd. ; the other is 12 rd. by 12 rd. How do the square rods of the two fields compare ? How much more fence would be required to enclose one field than the other ? * This means the product of a and 6. Observe that it is the numljer a (not a feet) that we multiply by the number b (not 6 feet). While it is probably true (see foot- note, p. 181) that the multiplicand always expresses measured quantity, it is also true that we often find the product of two factors mechanically. Indeed this is what we usually do in all multiplication of abstract numbers. In this case we find the product of a and 6 and know from former observations that this number equals the number of square feet iu the oblong. DIVISION 124. Division is (1) the process of finding how many times one number is contained in another number; or (2), it is finding one of the equal parts of a number. Note.— The word number as used above stands for measured magnitude. 125. The dividend is the number (of things) to be divided. Note.— Since in multipHcation the multipUcand and product must always be considered concrete (see foot-note, p. 181), then in division, the dividend, and either the divisor or the quotient, must be so regarded. 126. The divisor is the number by which we divide. Note. — The word number as used in Art. 126 may stand for meas- ured magnitude or for pure number, according to the aspect of the division problem. In the problem 324 h- 6, if we desire to find how many times 6 is contained in 324, the 6 stands for measured magni- tude — a number of things. But if we desire to find one sixth of 324, then the 6 is pure number, and is the ratio of the dividend to the required quotient. 127. The quotient is the number obtained by dividing. Note. — If the divisor is pure number, the quotient represents measured magnitude. If the divisor represents measured magnitude, the quotient is pure number. 128. The sign -^, which is read divided hy, indicates that the number before the sign is a dividend and the number following the sign a divisor. See notes 7 and 8, page 445. 191 192 COMPLETE ARITHMETIC. 129. Examples in Division. No. 1. No. 2. $5)$1565 5)$1665 313 $313 No. 3. No. 4. 2 bush.)246 bush. 2)246 bush. . 123 123 bush. No. 5. No. 6. 2a)6ab -\- 8ac - 12a 2)6ah-{-8ac-12a 36 +4c - 6 3ab-}-4ac- 6a 1. In example No. 1, we are required to find 2. In example No. 2, we are required to find 3. In example No. 3, we are required to find 4. In example No. 4, we are required to find 5. In example No. 5, we are required to find 6. In example No. 6, we are required to find Note.— Let it be observed that all the examples given on this page, indeed all division problems, may be regarded as requirements to find how many times one number of things is contained in another number of like things. Referring to example No. 2 given above : If one were required to find one fifth of 1565 silver dollars, he might first take 5 dollars from the 1565 dollars, and put one of the dol- lars taken in each of five places. He might then take another five dollars from the number of dollars to be divided, and put one dollar with each of the dollars first taken. In this manner he would continue to distribute fives of dollars until all the dollars had been placed in the five piles. He would then count the dollars in each pile. Observe, then, that one fifth of 1565 dollars is as many dollars as $5 is contained times in $1565. It is contained 313 times ; hence one fifth of 1565 dollars is 313 dollars. It is not deemed advisable to attempt such an explanation as the foregoing with young pupils ; but the more mature and thoughtful pupils may now learn that it is possible to solve all division problems by one thought process— finding how many times one nmnber of things is contained in another number of like things. *Fill the blank with the words, how many times five dollars are contained in $1565. tFill the blank with the words, one fifth of $1665. PART II. 193 Division— Simple Numbers. 130. Find the quotient of 576 divided by 4. '' Short Division." Explanation No. 1. 4)576 ^^® fourth of 5 hundred is 1 hundred with a remain- der of 1 hundred ; 1 hundred equals 10 tens ; 10 tens plus 7 tens are 17 tens. One fourth of 17 tens is 4 tens with a remainder of 1 ten ; 1 ten equals 10 units ; 10 units plus 6 units are 16 units. One fourth of 16 units is 4 units. Hence one fourth of 576 is 144. Explanation No. 2. Four is contained in 5 hundred, 1 hundred times, with a remainder of 1 hundred; 1 hundred equals 10 tens; 10 tens and 7 tens are 17 tens. Four is contained in 17 tens, 4 tens (40) times with a remain- der of 1 ten; 1 ten equals 10 units; 10 units and 6 units are 16 units. Four is contained in 16 units 4 times. Hence 4 is contained in 576, 144 times. 131. Find the quotient of 8675 divided by 25. "Long Division." Explanation. 25')8675('347 Twenty-five is contained in 86 hundred, 3 yg hundred times with a remainder of 11 hundred; 11 hundred equal 110 tens; 110 tens plus 7 tens equal 117 tens. Twenty-five is contained . in 117 tens 4 tens (40) times with a remainder 175 of 17 tens; 17 tens equal 170 units; 170 units 175 plus 5 units equal 175 units. Twenty-five is contained in 175 units 7 times. Hence 25 is contained in 8675, 347 times. 132. Problems. 1. 93492^49 5. 5904^328 2. 92169-^77 6. 7693-^ 157 3. 72855-^45 7. 8190-^-546 4 34694-^38 8. 12960 -^864 (a) Find the sum of the eight quotients. 117 100 194 COMPLETE ARITHMETIC. Division— Decimals. 133. Find the quotient of 785.65 divided by .5. Operation. Explanation. .5)785.6^5 First place a separatrix (v) after that figure ii\ 1 ^rr-i o the dividend that is of the same denomination aa the right-hand figure of the divisor — in this case after the figure 6. Then divide, writing the decimal point in the quotient when, in the process of division, the separatrix is reached — in this case after the figure 1. It was required to find how many times 5 tenths are contained in 7856 tenths. 5 tenths are contained in 7856 tenths 1571 times. There are yet 15 hundredths to be divided. 5 tenths are contained in 15 tenths 3 times; in 15 hundredths 3 tenths of a time. Note. — By holding the thought for a moment upon that part of the dividend which corresponds in denomination to the divisor, the place of the decimal point becomes apparent. 5 apples are contained in 7856 apples 1571 times. 5 tenths are contained in 7856 tenths 1571 times. 134. Solve and explain the following problems with special reference to the placing of the decimal point : 1. Divide 340 by .8 .8 )340.0 ^ 2. Divide 468.5 by .25 .25)468.50' 3. Divide 38.250 by 12.5 12.5)38.2^50 4. Divide 87 by 2.5 2.5)87.0' 5. Divide 546 by .75 .75)546.00' 6. Divide .576 by 2.4 2.4).5'76 7. 86 ^ .375 = 8. 94.5 ^ .8 = 9. 75 ^ .15= 10. 125 -^ .5 = 11. 12.5 ^ .05 = 12. 1.25 ^ .5 = (a) Find the sum of the twelve quotients. PART II. 195 Division— United States Money. 135. Divide $754.65 by $.27. Operation. $.27)$754.65X2795 54 214 189 256 243 135 135 Explanation- This means, find how many times 27 cents are contained in 75465 cents. 27 cents are contained in 75465 cents, 2795 times. Problem. At 27^ a bushel, how many bushels of oats can be bought for f 754. 65? As many bushels can be bought as $.27 is contained times in $754.65. It is con- tained 2795 times. 136. Divide $754.65 by 27. Operation. 27)$754.^65($27.^95 54 214 189 256 243 135 135 Explanation. Tljis means, find one 27th of $754.65. One 27th of $754.65 is $27.95. Note. — One might find 1 27th of $754.65 by finding how many times $27 is contained in $754.65. Problem. If 27 acres of land are worth $754,65, how much is one acre worth? 137. Divide $754.65 by .27. Operation. .27)$754.65'($2795 54 214 189 256 243 135 135 Explanation. This means, find 100 27ths of $754.65. One 27th of $754.65 is $27.95. 100 27ths of $754.65 is $2795. Note. — In practice, we find one 27th of 100 times $754.65. Problem. If .27 of an acre of land is worth $754.65, how much is 1 acre worth. at the same rate? 196 COMPLETE ARITHMETIC. Division— Denominate Numbers. 138. Divide 46 rd. 12 ft. 8 in. by 4. Operation. Explanation. 4)46 rd. 12 ft. 8 in. This means, find 1 fourth of 46 rd. 12 ft. 11 rd. lift. 5 in. ^^' „ ^^ i..« , • n ^ -.i One fourth of 46 rd. is 11 rd. with a re- mainder of 2 rd. ; 2 rd. equal 33 ft.; 33 ft. plus 12 ft. equal 45 ft. One fourth of 45 ft. equals 11 ft. with a remainder of 1 ft. ; 1 ft. eouals 12 in. ; 12 in. plus 8 in. equals 20 in. One fourth of 20 in. equals 5 in. One fourth of 46 rd. 12 ft. 8 in. equals 11 rd. 11 ft. 5 in. PROBLEM. The perimeter of a square garden is 46 rd. 12 ft. 8 in. How far across one side of it? 139. Miscellaneous. Tell the meaning of each of the following, solve, explain, and state in the form of a problem the conditions that would give rise to each number process : 1. Multiply 64 rd. 14 ft. 6 in. by 8. 2. Divide 37 rd. 15 ft. 4 in. by 5. 3. Divide $675.36 by $48. 4. Divide $675.36 by 48. 5. Divide $675.36 by .48. 6. Divide $675.36 by $4.8. 7. Divide $675.36 by 4.8. 8. Divide $675.36 by $.48. 9. Multiply $356.54 by .36. 10. Multiply $356.54 by 3.6. 11. Multiply $356.54 by 36. 12. Can you multiply by a number of dollars ? 13. Can you divide by a number of dollars? PART II. 197 Algebraic Division. 140. Examples. No. 1. No. 2. 4)12 + 5x4-8 a)12a + 2ah + 4a 3 + 5-2 12 + 2& + 4 No. 3. No. 4. 2)2^ + 3 X 2'-^ - 2 h)ah' + c^'^ + Sh 2^ + 3 X 2 - 1 . a&^ + c& + 3 1. Prove Nos. 1 and 3, by (1) reducing each dividend to its simplest form, (2) dividing it so reduced, by the divisor, and (3) comparing the result with the quotient reduced to its simplest form. 2. Verify No. 2 by letting a = 3, and b = 5. 3. Verify No. 4 by letting a - 3,b = 5, and c = 7. 141. (6xa,XctxaxaxcL)^(2xaxci) = 6a^-f- 2^^= 3al Observe that to divide one algebraic term by another we must find the quotient of the coefficients and the difference of the exponents. 142. Problems. 1. 6a'b -^2a= 3. Sa'b' -^ 2a = 2. 4:a*b' ^2a= 4. lOd'b' -^ 2a =^ 5. 2 a)6a'b + 4aV - 8a'b' + lOa'b' 6. Verify problem 5 by letting a = S and & = 5. 198 COMPLETE ARITHMETIC. Algebraic Division. 143. Problems. 1. Divide 4:a^x + 8a V + 6ax^ by 2ax. 2. Multiply the quotient of problem 1 by 2ax. 3. Verify problems 1 and 2 by letting a = 2 and x = S. 4. Divide Sah' + 6aV + 9a% by 3ah. 5. Multiply the quotient of problem 4 by Sah. 6. Verify problems 4 and 5 by letting a = 3 and h = 5. 7. Divide 2x^y -\- x?y^ — xy^ by xy. 8. Multiply the quotient of problem 7 by xy. 9. Verify problems 7 and 8 by letting x = 2 and 2/ = 3. 10. Divide 5ay - 2(2'?/' + dY by ft'?/. 11. Multiply the quotient of problem 10 by ct'y. 12. Verify problems 10 and 11 by letting a -1 and y = 2. 13. Divide Wx + h'x' - 36V by &;r. 14. Multiply the quotient of problem 13 by hx. 15. Verify problems 13 and 14 by letting & = 3 and x - 4:. Observe that when the divisor is a positive number, each term of the quotient has the same sign as the term in the dividend from which it is derived. 2)8 - 6 One half of + 8 is + 4 ; one half of - 6 is - 3. 16. 2x)4x' - Qx' + 8^' - 2x'' + 6x. PART II. Geometry. 144. Triangles. 199 llight Isosceles Isosceles Equilateral 1. A triangle has sides and angles. 2. A right triangle has one right angle ; that is, one angle of degrees. 3. An isosceles triangle has two angles that are equal and two sides that are equal. 4. An equilateral triangle has equal sides and equal angles. Fig. 5 Fig. 6 5. Cut from paper a triangle similar to the one shown in Fig. 5. Then cut it into parts as shown by the dotted lines. Ee-arrange the 3 angles of the triangle as shown in Fig. 6. Compare the sum of the 3 angles with two right angles as shown in Fig. 6. Convince yourself that the three angles of this triangle are together equal to two right angles, 6. Cut other triangles and make similar compaiisons, until you are convinced that the sum of the angles of any triangle is equal to two right angles. 200 COMPLETE ARITHMETIC. 145. Miscellaneous Review. 1. Ji in figure 1, the angle <x is a right angle, and the angle h is equal to the angle c, the angle h is an angle of how many degrees ? 2. If in figure 2, the angle d is an angle of 95° and the angle e is an angle of 40°, the angle / is an angle of how many degrees ? 3. If in figure 3, the angle x is an angle of 75°, the angle w is an angle of how many degrees ? 4. If in an oblong there are ah square feet, and the oblong is a feet long, it is feet wide, ab -^ a = 5. If in a rectangular solid there are ahc cubic feet, and the solid is a feet long and h feet wide, it is feet thick, ahc ^ ah = 6. Verify problems 4 and 5 by letting a = 3, & = 4, and c = 2. 7. There is a field that contains 1736 square rods; it is 28 rods long. How wide is the field ? 8. There is a solid that contains 4320 cubic inches; it is 24 inches long and 15 inches wide. How thick is the solid ? 9. How many square inches of surface in the solid de- scribed in problem 8 ? PEOPERTIES OF NUMBERS. To THE Teacher. — Under this head, number in the abstract is discussed with little or no distinction between numbers of things and pure number. It is dissociation and generalization, Mdthout wliich there could be little progress in the " science of number " or in the "art of computation." 146. Every number is fractional, integral, or mixed. 1. A fractional number is a number of the equal parts of some quantity considered as a unit; as, |-, .9, 5 sixths. 2. An integral number is a number that is not, either wholly or in part, a fractional number; as, 15, 46, ninety- five. 3. A mixed number is a number one part of which is integral and the other part fractional; as, 5|-, 27.6, 274|. 147. An exact divisor of a number is a number that is contained in the number an integral number of times. 5 is an exact divisor of 15. 5 is not an exact divisor of 1.5. 16| is an exact divisor of 100. 148. Every integral number is odd or even. 1. An odd number is a number of which two is not an exact divisor; as, 7, 23, 141. 2. An even number is a number of which two is an exact divisor; as, 8, 24, 142. 201 202 COMPLETE ARITHMETIC. Properties of Numbers. 149. Every integral number is prime or composite. 1. A prime number is an integral number that has no exact integral divisors except itself and 1 ; as, 23, 29, 31, etc. 2. Is two a prime number ? three ? nine ? 3. Name the prime numbers from 1 to 97 inclusive. Find their sum. 4. A composite number is an integral number that has one or more integral divisors besides itself and 1 ; as, 6, 8, 9, 10, 12, 14, 15, etc. 5. Name the composite numbers from 4 to 100 inclusive. Find their sum. 6. Is eight a composite number ? eleven ? fifteen ? (a) Find the sum of the results of problems No. 3 and No. 5. 150. To find whether an integral number is prime or composite. 1. Is the number 371 prime or composite? Operation. Explanation. 2)371 5)371 Beginning with 2 (the smallest prime Qp. . 7^ I number except the number 1), it is found by trial not to be an exact divisor of 371. 3)371 7)371 3 is not an exact divisor of 371. 123-1- 53 ^ ^^ ^^^^ ^^^ exact divisor of 371. 7 is an exact divisor of 371. Therefore 371 is a composite number, being composed of 53 sevens, or of 7 fifty-threes. Observe that we use as trial divisors only prime numbers. If 2 is not an exact divisor of a number, neither 4 nor 6 can be. Do you see why ? PART II. 203 Properties of Numbers. 2. Is the number 397 prime or composite ? Operation. Explanation. 2)397 3)397 By trial it is found that neither 2, 3, -, QQ -, Qo , 5, 7, 11, 13, 17, nor 19 is an exact divisor 0)oy7 /joy^ ;f^Q composite number between 2 and 79_|_ 56-}- 19 can be an exact divisor of 397; for 11^SQ7 1^'>SQ7 since one 2 is not an exact divisor of the number, several 2's, as 4, 6, 8, 12, etc., OD-f- 6\)-f- cannot be ; since one 3 is not an exact 17)397 19)397 divisor of the number, several 3's, as 6, 9, 2S-I- 20-1 ^'^'> ^^^*' ^^^^^^ot be ; since one 5 is not an exact divisor of the number, several 5's, as 10 and 15, cannot be; since one 7 is not an exact divisor of the number, two 7's (14) cannot be. No number greater than 19 can be an exact divisor of the number ; for if a number greater than 19 were an exact divisor of the number, the quotient (which also must be an exact divisor) would be less than 20. But it has aheady been proved that no integral number less than 20 is an exact divisor of 397. Therefore 397 is a prime number. Observe that in testing a number to determine whether it is prime or composite, we take as trial divisors prime num- bers only, beginning with the number two. Observe that as the divisors become greater, the quotients become less, and that we need make no trial by which a quo- tient will be produced that is less than the divisor. 3. Determine by a process similar to the foregoing whether each of the following is prime or composite: 127, 249, 257, 371. 151. Any divisor of a number may be regarded as a fac- tor of the number. An exact integral divisor of a number is an integral factor of the number. 204 COMPLETE ARITHMETIC. Properties of Numbers. 152. Prime Factors. 1. An integral factor that is a prime number is a prime factor. 5 is a prime factor of 30. 7 is a prime factor of and . 3 is a prime factor of and . 2 and 3 are prime factors of and and . 3 and 5 are prime factors of and and 2. Eesolve 105 into its prime factors. Operation. Explanation. 5)105 Since the prime number 5 is an exact divisor of 105, Qwi it is a prime factor of 105. Since tiie prime number 3 ' — ^ is an exact divisor of the quotient (21), it is a prime factor of 21 and 105. Since 3 is contained in 21 exactly 7 times, and since 7 is a prime number, 7 is a prime factor of 21 and of 105. Therefore the prime factors of 105 are 5, 3, and 7. Observe that if 7 and 3 are prime factors of 21 they must be prime factors of 105, for 105 is made up of 5 21's. 7 is contained 5 times as many times in 105 as it is in 21. Observe that every composite number is equal to the product of its prime factors. 105-5x3x7. 18=:3x3x 2. Observe that 2 times 3 times a number equals 6 times the number; 3 times 5 times a number equals 15 times the number, etc. Observe that instead of multiplying a number by 21, it may be multiplied by 3 and the product thus obtained by 7, and the same result be obtained as would be obtained by multiplying the number by 21. Why? PART II. 205 Properties of Numbers. 153. Multiples, Common Multiples, and Least Common Multiples. 1. A multiple of a number is an integral number of times tlie number. 30 and 35 and 40 are multiples of 5. 16 and 20 and 32 are multiples of 4. • 2. A common multiple of two or more numbers is an in- tegral number of times each of the numbers. 30 is a common multiple of 5 and 3. 40 is a common multiple of 8 and 10. is a common multiple of 9 and 6. is a common multiple of 8 and 12. 3. A common multiple of two or more integral numbers contains all the prime factors found in every one of the nnm- bers, and may contain other prime factors. 48 = 2x2x2x2x3. 150 = 2x3x5x5. A common mul- tiple of 48 and 150 must contain four 2's, one 3, and two 5's. It may contain other factors. 2x2x2x2x3x5x5 = 1200. 2x2x2x2x3x5x5x2 = 2400. 1200 and 2400 are common multiples of 48 and 150. 4. The least common multiple (1. c. m.) of two or more numbers is the least number that is an integral number of times each of the numbers. 40 and 80 and 120 are common multiples of 8 and 10 ; but 40 is the least common multiple of 8 and 10. 5. The least common multiple of two or more numbers contains all the prime factors found in every one of the num- bers, and no other prime factors. 206 COMPLETE ARITHMETIC. Properties of Numbers. 36 = 2 X 2 X 3 X 3. 120 = 2 X 2 X 2 X 3 X 5. The 1. c. m. of 36 and 120 must contain three 2's, two 3's, and one 5. 2 X 2 X 2 X 3x3x5 = 360, 1. c. m. of 36 and 120. 6. To find the 1. c. m. of two or more numbers: Resolve each number into Us prime factors. Take as factors of the I. c. m. the greatest number of 2's, 3's, 5's, 7's, etc., found in any one of the numbers. Example. Find the 1. c. m. of 24, 35, 36, and 50. Operation. 24 = 2x2x2x3. 35 = 5 X 7. . 36 = 2x2x3x3. 50 = 2 X 5 X 5. 2x2x2x3x3x5x5x7 = 12600, 1. c. m. Explanation. 24 has the greatest number of 2's as factors. 36 has the greatest number of 3's as factors. 50 has the greatest number of 5's as factors. 35 is the only number in which the factor 7 occurs. There must be as many 2's among the factors of the 1. c. m. as there are 2's among the factors of 24 ; as many 3's as there are 3's among the factors of 36 ; as many 5's as there are 5's among the factors of 50 ; as many 7's as there are 7's among the factors of 35 ; that is, three 2's, two 3's, two 5's, and one 7. Find the 1. c. m.: 7. Of 48 and 60. 11. Of 20, 30, and 40. 8. Of 60 and 75. 12. Of 40, 50, and 60. 9. Of 50 and 60. 13. Of 24, 48, and 36. 10. Of 30 and 40. 14. Of 25, 35, and 40. PART n. ^07 Algebra — Parentheses. 154. When an expression consisting of two or more terms is to be treated as a whole, it may be enclosed in a paren- thesis. ( 12 + (5 + 3) = ? ( 7a + (3a + 2a) = ? I 12 + 5 + 3 = ? ( 7a + 3a + 2a .-- ? Observe that removing the parenthesis makes no change in the results. M 2 _ (5 + 3) = ? ( 7a - (3a + 2a) = ? ( 12 - 5 - 3 = ? ( 7a - 3a - 2a = ? Observe the change in signs made necessary by the removal of the parenthesis. ( 12 - (5 - 3) = ? I 7a - (3a - 2a) = ? (12-5 + 3 = ? (7a-3a + 2a = ? Observe the change in signs made necessary by the removal of the parenthesis. A careful study and comparison of the foregoing prob- lems will make the reasons for the following apparent : I. If an expression within a parenthesis is preceded by the plus sign, the parenthesis may be removed without making any changes in the signs of the terms. II. If an expression within a parenthesis is preceded by a minus sign, the parenthesis may be removed; but the sign of each termin the parenthesis must be changed ; the sign + to - , and the sign - to +. 155. Eemove the parenthesis, change the signs if neces- sary, and combine the terms : 1. 15 _ (6 -f. 4) = 5. 15b - {12b - 46) = 2. 18 + (4 - 3) = 6. 18c + (9c - 3c) = 3. 27-(8 + 3)= 7. 24c^-(5^+3^) = 4. 45 + (12 -3)= 8. 36a;-(5aj + 4ic) = 208 COMPLETE ARITHMETIC. Algebra — Parentheses. 156. Multiplying an Expression Enclosed in a Parenthesis. 1. 6(7 + 4) =- ?* 6(7a + 4&) = ? Q(a -I- Z?) = ? Ans. 6a + 6b. a(b + c) = ? Ans. ah + ac. Observe that in multiplying the sum of two numbers by a third number, the sum may be found and multiplied; or each number may be multiplied and the sum of the products found. In the last three examples given above, substitute 5 for a, 3 for h, and 2 for c ; then perform again the operations indi- cated, and compare the results with those obtained when the letters were employed. 2. 6(7 - 4) = ? 6(7a - 45) = ? 6(a — h) = 1 Ans. 6a — 6b. a(b — c) = ? Ans. ah — ac. Observe that in multiplying the difference of two numbers by a third number, the difference may be found and multiplied ; or each number may be multiplied and the dilference of the products found. In the last three problems given above, substitute 5 for a, 3 for b, and 2 for c; then perform again the operations indi- catedi and compare the results with those obtained when the letters were employed. 157. Problems. li a = 6,b = S, and c = 2, find the value of the following : 1. S(a + b)-2(b-\-c). 2. 4(a + 2b) - 3(6 - c). 3. 2{2a -b) + 2(2h - c). * This means, that the sum of 7 and 4 is to be multiplied by six ; or that the sum of six 7's and six 4's is to be found. PART It. Geometry. 158. Triangles — Continued. 209 Right Triangle. 1. The sum of the angles of any triangle is equal to right angles or degrees. 2. In a right triangle there is one right angle. The other two angles are together equal to . 3. In a certain right triangle one of the angles is an angle of 40°. How many degrees in each of the other two angles ? Draw such a triangle. 4. Convince yourself by drawings and measurements that every equilateral triangle is equiangular. Equilateral Triangles. Equiangular Triangles. 5. Note that in every equiangular triangle each angle is one third of 2 right angles. So each angle is an angle of degrees. 6. If any one of the angles of a triangle is greater or less than 60, can the triangle be equiangular? a Can it be equilateral ? 7. If angle a of an isosceles triangle meas- ures 50°, how many degrees in angle &? In angle c ? 210 COMPLETE ARITHMETIC. 159. Miscellaneous Review. 1. I am thinking of a right triangle one of whose angles measures 32°. Give the measurements of the other two angles. Draw such a triangle. 2. I am thinking of an isosceles triangle ; the sum of its two equal angles is 100°. Give the measurement of its third angle. Draw such a triangle. 3. Let a equal the number of degrees in one angle of a triangle and h equal the number of degrees in another angle of the same triangle; then the number of degrees in the third angle is 180° — (a + ^)- If (^ equals 30, and h equals 45, how many degrees in the third angle ? 4. Name three common multiples of 16 and 12. 5. Name the hast common multiple of 16 and 12. 6. Find the sum of all the prime numbers from 101 to 127 inclusive. 7. Find the prime factors of 836. 8. With the prime factors of 836 in mind or represented on the blackboard, tell the following : {(£) How many times is 19 contained in 836 ? (6) How many times is 11 x 19 contained in 836 ? (c) How many times is 19 x 11 x 2 contained in 836? 160. Problems. Find the L c. m. 1. Of 18 and 20. 6. Of 36, 72, and 24. 2. Of 13 and 11. 7. Of 45, 81, and 27. 3. Of 24 and 32. 8. Of 33, 55, and 88. 4. Of 16 and 38. 9. Of 45, 65, and 85. 5. Of 46 and 86. 10. Of 3, 5, 7, and 11. (a) Find the sum of the ten results. DIVISIBILITY OF NUMBEES. 161. Numbers Exactly Divisible by 2; by 2|; by 3J; BY 5; by 10. 1. An integral number is exactly divisible by 2 if the right-hand figure is 0, or if the number expressed by its right-hand figure is exactly divisible by 2. Explanatory Note. — Every integral number that may be expressed by two or more figures may be regarded as made up of a certain number of tens and a certain number (0 to 9) of primary units; thus, 485 is made up of 48 tens and 5 units ; 4260 is made up of 426 tens and units ; 27562 is made up of 2756 tens and 2 units. But ten is exactly divisible by 2 ; so any number of tens, or any number of tens plus any number of twos, is exactly divisible by 2. 2. Tell which of the following are exactly divisible by 2, and why: 387, 5846, 2750, 2834. 3. Any number, integral or mixed, is exactly divisible by 2^- if the part of the number expressed by figures to the right of the tens' figure, is exactly divisible by 2^. 4. Show why the statement made in No. 3 is correct, employing the thought process given in the " Explanatory Kote" above. 5. Tell which of the following are exactly divisible by'2^, and why: 485, 470, 365, 472J, 3847^. 6. Any number, integral or mixed, is exactly divisible by 31 if the part of the number expressed by figures to the right of the tens' figure is exactly divisible by 3i-. 7. Tell which of the following are exactly divisible by ^, and why: 780, 2831, 576f, 742, 80. 211 212 COMPLETE ARITHMETIC. Divisibility of Numbers. 8. Any integral number is exactly divisible by 5 if it: J right-hand figure is or 5. Show why. 9. Any integral number is exactly divisible by 10 if ii% right-hand figure is . 162. Problems. 1. How many times is 2i contained in 582^?* 2. How many times is 2|^ contained in 375 ? 3. How many times is 2^ contained in 467|-? 4. How many times is 2i contained in 4680 ? 5. How many times is 3 J contained in 786|?t 6. How many times is 3^ contained 543^? 7. How many times is 3i contained in 8640? 8. How many times is 5 contained in 3885? 9. How many times is 5 contained in 1260? 163. Numbers Exactly Divisible by 25; by 33^; by 1 2|-; BY 16|; BY 20; by 50. 1. Any integral number is exactly divisible by 25 if its two right-hand figures are zeros, or if the part of the number expressed by its two right-hand figures is exactly divisible by 25. Explanatory Note. — Every integral number expressed by three or more figures may be regarded as made up of a certain number of hundreds and a certain number (0 to 99) of primary units; thus 4624 is madje up of 46 hundreds and 24 units ; 38425 is made up of 384 hundreds and 25 units; 8400 is made up of 84 hundreds and units. But a hundred is exactly divisible by 25 ; so any number of hundreds, or any tiumber of hundreds plus any number of 25's is exactly divisible by 25. * 2J is contained in 582} (4 X 58) + 1 times. Why ? 1 3i is contained in 786| (3 X 78) + 2 times. Why ? PART II. 213 Divisibility of Numbers. 2. Tell which of the following are exactly divisible by 25, and why : 37625, 34836, 27950, 38575. 3. Every number, integral or mixed, is exactly divisible by 33i, if that part of the number expressed by the figures to the right of the hundreds' figure is exactly divisible by 33^. 4. Show why the statement made in No. 3 is correct, employing the thought process given in the " Explanatory Note " under No. 1 on the preceding page. 5. Tell which of the following are exactly divisible by 33i, and why: 36466|, 2375, 46833^ 38900, 46820. 6. Any number, integral or mixed, is exactly divisible by 12|^, if the part of the number expressed by the figures to the right of the hundreds' figure, is exactly divisible by 12|^. Show why. 7. Tell which of the following are exactly divisible by 12^, and why: 375, 837^, 6450, 4329, 7467^. 8. Any number, integral or mixed, is exactly divisible by 16|, if 9. Tell which of the following are exactly divisible by 16f : 46331, 5460, 2350, 37400, 275831 2541 6f. 10. Any integral number is exactly divisible by 20 if the number expressed by its two right-hand figures is exactly divisible by 20. Show why. 11. Tell which of the following are exactly divisible by 20, and why: 3740, 2650, 3860, 29480, 3470. 12. Tell which of the following are exactly divisible by 50, and why : 2460, 3450, 6800, 27380, 25450. 214 COMPLETE ARITHMETIC. Divisibility of Numbers. 164. Problems. 1. How many times is 25 contained in 2450 ?* 2. How many times is 25 contained in 3775 ? 3. How many times is 33^ contained in 4666f ? f 4. How many times is 33|- contained in 343 3^ ? 5. How many times is 12^- contained in 4737 1^ ? 6. How many times is 12-|- contained in 3662|^ ? 7. How many times is 16| contained in 2533^? 8. How many times is 16| contained in 4550 ? 165. Numbers Exactly Divisible by 9. 1. Any number is exactly divisible by 9 if the sum of its digits is exactly divisible by 9. Explanatory Note. — Any number more than nine is a certain number of nines and as many over as the number indicated by the sum of its digits. Thus, 20 is two nines and 2 over; 41 is four nines and 4 -|- 1 over; 42 is four nines and 4 -f 2 over; 200 is twenty-two nines and 2 over ; 300 is thirty-three nines and 3 over ; 320 is a certain number of nines and 3 -|- 2 over; 321 is a certain number of nines and 3 -f 2 -f- 1 o/er. 326 is a certain number of nines and 3 -|- 2 + 6 over; but 3 -|- 2 -f- 6 = 11, or another nine and 2 over. 2. Eead the " Explanatory Note " carefully, and tell which of the following are exactly divisible by 9 : 3256, 4266, 2314, 2574. 166. Problems. 1. 4625 is a certain number of 9's and over. 2. 3526 is a certain number of 9's and over. 3. 2154 is a certain number of 9's and over. * 25 is contained in 2450 (4 x 24) + 2 times. Why? + 33J is contained in 46661 (3 x 46) -|- 2 times. Why ? PART II. 215 Divisibility of Numbers. 167. Prime Eactoks and Exact Divisors. 1. Any integral number is exactly divisible by each of its prime factors and by the product of any two or more of its prime factors. Thus, 30, (2x3x5), is exactly divisible by 2, by 3, by 5, and by (2 x 3), 6, and by (2 x 5), 10, and by (3 X 5), 15. 2. The exact integral divisors of 36, (2 x 2 x 3 x 3), are 2, 3, — , — , — , and . 168. Prime E actors'. Common Divisors, and Greatest Common Divisors. 1. Any prime factor or any product of two or more prime factors common to two or more numbers is a common divisor of the numbers. Thus, the numbers 30, (2 x 3 x 5), and 40, (2x2x2x5), have the factors 2 and 5 in common. So the common divisors of 30 and 40 are 2, 5, and 10, and the greatest common divisor is 10. Eule. — To Jind the greatest common divisor of two or more numbers, find the product of the prime factors common to the numbers. 2. Eind the g. c. d. of 50, 75, and 125. Operation No. 1. Opei -ation No. 2. 50 - 2 X 5 X 5. 5 50 75 125. 75 = 3 X 5 X 5. 125 = 5 X 5 X 5. 5 10 15 25. 2 3 5 5 X 5 = 25, g. c. d. 5 X 5 = 25, g. c. d 3. Eind the g. c. d. of 80, IOC ), 140. 4. Eind the g. c. d. of 48, 60, 72 5. Eind the g. c. d. of 64, 96, 256. 216 COMPLETE ARITHMETIC. Divisibility of Numbers. 6. Find the g. c. d. of 640 and 760. Operation. Explanation. 640)760(1 The number 760 is an integral number 640 of times the g. c. d., whatever that may he ; T^x/^^Q/r 90 is the number 640. We make an in- 600 complete division of 760 by 640 and have as a remainder the number 120. Since 40)120(3 640 and 760 are each an integral number 1^0 of times the g. c. d., their difference, 120, must be an integral number of times the g. c. d. ; for, taking an integral number of times a thing from an integral number of times a thing must leave an integral number of times the thing. Therefore, no number greater than 120 can be the g. c. d. But if 120 is an exact divisor of 640, it is also an exact divisor of 760, for it will be contained one more time in 760 than in 640. We make the trial, and find that 120 is not an exact divisor of 640 ; there is a remainder of 40. Since 600 (120 X 5) and 640 are each an integral number of times the g. c. d., 40 nmst be an integral number of times the g. c. d. But if 40 is an exact divisor of 120 it is an exact divisor of 600 (120 X 5) and 640 (40 more than 600) and 760 (120 more than 640). We make the trial, and find that it is an exact divisor of 120, and is therefore the g. c. d. of 640 and 760. Observe that any number that is an exact divisor of two numbers is an exact divisor of their difference. 169. From the foregoing make a rule for finding the g. c. d. of two numbers and apply it to the following : Find the g. c. d. : 1. Of 380 and 240. 6. Of 540 and 450. 2. Of 275 and 155. 7. Of 320 and 860. 3. Of 144 and 96. 8. Of 475 and 350. 4. Of 1728 and 288. 9. Of 390 and 520. 5. Of 650 and 175. 10. Of 450 and 600. (a) Find the sum of the ten results. PART II. 217 Algebra— Equations. 170. An equation is the expression of the equality of two numbers or combinations of numbers. Equations. (1) 2+4+6 = 3 + 5 + 4 (2) a + h + c = 40 -12 1. Every equation is made up of two members. The part of the equation which is on the left of the sign of equahty is called the first member ; the part on the right of the sign of equality, the second member. 2. If the same number be added to each member of an equation, the equality will not be destroyed. If X = 8, then ^' + 4 = 8 + 4. li a -\-b = 16, then a + & + c = 16 + c. 3. If the same number be subtracted from each member of an equation, the equality will not be destroyed. If ic = 8, then x - 3 = S - 3. If a + & = 16, then a-\-b — c = 16 — c. 4. If each member of an equation be multiplied by the same number, the equality will not be destroyed. If ic = 8, then 4x = 4: times 8, or 32. If a + & = 16, then 4a + 4& = 4 times 16, or 64. 5. If each member of an equation be divided by the same number, the equality will not be destroyed. If ^ = 8, then — = — , or 2. 4 4 If a + 6 = 16, then — H = — , or 4. 4 4 4 218 COMPLETE ARITHMETIC. Algebra— Equations. 7. Any term in an equation may be transposed from one member of the equation to the other ; but its sign must be changed when the transposition is made. If ic + 5 = 15, then x = 15 - 5, or 10. liy -6 = 27, then y = 27 + 6, or 33. li a -i-h-\-c = 18, then a -\- h = 18 - c. li x-\-y — z = 2b, then x -\- y = 25 -\- z. 171. To Find the Number for Which x Stands in an Equation in Which There Is No Other Unknown Number. Example No. 1. Equation, x -\- 2x -\- ?>x — 5 = Vd Transposing, x -\- 2x -{- ^ x = IZ -\- 5 Uniting, 6^ = 18 Dividing, x = ?> Example No. 2. Equation, 2x -\- "^x -\- % = 5x — 2a? + 18 Transposing, 2x -{- '^x — 5x -\- 2x = 18 — Q Uniting, 2a? = 12 Dividing, x = ^ Problems. Find the value of x. 1. a; + 4= 12 2. x^2,x = 8 3. 5x-2= 23 4. ZX - X = 4:4: 5. lx-^x = 144. 6. 3a; + 2^-4 = £c + 16 7. 5a; - 7 = 3a? + 5 8. lx-\-2x- x = ?,x-\-Z5 9. 5aj — 4:X — 3x -{- 6x = 44 10. 6^-8 -2a; = 3^ +5 (a) Find the sum of the ten results. PART II. 219 Geometry. 172. Quadrilaterals that are not Parallelograms. a c Trapezoid \ \ Trapezium 1. Two of the sides of a trapezoid are parallel and two are not parallel. In the trapezoid represented above the side ac is parallel to the side . 2. No two of the bounding lines of a trapezium are parallel. 3. In the trapezoid represented above no one of the angles is a right angle. Name the angles that are greater than right angles ; the angles that are less than right angles. 4. Draw a trapezoid two of whose angles are right angles. 5. Can you draw a trapezoid having one and only one right angle ? 6. Draw a trapezium one of whose angles is a right angle. 7. Can you draw a trapezium having more than one right angle ? 8. Every quadrilateral may be di- vided into two triangles. Eemember that the sum of the angles of two triangles is equal to four right angles. Observe that the sum of the angles of the two triangles is equal to the sum of the angles of the quadrilateral. So the sum of the angles of a quadrilateral is equal to four right angles. 220 COMPLETE ARITHMETIC. 173. Miscellaneous Review. 1. If two of the angles of a trapezoid are right angles and the third is an angle of 60°, how many degrees in the fourth angle ? Draw such a trapezoid.* 2. If the sum of three of the angles of a trapezium is 298°, how many degrees in the fourth angle ? Draw such a trapezium.* • 3. If one of the angles of a triangle is an angle of 80°, and the other two angles are equal, how many degrees in each of the other angles ? Draw the figure.* 4. If one of the angles of a quadrilateral is a right angle, and the other three angles are equal, what kind of a quad- rilateral is the figure ? 5. One of the angles of a quadrilateral is a degrees; another is h degrees ; the third is c degrees. How many degrees iti the fourth angle ? 6. The smallest angle of a triangle is x degrees ; another angle is 2 aj degrees, and the third is 3 a? degrees : Then x-\-1x-^'^x = 180. Find the value ^i x\ of 2 a? ; of 3 a?. 7. 643,265,245,350. Without performing the division tell whether this number is exactly divisible by 9 ; by 5 ; by 10; by 25; by 50 ; by 12|; by 18 ; by 6 ; by 15 ; by 30 ; by 90; byl6|.t 8. A number is made up of the following prime factors : 2, 2, 3, 3, 5, 7, 11. Is the number exactly divisible by 18 ? by 26 ? by 35 ? by 77 ? by 21 ? by 30 ? by 45 ? by 8 ? * It is not expected that this drawing will be accurate in its angular measure- ment—simply an approximation to accuracy, to aid the pupil in recognizing the comparative size of angles. t A careful study of pages 211-215 inclusive will enable the pupil to make the statements called for with little hesitation. FEACTIONS. 174. A fraction may be expressed by two numbers, one of them being written above and the other below a short horizontal line ; thus, |, H, f^f- 175. The number above the line is the numerator of the fraction ; the number below the line, the denominator of the fraction. 176. Kinds of Fractions. 1. A fraction whose numerator is less than its denomina- tor is a proper fraction. |, |, ||, are proper fractions. 2. A fraction whose numerator is equal to or greater than its denominator is an improper fraction. I^ 6, _2^i, are improper fractions. Note. — The fraction .7 is a proper fraction. 2.7 may be regarded as an improper fraction or as a mixed number. If it is to be con- sidered an improper fraction it should be read, 27 tenths; if a mixed number, 2 and 7 tenths. 3. Such expressions as the following are compound frac- tions : 3 of 6 2 nf 1 5 of "^ 4. A fraction whose numerator or denominator is itself a fraction or a mixed number, is a complex fraction. 2. 2 1 T> Tr-> :t-> are complex fractions. 4 3^ 2^' ^ 221 222 COMPLETE ARITHMETIC. Fractions. 5. Any fraction that is neither compound nor complex is a simple fraction. |, If, 14, are simple fractions. 6. A fraction whose denominator is 1 with one or more zeros annexed to it is a decimal fraction. y\, .7, .25, ^y^, are decimal fractions. Note 1. — The denominator of a decimal fraction maybe expressed by figures, or it may be indicated by the position of the right-hand figure of its numerator with reference to the decimal point. When the denominator is thus indicated, the fraction is called a decimal, and is said to be written decimally. Note 2.^A11 fractions that are not decimal are called common fractions. A decimal fraction when not " written decimally " (or thought of as written decimally) is usually classed as a common fraction. 7. A complex decimal is a decimal and a common fraction combined in one number. .7-|-, .25^, .056f, are complex decimals. 177. There are three aspects in which fractions should be considered. I. THE FRACTIONAL UNIT ASPECT. The numerator tells the number of things and the denomi- nator indicates their name. In the fraction ^ there are 5 things (magnitudes) called sevenths. In the fraction | there are five fractional units, each of which is one eighth of some other unit called the unit of the fraction. Note. — The function of the denominator is to show the number of parts into which the unit of the fraction is divided ; the function of the numerator, to show the number of parts taken. PART II. 223 Fractions. 11. THE DIVISION ASPECT. The numerator of a fraction is a dividend, the denomina- tor a divisor, and the fraction itself a quotient; thus, in the fraction f, the dividend is 5, the divisor 8, and the quo- tient |. Note. — In the case of an improper fraction, as |, it may be more readily seen by the pupil that the numerator is the dividend, the denominator the divisor, and the fraction (| = 2) the quotient; but the division relation is in every fraction, whether proper or improper, common or decimal, simple or complex. III. THE RATIO ASPECT.* The numerator of a fraction is an antecedent, the denom- inator a consequent, and the fraction itself a ratio; thus, in the fraction y^^, 7 is the antecedent, 10 the consequent, and ■^ the ratio. Note 1 — This relation may be more readily seen by the pupil in the case of an improper fraction. In the fraction J^, 12 is the ante- cedent, 4 the consequent, ^, or 3, the ratio. Note 2. — Every integral number as loell as every fraction is a ratio. The number 8 is the ratio of a magnitude that is 8 times some unit of measurement to a magnitude that is 1 time the same unit of measurement. 178. Eeduction of Fractions. 1. The numerator and the denominator of a fraction are its terms. 2. A fraction is said to be in its lowest terms when its numerator and denominator are integral numbers that are prime to each other. * This may be omitted until the book is reviewed. 224 COMPLETE ARITHMETIC. Fractions. 3. Reduce |-|-g- to its lowest terms. Operation. Explanation. 1 c\^^^ — ^^ Dividing each term of \^% by 10, we have ^^200 ~~ 20* 1 tenth as many parts, which are 10 times as ^n. A large. Dividing each term of \% by 4, we have 4)jTT = — . 1 fourth as many parts, which are 4 times as large. Hence, \^ = |. But 4 and 5 are prime to each other, and the fraction is in its lowest terms. Rule. — Divide each term of the fraction hy any common divisor except 1, and divide each term of the fraction thus obtained hy any common divisor except 1, and so continue until the terms are prime to each other. Reduce to lowest terms : 275 '^ -^375 520 156 ^ ^ 270 ^ ^ 340 *- -^210 (6)^ ^ ^ 180 (7) "^ ^ ^ 405 (8) ^^ ^ -^204 (9) ^^* (10) f (a) Find the sum of the ten results. J 4. Reduce | to higher terms — to 120ths. Operation. Explanation. 120 -^ 8 = 15. Ill T^ there are 15 times as many parts as there are in |, and the parts are 1 fifteenth 5 X 15 _ 2^ as large. Hence, /j^ = |. 8x15 120 * Divide each term by 12$. t Divide each term by J. t If the pupil has not had sufficient practice in addition of fractions to do this the finding of the sum may be omitted until the book is reviewed. PART II. 225 Fractions. Eeduce to higher terms — to 160ths. (1)1 (2)+i (3) A (4)ii (5) A (6)1 (7) A (8)U (9)H (10)11 (a) Find the sum of the ten results. 5. Two or more fractions whose denominators are the same, are said to have a common denominator. 6. Two or more fractions that do not have a common denominator may be changed to equivalent fractions having a common denominator. Example. f and I may be changed to 12ths, 24ths, or 36ths. 3-TT T-^T "S^TB T = "3"6 7. Two or more fractions that do not have a common denominator may be changed to equivalent fractions having their least common denominator. The 1. c. d. of two or more fractions is the 1. c. m. of the given denominators. Example. Change |i, ^, and |^ to equivalent fractions having their least common denominator. Operation. (1) The 1. c. m. of 30, 40, and 60 is 120. (2)120.30 = 4 3^.^ (3)120-.40 = 3 l^^^ (4) 120 -.60 = 2 |.^ 226 COMPLETE ARITHMETIC. Fractions. Eeduce to equivalent fractions having their 1. c. d. 1. 1^ and ^\. 6. ^\, |, and ^i. 2. U and il 7. ^\, ^. and ^. 3. II and i|. 8. ^\, f, and |f. 4 ^V and ^. 9. |f, ^\, and ||. 5. A and ||. 10. If A. and |f. (a) Find the sum of the twenty-five fractions.* 179. To Add Common Fractions. Rule. — Reduce the fractions if necessary to equivalent fractions having a common denominator, add their numerators, and write their sum over the common denominator. Example. Add H. U' and U. (1) The 1. c. m. of 45, 30, and 60 is 180. (2) H = tVj- U = ill- 1* = \U- (3) AV + iff + HI = «!• id the sum of — ^% and -^\. 6. 1^, J, and ^. TT and tV 7. ii, i, and T-V- il and ^V 8. il. i, and If. tV and if 9. A. 1. and If. A and ^. 10. A, i, and A- 1. 2. 3. 4. 5. (a) Find the sum of the ten sums.* (For a continuation of this work, see page 231.) * This may be omitted until the subject of fractions is reviewed. PAKT II. 227 Algebraic Fractions. a oc n 180. The expressions -, -, — -, are algebraic fractions. h 4: cd The above expressions are read, a divided by 5, x divided by 4, 6 divided by cd. 181. Eeduce to lowest terms: ah / a X h \ a^ ~ \a X d X cbj' 4a _ / 2 X 2 X a \ • 65" V2 X 3x &/' abc _ /a X h X c\ ' bcd~ \b X c xdj' ab -i- a b a^ -h a~ -a' 4a H- 2 2a 6b -^2 ~ ~3b abc fa X b X c\ abc -^ be a bed -^ be d Let a = 2, b = S, e = 5, and d = 7, and verify. Observe that to reduce a fraction to its lowest terms we have only to strike out the factors that are common to its numerator and denominator. a^'ir 4. -^. What factors are common to both numerator and a^e denominator? Reduce and verify. 2 1 5. -JL, AVTiat factors are common to both numerator and denominator ? Reduce and verify. a^_ xf__ 4a + 45 ^ • a'W ~ ' xY ~ ' 6e^M~ abc . ^ 2ax Zx + 6y ax 4:a X Iz 228 COMPLETE ARITHMETIC. Algebraic Fractions. 182. Keduce to higher terms : 1. Change -p- to a fraction whose denominator is ahc. ° DC 2a X a 2a^ Let a = 2, h = ^, and c = 5, and vei'ify tlie T^Xa^abc reduction. 2. Change ^ — to a fraction whose denominator is 2ay\ 3x X y 3xy Give any values you please to a, x, and y, Ony yr y~ 2ay^ ^^^ verify the reduction. 183. Keduce to equivalent fractions having a common denominator: X y Since the common denominator must be ^ ^^ d^^ exactly divisible by each of the given denomi- nators, it must contain all the prime factors * found in either of the given denominators. The new denominator must therefore be axaxhxd = a^bd ; a^bd -i- ab = ad ; a^bd -^ a^d = b. X X ad adx y xh hy ah X ad ~ a^hd d^d xh~ d^bd Give any values you please to a, h, d, x, and y, and verify. 4 Z The common denominator must contain the ah^ hc^ factors a, b, b, c, c. Reduce and verify. ^ xy ^ yz The common denominator is 5a. Reduce and 3. -^ and ^^ .. - 5 a verify. * Since the numerical values of the letters are unknown, each must be regarded as prime to all the others. The prime factors, then, in the first denominator are a and 6; in the second, a, a, and d. PART II. 229 Geometry. 184. Quadrilaterals. 1. All the geometrical figures on this page are quadrilaterals ; that is, each has four sides. 2. The first four figures are par- allelograms ; that is, the opposite sides of each flgitre are parallel. 3. The first two figures are rec- tangular; that is, their angles are right angles. 4. The first and third are equi- lateral ; that is, the sides are equal. 5. There is one equilateral rec- tangular parallelogram. Which is it? 6. There is one equilateral paral- lelogram that is not rectangular. Which is it ? 7. There is one rectangular par- allelogram that is not equilateral. Which is it ? 8. The sum of the angles of each figure on the page is equal to right angles. 9. Tell as nearly as you can the size of each angle of each figure. a b 1 c d Square. a L 2 c d ) 7 Oblong. r-j 1 / Rhombus. /a 4 1 / Rhomboid. a b\ Trapezoid. Trapezium. 230 COMPLETE ARITHMETIC. 135. Miscellaneous Review. 1. The difference of two numbers is 374-^; the smaller number is 243^-J. What is the larger number ? 2. The difference of two numbers is a; the smaller num- ber is h. What is the larger number ? 3. James had a certain number of dollars and John had three times as many ; together they had 196 dollars. How many had each ? (^ + 3a? = 196.) 4. William had a certain number of marbles ; Henry had twice as many as William, and George had twice as many as Henry; together they had 161. How many had each? {x-{-2x-\-4.x= 161.) 5. Divide 140 dollars between two men, giving to one man 30 dollars more than to the other. (ic + ^+ 30 - 140.) 6. By what integral numbers is 30 (2x3x5) exactly divisible besides itself and 1 ? 7. By what is ahc {a x h x c) exactly divisible besides itself and 1 ? (1) How many times is a contained in ahc ? (2) How many times is h contained in ahc ? (3) How many times is c contained in ahc ? (4) How many times is ah contained in ahc ? (5) How many times is ac contained in ahc ? (6) How many times is he contained in ahc ? Ohserve that a numher composed of three different prime factors has exact integral divisors. 8. Change | to 60ths. Is f more or less than |^ ? 9. Change | to lOOths. Change | to lOOths. 10. Change ^ to lOOths. Change | to lOOths. FEACTIONS. (Continued from page 226.) 186. To Subtract Common Fractions. jiULE. — Reduce the fractions if necessary to equivalent fractions having a common denominator, find the difference of their numerators, and write it over the common denominator. Example. From ^1 subtract ^j. . (1) The 1. c. m. of 25 and 35 is 175. (2) (3) Compare the following : 77 175ths-35 175ths = 42 175ths. 77 apples — 35 apples = 42 apples. Find the difference of — H = tVt- A = xVr i%V - t\V _ 43 - TT"g" 1. f and ^. 10. i and i. 2. t and A. 11. 1 and ^. 3. A- and |. 12. f and ^\. 4. H and I 13. 1 and ^V 5. 1 and ^. 14. i and ^V- 6. landi. 15. 1 and A- 7. i and |. . 16. 1 and f 8. i and i- 17. A and i. 9. i and f 18. If and i (a) Find the sum of the eighteen differences. 23X 232 COMPLETE ARITHMETIC. Fractions. 187. To Subtract one Mixed Number from another when THE Fraction in the Subtrahend is greater than the Fraction in the Minuend. Example. From 58| take 32f. Operation. Explanation. 58|- = 58^V ^4 ^^ greater than ^j, therefore we take 1 32-| = 324|- unit from the 8 units, change it to 24ths, and Difference "25^ ^^^ it to the 9 24ths. 2 units from 7 (8 — 1) units = 5 units. 3 tens from 5 tens — 2 tens. I. Find the difference of— 1. 24-1 and 16f. 6. 35| and 2^. 2. 29f and 15f 7. 28f and 14f. 3. 46i| and 18^ 8. 36/^ and 8^. 4. 52f and 31|. 9. 651- and 22|. 5. 47| and 18f. 10. 341- and 27^. (a) Find the sum of the ten differences. II. Reduce to simplest form — 1. 5J + 3i-5|. 2. 6|-3i + 4f 3. 2i-li + 3f. 4. 7| + 3|-1|. 5. 6^-3i+5f 7. 6f-2t,-l}-lf| + 2|. 8. 5J + 4| + 2i + 3| + 3i. (b) Find tlie sum of the eight results. PART II. 233 Fractions. 188. To Multiply a Fraction by an Integer. Multiply ^V by 6. Operation No. 1. Operation No. 2. 6 times -^^^ are || = If. 6 times ^-^^\= 1|. 1. Observe that by the first operation we obtain || ; that in || thei'e are 6 times as many parts as there are in ^^^ and that the parts ai'e of the same size as those in Z^. 2. Observe that by the second operation we obtain | ; that in | thei-e are the same number of parts as there are in ^''j, and that the parts are 6 times as great as those in /j. Note. — The 7 of ^^ may be regarded as a dividend ; the 24, as a divisor, and /j itself as a quotient. In ||, we have a dividend 6 times as great as that in g'j, the divisor remaining unchanged. In I we have a divisor 1 sixth as great as that in j\, the dividend remaining unchanged. Multiplying the dividend or dividing the divisor by any number multiplies the quotient by the same number. Rule. — To multvply a fraction hy an integer, multiply its numerator or divide its denominator hy the integer. I. Find the product. 1. A X 4. 5. f x8. 9. Ax 4. 2. A X 6. 6. *x9. 10. tV X 6. 3. iix5. 7. f X 8. 11. A X 5. 4. Wx7. 8. |x9. 12. 1 3 V 7 30 X /. (a) Find the sum of the twelve products. II. Find the product. 1. H x7. 5. 4f X 5. 9. 6.3 X 5. 2. H x6. 6. I|x4. 10. m X 4. 3. 7 7 '^0 x4. 7. 5|x5. 11. 4f x6. 4. 3.7 x5. 8. 8^x4. 12. 6^ x7. (b) Find the sum of the twelve products. 234 COMPLETE ARITHMETIC. Fractions. 189. To Divide a Fraction by an Integer. Divide f by 3. Operation No. 1. Operation No. 2. One third of f = f . One third of i = ^V One third of f = ^^ = 2. 1. Observe that by the first operation we obtain f ; that in | there are 1 third as many parts as there are in f , and that the parts are of the same size as those in |. 2. Observe that by the second operation we obtain /y ; that in /^ there are the same number of parts as there are in |, and that the parts are 1 third as great as those in f . Note 1. — The 6 of | may be regarded as a dividend; the 7 as a divisor, and the f itself as a quotient. In f we have a dividend 1 third as great as that in |, the divisor remaining unchanged. In g\ we have a divisor 3 times as great as that in |, the dividend remaining unchanged. Dividing the dividend or multiplying the divisor by any number divides the quotient by the same number. KuLE. — To divide a fraction by an integer, divide its •numerator or multiply its denominator by the integer. I. Find the quotient. (See p. 245, problems 15 and 16.) 1. A-4. 4. 1-^4. 7. H^4. 2. tV-5. 5. 1-^5. 8. il-5. 3. tV-20. 6. 1 - 20. 9. il-2( (a) Find the sum of the nine quotients. II. Find the quotient. (See p. 245, problems 17 and 18.) 1. 17i-^3. 4. 18^^ -^ 3. 7. l^ -^ 3. 2. 17| -H 4. 5. 18^3, ^ 4. 8. 16|- -^ 4. 3. 17i -^ 6. 6. 18^^ -^ ^• 9. 16^ H- 6. (b) Find the sum of the nine quotients. PART II. 235 Fractions. 190. To Multiply by a Fraction. $6 multiplied by 3, means, take 3 times $6. $6 x 3 = $18. $6 multiplied by 2, means, take 2 times $6. $6 x 2 = $12. $6 multiplied by 2^, means, take 2 J times $6 ; or 2 times $6 + |-of $6. $6x2|- = $15. $6 multiplied by |-, means, take ^ of $6. $6 x | = $3. $6 multiplied by |, means, take | of $6. $6 x f = $4. To THE Teacher. — Require the pupil to examine the preceding statements until he clearly understands that to multiply by a fraction is to take such part of the multiplicand as is indicated by the frac- tion. Thus : to multiply 48 by | is to take three fourths of 48 ; that is, three times 1 fourth of 4^. It will thus be clear that multiplication by a fraction involves both multiplication and division. Example I. Multiply 24 by |. 1 fourth of 24 is 6. 3 fourths of 24 are 18. Example III. Multiply 2 75 f by f. 1 fourth of 275| is 68395-. 3 fourths of 275f are 206yV ' EuLE. — To multijply hy a fraction, divide the multiplicand hy the denominator of the fraction and midtiply the quotient thus obtained by the mtmerator of the fraction. Observe that in practice we may, if more convenient, multiply the multiplicand by the numerator of the fraction, and divide the prod- uct thus obtained by the denominator. To multiply 12 by | we may take 3 times 1 fourth of 12 or 1 fourth of 3 t'mes 12, as we choose. Example II- Multiply f by |. 1 fourth of f is ^^. 3 f^ .rths of 1 are /^. Example IV. Multiply 346f by 2^. Two times 346f = 6924. 1 half of 346f = 1731. 6924 + 173|z = 866 Ans. 236 COMPLETE ARITHMETIC. Fractions. I. Find the product. (See p. 245, problems 19 and 20.) 1. 345 X f 4. 263 x f 7. 263 x f 2. 345 X yV* 5. 263 x |. 8. 576 x |. 3. 345 X i 6. 263 x f 9. 576 x |. (a) Find the sum of the nine products. II. Find the product. (See p. 245, problems 21 and 22.) I'-Xvi 4.5^1 75vl 2- t\ X A-t 5. f X |. 8. J X |. 3. -r\ X i. 6. I X |. 9. i X I- (b) Find the sum of the nine products. III. Find the product. (See p. 246, problems 23 and 24.) 1. 372^ X f 4. 523f x f 7. 523f x i- 2. 372J^ X yV 5. 523f x f. 8. 153i- x f. 3. 372^ X i- 6. 523f x f 9. 153|- x f. (c) Find the sum of the nine products. IV. Find the product. (See p. 246, problems 25 and 26.) 1. 462f X 2^. 6. 346-J X 3|. 2. 462| X 3yV 7. 346J X 2^. 3. 462f X 2f 8. 2751- X 4f 4. 346 J X 2i. 9. 2751- X 3^. 5. 346J X 3f. 10. 275J X 2^. (d) Find the sum of the ten products. * Take 3 times 1 tenth of 345, or 1 tenth of 3 times 345. t Lead the pupil to see that in problems of this kind the correct result may be obtained by " miiUiplyuig the mimerators together for a new numerator and the denomina- tors together for a new denominator"; that in so doing he divides the multiplicand by the denominator of the multiplier and multiplies the quotient so obtained by the numerator of the multiplier. PART II. Algebraic Fractions. 237 a c 1. c. d. = hd bd^h = d -^-^^ a X d b X d ad bd bd G X b d X b be bd ad be ad -\- be bd^ bd~ bd I + -. Led. = 21 21 -=-3 = 7. 2x7 3x7 14 2l 21 14 2l"^21 7 = 3. 15 29 5 X 3 _15 7x3~2r 21 Observe that in cases like the above, in which the denominators are prime to each other, the 1. c. d. is the product of the given denominators, and each new numerator may be found by multiplying the given numerator by the denominator of the other fraction. 191. Problems in Addition and Subtraction. a b ay bx ay -f bx X y xy xy xy Let a = 2, 6 = 3, ic = 5, ?/ = 7, and verify. a b ay bx ay — bx 2.-- X y xy xy xy Assign a numerical value to each letter and verify. 3. --- X y 4. Solve. — Then let £c = 5 and y = 1 and verify. X y 238 COMPLETE ARITHMETIC. Algebraic Fractions. 192. Problems in Multiplication and Division. Example I. a ac I'-'l-n L( *. Ct -L r- J 0X1-^C2X36., 3t a - 2, & - 5, and c = 3 ; then = .= = 1\. ODD Example II. a a 2 . 2 b^'-be 5^^~15 L( it a = 2,h = 5, and c = S : then — = — he 5x3 15 Example III. a c ac 2 3 6 • h^d^M 5 ^ 7 ~ 35 L( 3t a - 2, 5 - 5, c - 3, and 6^ - 7 ; then ^^2x3^ 6 od 5x7 35 I. Find the product and verify as above. 1. a^ ^ X ^ ax ^ V3 X c 3. xy 5. — - X 3a ¥ y^ "^ bx 2. -^.X X 4. xa 6. -x5 y' cd 7 II. Find the quotient and verify as above. 1. a'^ ^ X ^ ax ^ jj^c S, ^^y 5. ^ 3a b^ y^ *^ hx 2. X . ah — ■i-x 4. — y' cd o X ^ -5- a 6. - -^ 5 PART II. 239 Qeometry. 193. The Protractor. Measure angle c. 240 COMPLETE ARITHMETIC. 194. Miscellaneous Review. 1. A piece of land in the form of an equilateral triangle measures on one side 46^^^ rods. What is the distance around it ? 2. The perimeter of a piece of land that is an exact square is 246|- feet. How far across on one side ? 3. The length of a certain rectangular field is three times its breadth ; its perimeter is 360 rd. What is its breadth ? Its length ? 4. If f of the value of a farm is $2154, what is -| of the value of the farm ? Note. — If f of a certain number is 24, what is the number? What is I of the number ? 5. I spent f of my money and had $3.60 remaining, (a) How much did I spend ? (b) What I had remaining, equals what part of what I spent ? 6. Change ^^^ to an equivalent fraction whose denomi- nator is 30. 7. Change f to an equivalent fraction whose numerator is 30. a' 8. Change — to an equivalent fraction whose deuomi- nator is he. 9. Multiply I by f and multiply the product by 25. 10. Multiply 1^ by ^| and multiply the product by 25. 11. If -I of an acre of land is worth $36, how much are 37^ acres worth at the same rate ? 12. The rent of a house for 2 yr. 4 mo. was $840. What was the rate per year ? FRACTIONS. 395. To Divide by a Fraction. Example I. Divide 6 by |. Operation No. 1. Operation No. 2. 1^2 — 34- ^ H — 2-T 6 = V-. _ Q * 6 -f- 1 = 6 times f = -i/ = 9. Operation No. 1. 7 — 35 S - TO- Example II. Divide ^ by f . Operation No 2. 3 — 24 r - TIT- 35^24 TO" • T¥ 111 + ^^4' + 1 - f = l§ J - f = I of I = If = m. From the foregoing operations the following rules for dividing by a fraction are obtained : Rule I. — Reduce the dividend and the divisor to like frac- tional units, then divide the numerator of the dividend hy the numerator of the divisor. Rule II. — ''Invert the divisor and proceed as in multipli- cation" Observe that the inverted divisor shows the number of times the divisor is contained in 1 : then in 6 it is contained 6 times as many times ; in 4, 4 times as many ; in |, | as many ; in |, | as many, etc. * 18 thirds + 2 thirds = 9. $ 35 fortieths -+- 24 fortieths = IJl. 1 1 = g. 3 thirds + 2 thirds = IJ = e. g l = g. 5 fifths + 3 fifths = 1| = g. 241 242 COMPLETE ARITHMETIC. Fractions. I. Find the qUOtiert. ( see page 246, problems 27 and 28.) 1. 46 -^ I 4. 375 ^ f 7. 196 ^f 2. 46 ^ f . 5. 375 H- 1. 8. 196-^3. 3. 46-^1. 6. 375 H- f. 9. 196-^ |. (a) Find the sum of the nine quotients. II. Find the quotient. (See page 246, problems 29 and 30.) 1. |-|. 4. A-*- 7. H-l 2. 1-i 5. A-i 8. 4i-^l- 3. l-i*- 6. A-lf 9. ^T "^ T?- (b) Find the sum of the nine quotients. III. Find the quotient. (See page 246, problems 31 and 32.) 1. 5i*|. 4. 24|^|. 7. 19|-t- 2. 5J^|. 5. 24|^|. 8. 19|^f 3. 5i^|. 6. 24i^|. 9. 19| - f. (c) Find the sum of the nine quotients. IV. Find the quotient. (See page 246, problem 33.) 1. 325|--^2f 6. 174^-^-2 J. 2. 325|--f-2f. 7. 174^ -^2f 3. 325^^31 8. 174^-4-31. 4. 325^-^7 J. 9. 1741- -^ 7 J. 5. 3251- H-H. 10. 174^^ If (d) Find the sum of the ten quotients. PART II. - 243 Fractions. 196. To Eeduce Complex Fractions to Simple Fractions. I is read, | divided by |. f ^ 7 ^ 3 of s = 2 4 ^ e . T. Eeduce to their simplest forms. 5555 ^ T T 6 (a) Find the sum of the four fractions. Observe that a complex fraction may be reduced to a simple fraction by multiplying its numerator and denominator by some number that will in each case give an integral product. When this number can be easily discovered by inspection this is a convenient method of reduction : thus i = ^ 9 ~ Tl' II. Eeduce to their simplest forms. (See page 246, problem 34.) 6 1 K 11 1. g Z. ^ 6. ^ 4. 2 (b) Find the sum of the four fractions. III. Eeduce to their simplest forms. . 8 ^8 ^8 8 1-T 2.^ 3.^ 4.- ^ T ¥ "5 (c) Find the sum of the four fractions. IV. Eeduce to their simplest form. 1 l!i* 2 IM 3 ^ 4 ^ 100 100 100 100 ♦Divide the nxunerator and denominator by 12J. 244 COMPLETE ARITHMETIC. Fractions. 197. Pkactical Application of the Preceding Eules. Page 224, problems 1 and 2. 1. B owned a farm of 375 acres; he gave to his son 275 acres. What part of the farm did the son receive ? 2. Mr. L. earned $650 in one year; of this sum he ex- pended S520. What part of his earnings did he expend? Page 225, problems 1 and 2. 3. A lady owned ^ of an acre of land. How many 160ths of an acre did she own? 4. Benton walked ^^ of a mile. Express the distance he walked in 160ths of a mile. Page 226, Art. 179, problems 1 and 6. 5. In a certain furnace -^^ of a ton of coal was consumed in one day, and -^-^ of a ton the next day. What part of a ton was consumed in the two days ? 6. Mr. Luker has three lots of land ; in the first lot there are |-J of an acre ; in the second, i of an acre, and in the third, l^f of an acre. How many acres in all ? Page 231, Art. 186, problems 3 and 4. 7. Of -^^ of a mile of board fence i of a mile was burned. What part of a mile remained ? 8. Mr. Keynolds had put into the bank ^-J- of his annual salary; he drew from this money a sum equal to -J of his salary. What part of his salary remained in the bank ^ Page 232, Art. 187, L, problems 4 and 5. 9. From 52|- tons of hay, were sold and delivered 31 1 tons. How many tons remained of the unsold hay ? 10. On Monday James rode 47f mi. ; on Tuesday, 18| mi. How much further did he ride Monday than Tuesday ? PART II. 245 Page 233, Art. 188, I., problems 1 and 5. 11. If a street car makes a round trip in -^ of an hour, in how long a time can it make 4 such trips ? 12. If J yd. of ribbon are required to trim a hat, how much ribbon will be required to trim 8 such hats ? Page 233, Art. 188, II., problems 1 and 5. 13. At 3^ dollars a cord, what is the cost of 7 cords of wood ? 14. If Henry rides his wheel at the rate of 4| miles an hour, how far does he ride in 5 hours ? Page 234, Art. 189, I., problems 1 and 4. 15. If yV of a yd. of ribbon is cut into 4 equal pieces, what part of a yard is each piece ? 16. John hoes 4 rows of corn in | of an hour. In what part of an hour does he hoe 1 row ? Page 234, Art. 189, II., problems 1 and 5. 17. A horse traveled 17^ miles in 3 hours. What was his rate per hour ? 18. A farmer divided a field containing 18y3-g- acres into 4 equal lots. How many acres in each lot ? Page 236, I., problems 1 and 5. 19. At $345 an acre, what is the cost of |^ acre ? 20. At $263 an acre, what is the cost of f acre ? Page 236, II., problems 1 and 9. 21. A piece of land in the form of a rectangle is -{'^ of a mile long and |^ of a mile wide. The piece is what part of a square mile ? 22. At i a dollar per yard, what is the cost of |- of a yard of silk ? 246 COMPLETE ARITHMETIC. Page 236, III., problems 1 and 9. 23. A strip* of land 372|- rods long and ^ a rod wide con- tains how many square rods ? 24. At $153-|- an acre, find the cost of -|- of an acre. Page 236, IV., problems 1 and 8. 25. At $46 2 1 a mile, what is the cost of grading 2^^. miles of road? 26. How many square feet in a piece of land 275^^ ft. by 4J ft.? Page 242, I., problems 1 and 5. 27. At |- a dollar a bushel, how many bushels of potatoes can be bought for 46 dollars ? 28. At |- of a dollar a bushel, how many bushels of apples can be bought for 375 dollars? Page 242, II., problems 1 and 5. 29. At I of a dollar a yard, how many yards of cloth can be bought for -J of a dollar ? 30. At I of a dollar a yard, what part of a yard of cloth can be bought for -^\ of a dollar ? Page 242, III., problems 1 and 6. 31. If a rectangular diagram on the blackboard contains 5^ square feet and is f of a foot wide, how long is it? Make the diagram. 32. At Sf a bushel, how many bushels of meal can be bought for S24| ? Page 242, IV., problem 1. 33. A strip of land contains 325^ square rods, and is 2| rods wide. How long is it ? Page 243, II., problem 1. 34 . Five-sixths of an hour is what part of 5 hours ? PART II. 247 Algebraic Fractions. 198. Problems in Division with a Fraction for a Divisor. Example I. See page 241, Rule 11, and Observation. Let a = 1,1 - 2, and c = 3 ; then-— = — - — = -7r= 104 & 2 2 ^ Example TI. See page 241, Rule II, and Observation. a b ac X ' c ~ bx 10 • 3 " 20 ^^ Let a = 7,05 = 10,6 = 2, andc = 3; then-— = - ^t^ttt^^I bx 2 X 10 20 YH I. Find the quotient and verify as above. 1. x-^ 2.2/- X 4. b y 5. xy ^ - a 6. yz II. Find the quotient and verify as above. . a b 1. - -^ — X y 6. - -i- — y z ^ b X 0. — -i- — c y b^c_ X ' y 4. a b z X 6. c X d'" y 248 COMPLETE ARITHMETIC. Algebraic Fractions. 199. Miscellaneous Exercises. Example I. h ac A-h c c 5 5 5 Let a = 3,b = 2, and c = 5. c 5 5 5 I. Keduce to improper fractions and verify as above. 1. X + y 2-^+^ 3.^-1-1 Example II. ab 4- c c 11 _ „2 G 5 2 Let a r= 2, & =z 3, and c :i- 5; then a+-= 2 +-:= 3- So II. Keduce to mixed numbers and verify. 1. ax -{-b 2. bij + c s.'l+I III. Reduce and verify. 2 — ^- 3 4. A 2 PART II. 249 Geometry. 200. CONSTRUCTIOX PROBLEMS — TRIANGLES. 1. Draw a triangle. Make the side ah 3 inches long. Make the angle a, 45°. Make the angle h, 45°. Prove your work by measuring the angle c which should be an angle of degrees. Observe that if two angles of any triangle and the length of the included side are given, the triangle may he drawn. 2. Draw a triangle making one of the angles 40°, another 60°, and the included side 5 inches long. The third angle should measure degrees. Prove your work by measur- ing the third angle. Measure the sides carefully and observe that the longest side is opposite the largest angle, and the shortest side opposite the smallest angle. 3. Draw several triangles of different shapes and sizes. Convince yourself by measurement with the protractor that the sum of the three angles of any triangle is degrees. 4. Draw several triangles of different shapes and sizes. Convince yourself by measurement with a ruler that the sunt of two sides of any triangle is greater than the third side of the same triangle. 5. Attempt to draw a triangle whose sides are 6 inches, 3|^ inches, and 2^ inches. 250 COMPLETE ARITHMETIC. 201. Miscellaneous Review. 1. If |- of a cord of wood cost $4.50 how much will 27 cords cost at the same rate ? 2. If 2^ tons of coal cost $12.60, how much will 17|- tons cost at the same rate ? 3. A man owned Yy^^- acres of land; he sold 2 J acres. What fractional part of his land did he sell ? 4. The sum of two fractions is 1 -^^ ; one of the fractions is ^ . What is the other fraction ? 5. The product of two fractions is ^f ; one of the frac- tions is f . What is the other fraction ? 6. If a furnace consumes y^^ of a ton of coal a day, in how many days will 5^ tons be consumed ? 7. How many pounds of sugar at 4(f a pound must be given for 27|- pounds of butter at 23(f a pound? 8. How many pounds of coffee at S3^(f a pound must be given for 15 J dozen eggs at 20^ a dozen? 9. Which is the greater fraction, | or | ? 10. Multiplying both terms of a fraction by the same number does not change the value of the fraction. Does adding the same number to both terms of a fraction change the value ? 11. Dividing both terms of a fraction by the same num- ber does not change the value of the fraction. Does sub- tracting the same number from both terms of a fraction change the value ? 12. Change :|^ to a fraction whose denominator is 46. 13. Change ;^ to a fraction whose denominator is 15. FEACTIONS. 202. To Change Decimals to Common Fractions and Common Fractions to Decimals. Example I. Change .36 to a common fraction in its lowest terms. 36 9 .36= = — 100 25 Eeduce to common fractions. - 1. .45 3. .375 5. .55 7. .625 2. .045 4. .0375 6. .055 8. .0625 (a) Find the sum of the eight decimals. (b) Find the sum of the eight common fractions. Example II. Change | to a decimal. Operation. Explanation. 8)3.000 3 over 8 means 3 divided by 8. We therefore r;73 annex zeros to the numerator 3, and perform the .o I O , . . . division. One eighth of 30 tenths is 3 tenths with a remainder of 6 tenths, etc. Eeduce to decimals. 1. 2. i 3. 4. 5- i 6- fl 7. 8. 1 (c) Find the sum of the eight decimals. 251 252 COMPLETE ARITHMETIC. Fractions. Example III. Change f to a decimal. Operation. Explanation. 7)2.00 It will be observed that, however far this 90^1 division may be carried, there is always a re- mainder. The fact that there is a remainder 7)2.000 is indicated by writing the plus sign after the 9854- ^^^^ figure of the decimal. The first quotient may be read 28 hundredths, plus. 7)2.0000 Observe, too, that the error in the first .2857+ answer is less than 1 hundredth, since the true quotient is more than 28 hundredths and less than 29 hundredths. AVe may therefore say that the first result is true to hundredths ; the second, true to thousandths. Reduce to decimals, true to thousandths. 1. 4 4-tV 7. J 10. 1 2. f 5. ,3^ 8. 1 11. f 3. f 6-tV 9. i 12. i (a) Find the sum of the twelve common fractions. (b) Find the sum of the twelve decimals.* Determine which of the following fractions can be redu(ied to terminating decimals f and which cannot. 1. f 3. 1 5. 1 7. f 9. A 2. T 4. \ 6. t 8. A 10. tV Observe that if a fraction is in its lowest terms and the denom- inator contains any other prime factor besides 2's and o's, the fraction cannot be reduced to a "terminating" decimal. Can you tell why? * Observe that the difference between a and 6 must be less than 12 thousandths. Why? t See foot-note, page 253. PART II. 253 Fractions. 203. To Eeduce a Complex Decimal* to a Common Fraction. Example. Change .27^^^ to a common fraction. Operation. Explanation. 27^ Writing the denominator, M-e have •^ 'tt = -j^QQ 27t\ divided by 100. 273—300 27 r\ reduced to llths = \»A .3^(M)_ -5- 100 = yWg- = -j-V -T? - 100 = AVtt = rV Eeduce to common fractions. 1. .38i 4. .24^- 7. .611 2. .83^ 5. •35A 8. .16| 3. .45f 6. •40A 9. .544 (a) Find the sum of the nine common fractions. (b) Find the sum of the nine complex decimals. Note. — If. the division be carried sufficiently far in any non-ter- minating decimal there will be found a certain figure or set of figures that is constantly repeated : thus, we may have, .3666666, or .27272727, ar .5236236236. The part repeated is called a repetend, and may be «vTitten thus: .36, .27, .5236. It is a curious fact that the real denominator of any repetend is as many 9's as there are figures in the repetend ; .36 = .3§, .27 = f|, .5236 = .5f||. * Decimals that are complete without the annexation of a common fraction are said to be terminating decimals. .24 is a terminating decimal. .666666 + is a non- terminating decimal. A decimal with a common fraction annexed, as .33J, is some- times called a complex dednud. 254 COMPLETE ARITHMETIC. Fractions. 204. To Keduce a Fraction to Hundredths. Examples. J o 4)3.00 ^ p 100 75 ^r, 1.3 ^;orfof-=_or.75. II. I 3)2:00 rf 100^661^^ ^ .661 ^ 100 100 ^ -r-TT 7 8)7.00 „ n 100 874 o„, IV. ^V ^^^-^ ;or^Vofi^=-^or.02i. TO ^2f ^0 100 100 ^ I. Eeduce to hundredths. 1. ^ 6. 4- 11. 1 16. 3 -L 71 122 175 Ql «3 IQll 1R9 O. g- O. g^ 10. -f-fy 10. -j^ 4.3 Q2 14.'" 1Q1 5. k 10. I 15. ^3^ 20. ^ (a) Find the sum of the twenty decimals. (b) Find the sum of the twenty common fractions. II. Eeduce to hundredths. 1. 2. 3. (a) Find the sum of the nine decimals. (b) Find the sum of the nine common fractions. tVV 4-tV\ 7- T*,V 1 5 5. AV 8-tVt TT~S tVV 6- AV ^■^ PART II. 25o Fractions. III. Reduce to hundredths. Note. — Such fractions as the following may be easily reduced to hundredths by dividing the numerator and the denominator of each by that number which will change the denominator to 100.* 1. 2. 3. 4. (a) Find the sum of the twelve results. (b) Find the sum of the twelve common fractions. IV. Reduce to hundredths. Note. — Multiply the numerator and the denominator of each fraction by that number which will change the denominator to 100. 1. Ttnr 5. 45 40 9. AV AV 6. 4 10. AV ^ 7. 1 20 TOD 11. AV Ui 8. m 12. m 7 20 -i 7.i2 20 14 25 •^•fo 8. il 25 27 50 6. ** 12J 9.^ 50 3. (a) Find the sum of the nine decimals. V. Reduce to hundredths. 2. t'A 5- U 8- AV Q 1 5 4 fi 16 9 JL 9 6. (a) Find the sum of the nine decimals. * Every common fraction can be changed to hundredths by annexing two zeros to the numerator and dividing by the denominator; but this method of reduction is not always the most simple. 256 COMPLETE ARITHMETIC. Fractions. 205. Denominate Fractions. 1. One half inch is what part of a foot ? 2. Two and ^ inches are what part of a foot ? 3. Five and ^ inches are what part of a foot ? 4. One half foot is what part of a rod ? 5. Three and one half feet are what part of a rod ? 6. Ten and one half feet are what part of a rod ? 7. Sixty-four rods are what part of a mile ? 8. Mnety-six rods are what part of a mile ? 9. One hundred eighty rods are what part of a mile ? 10. One and one half quarts are what part of a peck ? 11. Two and one half quarts are what part of a gallon ? 12. Twenty-four quarts are what part of a bushel ? 13. Fourteen ounces are what part of a pound ? 14. Seven and one half ounces are what part- of a pound? 15. One and one fourth ounces are what part of a pound? 16. Six hundred pounds are what part of a ton ? 17. Four hundred fifty pounds are what part of a ton ? 18. Six hundred twenty-five pounds are what part of a ton f 19. Seventy-five square rods are what part of an acre ? 20. Forty -five square rods are what part of an acre ? 21. One hundred square rods are what part of an acre ? 22. Thirty-two cubic feet are what part of a cord ? 23. Fifty-six cubic feet are what part of a cord ? 24. One hundred cubic feet are what part of a cord ? 25. Seven and one half minutes are what part of an hour ? 26. Forty minutes are what part of an 8-hour day ? 27. Ninety minutes are what part of an 8-hour day ? PART II. 257 Algebraic Fractions. 206. Fractions in Equations. Example I. 2 Multiplying both members of the equation (see page 217, Art. 170, Statement 4) by 2, the denominator of the fraction in the equa- a? + 4a? = 60 Uniting terms, 5a; = 60 Dividing by 5, jc = 12 Example II. f + f + 5a; = 70 2 3 2r Multiplying by 2, x -^ — -f IQx = 140 o Multiplying by 3, Zx-^2x-\- 30aj = 420* Uniting terms, 35ic = 420 Dividing by 35, ic = 12 Problems. Find the value of x. 1. ^+^ 5 3 = 8. 3. - + 5a; = 64. 3 ^ X X = 6. 4. 2a;-- = 50. * Observe that the equation might have been cleared of fractions by multiplying both its members by 6, the 1. c. m. of 2 and 3. 258 COMPLETE ARITHMETIC. Algebra. 207. Problems Leading to Equations Containing One Unknown Quantity, Without Fractions. Example. John and Henry together have 60 oranges, and Henry has three times as many as John. How many has each ? Let X = the number John has, then Sx = the number Henry has, and x-\-Zx — the number they together have. Therefore x-\-Sx= 60. Uniting 4:X— 60. Dividing x = 16. Multiplying 3x = 45. John has 15 oranges and Henry has 45 oranges. Problems. 1. The sum of two numbers is 275, and the greater is four times the less. What are the numbers ? 2. Eobert has a certain sum of money and Harry has five times as much; together they have $216. How many dol- lars has each ? 3. One number is four times another, and their difference is 270. What are the numbers ? 4. Peter has a certain number of marbles and William has 8 more than Peter; together they have 96 marbles. How many has each ? 5. Sarah has a certain number of pennies and her sister has nine more than twice as many; together they have 93. How many has each ? 6. Two times Keuben's money plus three times his money equals 175 dollars. How many dollars has he ? PART II. 259 Qeometry. 208. Construction Problems — Triangles. 1. Draw a triangle whose base, ah, is 3 inches long. Make the angle a, 55° and the angle h, 35°. The angle c should be degrees. Measure the three sides. 2. Draw a triangle two of whose sides are equal. Measure and compare the angles opposite the equal sides. Observe that a triangle, two of whose sides are equal, has two angles equal; and conversely if two angles of a triangle are equal, two of the sides are equal. 3. If two triangles have the three sides of one equal to the three sides of the other, each to each, do you think the two triangles are alike in every respect ? 4. If two triangles have the three angles of one equal to the three angles of the other, each to each, do you think the two triangles are necessarily alike iji every respect ? 5. Draw two triangles, the angles of one being equal to the angles of the other, and the sides of one not being equal to the sides of the other. 6. Is it possible to draw a triangle whose sides are equal, but whose angles are unequal ? 7. Is it possible to draw a quadrilateral whose sides are equal but whose angles are unequal ? 260 COMPLETE ARITHMETIC. 209. Miscellaneous Review. 1. Without a pencil, change each of the following frac- tions to hundredths : -, 1, A, 1, 1, 1, A, _I_, A, 5 8 8 10 25 50 12i 33^ 16| 60 30 45 60 60 60 60 500' 200' 900' 150' 250' 125' 66f* 2. Butter that cost 25^ a pound was sold for 29^ a pound. The gain was equal to what part of the cost ? The gain was equal to how many hundredths of the cost ? 3. The taxes on an acre of land which was valued at $600 were $12. The taxes were equal to what part of the valuation ? The taxes were equal to how many hundredths of the valuation ? 4. Mr. Jones purchased 500 barrels of apples. He lost by decay a quantity equal to 75 barrels. What part of his apples did he lose ? How many hundredths of his apples did he lose ? 5. Kegarding a month as 30 days and a year as 360 days, what part of a year is 7 months and 10 days ? How many hundredths of a year in 7 months and 1 days ? How many thousandths of a year ? How many ten-thousandths of a year? 6. One cord 48 cubic feet is what part of 4 cords 16 cubic feet ? Change the fraction to hundredths ; to thousandths ; to ten-thousandths. 7. One mile 240 rods is what part of 3 miles 160 rods? Change the fraction to hundredths ; to thousandths ; to ten- thousandths. 8. From a bill of $175 there was a discount of $14. The discount is equal to how many hundredths of the' amount of the bill? PEECENTAGE. 210. Per cent means hundredth or hundredths. Per cent may be expressed as a common fraction whose denominator is 100, or it may be expressed decimally ; thus, 6 per cent = j-f-g- or .06; 28i- per cent = j^^ or .28|-; |- per cent = j^g or .00|. Note. — Instead of the words per cent sometimes the sign (%) is used ; thus 6 per cent may be written 6 %. 211. The base in percentage is the number of which hundredths are taken ; thus, in the problem, find 1 1 % of 600, the base is 600 ; in the problem, 16 ^s what per cent of 800 ? the base is 800 ; in the problem, 18 is 3% of what? the base is not given, but is to be found by the student. Observe that whenever the base is given in problems like the above, it follows the word of. 212. There are three cases in percentage and only three. Case I. To find some per cent (hundredths) of a number, as: find 15% of 600. Case II. To find a number when some per cent of it is given, as : 24 is 8% of what number ? Case III. To find what per cent one number is of another, as : 12 is what % of 400 ? Observe that a thorough knowledge of fractions is the necessary preparation for percentage. The work in percentage is work in fractions, the denominator employed being 100. 261 262 COMPLETE ARITHMETIC. Percentage. 213. Case I. Find 17 per cent (.17) of 8460. Note. — We may find f of a number by finding 3 times 1 fourth of it; that is, by multiplying it by |. So we may find .17 of a number by finding 17 times 1 hundredth of the number; that is, by multiplying by .17. Operation. Explanation. 84^60 .17 One per cent (1 hundredth) of 8460 is 84.60; 17 592 20 P®^ ^^"* (hundredths) of 8460 is 17 times 84.60, br 846^0 1438.20. 1438.20 Problems. 1. Find 17% of 6420 ; of 5252 ; of 31.40. 2. Find 35% of 6420 ; of 5252 ; of 31.40. 3. Find 43% of 6420; of 5252 ; of 31.40. 4. Find 25% of 6420; of 5252; of 31.40. 5. Find 50% of 6420 ; of 5252 ; of 31.40. 6. Find 30% of 6420; of 5252 ; of 31.40. 7. Find 35% of 6420; of 5252 ; of 31.40. 8. Find 65% of 6420; of 5252 ; of 31.40. (a) Find the sum of the twenty-four results. 9. A sold goods for B. As remuneration for his services he received a sum equal to 12 % of the sales. He sold $2146 worth of goods. How much did he receive ? 10. C is a collector of money. For this service he charges a commission of 6 % ; that is, his pay is 6 % of the amount collected. He collected for D $375. How much should he pay over to D, and how much should he retain as pay for collecting ? PART II. 263 Percentage. 214. Case II. 673.20 is 17 per cent (.17) of what number? Operation iN'o. 1. 17)673.20(39.60 51 100 163 153 3960. 102 102 Explanation, Since 673.20 is 17 hundredths of the number, 1 hundredth of the number is 1 seventeenth of 673.20, or 39.60 ; and 100 hundredths = 100 times 39.60, or 3960. Operation No. 2. .17)673.20X3960 51 163 153 102 102 Explanation. We may find 100 seventeenths of a number by finding 1 seventeenth of a hundred times the number. 100 times 673.20 is 67320. 1 seventeenth of 67320 is 3960. Observe that dividing by .17 is finding ^^ of the dividend, just as dividing by | is finding | of the dividend, and dividing by ^ is finding f of the dividend. Note. — Sometimes the process may be shortened by writing the per cent as a common fraction and reducing it to its lowest terms ; then using the reduced fraction instead of the one whose denomina- tor is 100. Problems. 1. 360 is 15% of what number? 2. 360 is 25% of what number? 3. 360 is 50% of what number? 4. 360 is 75% of what number? 5. 360 is 40% of what number? (a) Find the sum of the five resuliis. 264 COMPLETE ARITHMETIC. Percentage. Case II. — Continued. Problems. 1. $34.32 is 13 per cent of what? 2. $34.32 is 15 per cent of what? 3. $34.32 is 25 per cent of what? 4. $34.32 is 33i per cent of what ? 5. $34.32 is 50 per cent of what ? 6. $64.98 is 19 per cent of what? 7. $64.98 is 12 per cent of what ? 8. $64.98 is 20 per cent of what ? 9. $64.98 is 24 per cent of what ? 10. $64.98 is 25 per cent of what ? (a) Find the sum of the ten results. 11. A sold goods for B. As remuneration for his services A received a sum equal to 12% of the sales. He received $33.06. What was the amount of his sales ? How much money does B have left of what he received for the goods after paying out of it A's commission ? 12. C is a collector of money. For this service he charges a commission of 6 % ; that is, his pay is 6 % of the amount collected. His commission on a certain collection was $74.40. What was the amount collected ? How much should the man for whom he collected the money, receive ? 13. C collected a sum of money for D, deducted his com- mission of 6%, and paid the remainder of the sum collected to D. He paid D $350.15.* What was the sum collected ? How much money did C retain as his commission for col- lecting ? * 8350.15 is what % of the amount collected? PART II. 26b Operation No. 1. )625"^( 5400 900)625"^00(.69| 8500 8100 400 4 Percentage. 215. Case III. 625 is what per cent of 900 ? Explanation, 625 is lit of 900. This fraction may be changed to hundredths in the usual manner, viz., by performing the division indicated^ " carrying out " to hundredths only. 900 9 Operation No. 2. Explanation. 625 _ 25 36)25^00(.69-| = :69|< % 625 is f §§ or H of 900. 216 The fraction, ||, may be 340 changed to hundredths in the 324 usual manner. 16 4 36 ~ 9 Operation and Explanation No. 3. 625 625 is -— of 900. 900 625- 900- \:-z- "•-">' Operation and Explanation No. 4. One hundredth of 900 is 9 ; then 625 is as many hundredths of 900 as 9 is contained times in 625. 9 is contained in 625, 69f times ; so 625 is Q9i-% (hundredths) of 900. Problems. What per cent (how many hundredths) of 800 is — (1)250? (2)375? (3)475? (4)350? (5)150? (a) Find the sum of the five results. 266 COMPLETE ARITHMETIC. Percentage. Case III. — Continued. Problems. What % is— 1. 32 of 600 ? 6. 50 of 750 ? 11. 40 of 325 ? 2. 54 of 600? 7. 90 of 750? 12. 50 of 325 ? 3. 75 of 600 ? 8. 85 of 750 ? 13. 62 of 325 ? 4. 95 of 600 ? 9. 80 of 750 ? 14. 78 of 325 ? 5. 344 of 600? 10. 445 of 750? 15. 95 of 325? (a) Find the sum of the first five results. (b) Find the sum of the second five results. (c) Find the sum of the third five results. What % is — 1 6. $.76 of $38 ? (What is one % of $38 ?) 17. $9.02 of $225.50 ? (What is one % of $225.50?) 18. $17.13 of $342.60 ? (What is one ^,of $343.60?) 19. $58.45 of $835 ? (What is one % of $835?) 20. $1.29 of $6.45 ? (What is one % of $6.45?) 21. $113.10 of $754 ? (What is one % of $754?) 22. $21 of $175 ? (What is one % of $175?) 23. A sold goods for B to the amount of $346.25 ; he received as his commission for selling the goods, $41.55. His commission was what per cent of the sales ? 24. C collected for D $643.50 ; he retained as his com- mission for collecting the money, $38.61. His commission was what per cent of the sales ? 25. F purchased 256 barrels of apples; he lost by decay a quantity equal to 16 barrels. What % of his apples did he lose ? PART II. 267 Algebra. 216. Problems Leading to Equations Containing Fractions. If to i of Bemie's money you add ^ of his money the sum is $63. How much money has he ? Let X = the number of dollars he has. Then | + X = 63. Multiplying by 3, Zx -+4 = 189 Multiplying by 4, 4: x -\- S x = 756* Uniting terms, 7 £C = 756 Dividing by 7, (K = 108 Thus we find that Bemie had 108 dollars. 1. If to ^ of a certain number you add \ of the same number, the sum is 45. What is the number ? 2. A lady upon being asked her age replied : If from ^ of my age you subtract \ of my age the remainder will be 9 years. How old was she ? 3. The sum of two numbers is 72, and the less number equals |- of the greater number. What are the numbers ? Let X = the greater number. 4. The sum of two numbers is 75, and the less number equals f of the greater number. What are the numbers ? 5. The difference of two numbers is 30, and the less number equals | of the greater number. What are the numbers ? * Observe that the equation might have been cleared of fractions by multiplying both its members by 12, the 1. c. m. of 3 and 4. 268 COMPLETE ARITHMETIC. Algebra. 217. Miscellaneous Problems. 1. Harry has some marbles; Joseph has 8 more than i as many as Harry; together they have 68. How many has each ? . 2. William has" 12 more than i as many cents as Lucius; together they have 87 cents. How many cents has each ? 3. If to the half of a certain number you add a third of the number, the sum will be 5 more than 3 fourths of the number. What is the number ? Let X = the number, Then -A 5 = — 2 3 4 4. If to 1^ of a number you add -i^ of the same number, the sum will be 12 more than | of the number. What is the number ? 5. If to a certain number you add 5 times the number, the sum will be 24 more than 4 times the number. What is the number ? 6. If to |- of a number you add i of the number the sum will be a. What is the number ? Let X = the number. Then - + - = a 2 3 Multiplying by 6, 3« + 2a3 = 6a Uniting 6x = 6a Dividing by 5 x = — 5 Observe that yoii may put in the place of a any number you please ; hence any number is g of the sum of its half and third. PART II. 269 Geometry. 218. Construction Problems — Quadrilaterals. 1. Draw a rhombus. Make the angle a, 110°. Use the protractor for measuring the angle a only. Make ah and ac each 2 inches. Then draw hd parallel to ac and cd paral- lel to ab. Prove your work by measuring the other angles. Ohserve that when one side and one angle of a rhombus are given the rhombus may be drawn. 2. If the angle a of a rhomboid is 110°, how many de- grees in angle b ? In angle c ? In angle d ? ' 3. Draw a trapezoid. Make the side a6 3 inches long. Make the angle a, 110 degrees, and the line ac 2 inches long. Make the angle &, 90 degrees ; and the line bd of indefinite length, but long enough to form the side bd. Draw cd par- allel to ab. Prove the work by finding the sum of the four angles. 4. Draw a trapezium. Make the angle a, 100° ; the line ab 3 inches long, and the line ac 2 inches long. Make the angle b, 120°, and the side bd of indefinite length. Make the angle c, 85°. Angle d should be an angle of how many degrees ? Prove the work by measuring angle d. 5. Draw four lines that together make a very irregular trapezium ; then using the protractor, find the sum of the four angles. 270 COMPLETE ARITHMETIC. 219. Miscellaneous Review. 1. If I lose ^\ of my money/what % of my money do I lose? 2. Mr. Button spent $5 of the $12 which he had earned. What per cent of what he earned did he spend ? 3. Mr. Thomas earns $150 per month. The monthly rent of the house in which he lives is equal to 15% of what he earns. How much is his rent per year ? 4. Mr. Jones purchased 500 barrels of apples. He lost by decay a quantity equal to 75 barrels. What per cent of the apples purchased remained sound ? 5. Mr. Brown borrowed $625. He used 87% of this money to pay debts. How much of the money borrowed did he have left ? 6. Mr. Green paid $24.60 for a suit of clothes. If this was exactly 15% of his monthly earnings, how much does he earn per month ? 7. Mr. White earns $1500 per year. He spends $312 for board, $200 for clothes, $75 for books and papers, $80 for traveling and amusements; gives to his church $40, for charitable purposes, $35, and all his other expenses amount to $23. The remainder of his salary he puts into a savings bank. What per cent of his salary does he save ? 8. If to ^ of my money you add ^ of my money the sum will be $63. How much money have I ? 9. The sum of two numbers is 120, and the less number is "I of the greater. What are the numbers ? 10. If one angle of a rhomboid is 80°, what is the size of each of the other angles ? 11. Three twenty-fifths is what % of f ? PEECENTAGE. 220. Problems in Case III, somewhat disguised. Example No. 1. 75 is how many % more than 60 ? 75 is 15 more than 60 ; so the percentage part of this problem is, 15 is what % of 601 Ans. 25^. Therefore 75 is 25^ more than 60 i that is 75 is 25 hundredths of 60 more than 60. Example No. 2. 60 is how many % less than 75 ? 60 is 15 less than 75 ; so the percentage part of this problem is, 15 is what % of 75? Ans. 20^. Therefore 60 is 20^ less tharr 75; that is, 60 is 20 hundredths of 75 less than 75. Observe that in example No. 1, 60 is the base, and that in example No. 2, 75 is the base ; and that in problems of this kind the base always follows the word than. Problems. 1. 60 is how many per cent more than 45 ? 2. 45 is how many per cent less than 60 ? 3. 150 is how many per cent more than 125 ? 4. 125 is how many per cent less than 150 ? 5. 225 is how many per cent more than 200 ? 6. 200 is how many per cent less than 225 ? 7. James has $345 ; Peter has $414. Peter has how many per cent more than James? James has how many per cent less than Peter ? 271 272 COMPLETE ARITHMETIC. Percentage. 221. Application of Art. 220 to "Loss and Gain." Find the per cent of loss or gain in each of the following:* 1. Bought for 25^ and sold for 30^. 2. Bought for 25^ and sold for 23^. 3. Bought for 25^ and sold for 27^. 4. Bought for 25^ and sold for 21^. 5. Bought for $40 and sold for $48. 6. Bought for $40 and sold for $35. 7. Bought for $40 and sold for $52. 8. Bought for $40 and sold for $36. 9. Sold for 65^ that which cost 50^. 10. Sold for 18^ that which cost 20^. IJ. Sold for 30^ that which cost 36f 12. Sold for 35^ that which cost 30^. 13. Sold for $50 that which cost $40. 14. Sold for $40 that which cost $50. 15. Sold for $30 that which cost $40. 16. Mr. Watson bought 60 lbs. of tea at 32^ a pound and sold it at 50^. What was his per cent of gain if he sold as many pounds as he bought ? If he lost 4 lb. by " down- weights " and wastage, how much money did he gain ? What was his real per cent of gain ? 17. Mr. Jenkins bought gloves at $4.50 per dozen and sold them at 50^ a pair. What was his per cent of gain ? 18. Mr. Warner bought apples at 40^ a bushel. He lost 25% of them by decay and sold the remainder at 50^ a bushel. Did he gain or lose by the transaction ? What was his per cent of gain or loss ? * In speaking of the per cent of loss or of gain the cost is regarded as the hose unless otherwise specified. PART II. 273 Percentage. 222. Percentage Problems under Case I, in which the PER cent is more THAN 100. Find 175% (III or 1.75) of $632.60. Operation No. 1. Explanation. $6'32.60 One % of ^632.60 is 16.326. 1.75 175^ of 1632.60 is 175 times |6.326. or $31.6300 '^1107.05. $442 820 Observe that b% of $632.60 is 131.63; that $632 60 "^^^ °^ $632.60 is 1442.82 ; that 100^ of $632.60 is $632.60. $1107.0500 Operation No. 2. 175% = \U = T- i of $632.60 = $158.15. I of $632.60 = 7 times $158.15, or $1107.05. Problems. 1. Find 175% of 356; of 276; of 540.20. 2. Find 155% of 356; of 276; of 540.20. 3. Find 145% of 356; of 276; of 540.20. 4. Find 125% of 356; of 276; of 540.20. 5. Find 200% of 356; of 276; of 540.20. 6. Find 150% of 356; of 276; of 540.2a. 7. Find 250% of 356 ; of 276 ; of 540.20. (a) Find the sum of the twenty-one results. 8. David's money is equal to 150% of Henry's money. Henry has $240. How much has David ? 9. If goods cost $260, and the profit on them is 125%, what is the selHng price ? 10. If a Chicago lot at the beginning of a certain year was worth $8000, and during the year increased in value 250%, what was it worth at the end of the year ? 274 COMPLETE ARITHMETIC. Percentage. 223. Percentage Problems under Case II, in which THE PER CENT IS MORE THAN 100. $1107.05 is 175% of how much money? Operation No. 1. 175mi07\05(S6.326 1050 ~570 525 100 $632.60 455 350 1050 1050 Explanation, Since $1107.05 is 175 hundredths of the money, 1 hundredth of the money is one 176th of $1107.05, or $6,326; and XOO hundredths, 100 times 16.326, or $632.60. Operation No. 2. 1.75W107.05X$632.60 1050 570 525 455 350 1050 1050 Explanation. We may find one hundred 175ths of a number by finding one 175th of 100 times the number. One hundred times $1107.05 is $110705. One 175th of $110705 is $632.60. Operation and Explanation No. 3. 175^ = ig§ = |. If $1107.05 is 7 fourths of the money, 1 seventh of $1107.05, or $158.15, is 1 fourth of the money, and 4 fourths, or the whole of it, is 4 times $158.15, or $632.60 Problems. 1. 43.50 is 125% of what number? 2. 65.10 is 150% of what number? 3. 48.50 is 200% of what number? 4. 59.20 is 250% of what number? 5. 29.44 is 115% of what number? (a) Find the sum of the five answers. PART II. 275 Percentage. 224. Percentage Problems under Case III, in which THE PER CENT IS MORE THAN 100. $1107.05 is what per cent of $632.60 ? Operation. Explanation. $6.326)$1107.050X175 6326 One per cent of $632.60 is $6,326 ; AHAA.^ then $1107.05 is as many per cent of 44282 $632.60 as $6,326 is contained times in $1107.05. It is contained 175 times, ^1630 so ^1107.05 is 175^ of 632.60. 31630 Note. — The above problem may be solved as a similar problem is solved on page 265, operation No. 1. $1107.05 is HB:U of $632.60. Perform the division indicated, carrying out to hundredths only. The quotient will be 1.75 or i§^ = 175^. Problems. 1. What per cent of 845 is 2112.5 ? 2. What per cent of 845 is 1056.25 ? 3. What per cent of 845 is 1352 ? 4. What per cent of 845 is 1183 ? 5. What per cent of 845 is 2746.25 ? (a) Find the sum of the 5 answers. 6. Eeuben has $2420 ; Bernie has $4961. Bernic's money equals how many per cent of Eeuben's money ? 7. A certain house cost $6425. The lot upon which it stands cost $2325. (a) The cost of the house equals how many per cent of the cost of the lot ? (b) The cost of the lot equals how many per cent of the cost of the house ? 276 COMPLETE ARITHMETIC. Percentage. 225. Miscellaneous Problems. 1. Find 1-%* of 632 ; of 356 ; of 272. 2. Find |% of 632 ; of 356 ; of 272. 3. Find .25 %t of 632 ; of 356 ; of 272. (a) Find the sum of the nine results. 4. Find 3|-% of 496 ; of 532 ; of 720. 5. Find 4J% of 496 ; of 532 ; of 720. 6. Find 1.75% of 496 ; of 532 ; of 720. (b) Find the sum of the nine results. 7. What part of 94 is 11 ? J What per cent ? 8. What part of 94 is 36 ? What per cent ? 9. What part of 94 is 47 ? What per cent ? (c) Find the sum of the three " per cents." 10. 15 is 3% of what number ? 4% ? 5% ? 11. 24 is 3% of what number ? 4% ? 5% ? 12. 42 is 3% of what number ? 4% ? 5% ? (d) Find the sum of the nine answers. 13. 375 is 125% (||-|-) of what number? 14. 436 is 109% (Iff) of what number? 15. 598 is 115% (|i|) of what number? (e) Find the sum of the three answers. 16. 544 is 15% less than what number? 17. 545 is 25% more than what number? 18. 510 is 170% of what number ? (f) Find the sum of the three answers. * This means J of 1 per cent. t This means find 25 hundredths of 1 per cent. t Answer with a common fraction in its lowest terms. PART II. 277 Algebra. 226. To Find Two Numbers when their Sum and Difference are Given. 1. The difference of two numbers is 4 and their sum is 20. What are the numbers ? Let X — the smaller number, then x-\- Ai — the larger number, and a? + ic + 4 - 20. Transposing, ^ + ^ = 20 — 4. Uniting, 2 a? = 16. Dividing, a? = 8, the smaller number. ic + 4 — 12, the larger number. 2. The difference of two numbers is 9 and their sum is 119. What are the numbers ? 3. The difference of two fractions is ^V ^^^ their sum is \. What are the fractions ? 4. The difference of two numbers is d and their sum is s. What are the numbers ? Let X = the smaller number, then X -\- d = the larger number, and X -\- X -\- d = s Transposing, x -\- x = s — d Uniting, 2x = s — d Dividing, x = 2 Observe that any number you please may be put in the place of s, and any number less than s in the place of d ; so when the sum and the difference of two numbers are given, the smaller number may be found by subtracting the difference from the sum and dividing the remainder by 2. 278 COMPLETE ARITHMETIC. Algebra. 227. Another Method of Finding Two Numbers when THEIR Sum and Difference are Given. 1. The difference of two numbers is 17 and their sum is 69. What are the numbers ? Let X = the larger number, then X — 11 = the smaller number, and x-\- X — 11 = ^^. Transposing, x -\- x = 69 + 17. Uniting, 2x = 86. Dividing, ic — 43, the larger number. X — 11 = 2^, the smaller number. 2. The difference of two numbers is 8.4 and their sum 75.6. What are the numbers ? 3. The difference of two numbers is d and their sum is s. What are the numbers ? Let a? r= the larger number, then X - d — the smaller number, and X -\- X — d = s Transposing, x -\- x = s -\- d Uniting, 2^ = s + c? s^d Dividing, • x — ^' 2 Observe that any number you please may be put in the place of s, and any number less than s in the place of d ; so when the sum and difference of two numbers are given, the larger may be found by add- ing the difference to the sum and dividing the amount by 2. 4. A horse and a harness together are worth $146, and the horse is worth $74 more than the harness. Find the value of each. PART II. 279 Geometry. 228. How Many Degrees m Each Angle of a Kegular Pentagon ? Fig.l. Pig. 2.. 1. Every regular pentagon may be divided into equal isosceles triangles. 2. The sum of the angles of one triangle is equal to right angles ; then the sum of the angles of 5 triangles is equal to right angles. 3. But the sum of the central angles in figure 2, (a + & + c-\- d-\- e) is equal to right angles ; then the sum of all the other angles of the five triangles is equal to 10 right angles, less 4 right angles, or 6 right angles = 540°. But the angular space that measures 540°, as shown in figure 2, is made up of 10 equal angles, so each one of the angles is 1 tenth of 540° or 54°. Two of these angles, as 1 and 2, make one of the angles of the pentagon ; therefore each angle of the pentagon measures 2 times 54° or 108°. 4. Using the protractor construct a regular pentagon as follows : (a) Draw two lines that meet in a point, each line being 2 inches long and the angular space between them being 108°. (b) Regarding the two lines as two sides of a regular pentagon, draw two more sides each 2 inches long and joining those already drawn at an angle of 108°. (c) Complete the figure by drawing the fifth side, and prove your work by measuring the last line drawn ^nd the other two angles. 280 COMPLETE ARITHMETIC. 229. Miscellaneous Review. Remembering that in speaking of the per cent of loss or gain, the cost is the base unless otherwise specified, tell the per cent of loss or gain in each of the following : 1. Bought for 2 and sold for 3. 2. Bought for 3 and sold for 2. 3. Bought for 4 and sold for 5. 4. Bought for 5 and sold for 4. 5. Bought for 5 and sold for 6. 6. Bought for 6 and sold for 5. 7. Bought for 8 and sold for 10 ; for 12. 8. Bought for 8 and sold for 14 ; for 16. 9. Bought for 8 and sold for 18 ; for 20. 10. Bought for 8 and sold for 4 ; for 2. 11. Mr. Parker sold goods at a profit of 25%; the amount of his sales on a certain day was S24.60. How much was his profit ? 12. Mr. Jewell sold goods at a loss of 25%; the amount of his sales on a certain day was $24.60. How much was his loss. 13. By selling a horse for $156 there was a loss to the seller of 20%. What would have been his gain per cent if he had sold the horse for $234 ? 14. A bill was made for wood that was supposed to be 4 feet long. It was afterwards found to be only 46 inches long. What % should be deducted from the bill ? 15. The marked price on a pair of boots was 25% above cost. If the dealer sells them for 25% less than the marked price will he receive more or less than the cost of the boots ? 16. If by selling goods at a profit of 12% a man gains $6.60, what was the cost of the goods ? PEKCENTAGE. 230. Discounting Bills. Many kinds of goods are usually sold " on time " ; that is, the buyer may have 30, 60, or 90 days in which to pay for them. If he pays for such goods at the time of purchase, or within ten days from the time of purchase, his bill is " discounted " from 1 ^ to 6 ^, accord- ing to agTeement ; that is, a certain part of the amount of the bill is deducted from the amount. Example. Mr. Smith bought of Marshall Field & Co. a bill of goods amounting to $350.20. The discount for immediate pay- ment ("spot cash") was 1%. How much must he pay for the goods ? 1% of $350.20 is $3.50. $350.20 - $3.50 = $346.70. Problems. "Figure the discounts" on the following bills: 1. Bill of $324.37, discounted at 2%. 2. Bill of $276.45, discounted at 1%. 3. Bill of $356.50, discounted at 3%. 4. Bill of $536.50, discounted at 6%. 5. Bill of $561.80, discounted at 4%. (a) Find the sum of the five bills before they are dis- counted. (b) Find the sum of the discounts. (c) Find the sum of the five bills after discounting. 281 282 COMPLETE ARITHMETIC. Applications of Percentage. 231. Discounts from List Price. Dealers in hardware, rubber boots and shoes, belting, rubber hose, and many other kinds of goods, sell from a list price agreed upon by the manufacturers. The actual price is usually less than the list price. " 20^ off " means that the list price is to be discounted 20^. " 20 and 10 off " means that the list price is to be discounted 20^^, and what remains is to be discounted 10^. Sometimes as many as nine successive discounts are allowed. Observe that in computing these the base changes with each discount. Problems. Find the actual cost of — 1. 500 ft. f-inch gas pipe (list, 7^ per ft.) at 50 and 10 off. 2. 350 ft. i-inch gas pipe (list, 8^ per ft.) at 50 and 10 off. 3. 200 ft. 1^-inch gas pipe (list, 26^ per ft.) at 55 and 10 off. 4. 260 ft. 2 -inch gas pipe (list, 35^' per ft.) at 55 and 10 off. 5. 48 ^-in. elbows (list, 7^ each) at 65 and 20 off. 6. 36 J-in. elbows (list, 9^ each) at 65 and 20 off. (a) Find the entire cost of the six items. 7. Find the cost of 12 pairs men's rubber boots (list, $3.00 per pair) at 25 and 10 off. Find the cost of the same at 35 off. Why are the results unlike ? 8. Which is the lower price, 50 and 10 off or 60 off ? 9. Bought for 40 off from list price and sold for 10 off from list price. What was my gain per cent ? 10. Bought for 70 off from Hst and sold for 50 and 20 off from list. Pid I lose or gain and how msLHj per cent ? PART II. 283 Applications of Percentage. 232. Selling "on Commission." When goods are sold " on commission " the selling price is the base ; that is, the seller receives a certain per cent of the selling price as remuneration for services. Commission is the sum paid an agent, or commission merchant, for transacting business. Problems. 1. At 40%, what is the commission for selling $275 worth of books ? If the salesman sells and collects for 40 % of the selling price, how much of the $275 will he retain and how much "pay over" to the man for whom he sells the books ? 2. While selling books on a commission of 40% my com- mission amounted to $56. What was the selling price of the books ? If I not only sold but collected for 40 % of the selling price, how much money should I "pay over" to my employer ? 3. A real estate agent sold a house and lot for $4250. If his commission is 5%, how much should he receive for his services ? 4. A real estate agent sold a piece of property upon which his commission at 5% amounted to $275. What was the selling price of the property. How much should the owner receive for the property after deducting the commission ? 5. A commissiori merchant sold 2140 lbs. of butter at 23^ a pound. After deducting his commission of 5% and paying freight charges of $36.50, and storage charges of $21.40, how much should he send to the man for whom he made the sale ? 284 COMPLETE ARITHMETIC. Applications of Percentage, 233. Taxes. A tax is a sum of money paid for public purposes. A tax on property is reckoned at a certain per cent of the assessed value of the property. The assessed value may or may not be the real value. It is often much below the real value. Problems. 1. Mr. Hardy has a farm of 240 acres which he values at $24000. Its assessed value is $22 per acre. If his state tax is |-%, his county tax 1^%, his town tax ^%,his school tax 2%, and his special road and bridge tax 1%, how much money must he pay as taxes on his farm ? 2. The assessed value of the taxable property of a certain school district is $176,242.25. If the school tax is 2|-% and the collector receives 2 % of the amount collected as his commission, and collects the entire amount of the tax, how much should the district officers receive from this source for school purposes ? 3. The assessed value of Mr. Eandall's property is $3400. At the rate of 15 mills on a dollar,* how much tax must he pay? 4. The assessed value of the property of a district of a certain city is $250,000. (a) What must be the per cent of taxation to raise $10,000 ? (b) What will be the net sum realized for public purposes if the collector is able to collect only 95% of this tax and he receives for his services 2% of the amount collected ? 5. Mr. Evans's tax is $35.60 ; the rate of taxation is 2|^%. What is the assessed value of his property ? * " 15 mills on a dollar " is the same as 11%. TART II. 285 Applications of Percentage. 234. Insurance is a guaranty by one party to pay a cer- tain sum to another party in the event of loss or damage. The policy is the written contract given by the under- writer to the insured. The premium is the sum paid for insurance. Problems. 1. A store valued at $7500 was insured for $5000 for 1 year. The rate of insurance was 2%. What was the amount of the premium ? 2. A stock of goods valued at $10000 was insured for $5000. A fire occurred, but part of the goods were saved. It was found that the entire loss to the owner of the goods was $4750. (a) How much should he receive? (b) How much should he receive if the loss were $5750 ?* 3. An insurance agent offers to insure my farm buildings for $3500 for 1 year at 1%, or for 5 years at 3% ; the entire premium in either case to be paid in advance, (a) If I accept the first proposition, how much is the premium to be paid ? (b) How much if I accept the second ? 4. What is the rate of insurance on the nearest store and stock of goods ? On farm property ? On village or city property other than stores ?t 5. A large building was insured in one company for $25000, in another company for $15000. It was damaged by fire to the extent of $12800. How much of the damage should each company pay ? J * In case of total loss the owner would receive $5000. In case of partial loss the owner should receive the full amount of the loss, provided it does not exceed $5000. t Any insurance agent will be willing to answer these questions for you. X The companies must share the loss in proportion to the amount of insurance carried by them. 286 COMPLETE ARITHMETIC. 235. Miscellaneous Problems in Applications of Percentage. 1. A dealer who had marked goods 50% above the cost, sold them after deducting 10% from the marked price. His profit on that sale was what per cent of the cost of the goods ? 2. By selling a suit of clothes for $7.20 I would lose an amount equal to 1 % of the cost. For what must I sell the suit to gain a sum equal to 10% of the cost ? 3. I sold goods at a loss of 7%. My actual loss was $3.50. What was the cost of the goods ? 4. The real value of a stock of goods was $8250. They were insured for $5500. A fire occurred and the salvage amounted to only $575. If the insurance company promptly settles in accordance with the above facts what is the actual loss to the owner of the goods ? 5. If I sell goods on a commission of 12|-%, what must bt^ the amount of my sales in order that I may receive an annual salary of $2500 ? 6. A school numbers 140 pupils. The absence for one week was as follows : Monday, 3 days ; Tues., 5 days ; Wed., 4 days ; Thurs., 5 days ; Friday, 3 days, (a) What was the per cent of absence ? (b) What was the per cent of attend- ance? 7. Sold a horse for $120 and gained 25%. What did the horse cost me ? 8. When the cost is f of the selling price what is the gain per cent ? 9. When the selling price is f of the cost what is the loss per cent ? PART II. 2sr Algebra. 236. Miscellaneous Problems. 1. In a school there are 896 pupils. There are three times as many boys as girls. How many girls ? How many boys ? 2. A man had 235 sheep. In the second flock there were 15 more sheep than in the first. In the third flock there were 20 fewer than in the first. How many sheep in each flock ? Note — Let x = the number in the first flock ; then x -f- 15 = the number in the second, and x — 20, the number in the third. 3. A man owns three farms. In the second there are half as many acres as in the first. In the third there are twice as many acres as in the first. In all there are 560 acres. How many acres in each farm ? 4. In an apple and pear orchard containing 296 trees, there were 5 more than twice as many apple trees as pear trees. How" many of each kind ? 5. To a number I add one half of itself and 15, and have 150. What is the number? 6. From three times a number I subtract -| of the number and 5, and have 37 remaining. What is the number ? /2x Note. — Let x = the number ; then 3a: —(—-{- 5) = 37. On ^ o removing the parenthesis, what signs must be changed? See page; 207, IL 7. If to three times a number I add ^ of the number and 18, the sum will be 238. What is the number ? 8. Two thirds of a number is equal to the number decreased by 56. What is the number? 288 COMPLETE ARITHMETIC. Algebra. 237. Miscellaneous Problems. 1. A is 50 years old. B is 20 years old. In how many years will A be only twice as old as B ? Note. — Let x = the number of years ; then (20 -\-x) X 2 = 50 -f x. 2. Find four consecutive numbers whose sum is 150. Note. — Let x = the first ; then x -\-l = the second ; a; -|- 2 = the third, etc. 3. Find three consecutive numbers whose sum is 87. 4. Two numbers have the same ratio as 2 and 3. and their sum is 360. What are the numbers ? 5. Two numbers have the same ratio as 3 and 4, and their sum is 168. What are the numbers ? 6. Two numbers have the same ratio as 2 and 5, and their difference is 87. What are the numbers ? 7. A has $350. B has $220. How many dollars must A give to B so that each may have the same sum ? Note. — Let x =. the number of dollars that must be given by A to B ; then 220 + a; = 350 - x. 8. C has $560. D has $340. How many dollars must C give to D so that each may have the same sum ? 9. E has $630. F has $240. How many dollars must E give to F so that E will have exactly twice as many dollars asF? 10. The fourth and the fifth of a certain number are together equal to 279. What is the number ? 11. The difference between 1 fourth and 1 fifth of a cer- tain number is 28. What is the number ? PART II. 289 Qeometry. 238. How MANY Degrees in each Angle of a Eegular Hexagon ? Fig. 1. Fig .2. / A\ ^ \ /^ a>? ^ TV \ f/e \^ 7 \ f/l 1^9/ ^ 1. Every regular hexagon may be divided into equal isosceles triangles. 2. The sum of the angles of one triangle is equal to right angles, then the sum of the angles of 6 triangles is equal to right angles. 3. But the sum of the central angles in Fig. 2 (a + & + c -i- d -\- e -\- f) is equal to right angles ; then the sum of all the other angles of the six triangles is equal to 12 right angles less 4 right angles, or 8 right angles = 720°. But the angular space that measures 720°, as shown in Fig. 2, is made up of 12 equal angles ; so each one of the angles is one 12th of 720°, or 60°. Two of these angles, as 1 and 2, make one of the angles of the hexagon ; therefore each angle of the hexagon measures 2 times 60°, or 120°. 4. Using the protractor, construct a regular hexagon, mak- ing each side 2 inches long. Observe that since all the angular space about a point is equal to 4 right angles, or 360°, and since the space around the central point of the hexagon is divided into 6 equal angles, each of these angles is an angle of (360° -v- 6) 60°. But each of the other angles of these triangles has been shown to be an angle of 60° ; so each triangle is equiangular. Are the triangles equilateral ? 290 COMPLETE ARITHMETIC. 239. Miscellaneous Review. 1. A man buys goods for $60 and sells them for $75. He gains dollars. (a) The gain equals what part of the cost ? What % ? (b) The gain equals what part of the selling price ? What per cent ? (c) The cost equals what part of the selling price ? What per cent ? 2. When the cost is 2 thirds of the selluig price what is the per cent of gain ? 3. When the selling price is 2 thirds of the cost what is the per cent of loss ? 4. Bought for $200 and sold for $300. What was the per cent of gain ? 5. Bought for $300 and sold for $200. What was the per cent of loss ? 6. A tax of 15 mills on a dollar was levied in a certain town, the assessed value of the taxable property being $475,250. If 5% of the tax is non-collectable and if the collector is allowed 2% of the amount collected, for his services, how much will be realized from the levy ? 7. Which is the greater discount, "20 and 10 and 5 off" or "35 off"? 8. A sold goods for B on a commission of 15%. His sales for a certain period amounted to $780. If the goods cost B exactly $600 was B's net profit more or less than 10% ? 9. A offers rubber boots at " 50 and 20 off " ; B offers them at " 20 and 50 off." The quality and list price bemg the same, which offer shall I accept ? INTEEEST. J40. Interest is compensation for the use of money. 241. The money for which interest is paid is called the principal. 242. The principal and interest together are called the amount. Note. — Interest is usually reckoned in per cent, the principal being the base ; that is, the borrower pays for the use of money a sum equal to a certain per cent of the principal. When a man loans money " at 6% " he expects to receive back the principal, and a sum equal to 6% of the principal for every year the money is loaned and at that rate for fractions of years. Example. Find the interest of $257 for two years at 6%. Operation. Explanation. $2^57 .12 The interest of any sum for 2 years at Q% is 12 hundredths of the principal. One hundredth of $257 is 12.57, and 12 hundredths of $257 is 12 times $2.57 or $30.84. 5.14 25.7 $30.84 1. Find the interest of $242 for 3 yr. at 7%. 2. Find the interest of $375 for 2 yr. at 6%. 3. Find the interest of $146 for 1 yr. at 5%. 4. Find the interest of $274 for 3 yr. at 5%. 5. Find the interest of $375 for 2 yr. at 8%. 6. Find the interest of $864 for 3 yr. at 7%. (a) Find the sum of the six results. 291 292 COMPLETE ARITHMETIC. Interest. 243. To Compute Interest for any Number of Years AND Months. Note. — The interest for 1 month is 1 twelfth as much as it is for 1 year ; for 2 months, 2 tweKths or 1 sixth, etc. Example. Find the interest of $324.50 for 2 yr. 5 mo. at 6%. Operation and Explanation No. 1. Interest of 1324.50 for 1 year at 1 ^ = ^3.245 Interest of ^324.50 for 1 year at 6^ = $19,470 Interest of $324.50 for 2 years at 6 ^ = $38,940 Interest of $324.50 for 1 mo. aiQfc^ $1.6225 Interest of $324.50 for 5 mo. at 6 ^ = 8.1125 Interest of $324.50 for 2 yr. 5 mo. at 6 ^ = $47.0525 Operation No. 2. Explanation. 2 yr. 5 mo. = 2^^ years. The interest of any sum for 2 yi-. 2—5 times 06 = 14^ ^ ™^°' ^^ ^^^' hundredths of the prin- ^ ^ ] ~ ' ^' cipal. $3'24.50 1 hundredth of $324.50 is $3.2450 •14j - I hundredth of $324.50 is $1.6225 1.6225 4 hundredths of $324.50 is 12.9800 12 9800 10 hundredths of $324.50 is 32.450 32 450 14| hundredths of $324.50 is $47.0525 $47.0525 Problems. 1. Find the interest of $325.40 for 1 yr. 6 mo. at 7%. 2. Find the interest of $420.38 for 2 yr. 10 mo. at 6% 3. Find the interest of $221.60 for 2 yr. 3 mo. at 6%. 4. Find the interest of $145.20 for 1 yr. 9 mo. at 5%. 5. Find the interest of $340.10 for 3 yr. 1 mo. at 4%. (a) Find the sum of the five results. PART I. 293 Interest. 244. To Compute Interest for any Number of Years, Months, and Days. Note. — In computing interest, 30 days is usually regarded as 1 month, and 360 days as 1 year; so each day is g^j, of a month or gju of a year. Example. Find the interest of $256.20 for 2 yr. 7 mo. 13 days at 6%. Operation and Explanation No. 1. Interest of $256.20 for 1 yr. at 1 ^ = $2.5620 Interest of $256.20 for 1 yi\ at 6 ^ = $15.3720 Interest of $256.20 for 2 yr. at 6 ^ = $30.7440 Interest of $256.20 for 1 mo. at 6^ = $1.2810 Interest of $256.20 for 7 mo. at 6^ = $8.9670 Interest of $256.20 for 1 da. at 6^ = $.04270 Interest of $256.20 for 13 da. at 6^ = .5551 Interest of $256.20 for 2 yr. 7 mo. 13 da. at 6^ = $40.2661 Operation No. 2. Explanation. 2 yr. 7 mo. 13 da. 2f |f yr. 2 11 3. times .06 = .15M The interest of any sum tor 2 yr. 7 mo. 13 da. at Q% is .15f§ of the prin- cipal. 3' 6 ^■^^"^'^ • " " — •-^'^60 $2^56.20 •l^ef 1 hundredth of $256.20 is $2.5620 1.8361 |§ of 1 hundredth of $250.20 is $1.8361 12.8100 5 hundredths cf $256.20 is $12.81 25.620 10 hundredths of $256.20 is $25.62 $40.2661 15f^ hundredthsof $256.20 is$40.2661 Problems. 1. Find the interest of $350.40 foi 2 yr. 5 mo. 7 da. at 6%. 2. Find the interest of $145.30 for 1 yr. 7 mo. 10 da. at 8%. 3. Find the interest of $1 74.20 for 2 yr. 3 mo. 1 5 da. at 7 %. (a) Find the sum of the three results. 294 COMPLETE ARITHMETIC. 245. The '^six'per^cent method." Note.— If the teacher so prefers, the problems on the three pre- ceding pages, as well as those that follow, may be solved by the " six- per-cent method." Explanatory. The interest at 6% for 1 yr. = .06 of the principal. The interest at Q% for 1 mo. = ^V of .06, or .005 of the principal. The interest at Q% for 1 day = g^^ of .005, or .000^ of the principal. Reading Exercise. 1. Interest for 1 yr. at 6% = lOOths of the principal; for 2 yr. = lOOths ; for 3 yr. = lOOths ; for 1 mo. = lOOOths ; for 2 mo. = lOOOths for 3 mo. = lOOOths ; for 4 mo. ^ lOOOths for 5 mo. ■= lOOOths ; for 6 mo. = lOOOths for 1 da. = 1000th ; for 3 da. = 1000th ; for 6 da. = 1000th ; for 12 da. = lOOOths ; for 18 da. = lOOOths ; for 24 da. = lOOOths ; for 25 da. = lOOOths ; for 27 da. = lOOOths. Find the interest of $243.25 for 2 yr. 5 mo. 18 da. at 6%. Operation and Explanation. Interest for 2 yr. = .12 of the principal. Interest for 5 mo.= .025 of the principal. Interest for 18 da. = .003 of the principal. Total interest = .148 of the principal. 1243.25 X .148 = $36,001. (1) $36,001 plus i of $36,001 = int. of same prin. for the same time at 7^. (2) $;36.001 less I of $36,001 = int. of same prin. for the same time at 5^. (3) How find the interest at 8^ ? At 9^? At 4^? At 3^? PART II. 295 Interest. 246. Problems to be Solved by the "Six Per Cent Method." 1. Find the interest of $265 for 1 yr. 3 mo. 13 da. at 6%.* 2. Find the interest of $346 for 2 yr. 5 mo. 20 da. at 6%.t 3. Find the interest of $537 for 1 yr. 7 mo. 10 da. at 6%4 4. Find the interest of $428 for 3 yr. 3 mo. 14 da. at 6%. 5. Find the interest of $150 for 1 yr. 6 mo. 15 da. at 6%. (a) Find the sum of the five results. 6. Find the interest of $245.30 for 6 mo. 18 da. at 7%. 7. Find the interest of $136.25 for 8 mo. 10 da. at 7%. 8. Find the interest of $321.42 for 5 mo. 15 da. at 7%. 9. Find the interest of $108.00 for 10 mo. 8 da. at 7%. 10. Find the interest of $210.80 for 7 mo. 21 da. at 7%. (b) Find the sum of the five results. 11. Find the amount of $56.25 for 2 yr. 4 mo. 2 da. at 6%. 12. Find the amount of $31.48 for 1 yr. 5 mo. 11 da. at 6%. 13. Find the amount of $55.36 for 2 yr. 8 mo. 12 da. at 6 %. 14. Find the amount of $82.75 for 2 yr. 10 mo. 8 da. at 6%. 15. Find the amount of $27.35 for 1 yr. 1 mo. 1 da. at 6%. (c) Find the sum of the five results. 16. Find the amount Of $875 for 3 yr. 8 mo. 15 da. at 5%. 17. Find the amount of $346 for 2 yr. 6 mo. 12 da. at 4%. 18. Find the amount of $500 for 1 yr. 7 mo. 18 da. at 3%. 19. Find the amount of $600 for2 yr. 5 mo. 21 da. at 8%. 20. Find the amount of $825 for 1 yr. 9 mo. 24 da. at 9%. (d) Find the sum of the five results. *lyr. .06 t2yr. .12 nyr. .06 3 mo. .015 5 mo. .025 7 mo. .035 13 da. .002^ 20 da. .003J 10 da. .0011 Total .077J Total .148§ Total .096f 296 COMPLETE ARITHMETIC. 247. To Find the Time Between Two Dates. Example. How many years, months, and days from Sept. 25, 1892, to June 10, 1896? The Usual Method. 1896-6-10 From the 1896th yr., the 6th mo., and 1892 - 9 - 25 the 10th day, subtract the 1892nd yr., the 9th mo., and the 25th day. Regard a month as 30 days. A Better Method. 3-8-15 From Sept. 25, 1892, to Sept. 25, 1895, is 3 years. From Sept. 25, 1895, to May 25, 1896, is 8 months. From May 25, 1896, to June 10, 1896, is 16 days. Note.— The two methods will not alwaj'S produce the same results. The greatest possible variation is two days. Find the time from Jan. 22, 1895, to March 10, 1897, by each method, and compare results. The difference arises from the fact that the month as a measure of time is a variable unit— sometimes 28 days, sometimes 31. The " usual method" regards each month as 30 days ; the " better method " counts first the whole years, then the whole months, then the days remaining. By the " usual method," the time from Feb. 28, 1897, to March 1, 1897, is 3 days ; by the " better method," it is 1 day. Problems. Find the time by both methods and compare the results. 1. From March 15, 1894, to Sept. 10, 1897. 2. From March 15, 1894, to Sept. 20, 1897. 3. From May 25, 1895, to Sept. 4, 1898. 4. From May 25, 1895, to Oct. 4, 1898. 5. From June 28, 1894,. to Mch. 1, 1898. 6. From June 28, 1894, to May 1, 1898. 7. From Jan. 10, 1892, to Jan. 25, 1898. 8. From Jan. 10, 1892, to Dec. 25, 1898. 9. From April 15, 1893, to Aug. 15, 1898. PART II. 297 248. Algebra Applied to Problems m Percentage. Example. 75 is 15 per cent of what number ? Let X = the number sought. Tlien or x, or = 75. 100 100 Multiplying by 100, 15 x = 7500. Dividing by 15, a? = 500. Ans. Problems. 1. 56 is 2 per cent of what number? 2. 45 is 5 per cent of what number ? 3. 60 is 12 per cent of what number ? 4. 37 is 4 per cent of what number ? 6. 53 is 8 per cent of what number ? 6. n is r per cent of what number ? Let X = the number sought. Then = n 100 Multiplying by 100, rx = 100 n Dividing by r x = * 7. James has $54.20, and James's money equals 40 per cent of Henry's money. How much money has Henry ? 8. Mr. WilHams's annual expenses are $791.20 ; this is 92 per cent of his annual income. How much is his annual income ? ♦Observe that every problem on this page can be solved by this formtUa. 298 COMPLETE ARITHMETIC. 249. Algebra Applied to Peoblems in Percentage. Example. 60 is what per cent of 75 ? Let X = the per cent (number of hundredths). Then — of 75, or —= 60. 100 100 Multiplying by 100, 75x = 6000 Dividing by 75, x = 80. Ans. Problems. 1. 180 is what per cent of 200 ? 2. 17 is what per cent of 340 ? 3. $87.50 is what per cent of $250 ? 4. 81 is what per cent of 540 ? 5. $75.60 is what per cent of $630 ? 6. n is what per cent of & ? * Let X - - the per cent. Then — of b, or 100 bx 100 = n. Multiplying by 100, bx. - IOOtz, Dividing by b, X - IOOti^ 7, Solve the first five problems in four ways : (1) Let X = the per cent and solve as the " example " is solved. (2) Using one hundredth of each base as a divisor and the other number mentioned in the problem as a dividend, find the quotient. (3) Find what part the first number mentioned in each problem is of the base, and change the fraction thus obtained to hundredths. (4) Apply the formula, = x. * The b in problem 6 and in the formula may be thought oi as standing for the base. 250. PART II. Geometry. Some Interesting Facts about Squares, Triangles, and Hexagons. 299 1. Four equal squares may be so joined as to cover all the space about a point. Each angle whose vertex is at the central point of the figure is an angle of 90°. Four times 90° = degrees. 2. Six equal equilateral triangles may be so joined as to cover all the space about a point. Each angle whose vertex is at the central point of the figure is an angle of 60°. Six times 60° = degrees. Cut from paper 6 equal equilateral triangles and join them as shown in the figure. 3. Three equal hexagons may be so joined as to cover all the space about a point. Each angle whose vertex is at the central point of the figure is an angle of 120°. Three times 120° = degrees. Cut from paper 3 equal hexagons and join them as shown in the figure. 4. Since every regular hexagon may be divided into six equal equilateral tri- angles, as shown in the figure, it follows that the side of a regular hexagon is exactly equal to the radius of the circle that circumscribes the hexagon. 300 COMPLETE ARITHMETIC. 251. Miscellaneous Reviews. 1. Find the interest of $250 from Sept. 5, 1896, to Jan. 17, 1898, at 6%. 2. Find the amount of $340 from April 19, 1895, to Oct. 1, 1896, at 6%. 3. Find the amount of $500 from May 20, 1897, to Feb. 28, 1898, at 5%. 4. Find the amount of $630 from July 1, 1896, to Nov. 1, 1896, at 7%. 5. Find the amount of $800 from Jan. 1, 1897, to Jan. 25, 1897, at 6%. 6. If a 60-day bill of $400 is discounted 2% for cash, how much ready money vi^ill be required to pay the bill ? 7. Find the amount of $392 for 60 days (two months) at 6%. Note. — Observe that 8392 is the answer to problem 6. The result of problem 7, then, is the amount that the goods mentioned in problem 6 would cost at the end of 60 days if the purchaser bor- rowed the money at Q% to pay for them. Compare this result with $400. How much does the purchaser of the bill save by borrowing the money at 6% to pay the bill instead of letting the bill run 60 days and then paying $400 ? 8. Find the cost of goods, the list price being $46, and the discounts "50 and 10 off' and 2 off for cash." 9. My remuneration for selling goods on a commission of 40% amounted to $56.20. How much should the man for whom the goods were sold receive ? Note.— $56.20 is 40^ of the selling price of the goods. The man for whom the goods were sold should receive 60^ of the selling price. When goods are sold on a commission of 40^, what the agent receives equals what part of what the employer receives? What the employer receives is how many times what the agent receives ? PEOMISSOEY NOTES. 252. When a man borrows money at a bank he gives his note for a specified sum to be paid at a specified time, and he receives therefor, not the sum named in the note, but that sum less the interest upon it from the time the note is given to the bank officials to the time the note is due. 253. The acceptance of a note by the bank officials and the payment of a sum less than it will be worth at maturity is called " discounting the note!' 254. The proceeds of a note is the sum paid for it. As a rule, the notes discounted at a bank are those which bear no interest, and the date of discounting is usually the same as the date of the note, though not necessarily so. Example. Find the discount and proceeds of the following: S800. Jacksonville, III., Jan. 10, 1898. Sixty days after date I promise to pay James Kice, or order, eight hundred dollars, value received. Arthur Williams. Discounted at 6%, Jan. 10, 1898. Erom the date of discount to the date of maturity it is 60 days. Interest of $800 for 60 days = $8.00. Proceeds of note = $800 ~ $8 : Observe that the bank receives the interest on $800 for two months at Q% for the use of $792 for two months. The actual rate of interest is therefore a Httle more than the rate named. 301 302 COMPLETE ARITHMETIC. 255. Problems in Bank Discount. 1. $375.00. Chicago, III., Apr. 5, 1898. Thirty days after date I promise to pay to the order of John Smith, three hundred seventy-five dollars, at the Union National Bank. Value received. James White. Discounted April 5, at 6 %. Find proceeds. 2. $450.00. Aurora, III., March 10, 1898. Sixty days after date I promise to pay to Wm. George, or order, four hundred fifty dollars, at the Old Second National Bank. Value received. F. D. Winslow. Discounted March 10, at 7 %. Find proceeds. 3. $2300.00. Waukegan, III., Feb. 9, 1898. Ninety days after date I promise to pay to the order of John Mulhall, two thousand three hundred dollars, at the Security Savings Bank. Value received. Chas. Whitney. Discounted Feb. 9, at 7 ^. Find proceeds. 4. $5000.00. Serena, III., Jan. 20, 1898. Sixty days after date I promise to pay to John Parr, or order, five thousand dollars. Value received. Harry Brown. Discounted Jan. 20, at 7 %. Find proceeds. 5. $3500. Boston, Mass., April 12, 1898. Sixty days after date I promise to pay to W. J. But- ton, or order, three thousand five hundred dollars. Value received. Harry Wilson. Discounted Apr. 24, at 6 %. Find proceeds. PART II. 303 Promissory Notes. 256. The Discounting of Interest-Bearing Notes. Whenever a note is presented at a bank to be discounted, its value at maturity is regarded as the sum upon which the discount is to be reckoned. When the note is not interest-bearing, its face value is its value at maturity. In the discounting of short-time notes at banks it is customary to find the exact number of days from the date of discounting to the date of maturity and to regard each day as ^l^ of a year. Example. $750.00. Austin, III., Jan. 1, 1898. Six months after date, I promise to pay to the order of N. D. Gilbert, seven hundred fifty dollars, with interest at the rate of six per cent per annum. 0. T. Bright. Discounted at a bank, May 10, 1898, at 7%. Value of the note at maturity, $772.50 Discount, 52 days at 7%, 7.81 Proceeds, $764.69 1. $540.00. ToPEKA, Kansas, Dec. 10, 1897. Five months after date, I promise to pay to the order of J. C. Thomas, five hundred forty dollars, with interest at the rate of six per cent per annum. Hiram Baker. Discounted at a bank, April 1, 1898, at 7^. 2. $325.00. Earlville, III., Feb. 1, 1898. Six months after date, I promise to pay to the order of Wm. E. Haight, three hundred twenty-five dollars, with interest at the rate of six per cent per annum. Chas. Hoss Discounted at a bank, June 1, 1898. 304 COMPLETE ARITHMETIC. 257. Partial Payments on Notes. A partial payment is a part payment made upon a note before the time of final settlement. There are two methods in common use of finding the value of a note at maturity, upon which one or more partial payments have been made. The first method is often applied to computation of " short-time notes," such as run one year or less. A formal state- ment of this method is called the — Merchants' Eule. — Find the amount of the face of the note from the date to mat^irity. Then find the amount of each payment from the time it was made to the maturity of the note. From the amount of the face of the note subtract the sum of the amounts of the payments. $450. Waukegan, III., Jan. 1, 1897. One year after date, I promise to pay to the order of Wm. E. Toll, four hundred fifty dollars, with interest at six per cent. Value received. Jerome Biddlecom. Part payments were made upon this note as follows; March 1, 1897, $250; May 1, 1897, |150. Amount of the face of the note for one year, $477.00 Amount of $250, March 1, '97, to Jan 1, '98, $262.50 Amount of $150, May 1, '97, to Jan. 1, '98, 156.00 $418.50 Value at maturity, $ 58.50 1. $1000.00. Springfield, III., Jan. 1, 1898. Eight months after date, I promise to pay to the order of Fred H. Wines, one thousand dollars, with interest at six per cent. Value received. J. H. Freeman. Part payments were made upon this note as follows : Apr. 1, 1898, $350; June 15, 1898, $240. How much was due on this note at maturity, Sept. 1, 1898? PART II. 305 Promissory Notes. 258. A decision of the United States Supreme Court many years ago led to the very general adoption in the solution of " partial payment problems," of what is now known as the United States Eule. — Find the amount of the principal at the time of the first payment. Subtract the first payment from this amount. The remainder is a new principal, upon which find the amount at the time of the second payment. Sub- tract the second payment from this amount. Continue this pro- cess to the time of settlement. The last amount is the sum due. An exception to the foregoing rule is required when there is a pay- ment which is less than the interest due at the time the payment is made. In such case the payment is treated as though made at the time of the next payment ; and if the two payments together do not equal the interest due, the sum of both is again carried forward. $950. Petersburg, III., Jan. 1, 1895. « On or before Jan. 1, 1898, 1 promise to pay to N. W. Branson, or order, nine hundred fifty dollars, with interest at six per cent. Value received. B. Laning. Part payments were made on this note as follows: July 1, 1895, $150; Jan. 1, 1896, $200; Jan. 1, 1897, $250. Find the amount due Jan. 1, 1898. Amount of $950, July 1, 1895 (6 mo.), $978.50 Subtract first payment, 150.00 New principal, $828.50 Amount of $828.50, Jan. 1, 1896 (6 mo.), $853.35 Subtract second payment, 200.00 New principal, $653.35 Amount of $653.35, Jan. 1, 1897 (1 year), $692.55 Subtract third payment, 250.00 New principal, $442.55 Amount of $442.55, Jan. 1, 1898 (1 year), $469.10 306 complete arithmetic. 259. Problems in "Partial Payments." 1. Date of note, Jan. 1, 1896. Pace of note, $800. Kate of interest, six per cent. Payment May 1, 1896, $125. Payment Dec. 1, 1896, $230. Find amount due Jan. 1, 1898. 2. Date of note, July 10, 1896. Face of note, $500. Rate of interest, six per cent. Payment Dec. 22, 1896, $200. Payment July 15, 1897, $200. Find amount due Jan. 1, 1898. ' 3. Date of note, Jan. 1, 1897. Face of note, $600. Eate of interest, six per cent. Payment Sept. 1, 1897, $100. Find amount due Jan. 1, 1898. Note. — Solve problem 3 by the Merchants' Rule and by tha United States Rule. The answer obtained by the latter will be 48 cents larger than the answer obtained by the former. Observe that the Merchants' Rule is based upon the supposition that interest is not due until the time of settlement of the note, while the U. S. Rule is based upon the supposition that interest is due whenever a payment is made. By applying the latter rule to problem 3, $24 of interest must be paid Sept. 1 ; by applying the former rule to the same problem the entire $100 paid Sept. 1 applies in payment of principal. The answer, then, by the U. S. Rule must be greater than the answer by the Merchants' Rule, by the interest on |24 for four months, or 48 cents. PART II. 307 Algebra. 260. Algebra Applied to Some Problems in Interest Example. What principal at 6% will gain $96 in 2 yrs.? Let X = the principal. Since the interest at 6% for 2 years equals -^^ of the principal then 11., or 1^ = 96. 100 100 Multiplying by 100 12ic = 9600 X = 800.* Problems. 1. What principal at 6% will gain $67.50 in 2 years 6 months ? 2. What principal at 6% will gain $27.20 in 1 year 4 months ? 3. What principal at 7% will gain $87.50 in 2 years 6 months ? 4. What principal at 5% will gain $187.50 in five years ? 5. What principal at 8% will gain $64 in 1 year 3 months ? 6. What principal at 6% will gain $61.20 in 2 years 6 months 1 8 days ? 153 153a; Let X = the principal, then x, or = $61.20. ^ ^ 1000 1000 (a) Find the sum of the six answers. * What principal at a% will gain 6 dollars in c years? 308 COMPLETE ARITHMETIC. 261. Algebra Applied to Some Problems in Interest. Example. What principal at 6% will amount to $828.80 in 2 years ? Let X — the principal. 1 2r Then x^ J_:= 828.80. 100 Multiplying by 100, 100^ + 12^ = 82880. Uniting, 112.^? = 82880 X = 740.* Problems. 1. What principal at 6% will amount to $368 in 2 years 6 months ? f 2. What principal at 6% will amount to $588.30 in 1 year 10 months? 3. What principal at 5% will amount to $393.75 in 2 years 6 months ? 4. What principal at 5% will amount to $287.50 in three years ? 5. What principal at 6% will amount to $458.80 in 2 years 5 months and 12 days? 147 14:7x Let X - the principal ; then -r-rr^> or Yo^(\ ~ ^^^ interest, 147a; and X A = the amount, $458.80. 1000 fa) Find the sum of the five answers. * What principal at ai will amount to 6 dollars in c years? f To THE Pupil.— Prove each answer obtained by finding its amount for the given time at the given rate IPAKT It. 309 Geometry. 262. The Area of a Eectangle. 1. One side of every rectangle may be regarded as its base. The side perpendicular to its base is its altitude. 2. The number of square units in the row of square units next to the base of a rectangle, taken as many times as there are linear units in its altitude, equals the number of square units in its area. In the figure given, we have 4 sq. units x 3 = 12 sq. units. Note 1. — In the above, it is assumed that the base and altitude are measured by the same linear unit, and that the square unit takes its name from the linear unit. Note 2. — In. the actual finding of the area of rectangles for prac- tical purposes, the work is done mainly with abstract numbers and the proper interpretation is given to the result. There can be no serious objection to the rule for finding the area of rectangles as given in the old books, provided the pupil is able to interpret it. Rule. — To find the area of a rectangle, " multiply its base hy its altitude.'' Problems. 1. Find the area of the surface of a cubical block whose edge is 9 inches in length. 2. Find the area in square yards of a rectangular piece of ground that is 36 feet by 45 feet. 3. Find the area in acres of a rectangular piece of land that is 92 rods by 16 rods. 4. Find the area in square rods of a piece of ground that is 99 feet by 66 feet. 310 COMPLETE ARITHMETIC. 263. Miscellaneous Review. 1. Clarence Marshall wished to borrow some money at a bank. He was told by the president of the bank that they (the bank officials) were "discounting good 30-day paper" at 7%. Mr. Marshall's name being regarded as "good," he drew his note upon one of the forms in use at the bank, for $1000 payable in thirty days without interest. On the presentation of this note to the cashier, how much money should he receive ? 2. If Mr. Marshall's note described in problem 1 was dated April 10, 1898, (a) when must it be paid? (b) How much money will he pay when he " takes up " the note ? (c) Does he pay for the use of the money borrowed, at the rate of exactly 7 % per annum ? 3. If a bank is discounting at 7%, how much should be given for a note of $200 due in two months from the time it is discounted and bearing interest for the two months at the rate of 6 % per annum ? 4. Find the value at the time of settlement of the follow- ing note : Date of note, Apr. 1, 1896. Face of note, $300. Eate, 6%. Payment, Aug. 1, 1896, $75. Payment, Apr. 1, 1897, $80. Settled, Aug. 1, 1898. 5. What principal at 6% will gain $6 in 1 year 4 months ? 6. What principal at 6% will amount to $81 in 1 year 4 months ? 7. Find the area, in square feet, of a walk 4 feet wide around a rectangular flower-bed that is 40 feet long and 12 feet wide. STOCKS AND BONDS. 264. Some kinds of business require so much capital that many persons combine to provide the necessary money. Such a combination of men organized under the laws of a State, the capital being divided into shares, is known as a corporation, or stock-company. Those who own the shares are called stock-holders. The stock-holders elect from their own number certain men to manage the business. These man- agers are called directors. 265. The nominal value of a share is its face value ; that is, the sum named on its face. Large corporations, usually, though not always, divide their capital into $100 shares. If the business is prosperous, shares may sell on the market for more than their nominal value. The stock is then said to be " above par," or " at a premium.'^ If the business does not prosper, the shares may sell on the mar- ket for less than their nominal value. The stock is then said to be " helow par,*' or " at a discount." 266. If the business is profitable, a part or all of the earn- ings is periodically divided among the stock-holders. The sum divided is called a dividend. Dividends are always reckoned on the nominal or par value of the stock. If a corporation declares a 2% dividend, it pays to each stock-holder a sum equal to 2% of the nominal value of the stock which he owns. 267. The kinds of business which are usually conducted by corporations, are: The mining of coal, silver, gold, etc.; the operation of gas works, railroads, large manufacturing establishments of all kinds, creameries, etc. 3U 312 COMPLETE ARITHMETIC. Certificate of Stock. / /: 1. Examine the above certificate. What part of the entire stock of the Werner School Book Company would the owner of one hundred shares have ? 2. A 3% dividend would require how much money from the treasury of the Company ? How much money should the owner of one hundred shares receive ? 3. If a 4% dividend is declared how much money should the owner of seven hundred and fifty shares of stock receive as his share of the dividend ? PART II. 313 Stocks. 268. History of a Stock Company. The farmers of a certain community agreed to combine in the building and management of a creamery. It was deter- mined that a capital of $5000 was necessary. This was divided into 50 shares of $100 each. Men then came for- ward and contributed as follows • A took 5 shares and paid in B took 3 shares and paid in C took 6 shares and paid in D took 4 shares and paid in E took 7 shares and paid in $700. F took 2 shares and paid in $200. G took 10 shares and paid in H, T, J, K, L, M, N, O, P, Q, R, S, and T took 1 share each and paid in $100 each. (a) At the end of the first year the directors declared an 8% dividend. How much did A receive ? B ? C ? K ? (b) Wliat was the entire amount of the money divided among the stock-holders at the end of the first year ? (c) Soon after this dividend was paid, A sold his stock to X at a premium of 10%. How much did A receive for his stock ? (d) At the end of the second year the profits were found to be comparatively small, and the directors could pay a div- idend of only 3%. How much did X receive ? B ? C ? K ? (e) What was the entire amount of the money divided among the stock-holders at the end of the second year ? (f) Soon after this dividend was paid, B, C, D, E, and F sold their stock to Y at a discount of 10%. How much did this stock cost y ? 314 COMPLETE ARITHMETIC. History of a Stock Company — Continued. (g) At the end of the third year, the profits were so small that no dividend was declared. The stock-holders became disheartened and many of them offered to sell their stock at a large discount. Z appeared in the market and bought at "50^ on the dollar" all the stock of the company except that owned by X and Y. How much did this stock cost Z? (h) At the end of the fourth year, the directors, X, Y, and Z, declared a 10% dividend. How much money was divided and how much did each receive ? (i) Before the close of the fifth year, the property burned and the lot upon which it stood was sold. After the insur- ance money had been received, the book accounts collected, and all debts paid, there remained in the treasury of the company $4350. How much of this money should each stock-holder, X, Y, and Z, receive ? (j) Did this creamery enterprise prove a good investment forX? ForY? ForZ? For A? For B?. For M? Miscellaneous Problems. 1. The directors of a company whose capital is $50000 determined to distribute among the stock-holders $2500 of profits, (a) A dividend of what per cent shall be declared ? (b) How much will a man receive who owns 15 lOO-doUar shares ? 2. A company whose capital is $75000 pays a dividend of 3%. (a) How much money is divided among the stock- holders ? 3. Mr. Steele owns 20 shares ($100) in the C, B. & Q. R R. He receives as his part of a certain dividend $110. What is the per cent of the dividend ? PART II. 315 Bonds. 269. A bond is a very formal promissory note given by a government or other corporate body, as a railway or a gas company, for money borrowed. Bonds usually have attached to them small certificates called coupons. These are really little notes for the interest that will be due at different times Thus, a 10-year bond for $1000 with interest at 6% payable semi-annually may have 20 coupons attached, each calling for $30 of interest. 270. Money invested in bonds yields a specified income ; but the income from money invested in stocks depends upon the profits of the company. 271. Bonds, like stocks, are sometimes sold for more than their face value. They are then said to be " above par" or " at a premium." Like stocks, too, they are sometimes " helow par" or " at a discount." 1. What is the semi-annual interest on two lOOO-doUar U.S. 5% bonds? 2. What sum should be named on each coupon of a 1000- doUar city bond if the interest is payable annually at the rate of 7% ? 3. To raise the money to build a court-house, a certain county issued $50000 worth of 6% ten-year bonds. These sold upon the market at 2% premium, (a) How much money was received for the bonds ? (b) How much did A pay, who bought three lOOO-doUar bonds? (c) If the in- terest was payable semi-annually, how much should A receive each 6 months on this investment ? 4. Has the county or city in which you live any "bonded indebtedness " ? If so, how much, and what is the rate of interest ? 316 COMPLETE ARITHMETIC. City Bond with Coupons Attached. ■ro oi'WKcmoi Mixsrm/ioma Note. — Bonds are made in great variety both as to form and content; but in all, indebtedness is acknowledged, and the amount, rate of interest, and time of payment for both principal and interest, named. The above is a very short and concise form of Bond (much reduced in size) and is an exact copy of one prepared for actual use. 1. Examine the above Bond. If the time it is to run is five years, how many coupons should be attached? 2. If the Bond is dated Jan. 1, 1898, what date should be written in each coupon? 3. If the face of the Bond is $100 and the rate 5%, what sum should be written in each coupon ? PART II. 317 272. Algebra Applied to Some Problems m Interest. Example. At what rate per cent will $500 gain $55 in 2 yrs.?* Let X = the rate, then — — of 500, or , or 10^ = the interest, and lOeZ? = 55 Dividing x = 6^- Problems. 1. At what rate per cent will $450 gain $72 in 2 years? 2. At what rate per cent will $320 gain $48 in 3 years ? 3. At what rate per cent will $560 gain $84 in 2 years 6 months ? 4. At what rate per cent will $600 gain $75 in 2 years 6 months ? 5. At what rate per cent will $600 gain $114 in 2 years 4 months 15 days ? 2 yr. 4 mo. 15 da. = 2|- years. Let X = the rate. Then ^ of 600 = 114. Note. — Problem 5 may be solved arithmetically by finding the interest of $600 for 2 yr. 4 mo. 15 da. at Q%. Divide this interest by 6 (to find the interest at 1%) and find how many times the quo- tient is contained in |114. ♦ The arithmetical solution of this problem is as follows : The interest of $500 for 2 years at 1^ is igj, of $500. ih of S500 = $10. To gain $55 in. 2 years, $500 must be loaned at as many per cent as $10 is contained times in $55. It is contained 5| times ; so $500 must be loaned at 5i % to gain $55 in 2 years. Observe that by this method we divide the given interest by the interest of the principal for the given time at one per cent. 318 COMPLETE ARITHMETIC. Algebra. 273. Algebra Applied to Some Problems in Interest. Example. In how long a time will $650 gain $97.50 at 6%? Let X — the number of years, then — of 650 = 97.50 100 Simplifying, 39^ = 97.50 Dividing, ^ = 2.5* Problems. 1. In how long a time will $400 gain $30 at 5%? 2. In how long a time will $600 gain $96 at 6%? 3. In how long a time will $800 gain $68 at 6%? 4. In how long a time will $500 gain $56 at 6%? 5. In how long a time will $400 gain $29 at 6%? Review Problems. 6. What principal at 8% will gain $124.80 in 3 years? 7. What principal at 7% will amount to $410.40 in 2 years ? 8. At what rate per cent will $900 gain $72 in 2 years? 9. In how long a time will $1000 gain $160 at 6 per cent? 10. What p/incipal at 5% will amount to $736 in 3 years ? To the Pupil. — Prove each answer by finding the interest on the given principal at the given rate for the time obtained. * The arithmetical solution of this problem is as follows : The interest of $650 for one year at % is 839. As many years will be required to gain $97.50 as $39.00 is con- tained times in $97.50. It is contained 2\ times ; so in 2J years $650 will gain $97.50. Observe that by either method we divide the given interest by the interest of the principal for 1 year at the aiven rate. PART II. 319 Geometry. 274. The Akea of a Ehomboid * 1. One side of a rhomboid may- be regarded as its hase. The per- pendicular distance from the base to the opposite side is its altitude. 2. Convince yourself by measurements and by paper- cutting that from every rhomboid there may be cut a tri- angle, (abc), which when placed upon the opposite side, {def), converts the rhomboid into a rectangle (adeh). Observe that the base of the rectangle is equal to the base of the rhomboid, and the altitude of the rectangle equal to the altitude of the rhomboid. 3. A rhomboid is equivalent to a rectangle having the same base and altitude. Hence, to find the area of a rhom- boid, find the area of a rectangle whose hase and altitude are the same as the hase and altitude of the rhomhoid. Or, as the rule is given in the older books, — " Multiply the hase hy the altitude!' Problem. — If the above figure represents a piece of land, and is drawn on a scale of -J- inch to the rod, how many acres of land ? * The statements upon this page apply to the rhombus as well as to the rhomhoid. 320 COMPLETE ARITHMETIC. 275. Miscellaneous Review. 1. Mr. Watson purchased 15 shares of C, B. & Q. R R stock at 12% discount, (a) How much did he pay for the stock ? (b) When a 3 % dividend is declared and paid, how much does he receive ? * 2. James Cooper bought 12 shares of stock in the Sugar Grove Creamery at 8 % below par, and a few days after sold the stock at 5% above par. How much more did he receive for the stock than he gave for it ? 3. A certain city borrowed a large sum of money and issued therefor 10-year 5% bonds with the interest payable semi-annually, (a) How many coupons were attached to each bond? (b) On a Si 000 bond, each coupon should call for how much money ? 4. Sometimes such bonds as those described in problem 3 are offered for sale to the highest bidder, in "blocks" of $10000, $20000, or $50000. If a $20000 "block" is "bid off" at 2^% premium, how much should the city receive for the block"? 5. What must be the nominal value of 5% bonds that will yield to their owner an annual income of $750 ? Let X = the nominal value ; then — = $750. 100 6. What must be the nominal value of 4% bonds that will yield to their owner an annual income of $720 ? 7. A owns $6000 of 5% bonds; B owns $8000 of 4|-% bonds. How much greater is the annual income from B's bonds than from A's ? 8. Find the area of a piece of land in the form of a rhom- boid, whose base is 32 rods and whose altitude is 15 rods. * The par value of each share of stock mentioned on this page is $100. EATIO. 276. Ratio is relation by quotient. The two numbers (magnitudes) of which the ratio is to be found are called the terms of the ratio. The first term is called the antecedent and the second term the consequent. The ratio is the quo- tient of the antecedent divided by the consequent. The usual sign of ratio is the colon. It indicates that the ratio of the two numbers between which it stands is to be found, the number preceding the colon being the antecedent, and the number following it, the consequent. The expression, 12 : 4 = 3, is read, the ratio of 12 to 4 is 3. Exercise. Eead and complete the following : 1. 12:4- 4:12 = 12:2 = 2. 18:9 --= 9:18 = 18 : 6 = 3. 15:5 = 5:15 = 15:10 = Note. — It will be observed that the sign of ratio is the sign of division (-=-) with the line omitted. 277. Every integral nitmher is a ratio. The number 4 is the ratio of a magnitude 4 (inches, ounces, bushels) to the measuring unit 1 (inch, ounce, bushel). The number 7 is the ratio of 7 yards to 1 yard ; of 7 dollars to 1 dollar, or of 7 seconds to 1 second, etc. Note. — The ratio aspect of numbers is not the aspect most fre- quently uppermost in consciousness ; neither ought it to be. But the pupil should now see that number is ratio ; that while it implies aggregation and often stands in consciousness for magnitude, its essence is relation — ratio. 321 322 COMPLETE ARITHMETIC. Ratio. 278. Every fractional number is a ratio. The fraction |- is the ratio of the magnitude 3 to the magnitude 4. So ^, (3), is the ratio of 12 to 4. Observe that in every case the terms of a ratio may be written as the terms of a fraction ; the ante- cedent becoming the numerator and the consequent the denominator of the fraction. The fraction itself is the ratio. Exercise L the fraction to its simplest form. 1. The ratio of 20 to 6 is \^- = -i/ = 3^. 2. The ratio of 6 to 20 is ^V = _3_. 3. The ratio of 7 to 5 is — ; of 5 to 7, — . 4. The ratio of 12 to 1 is — ; of 1 to 12, - Exercise II. 1. f is the ratio of 5 to 7 ; of 10 to 14 ; of 15 to 21, etc. 2. ^ is the ratio of — to — ; of — to — ; of — to — , etc. 3. |- is the ratio of — to — ; of — to — ; of — to — , etc. 4. 8 is the ratio of 8 to 1 ; of — to — ; of — to — , etc. Exercise III. Make the necessary reduction and find the ratio : * 1. Of 2 feet to 8 inches. 2. Of 3 yards to 6 inches. 3. Of 6 rods to 3 yards. 4. Of 2 rods 5 yards to 1 yard 1 foot. * The comparison of two magnitudes involves their measurement by the same standard. To compare feet with inches, the inches may be changed to feet or the feet to inches, or both may be changed to yards. PART II. 323 Ratio. 279. Not only is number itself ratio, but a large part of the work in arithmetic is merely the changing of the form of the expression of ratios. Exercise IV. (Reducing fractions to their lowest terms.) 1. Express the ratio of 30 to 40 in its simplest form. 2. Express the ratio of 560 to 720 in its simplest form. 3. Express the ratio of ^ 5 to 875 in its simplest form. 4. Express the ratio of j min. to 2 hours in its simplest form. 5. Express the ratio of 1 lb. 4 oz. to 5 lb. 8 oz. in its sim- plest form. Exercise V. (Reducing improper fractions to integers.) 1. Express the ratio of 400 to 50 in its simplest form. 2. Express the ratio of 375 to 25 in its simplest form. 3. Express the ratio of 256 to 16 in its simplest form. 4. Express the ratio of 3 hours 20 minutes to 50 minutes in its simplest form. Exercise VI. (Reducing complex fractions to simple fractions.) 1. Express the ratio of ^ to -| in its simplest form. 2. Expr(;ss the ratio of |- to f in its simplest form. 3. Express the ratio of 2|- to 8|- in its simplest form. 4. Express the ratio of -|- of an inch to 1 foot in its sim- plest form. Note. — Observe that the denominator in fractions corresponds to the consequent in ratio. 324 COMPLETE ARITHMETIC. Ratio. Exercise VII. (Changing common fractions to decimals.) 1. Express the ratio of 3 to 4 (J), in hundredths. 2. Express the ratio of 20 to 50, in tenths. 3. Express the ratio of 30 to 80, in thousandths. 4. Express the ratio of 50 sq. rd. to 1 acre 40 rd., in hundredths. Exercise VIII. (Finding what per cent one nvimber is of another. ) 1. Express the ratio of 15 to 20, in hundredths. 2. Express the ratio of 14 to 200, in hundredths. 3. Express the ratio of 17 to 25, in hundredths. 4. Express the ratio of 16 to 33^, in hundredths. 5. Express the ratio of 27 to 500, in hundredths. Exercise IX. (Changing " per cent " to a common fraction in its lowest terms, or to a whole or mixed number.) 1. A's money equals 40% of B's money, (a) Express the ratio of A's money to B's money in the form of a fraction in its lowest terms, (b) Express the ratio of B's money to A's money in its simplest form. 2. One number is 50% more than another number, (a) Express the ratio of the; smaller to the larger number in the form of a fraction in its lowest terms, (b) Express the ratio of the larger to the smaller number in its simplest form. Note. — Observe that the base in percentage corresponds to the consequent in ratio. PART II. 325 Ratio. Exercise X. The specific gravity of a liquid or solid is the ratio of its weight to the weight of the same bulk of water. 1. A cubic foot of water weighs 62|- lb. A cubic foot of cork weighs 15 lb. What is the ratio of the weight of the cork to the weight of the water ? Express the ratio in hun- dredths. What is the specific gravity of cork ? 2. A certain piece of limestone weighs 37 ounces. Water equal in bulk to the piece of limestone weighs 15 ounces. What is the ratio of the weight of the limestone to the weight of the water? What is the specific gravity of the limestone ? 3. A certain bottle holds 10 ounces of water or 9^ ounces of oil. What is the ratio of the weight of the oil to the weight of the water? Express the ratio in hundredths. What is the specific gravity of the oil? Note. — Observe that in specific gravity problems, the weight of water corresponds to the consequent in ratio problems. 280. Miscellaneous Questions. 1. What is the ratio of a unit of the first integral order to a unit of the first decimal order ? 2. What is the ratio of a unit of any order to a unit of the next lower order ? 3. What ratio corresponds to 6 per cent ? 4. What is the ratio of a dollar to a dime ? Of a dime to a cent ? Of a cent to a mill ? 5. What is the ratio of 1 to y^^-? Of 1 tenth to 1 hun- dredth? 6. What is the ratio of a rod to a yard? Of a yard to a foot? Of a foot to an inch? 326 COMPLETE ARITHMETIC. Ratio. 281. Some Old Peoblems in New Forms* 1. What is the ratio of the area of a 2 -inch square to the area of a 6-in. square ? * Of a 6-in. square to a 2-in. square ? 2. What is the ratio of the perimeter of a 2 -in. square to the perimeter of a 6-in. square ? Of the perimeter of a 6-in. square to the perimeter of a 2-in. square ? 3. What is the ratio of the area of a 3-ft. square to the area of a 6-ft. square ? Of a 6-yd. square to a 3-yd. square ? 4. What is the ratio of the perimeter of. a 3-ft. square to the perimeter of a 6-ft. square ? Of the perimeter of a 6 ft. square to the perimeter of a 3-ft. square ? 5. What is the ratio of the solid content of a 2 -inch cube to the solid content of a 6-in. cube ? Of a 6-in. cube to a 2-in. cube ? 6. What is the ratio of the surface of a 2-in. cube to the surface of a 6-in. cube ? Of the surface of a 6-in. cube to the surface of a 3-in. cube ? 7. What is the ratio of the solid content of a 3-ft. cube to the solid content of a 6-ft. cube ? Of a 6-yd. cube to a 3-ft. cube ? 8. What is the ratio of the surface of a 3-ft. cube to the surface of a 6-ft. cube ? Of the surface of a 6-yd. cube to the surface of a 3-yd. cube ? 9. What is the ratio of a square inch to a square foot ? Of a cubic inch to a cubic foot ? * If pupils image the magnitudes compared, they will find no difl&culty in the solution of these problems. PART II. 327 Algebra. 282. Algebra Applied to Some Problems in Eatio. Example I. The consequent is c ; the ratio is r. What is the ante- cedent ? Let X = the antecedent. Then - = r. c and X = cr, the antecedent. Fro7n the above, learn that the antecedent is always equal to the product of the consequent and the ratio. 1. Consequent 75; ratio 11. Antecedent? 2. Consequent 92; ratio |-. Antecedent? 3. Consequent .56 ; ratio ^. Antecedent? Example II. The antecedent is a ; the ratio is r. What is the conse- quent ? Let X = the consequent. Then - = r. X and a = rx, or rx = a dividing by r, x = -, the consequent. From the above, learn that the consequent is always equal to the quotient of the antecedent divided by the ratio. 1. Antecedent 75 ; ratio 5. Consequent ? 2. Antecedent 9 6 ; ratio f. Consequent? 3. Antecedent f ; ratio |. Consequent? 328 COMPLETE ARITHMETIC. 283. To Find Two Numbees when theie Sum and Eatio are Given. Example. The sum of two numbers is 36 and their ratio is 3. What are the numbers ? Let X = the smaller number. Then Sx = the larger number, and x-\-3x = 3Q 4^ = 36 X = 9, the smaller number. Sx = 27, the larger number. Problems. 1. The sum of two numbers is 196, and their ratio is 3. What are the numbers ? 2. The sum of two numbers is 294, and their ratio is 2^. What are the numbers ? 3. The sum of two decimals is .42, and their ratio is 2^. What are the decimals ? 4. The sum of two numbers is s, and the ratio of the larger to the smaller is r. What are the numbers ? Let X = the smaller number. Then rx = the larger number, and rx-{- x = 8 or (r -\-l) X = s s ^ Dividing, x r + 1 * Observe that any number you please may be put in the place of s, and any number greater than 1 in the place of r; therefore when the sum of two numbers and the ratio of the larger to the smaller are given, the smaller number may be found by dividing the sum by the ratio plus 1. PART II, 329 Geometry. 284. The Area of a Triangle. 1. One side of a triangle may be regarded as its base. The perpendicular distance from its base (or from its base extended) to the opposite angle is its altitude. 2. What is the altitude of the first of the above triangles ? Of the second ? Of the third ? 3. Convince yourself by measure- ment and by paper cutting that every triangle is one half of a parallelogram having the same base and the same altitude as the triangle. 4. To find the area of a triangle, Find the area of the parallelogram having the same base and altitude, and take one half of the result. Problem. — If the above figure represents a piece of land, and is drawn on a scale of ^ inch to the rod, what part of an acre of land does it represent ? 330 COMPLETE ARITHMETIC. 285. Miscellaneous Review. ' 1. The specific gravity of granite is 2.7.* How much does a cubic foot of granite weigh ? 2. A certain vessel is exactly large enough to contain 1000 grains of water. It will contain only 700 grains of petroleum. What is the specific gravity of the petroleum ?f 3. The specific gravity of gold is 19.3. How much does a cubic foot of gold weigh ? 4. A cubic foot of sulphur weighs 125 lbs. What is the specific gravity of sulphur ? 5. A cubic foot of steel weighs 487.5 lbs. What is the specific gravity of steel ? 6. What is the ratio of 1 bu. to 1 pk.? Of 1 pk. to 1 qt.? 7. What is the ratio of $37^ to $15 ? Of $15 to $37|-? 8. What is the area of a rhomboid whose base is 16 inches and whose altitude is 1 6 inches ? 9. Is the rhomboid described in problem 8, equilateral ? 10. The ratio of the perimeter of one square to the perim- eter of another square is 4. What is the ratio of the areas of the two squares ? 11. Draw three triangles, the base of each being 4 inches and the altitude of each being 2 inches. Make one of them a right-triangle; another, an isosceles triangle, and the third having angles unhke either of the other two. Wliat can you say of the area of the right-triangle as compared with each of the others ? ♦ See page 325, exercise 10. t This means, what is the ratio of the weight of the petroleum to the weight of the same hulk of water? PEOPORTION. 286. The terms of a ratio are together called a couplet. Two couplets whose ratios are equal are called a proportion. The two couplets of a proportion are often written thus: 6 : 18 = 10 : 30, and should be read, the ratio of 6 to 18 equals the ratio of 10 to 30. Couplets are sometimes written thus: 20 : 4 : : 50 : 10, and read, 20 is to 4 as 50 is to 10.* 287. To Find a Missing Term in a Proportion. Example I. 36 : 12 ::.??: 25. The ratio of the first couplet is 3 ; that is, the antecedent is 3 times the consequent. Since the ratios of the couplets are equal, the ratio of the second couplet must be 3, and its antecedent must be 3 times its consequent. Three times 25 = 75, the missing term. Problems. Find the missing term. 1. 90:45::^:180. 4 20:60 = ^^:225. 2. 48:12::^:150. 5.30:50=^:175. 3. 75 : 30 : : « : 140. 6. 90 : 20 = « : 140. ' *The ratio sign (:) may be regarded as the sign of division (-s-) with the hori- zontal line omitted, and the proix^rtion sign (: :) the sign of equality ( = ) with an erasure through its center, thus :(= = ). 331 332 COMPLETE ARITHMETIC. Proportion. Example IL 36 : 12 = 48 : x. Since the ratio of the first couplet is 3, the ratio of the second couplet must be 3, and x must equal 1 third of 48. 1 third of 48 is 16. Problems. Find the missing term. 1. 84:21 = 172:^. 4. 20 : 60 : : 120 :a?. 2. 96:16:: 45:^. 5. 25:35= 45:^. 3. 75:30 = 125:^. 6. 50 : 25 : : 14 :^. Example III. 36:^ = 45:15. The ratio of each couplet is 3 ; so each consequent must be 1 third its antecedent, and a:, 1 third of 36, or 12. Problems. Find the missing term. 1.54:^= 90:30. 4. 18 :^' : : 65 : 195. 2. 75:^::125:25. 5. 50:^ = 12: 18. 3. 50:a^= 40:16. 6. 35:^:: 21: 3. Example IV. ^ : 12 = 100 : 25. The ratio of each couplet is 4 ; so each antecedent must be 4 times its consequent, and a:, 4 times 12, or 48. Problems. Find the missing term. 1. ic:16:: 51:17. 4. ^:96 = 23:92. 2. ic:22 = 76:19. 5. aj:40 :: 36 :48. 3. ^:11::24: 3. 6. x:27 = 42:U. PART II. 333 288. Practical Problems. 1. If 75 yd. of cloth cost $115.25, how much will 15 yd. cost at the same rate ? 75 yd.: 15 yd. =$115.25: a-. 2. If 2|- acres of land cost $76.20, how much will 15 acres cost at the same rate ? 3. If 7 tons of coal can be bought for $26, how many tons can be bought for $39 ? 7 tons : x tons : : <|26 : f 39. 4. If 36 lb. coffee can be bought for $7, how many pounds can be bought for $17 J? 5. If sugar sells at the rate of 18 lb. for $1, how much should 63 lb. of sugar cost ? 6. If a post 6 ft. high casts a shadow 4 feet long, how high is that telegraph pole which at the same time and place casts a shadow 20 feet long ? 7. If a post 5 feet high casts a shadow 8 feet long, how high is that steeple which casts a shadow 152 feet long ? 8. If a train moves 50 miles in 1 hr. 20 min.,at the same rate how far would it move in 2 hours ? 9. If a boy riding a bicycle at a uniform rate goes 12 miles in 1 hr. 15 min., how far does he travel in 25 minutes ? To THE Teacher. — After the pupil has solved the above prob- lems by making use of the fact of the equality of the ratios, he should solve them by an analysis somewhat as follows : Prob. 1. Since 75 yd. cost $115.25, 1 yd. costs ^^ of $115.25 ; but 15 yd. cost 15 times as much as 1 yd., so 15 yd. cost 15 times -^^ of $1 ,^5.25- 334 COMPLETE ARITHMETIC. 289. Magnitudes Which Are Proportional to the Squares of Other Magnitudes. The areas of two squares are to each other as ■ the squares of their lengths. The areas of two circles are to each other as the squares of their diameters. Observe that the ratio of the areas of the above squares is | (or |). But the area of each circle is about | (more accurately, .785-|-) of its circumscribed square ; so the ratio of the areas of the circles is | (or |). 1. The area of a 6 -inch circle is how many times as great as the area of a 3-inch circle ? 2. If a 4-inch circle of brass plate weighs 3 ounces, how much will a 6 -inch circle weigh, the thickness being the same in each case ? 3. If a piece of rolled dough 1 foot in. diameter is enough for 17 cookies, how many cookies can be made from a piece 2 feet in diameter, the thickness of the dough and the size of the cookies being the same in each case ? 4. If a piece of wire ^ of an inch in diameter will sustain a weight of 1000 lbs., how many pounds will a wire ^ of an inch in diameter sustain ? PART II. 335 Proportion. 290. Magnitudes Which Ake Proportional to the Cubes of Other Magnitudes. The solid con- tents of two cubes are to each other as the cubes of their lengths. The solid con- tents of two spheres are to each other as the cubes of their diameters. Observe that the ratio of the solid contents of the above cubes is /y (or V). But the solid content of each sphere is about ^ (more accurately, .5236—) of its circumscribed cube; so the ratio of the solid contents of the spheres is g^ (or ^). 1. The solid content of a 6-inch sphere is how many times as great as the soHd content of a 3-inch sphere ? 2. If a 4-inch sphere of brass weighs 10 lbs., how many pounds will a 6-inch sphere of brass weigh ? 3. If a sphere of dough 1 foot in diameter is enough for 20 loaves of bread, how many loaves can be made from a sphere of dough 2 feet in diameter ? 4. If the half of a solid 8-inch globe weighs 4 lbs, how much will the half of a solid 5 -inch globe weigh, the material being of the same quahty ? 336 COMPLETE ARITHMETIC. 291. Magnitudes Which Are Inversely Proportional TO Other Magnitudes or to the Squares of Other Magnitudes. Example. If 5 men do a piece of work in 16 days, how long will it take 8 men to do a similar piece of work ? Operation and Explanation. It is evident that the time required will be inversely proportional to the number of men employed ; that is, if twice as many men are employed, not twice as much, but |^ as much time will be required. Hence the proportion is not 5 : 8 == 16 : x, but, 5 : 8 = a: : 16 ; hence, 5:8=10:16. The interpretation of the above equation is, if 5 men can do a piece of work in 16 days, 8 men can do it in 10 days. 1. If 4 men can do a piece of work in 20 days, how long will it take 5 men to do a similar piece of work ? 2. If 8 men can do a piece of work in 12 days, how long will it take 3 men to do a similar piece of work ? It can be shown that the intensity of light upon an object dimin- ishes as the square of the distance between the luminous body and the illuminated object increases ; that is, if the distance be twice as great in one case as in another, the intensity is not twice as great, not I as great, but i as great; if the distances are as 2 to 3 the intensities are, not as 2 to 3, not as 3 to 2, but as 9 to 4. The intensity at 2 feet is | as great as at 3 feet. 3. Object A is 15 feet from an incandescent electric light. Object B is 20 feet from the same light. Object C is 30 feet from the same light, (a) How does the intensity of the L'ght at B compare with the intensity at A ? (b) How does the intensity at C compare with the intensity at A ? PART II. 337 Algebra. 292. To Find the Missing Term of a Proportion Without Finding the Eatio. The first and fourth terms of a proportion are called the extremes, and the second and third terms, the means; thus, in the proportion 12:6 = 8: 4, 12 and 4 are the extremes and 6 and 8 are the means. Observe that in the following proportions the product of the means equals the product of the extremes : 6:3 = 8:4; then 6x4 = 3x8 ^:i = 4:2; then | x 2 = | x 4 Let a :h = G :d, stand for any proportion. a G *«^ ^^^" ^ = d Clearing of fractions, ad = Ig But a and d are the extremes and h and g the means ; hence, in any proportion in which abstract numbers are em- employed, the product of the means equals the product of the extremes. Example I. 30 :20 = 18:^. 30a; = the product of the extremes. 20 X 18 = the product of the means. Then 30£C = 20 x 18, or 360, and X = 12. Example II. 10:25 = ^:50. Then 25^ = 10 x 50, or 500, and ic = 20. 338 COMPLETE ARITHMETIC. Algebra. Example III. 40:^ = 25:5. Then 25^ = 40 x 5, or 200, and ^ = 8. Example IV. ^: 35 =4:28. Then 2Sx = 35 x 4, or 140, and X — h. Find the missing terms : 1. 24:72 =ic:69. 2. 45:12 = lb:x. 3. 35 : a? = 14 : 40. 4. rc:70 = 3:21. 9. .6 : .8 = 15 :x. 5. 55:25 =^:10. 10. .25:5 =^:40. 11. If 8 acres of land cost $360, how much will 15 acres cost at the same rate ? 8:15 = 360: ;^\*' 12. If 12 horses consume 3500 lb. of hay per month, how many pounds will 1 5 horses consume ? 13. If 11 cows cost $280.50, how many cows can be bought for $433.50 at the same rate? * Observe that in the solution of concrete problems by the method here given the numbers must be regarded as abstract. It would be absurd to talk or think of finding the product of 15 acres and 360 dollars and dividing this by 8 acres. It is true, however, that the ratio of 8 acres to 15 acres equals the ratio of 860 dollars to x dollars. It is also true that in the proportion 8 : 15 = 360 : x, the product of the means is equal to the product of the extremes. 6. 1 :4 = a;:30. 7. X ; : f = 40 : 6. 8. i :f = ^:8. PART II. 339 Geometry. 293. The Area of a Trapezoid. 1. Convince yourself by measurement and by paper cut- ting that from every trapezoid there may be cut a triangle (or triangles) which when properly adjusted to another part (or parts) of the trapezoid, will convert the trapezoid into a rectangle. 2. Convince yourself that the rectangle made from a trape- zoid is not so long as the longer of the parallel sides of the trapezoid, and not so short as the shorter of the parallel sides of the trapezoid — that its length is midymy between the lengths of the two parallel sides of the trapezoid. Note. — Observe that the length of the rectangle thus formed may be found by adding half the difference of the parallel sides of the trapezoid to its shorter side, or by dividing the sum of its parallel sides by 2. 3. To find the area of a trapezoid, find the area of the rectangle to which it is equivalent, or, as the rule is usually given, — "Multiply one half the sum of the parallel sides by the altitude" 4. Find the area of a trapezoid whose parallel sides are 10 inches and 15 inches respectively, and whose altitude is 8 inches. 5. How many acres in a trapezoidal piece of land, the parallel sides being 28 rods and 36 rods respectively, and the breadth (altitude) 25 rods ? 340 COMPLETE AKITHMETIC. 294. Miscellaneous Review. 1. If 3 men can build 72 feet of sidewalk in a day, how many feet can 4 men build ? 2. If 3 men can do a piece of work in 12 hours, in how many hours can 4 men do an equal amount of work ? 3. If a piece of land 8 rods square is worth $500, how much is a piece of land 1 6 rods square worth at the same rate ? 4. If a ball of yarn 3 inches in diameter is enough for a pair of stockings, how many pairs of stockings can be made from a ball 6 inches in diameter ? * 5. If a grindstone 12 inches in diameter weighs 40 lb., how much will a grindstone 18 inches in diameter weigh, the thickness and quality of material being the same ? 6. The opening in an 8-inch drain tile is how many times as large as the opening in a 2 -inch drain tile ? f 7. Find the area of a rhomboidal piece of land whose length (base) is 64 rods and whose width (altitude) is 15 rods. 8. Find the area of a trapezoidal piece of land, the length of the parallel sides being 44 rods and 52 rods respectively, and the width (altitude) being 18 rods. $ 9. Find the area of a triangular piece of land whose base is 42 rods and whose altitude is 20 rods. * Compare a 3-inch cube and a 6-inch cube. Remember that a 3-inch sphere is a little more than half of a 3-inch cube, and a 6-inch sphere a little more than one half of a 6-inch cube. t Compare a 6-inch square with a 2-inch square. Remember that a 2-inch circle is about 2 of a 2-inch square, and an 8-inch circle about 3 of an 8-inch square. % Draw a diagram of the land on a scale of i inch to the rod. POWERS AND EOOTS. 295. A product obtained by using a number twice as a factor is called the second power or the square of the num- ber; thus, 25, (5 X 5), is the second power, or the square of 5. Note. — Twenty-five is called the second power of 5, because it may be obtained by using 5, twice as a factor. It is called the square of 5, because it is the number of square units in a square whose side is 5 linear units. 1. What is the second power of 2 ? 8 ? 3 ? 5 ? 2. What is the square of 4 ? 7? 1? 6? 9? 10? 11'=:? 12' = ? 13'=? 14' = ? 15' = ? 16' = ? 17' = ? 18' = ? (a) Find the sum of the eighteen squares. 296. The square root of a number is one of the two equal factors of the number. The radical sign, V, (without a figure above it) indicates that the square root of the number following it, is to be taken ; thus V 64, means the square root of 64. 1. What is the square root of 144 ? 81 ? 49 ? 2. What is the square root of 36 ? 25 ? 16 ? V9 = ? ^64 = ? 'v/121 = ? V100 = ? V'4 = ? VI = ? V400 = ? V169 = ? (b) Find the sum of the fourteen results. 341 342 COMPLETE ARITHMETIC. Powers and Roots. 297. Any number that can be resolved into two equal factors is a perfect square. 1. Tell which of the following are perfect squares and which are not : 9, 10, 12, 16, 18, 25, 32, 36. Note. — It is a curious fact that no number, either integral or mixed, can be found which, when multiplied by itself, will give as a product 10, or 12, or 14, or any number that is not 2^, perfect square. 2. Any integral number that is a perfect square is com- posed of an even number of like prime factors ; that is, its prime factors are an even number of 2's, 3's, 5's, 7's, etc. 3. Tell which of the following are perfect squares : 144,(2 X 2x2x2x3x3); 250,(2 x5 x 5x5); 225, (5x5x3x3). Rule. — To find the square root of an integral numher^ that is a perfect square, resolve the member into its prime factors and take half of them as factors of the root ; that is, one half as many 2's, 3's, or 5's, etc., as there are 2's, 3's, or 5's, etc., in the factors of the number. 4. Find the square root of 1225. 1225 = 5x5x7x7. V1225 = 5 x 7 = 35. 5. Find the square root of 441 ; of 400. 6. Find the square root of 576 ; of 324. 7. Find the square root of 784; of 2025. 8. Find the square root of 625 ; of 3025. (a) Find the sum of the last eight results. PART II. 343 Powers and Roots. 298. The Square of Common Fractions. 1. The square of |, (^ x J), is . Note. — A square whose side is | (of a Hiiear unit) has an area of i (of a square unit). Show this by diagram. 2. Answer the following and illustrate by diagram if necessary : ay = ? (})' = ? ay = ? ay = ? (a) Find the sum of the eight results. 3. A square of sheet brass whose edge is -^^ of a foot is what part of a square foot ? 299. The Square Eoot of Common Fractions. 1. The square root of ^ is . Note 1. — A square whose area is i% (of a square unit) is | (of a linear unit) in length. Show this by diagram. Note 2. — Only those fractions are perfect squares which, when in their lowest terms, have perfect squares for numerators and perfect squares for denominators. 2. What is the square root of || ? Of ^ f ? Of -J- (b) Find the sum of the seven results. 3. The area of a square piece of sheet brass is -f^^ of a square foot. What is the length of the side of the square ? 4. How long is the side of a square of zinc the area of which is -|| of a square yard ? 344 COMPLETE ARITHMETIC. Powers and Roots. 300. The Square of Decimals. 1. The square of .5 is — — . Note. — A square whose side is .5 (of a linear unit) has an area of .25 (of a square unit). Show this by diagram. 2. Answer the following and illustrate by diagram if nec- essary : .1' = ? .2' = ? .3'' = 2 .42 = ? .5* = ? .6' = ? .7' = ? .8' = ? 1.2' = ? 1.5' = ? 1.6' = ? 1.8' = ? (a) Find the sum of the twelve results. 3. A square of sheet brass whose edge is .9 of a foot is what part of a square foot ? 301. The Square Boot of Decimals. 1. The square root of .25 is — — . Note 1. — A square whose area is .25 (of a square unit) is .5 (of a linear unit) in length. Show this by diagram. Note 2. — Only those decimals are perfect squares which, when in their lowest decimal terms, have numerators that are perfect squares and denominators that are perfect squares. The decimal denomina- tors that are perfect squares are 100, 10000, 1000000, etc. 2. What is the square root of -^\%- ? Of .36 ? Of .64 ? \/||4 = ? ^1.44 = ? V2.25 = ? ^6.25=? (b) Find the sum of the seven results. 3. How long is the edge of a square of zinc whose area is 4.84 square feet ? * *4.84feetis}8Jfeet. PART II. 345 Powers and Roots. 302. A product obtained by using a number three times as a factor is called the third power, or the cube, of the number; thus, 125 (5x5x5) is the third power, or the cube, of 5. Note. — One hundred twenty-five is called the third power of 5, because it may be obtained by using 5 three times as a factor. It is called the cube of 5 because it is the number of cubic units in a cube whose edge is 5 linear units. V = 1 3' = 27 5' = 125 V = 343 9=* = 729 1. Find the cube of 12 ; of 13 ; of 14 ; of 15. 16':=? 17' = ? 18' = ? 19'=? 20' = ? (a) Find the sum of the nine results. 303. The cube root of a number is one of its three equal factors. The radical sign with a figure 3 over it indicates that the cube root of the number following it is to be taken; thus, 'y/512, means, the cube root of 512. Rule. — To find the cuhe root of an integral number that is a perfect cube, resolve the member into its prime factors and take one third of them as factors of the root ; that is, one third as many 2's, 3's, or 5's, etc., as there are 2's, 3's, or 5's in the factors of the number. 1. Find the cube root of 216. 216 = 2x2x2x3x3x3. \/216 = 2x3 2. Find the cube root of 1728; of 3375; of 2744; of 10648; of 5832. (b) Find the sum of the five results. 346 COMPLETE ARITHMETIC. Powers and Roots. 304. Miscellaneous Problems. 1. Square 42. Then resolve the square of 42 into its prime factors and compare them with the prime factors of 42. 2. Cube 42. Then resolve the cube of 42 into its prime factors and compare them with the prime factors of 42. 3. Square 45. Then resolve the square of 45 into its prime factors and compare them with the prime factors of 45. 4. Cube 45. Then resolve the cube of 45 into its prime factors and compare them with the prime factors of 45. 5. Divide the cube of 15 by the square of 15. 6. Divide the cube of ^ by the square of ^. 7. Divide the cube of .7 by the square of .7. 8. Divide the cube of 2.5 by the square of 2.5. 9. Find the square root of 5 x 5 x 7 x 7. 10. Find the cube root of 3x3x3x5x5x5x7x7 X 7. 11. Find the square root of each of the following perfect squares : (1) 3025 (2) 4225 (3) 5625 (4) 7225 (^)Ui (^)^% iV^'A (8)^^A (9) .64 (10) .0064 (11) .0625 (12) 2.56 * 12. Find the cube root of each of the following perfect cubes : (1) 1728 (2) 15625 (3) 3375 (4) 9261 (5)tIt (6) /tV (7) ,Vj (8)TiT ♦ Think of this number as fgg. PART II. 347 Algebra. 305. To Find the Square Eoot of Numbers Eepresented BY Letters and Figures. Explanation. Since the square root of a number is one of its two equal factors, the square root of a\ (a x a x a x a), is a^, {a x a). The square root of a^ is a. The square root of a^ is a^. Let a = 3, and verify each of the foregoing statements. 1. v"^ = ? V"F = ? V"^ = ? Verify. 2. V «^ = ? VaW = ? V"^' = ? Verify. 4. V^5^y = ? • V36a?y = ? V 49^ = ? 5. Let a = 2, h = 3, and c = 5, and find the numerical value of each of the following : (1) V«^' (2) \/«V (3) VW" (4) V«'^' (5) Va'^'c' (6) V(^'bV (a) Find the sum of the six results. 6. Let a = 2, b = 3,x = 5, and y = 7, and find the numer- ical value of each of the following : (1) aV'xY ' (2) bV^ (3) aV'^' (4) 3aVx' (5) 4:bVy' (6) baV^Y (b) Find the sum of the six results. 348 COMPLETE ARITHMETIC. Algebra. 306. To Find the Cube Eoot of Numbers Eepresented BY Letters and Figures. Explanation. Since the cube root of a number is one of its three equal factors, the cube root of a^, (a x a x a x a x a X a), is a\ (a X a). The cube root of a^ is a. The cube root of a^ is a^ Let a = 2, and verify each of the foregoing statements. 1. y/V = ? i/b' = ^. \/h' = l Verify. 2. </«'&' = ? ^a%'=:'i. 's/a'h' = ? Verify. 3. V8a« = ? ^27b'=l V64a« = ? Verify. 4. Let a = 2, h = 3, and c = 5, and find the numerical values of each of the following : (1) y/a'b' (2) y/8c' (3) </27^>^^ (4) 2Va'¥ (5) 3V&V (6) 4v^6V (a) Find the sum of the six results. 307. Miscellaneous Problems. Let a = 2,h = 3, X = 6, and y = 7, and find the numer- ical value of each of the following : (1) ah + V^Y (2).abVxy (3) 2a^x\/ (4) ab + ^xY (5) ab^xY (6) 2by/ xY (7) \Va'y' (8) VK^'* (9) IV aV (b) Find the sum of the nine results. * The factors of this number are J, \, a, o, o, a, &, h. PART II. 349 D E 3 4 1 2 A B C Qeometry. 308. The Square of the Sum of Two Lines. 1. Study the diagram and observe — (1) That the hne AG i^ the sum of the lines AB and BG. (2) That the square, 1, is the square oiAB. (3) That the rectangle, 2, is as long as ^^ and as wide as BG. (4) That the rectangle, 3, is as long as AB and as wide as BG. (5) That the square, 4, is the square of BG. (6) That the square, AGED, is the square of the sum of AB and BG. 2. SiQce a similar diagram may be drawn with any two lines as a base, the following general statement may be made : The square of the sum of two lines is equivalent to the square of the first plus twice the rectangle of the two lines plus the square of the second. 3. If the hne AB is 10 inches and the line BG 5 inches, how many square inches in each part of the diagram, and how many in the sum of the parts ? 4. Suppose the hne AB w> equal to the hne BG ; what is the shape of 2 and 3 ? 5. In the light of the above diagram study the following: 14' = 196. (10 + 4/ = W+ 2 (10 X 4) + 4' = 196. 16' = ? (10 + 6)' = 25'=? (20 + 5)' = 350 COMPLETE ARITHMETIC. 309. Miscellaneous Review. 1. What is the square root of a^V^I What is the square root of 3x3x5x5? 2. What is the cube root of dW What is the cube root of 2x2x2x7x7x7? 3. What is the square root of a^V^I What is the square root of 5^ x 3"? 4. What is the cube root of a%^'i What is the cube root of 3' x 5'? 5. The area of a certain square floor is 784 square feet. How many feet in the perimeter of the floor ? 6. The area of a certain square field is 40 acres. How many rods of fence will be required to enclose it ? 7. The solid content of a certain cube is 216 cubic inches. How many square inches in one of its faces ? 8. If there are 64 square inches in one face of a cube, how many cubic inches in its solid content ? 9. The square of (30 + 5) is how many more than the square of 30 plus the square of 5? 10. The square of (40 + 3) is how many more than the square of 40 plus the square of 3 ? 11. The square of a is a^; the square of 2a is 4al The square of two times a number is equal to how many times the square of the number itself ? 12. The square of an 8-inch line equals how many times the square of a 4-inch line ? SQUARE ROOT. 310. To Find the Approximate Square Root OF Numbbrs THAT ARE NOT PERFECT SQUARES. Find the square root of 1795. Regard the number as representing 1795 1-inch squares. These are to be arranged in the form of a square, and the length of its side noted. 100 1-inch squares = 1 10-inch square. 1700 1-inch squares = 17 10-inch squares. But 16 of the 17 10-inch squares can be arranged in a square that is 4 by 4; that is, 40 inches by 40 inches. See diagram. After making this square (40 inches by 40 inches) there are (1700 - 1600 -f 95) 195 1-inch squares remaining. From these, additions are to be made to two sides of the square already formed. Each side is 40 inches; hence the additions must be made upon a base line of 80 inches. These addi- tions can be as many inches wide as 80 is contained times in 195.* 195 ^ 80 = 2+. The additions are 2 inches wide. These will require 2 times 80, -j- 2 times 2, = 164 square inches. After making this square (42 in. by 42 in.) there are (195 — 164) 31 square inches remaining. If further additions are to be made to the square, the 31 square inches must be changed to tenth-inch squares. In each 1-inch square there are 100 tenth-inch squares; in 31 square inches there are 3100 tenth-inch squares. From these, additions are to be made upon two sides of the 42-inch square. 42 inches equal 420 tenth-inches. The additions must be made upon a base line (420 X 2) 840 tenth-inches long. These additions can be as many tenth-inches wide as 840 is contained times in 3100. 3100 -^ 840 = 3 -f . The additions are 3 tenth-inches wide. These will require 3 times 840, -|- 3 times 3, =: 2529 tenth-inch squares. After making this square (42.3 by 42.3) there are (3100 - 2529) 571 tenth-inch squares remaining. (If further additions are to be made to the square, the 571 tenth-inch squares must be changed to hundredth- inch squares.) The square root of 1795, true to tenths, is 42.3. * Allowance must be made for filling the little square shown at the upper right- hand corner of the diagram. 351 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 352 COMPLETE ARITHMETIC. Square Root. IS'oTE. — Pupils who have mastered the work on the preceding page will have no ditficulty in discovering that the same result may be obtained by the following process : Find the square root of 1795. Operation. 1795.^42.36 16 40 X 2 = 80* •2 195 164 420 x2 = 840 3 3100 2529 4230 X 2 = 8460 6 57100 50796 6304 Rule. 1. Beginning vjith the decimal point, group the fig- ures as far as possible into periods of two figures each. 2. Find the largest square in the left-hand period^ and place its root at the right as the first figure of the com- plete root. 3. Subtract the square from the left-hand period and to the difference annex the next period. Regard this as a dividend. 4. Take 2 times 10 times the root already found as a trial divisor, and find how many times it is contained in the divi- dend. Write the quotient as the second figure of the root, and also as a part of the divisor. Multiply the entire divisor by the second figure of the root, subtract the product from the dividend, and proceed as before. Problems. Find the approximate square root (true to tenths) : (1) 875. (2) 1526. (3) 2754. (4) 4150. (5) 624. (6) 624.7. (7) 62.47. (8) 6.24. (a) Find the sum of the eight results. * The entire divisor is 80 and 2; that is, 82. t The left-hand period may consist of either one or two figures. PART II. 353 Square Root. 311. To Find the Approximate Squaee Eoot of Decimals That Are Not Perfect Squares. Find the square root of .6. Regard the number as representing .6 of a 1-inch square. .6 of a 1-inch square = 60 tenth-inch squares. But 49 of the 60 tenth-inch squares can be arranged in a square that is 7 by 7 ; that is, 7 tenths of an inch by 7 tenths of an inch. After making this square there are (60 — 49) 11 tenth-inch squares remaining. If ad- ditions are to be made to the square, the 11 tenth-inch squares must be changed to hun- dredth-inch squares. In each tenth -inch square there are 100 hundredth-inch squares; in 11 tenth-inch squares there are 1100 hundredth-inch squares. From these, addi- tions are to be made upon two sides of the .7-inch square. ' .7 = 70 hundredths. The additions must be made upon a base line (70 X 2) 140 hundredth-inches long. These additions can be as many hundredth-inches wide as 140 is con- tained times in 1100. 1100 -^ 140 = 7+. The additions are 7 hundredth-inches wide. These will require 7 times 140, -f 7 times 7, = 1029 hundredth-inch squares. , After making this square (.77 by .77) there are (1100 - 1029) 71 hundredth-inch squares remaining. (If further additions are to be made to the square the 77 hundredth-inch squares must be changed to thousandth-inch squares.) The square root of .6, true to hun- dredths, is .77. Note. — The work on this page should be first presented orally by the teacher. It must be given very slowly. Great care must be taken that pupils image each magnitude when its word-symbol is spoken by the teacher. Any attempt to move forward more rapidly than this can be done by the slowest pupil will result in failure so far as that pupil is concerned. 354 COMPLETE ARITHMETIC. Square Root. Note. — Pupils who have mastered the work on the preceding page will readily understand the following process. See rule on page 352. 1. Find the square root of .6. Operation. '6000(.774 49 70 X 2 = 140 1100 7 1029 Observe — 1. That in grouping deci- mals for the purpose of ex- tracting the square root it is necessary to begin at the deci- mal point. 2. That the square root of any number of hundredths is a number of tenths; the square root of any number of ten-thousandths is a number of hun- dredths^ etc. 11^x2 = 1540 4 7100 6176 924 2. Find the square root of 54264.25. 54264.^25(232.946 4 20 X 2 = 40 U 3 11 t2 29 230 X 2 = 460 ] 2 L364 924 2320 X 2 = 4640 9 440.25 418.41 23290 X 2 = 4658( = 4658^ ) 21.8400 t 18.6336 232940 X 2 ?0 3.206400 6 2.795316 Observe that the trial divisor is al- ways 2 times 10 times the part of the root already found. .411084 PART II. 355 Square Root. 312. The following numbers are perfect squares. Find their square roots by both the factor method and the method given on the four preceding pages. (1) 6889 (2) 841 (3) 71824 (4) 1849 (5) 729 (6) 60516 (a) Find the sum of the six results. (7) AV (8) III (9) in (10) -V// (11) %V- (12) m (b) Find the sum of the six results. (13) .81 (14) .0625 (15) .04 (16) 1.21 (17) .7921 (18) .0004 (c) Find the sum of the six results. 313. Miscellaneous. 1. The square of a number represented by one digit gives a number represented by or digits. 2. The square of a number represented by two digits gives a number represented by or digits. 3. The square of a. number represented by three digits gives a number represented by — — or digits. 4. The square root of a perfect square represented by one or two digits is a number represented by • digit. 5. The square root of a perfect square represented by three or four digits is a number represented by = digits. 356 COMPLETE ARITHMETIC. Square Root. 314. Miscellaneous Problems. 1. What is one of the two equal factors of 9216 ? 2. What is one of the four equal factors of 20736 ? * 3. If 7921 soldiers were arranged in a soKd square, how many soldiers would there be on each side ? 4. How many rods of fence will enclose a square field whose area is 40 acres ? 5. How many rods long is one side of a square piece of land containing exactly one acre ? f 6. If the surface of a cubical block is 150 square inches, what is the length of one edge of the cube ? 7. How many rods of fence will enclose a square piece of land containing 4 acres 144 square rods ? 8. Find the side of a square equal in area to a rectangle that is 15 ft. by 60 ft. 9. Compare the amount of fence required to enclose two fields each containing 10 acres : one field is square, and the other is 50 rods long and rods wide. 10. Find the area of the largest possible rectangle having a perimeter of 40 feet. 11. If a square piece of land is ^ of a square mile, how much fence will be required to enclose it ? * To find one of the four equal factors of a number (the 4th root) extract the square root of the square root. Why ? What is the fourth root of 81 ? t Find the answer to problem 5, true to hundredths of a rpd. PART II. 357 Algebra. 315. Square Eoot and Area. 1. If a piece of land containing 768 square rods is three times as long as it is wide, how wide is it ? * Let X — the width, then 3 ^ = the length, and X (3 x) or %^ — the area. 3 ^' = 768 x^ = 256 X —\^ 2. If a certain room is twice as long as it is wide, and the area of the floor 968 square feet, what is the length and the breadth of the room ? 3. One half of the length of Mr. Smith's farm is equal to its breadth. The farm contains 80 acres. How many rods of fence will be required to enclose it ? 4. Each of four of the faces of a square prism is an oblong whose length is twice its breadth. The area of one of these oblongs is 72 square inches. What is the solid content of the prism. 5. The width of a certain field is to its length as 2 to 3. Its area is 600 square rods. The perimeter of the field is how many rods ? 6. If \ of the length of an oblong equals the width and its area is 768 square inches, what is the length of the oblong ? 7. If to 2i^ times the square of a number you add 15 the sum is 375. What is the number? *To solve this problem arithmetically, one must discern that this piece of land can be divided into three equal squares, the side of each square being equal to the width of the piece. 358 COMPLETE ARITHMETIC. Algebra. 316. Square Eoot and Proportion. When the same number forms the second and the third term of a proportion it is called a mean proportional, of the first and the fourth term ; thus, in the proportion 3 : 6 : : 6 : 12, 6 is a mean proportional of 3 and 12. Example. In the proportion 12 : .^ : : ^ : 75, find the value of x. Since the product of the means equals the product of the extremes, x times x equals 12 times 75, or, x' = 900. X = 30. Find the value of x in each of the following proportions : 1. 9:x::x:16. 4. 12:^::^:48. 2. 16:x::x:25. 5. 5 :^: :;z;: 125. * 3. 8:x::x:S2. 6. S6:x::x:4:9. (a) Find the sum of the six mean proportionals. 7. An estate was to be divided so that the ratio of A's part to B's would equal the ratio of B's part to C's. If A received $8000 and C received $18000, how much should B receive ? 8. Find the mean proportional of f and 1|. 9. The ratio of the areas of two squares is as 4 to 9. What is the ratio of their lengths ? 10. The area of the face of one cube is to the area of the face of another cube as 16 to 25. What is the ratio of the solid contents of the cubes ? PART II. 359 Geometry. Fig. 1. V\A W X> \ B 317. Eight-Triangles. 1. The longest side of a right-triangle is the hypothenuse. Either of the other sides may be re- garded as the base, and the remaining side as the perpendicular. 2. Convince yourself by examination of the figures here given, and by careful measurements and paper cutting, that the square of the hypothenuse of a right-triangle is equiva- lent to the sum of the squares of the other two sides. Figures 2 and 3 are equal squares. If from figure 2, the four right-triangles, 1, 2, 3, 4, be taken, H, the square of the hypothenuse, re- mains. If from figure 3, the four right-tri- angles (equal to the four right-triangles in figure 2) be taken, B, the square of the base, and P, the square of the perpendicular, re- main. When equals are taken from equals the remainders are equal, therefore the square, H, equals the sum of the squares B and P. Fig. 2. Fig. 3. 2.^^ P B 3/ 3. To find the hypothenuse of a right-triangle when the base and perpendicular are given : Square the base ; square the perpendicular ; extract the square root of the sum of these squares. 360 COMPLETE ARITHMETIC. 318. Miscellaneous Review. 1. Find approximately the diagonal of a square whose side is 20 feet.* 2. Find approximately the distance diagonally across a rectangular floor, the length of the floor being 30 feet and its breadth 20 feet. 3. How long a ladder is required to reach to a window 25 feet high if the foot of the ladder is 6 feet from the building and the ground about the building level ? 4. If the length of a rectangle is a, and its breadth h, what is the diagonal ? 5. The base of a right triangle is 40 rods and its perpen- dicular, 60 rods, (a) What is its hypothenuse ? (b) What is its area ? (c) What is its perimeter ? 6. The area of a certain square piece of land is 2^ acres, (a) Find (in ¥ods) its side, (b) Find its perimeter, (c) Find its diagonal, true to tenths of a rod. 7. The length of a rectangular piece of land is to its breadth as 4 to 3. Its area is 30 acres, (a) Find its breadth, (b) Find its perimeter, (c) Find the distance diagonally across it. 8. A certain piece of land is in the shape of a right-tri- angle. Its base is to its altitude as 3 to 4. Its area is 96 square rods, (a) Find the base, (b) Find the altitude, (c) Find the perimeter. 9. Find one of the two equal factors of 93025. *From the study of right-triangles on page 359 it may be learned that the diagonal of a square is equal to the square root of twice the square of its side. METEIC SYSTEM. Note. — The teacher should present this subject orally before attempting the pages that follow. It will be well if this oral work can be commenced many weeks before this page is reached in the regular work of the class. A meter stick, a liter measure, and metric weights should be provided and each pupil should weigh and meas- ure until he can easily think quantity in the units of this system without reference to the units of any other system. 319. All units in the metric system of measures and weights are derived from the primary unit known as the meter. When the length of the primary unit of this system was deter- mined it was supposed to be one ten-millionth of the distance from the equator to the pole. A pendulum that vibrates seconds is nearly one meter long. In the names of the derived units of this system the prefix deka means 10; hekto means 100; kilo means 1000; myria means 10000; deci means tenth ; centi means hundredth ; milli means thousandth. 320. Linear Measure. 10 millimeters (mm.) = 1 centimeter (cm.).* 10 centimeters = 1 decimeter (dm.). 10 decimeters = 1 meter (m.). 10 meters — 1 dekameter (Dm.). 10 dekameters = 1 hektometer (Hm.). 10 hektometers = 1 kilometer (Km.). 10 kilometers = 1 myriameter (Mm.). * In the common pronunciation of tliese words the primary accent is on the first syllable and a secondary accent on the penultimate syllable ; thus, cen'tim6ter. In the better pronunciation the accent is on the vowel preceding the letter m, that is, on the antepenultimate syllable ; thus, centim'eter, dekam'eter, etc. 362 . COMPLETE ARITHMETIC. Metric System. 321. The names of the units of surface measurement are the same as those used for hnear measurement, combined with the word square; thus, a surface equivalent to a square whose side is a meter is 1 square meter. The pupil, if properly taught to this point, will be able, without difficulty, to fill the blanks iu the table of — Squaee Measure. 100 square millimeters (sq. mm.) = 1 square centimeter (sq. cm.). square centimeters = 1 square decimeter (sq dm.). square decimeters = 1 square meter (sq. m.). square meters = 1 square dekameter (sq. Dm.). square dekameters = 1 square hektometer (sq. Hm.). square hektometers , = 1 square kilometer (sq. Km.). square kilometers = 1 square myriameter (sq. Mm.). Note. — The special unit of surface measure for measuring land is equivalent to a square whose side is ten meters. Thigi unit is called an ar. 100 centars (ca.) = 1 ar (a.). 100 ars =1 hektar (Ha.). Exercise. 1. In a square decimeter there are sq. cm. 2. In 2 square decimeters there are sq. cm. 3. In a 2 -decimeter square there are sq. cm. 4. In a square meter there are sq. dm. 5. In 2 square meters there are sq. dm. 6. In a 2 -meter square there are sq. dm. 7. In a square meter there are sq. cm. 8. In 2 square meters there are sq. cm. 9. In a 2 -meter square there are sq. cm. PART II. 363 Metric System. 322. The names of the units of volume measurement are the same as those used for linear measurement, combined with the word cuMc; thus, a volume equivalent to a cube whose edge is a meter is 1 cubic meter. The pupil should be able easily to fill the blanks in the table of — Volume Measure. 1000 cubic milhmeters (cu. mm.) = 1 cubic centimeter (cu. cm.).* cubic centimeters = 1 cubic decimeter (cu. dm.). —— cubic decimeters = 1 cubic meter (cu. m.). cubic meters = 1 cubic dekameter (cu. Dm.). cubic dekameters = 1 cubic hektometer (cu. Hm.). Note 1. — The special unit of capacity for measuring liquids, grain, small fruits, etc., is the liter. It is equal to 1 cubic decimeter. 10 liters (1.) = 1 dekaliter (Dl.), and 1 tenth of a liter = 1 deciliter (dl.), etc. Note 2. — The special unit for measuring wood is the ster. It is equal to 1 cubic meter. Exercise. 1. A cubic meter equals liters. 2. A cubic meter equals cubic decimeters. 3. A cubic meter equals cubic centimeters. 4. A cubic decimeter equals cubic centimeters. 5. Two cubic decimeters equal cubic centimeters. 6. A 2-decimeter cube equals cubic centimeters. 7. A 5-centimeter square equals square centimeters. 8. A 5 -centimeter cube equals cubic centimeters. 9. One tenth of a liter equals cubic centimeters. 10. One deciliter equals cubic centimeters. * The abbreviation cc. is often used for cubic centimeter. 364 COMPLETE ARITHMETIC. Metric System. 323. The primary unit of weight is the gram. This equals the weight of one cubic centimeter of pure water. Weight. 10 miUigrams (mg.) = 1 centigram (eg.). 10 centigrams = 1 decigram (dg.). 10 decigrams = 1 gram (g.). 10 grams = 1 dekagram (Dg.). 10 dekagrams = 1 hektogram (Hg.). 10 hektograms = 1 kilogram (Kg.). Note. — The special miit for the weight of very heavy articles is the tonneau. It equals the weight of a cubic meter of pm'e water, or 1000 kilograms. Exercise. 1. The weight of 1 liter of water is grams. 2. The weight of 6 cubic centimeters of water is . 3. The weight of a cubic decimeter of water is . 4. One kilogram of water equals cubic centimeters. 5. One hektogram of water equals cc. 6. One dekagram of water equals cubic centimeters. 7. If the specific gravity of iron is 7.5, what is the weight of a cubic centimeter of iron ? 8. If the specific gravity of cork is ^, what is the weight of a cubic decimeter of cork ? 9. If the specific gravity of oil is .9, what is the weight of a liter of oil ? 10. What is the weight of a cubic meter of stone whose specific gravity is 2.5 ? 11. What is the weight of a cubic decimeter of wood whose specific gravity is .8 ? PART II. 365 Metric System.' 324. Miscellaneous Problems. 1. Find the area of a rectangular surface that is 1 meter long and 6 decimeters wide. Make a diagram of this surface upon the blackboard. 2. Find the area of a rectangular surface that is 2 deci- meters long and 5 centimeters wide. Make a diagram of this surface on your slate or paper. 3. Find the solid content of a 5-centimeter cube. A 5- centimeter cube is what part of a cubic decimeter ? 4. Find the solid content of a 4-decimeter cube. A 4- decimeter cube is what part of a cubic meter ? 5. Find the entire surface of a 4-centimeter cube. The surface of a 4 centimeter cube is what part of a square deci- meter ? 6. Find the area of a rectangular surface that is 2.4 yards by 5 yards ; of a rectangular surface that is 2.4 meters by 5 meters. 7. Which is the larger of the two surfaces described in problem 6 ? 8. Find the area of a rectangular surface that is 3.5 yards by 2.5 yards ; of a rectangular surface that is 3.5 meters by 2.5 meters. 9. Find the volume of a rectangular solid that is 3.4 feet by 3 feet by 2 feet ; of a rectangular solid that is 3.4 meters by 3 meters by 2 meters. 10. Find the volume of a rectangular solid that is 3.5 meters by 2.3 meters by 4.6 meters. 11. What is the weight of a cubic decimeter of wood whose specific gravity is .5 ? 366 COMPLETE ARITHMETIC. • Metric System. 325. Miscellaneous Problems. 1. Estimate in meters the width of the lot upon which the school building stands. Measure it. 2. Estimate in centimeters the width of your desk. Measure it. 3. Estimate in square centimeters the area of a sheet of paper. Measure and compute. 4. Estimate in square meters the area of the blackboard. Measure and compute. 5. Estimate the number of cubic meters of air in the school room. Measure and compute. 6. Estimate in grams the weight of a teaspoonful of water. Weigh it.* 7. Estimate in kilograms your own weight. 8. Estimate in liters the capacity of a water pail. 9. Estimate in kilograms the weight of a gallon of water. 10. Estimate in kilometers the distance from the school- house to your home. 326. Table of Equivalents. Meter a little more than 1 yard . . . 39.37 inches. Kilometer .... nearly | of a mile 3280.8-f- feet. Decimeter .... nearly 4 inches 3.937 inches. Ar nearly ^ of an acre ..... 3.954 sq. rd. Ster a little more than I cord . . . 35.3-|- cu. ft. Liter a little more than 1 liquid quart, 1.056 -f qt. Gram nearly 15^ grains 15.4+ grains. Kilogram nearly 2 J pounds 2.204-|- lb. *Every school should be provided with scales, weights, and measures. PART II. 367 Algebra. 327. Metric Units in Algebraic Problems. 1. I am thinking of a rectangular surface. Its length is 5 times its breadth. Its area is 45 square decimeters. How long and how wide is the surface ? * 2. I am thinking of a triangular surface. Its base is three times its altitude. Its area is 8.64 square meters. What is the length of its base ? 3. I am thinking of a cube whose entire surface is 150 square centimeters. What is the length of one of its edges ? 4. The perimeter of a certain rectangle is 20.4 meters. Its length is twice its breadth, (a) Find its length and breadth, (b) Find its area. 5. The difference in the weight of two lead balls is 24 grams. The united weight of the two balls is 1 kilogram, (a) Find the weight of each ball, (b) Does the heavier ball weigh more or less than 1 pound ? 6. A merchant had three pieces of lace. In the second piece there were twice as many meters as in the first. In the third piece there were 6 meters more than in the second. In the three pieces there were 106 meters, (a) How many meters in each piece ? (b) Were there more or less than 53 yards in the second piece ? 7. John weighs 3.6 kilograms more than Henry. To- gether they weigh 83.6 kilograms, (a) Find the weight of each boy. (b) Does John weigh more or less than 90 pounds ? * Let X = the number of decimeters in the breadth of the surface. 368 COMPLETE ARITHMETIC. Algebra. 328. Metric Units in Algebraic Problems. 1. A ball rolling down a perfectly smooth and uniformly inclined plane rolls 3 times as far the 2nd second as the 1st; 5 times as far the 3rd second as the 1st; 7 times as far the 4th second as the first. If in 4 seconds it rolls 192 deci- meters (a) how far did it roll in the 1st second ? (b) In the 4th second ? (c) Did it roll more or less than 48 inches in the first second ? 2. I am thinking of a right-triangle. Its altitude is to its base as 3 to 4. The sum of its altitude and base is 14 centi- meters, (a) Find the altitude, (b) Find the base, (c) Find the area, (d) Find the hypothenuse. (e) Is the hypothenuse more or less than 4 inches ? 3. A freely falling* body falls three times as far the 2nd second of its fall as it does the 1st second. In two seconds it falls 19.6 meters, (a) How far does it fall in the 1st second? (b) In the 2nd second? 4. A freely falling body falls 3 times as far the 2nd minute of its fall as it does the 1st minute. In two minutes it falls 70560 meters, (a) How far does it fall in the 1st minute ? (b) In the 2nd minute? (c) 70560 meters equals how many kilometers? (d) 70560 meters equals (approximately) ho\i» many miles? 5. A freely falling body falls 3 times as far the 2nd half- second as it does the 1st half-second. In one second it falls 4.9 meters, (a) How far does it fall in the 1st half-second ? (b) In the 2nd half -second ? * A freely falling body is a body falling in a perfect vacuum. PART II. 369 Geometry. 329. The Circumference of a Circle. 1. Cut a 3-iiich circle from cardboard. By rolling it upon a foot rule, measure its circumference. 2. Measure the diameter of a bicycle wheel; then by rolHng it upon the ground or upon the school-room floor, measure its circumference. 3. In a similar manner measure the diameters and the circumferences of other wheels until you are convinced that the circumference of a circle is a little more than times its diameter. 4. The circumference of a circle is nearly 3^ times the diameter; more accurately, it is 3.141592+ times the diameter. I^OTE. — It is a curious fact that the diameter of a circle being given in numbers, it is impossible to express in numbers its exact circumference. The circumference being given in numbers, it is impossible to express in numbers its exact diameter. In other words, the exact ratio of the circumference to the diameter is not expressible. 5. Find the approximate circumference of a 5-inch circle ; of a 7-inch circle; of a 10-inch circle.* 6. Find the approximate diameter of a circle that is 6 ft. in circumference.* 7. The circumference of a 6-inch circle is how many times the circumference of a 3-inch circle ? 8. The diameter of a circle whose circumference is 12 inches is what part of the diameter of a circle whose circum- ference is 24 inches ? * In the solution of such problems as these, the pupil may use, as the approxi- mate ratio of the circumference to the diameter, 3.14. CONTENTS— PART III, Denominate Numbers, Linear Measure, Surface Measure, Volume Measure, Capacity, - - - - Weight, _ . . . Time, . . . .. Circular Measure, - Longitude and Time, Value Measure, Short Methods, Multiplication, Division, - - - - Cancellation, Miscellaneous, - Practical Approximations, Miscellaneous Examination Problems, Explanatory Notes, Pages 371-400 371,372 373-380 381-390 391, 392 393-395 396,397 398 399 400 401-414 401-409 410 411, 412 412-414 415-420 421-442 443-446 370 PART III. DENOMINATE NUMBEKS. Linear Measure. Note. — Pupils who have mastered the Elementary Book and the preceding pages of this book have had much practice in the use of denominate numbers. In part to provide for ready reference and in part to give further application of the principles already presented, the subject is here treated as a whole. 331. The English and United States standard unit of length is the Imperial yard arbitrarily fixed by Act of Parliament and afterward adopted in the United States. It is about 14x1^1" ^^ ^^® length of a pendulum that vibrates once a second at the level of the sea in the latitude of London. It is ff f f of a meter. Table. 12 inches (in.) = 1 foot (ft.). 3 feet = 1 yard (yd.). 5 1 yards = 1 rod (rd.). 16^ feet = 1 rod. 320 rods = 1 mile (mi.). 1760 yards = 1 mile. 5280 feet = 1 mile. 1 fathom (used in measuring the depth of the sea) = 6 feet. 1 knot (used in navigation) = 1.15+ miles. 1 league (used in navigation) = 3 knots. 1 hand (used in measmdng the heights of horses) = 4 inches. 1 chain (used by civil engineers) = 100 feet. 1 chain (used by land surveyors) = 66 feet. 1 pace (used in measuring approximately) = ^ of a rod. 1 barleycorn (used in grading length of shoes) = J of an inch. 1 furlong (a term nearly obsolete) = ^ of a mile. 371 372 COMPLETE ARITHMETIC. Denominate Numbers— Linear Measure. Exercise. 1. Mont Blanc is 15810 feet, or about miles high. 2. Mt. Everest is 29000 feet, or about miles high. 3. Commodore Dewey opened fire on the enemy at a distance of 5000 yards, or about miles. 4. My horse, measured over the front feet, is 16^ hands, or feet inches high. 5. The vessel seemed to be about three leagues, or miles distant. 6. On sounding, they found the depth of the water to be 15 fathoms, or feet. 7. The cruiser made 20 knots, or about miles, an hour. 8. The length of the lot was 36 paces, or about rods. 9. 10000 feet is nearly miles. 10. 15000 feet is nearly miles. 11. 1000 yards is about of a mile. 12. 100 feet is rods foot. 13. 200 feet is - — rods feet. 14. 300 feet is rods feet. 15. A kilometer is about rods. 16. A Civil Engineer's chain is rods foot. Problems. 1. A seven-foot drive wheel of a locomotive makes how many revolutions to the mile ? 2. Which is the longest distance, 5 miles 319 rods 16 feet 6 inches, 5 miles 319 rods 5 yards 1 foot 6 inches, or 6 miles ? 3. Eeduce 40 rd. 4 ft. 5 in. to inches. PART III. 373 Denominate Numbers — Surface Measure. 332. The standard unit of surface measure is a square yard which is the equivalent of a 1-yard square. This unit, like the square foot, square inch, square rod, and square mile, is derived from the corresponding unit of linear measure. Table. 144 square inches (sq. in.) = 1 square foot (sq. ft.). 9 square feet = 1 square yard (sq. yd.). 30i square yards = 1 square rod (sq. rd.). 272 1 square feet = 1 square rod. 160 square rods = 1 acre (A.). 4840 square yards = 1 acre. 43560 square feet = 1 acre. 640 acres = 1 square mile (sq. mi.). Exercise. 1. Show by a drawing that there are 144 square inches in a 1-foot square. 2. Show by a drawing that there are 9 square feet in a 1-yard square. 3. Show by a drawing that there are 30^ square yards in a 1-rod square. 4. Estimate the number of square yards of blackboard in the room ; the number of square feet of blackboard. 5. Estimate the number of square feet in the floor of the schoolroom ; the number of square yards. 6. Estimate the square yards of plastering on the walls of the schoolroom. 7. Estimate the number of square rods in the schoolhouse lot. Is the lot more or less than ^ of an acre ? 374 COMPLETE ARITHMETIC. Denominate Numbers— Surface Measure. 333. In the measurement of land it is more convenient to use a decimal scale ; hence the invention of the Gunter Chain. This chain is 4 rods long and is divided into 100 links. Observe that links are hundredths of chains. Observe that square chains are tenths of acres. 1. Land, 3 chains by 4 chains contains acres. 2. Land, 5 chains by 4 chains contains acres. 3. Land, 3 chains by 8 chains contains acres. 4. Land, 5 chains by 7 chains contains acres. 5. Land, 8 chains by 6 chains contains acres. 6. Two chains 35 links equals chains. 7. Two chains 75 links equals chains. 8. Two chains 5 links equals chains. 9. Two chains 9 links equals chains. 10. Land, 4 ch. by 4.50 ch. contains acres. 11. Land, 5 ch. by 3.20 ch. contains acres. 12. Make a rule and find the number of acres in each of the following : (1) Land, 12 chains 35 links by 9 chains 50 links. (2) Land, 21 chains 8 links by 12 chains 30 links. (3) Land, 32 chains 25 links by 15 chains 6 links. (a) Find the sum of the area of the ten pieces of land described on this"^age. To THE Teacher. — A rod is exactly 25 hnks. A foot is about \\ links. Hence rods and feet can be easily changed to chains and links by regarding each 4 rods as 1 chain and each additional rod as 25 links and each additional foot as 1^ links. The error in any one measurement never exceeds 2 inches. 9 rd. 12 ft. = 2 chains 43 (25 + 18) links. PART III. 375 Denominate Numbers— Surface Measure. 334. To determine the amount of carpet necessary for a given room several minor problems must be solved which can be best studied by means of an — Example. 1. How many yards of carpet must be purchased for a room 1 6 ft. by 2 ft. if the carpet is 1 yd. wide ? (1) How many breadths will be necessary if the carpet is put down lengthwise of the room ? How much must be cut off or turned under from one breadth in this case ? (2) How many breadths will be necessary if the carpet is put down crosswise of the room? How much must be cut off or turned under from one breadth in this case ? (3) Make two diagrams of the room on a scale of 1 inch to the foot and show the breadths of carpet in each case. (4) How many yards must be purchased in each case ? (5) If in the first case there is no waste in matching the figure and in the second case there is a waste of 8 inches on each breadth exce;pt the first, which plan of putting down the carpet will require the greater number of yards ? (6) If the carpet costs 90^ a yard and the conditions are as stated in No. 5, what is the cost of the carpet in each case ? 2. How many yards of carpet must be purchased for a room 16 ft. by 20 ft. if the carpet is f of a yard wide and there is no waste in matching the figure ? 3. How many yards of carpet must be purchased for a room that is 15* ft. 6 in. by 16 ft. 4 in. if the carpet is f of a yard wide, is put down lengthwise of the room, and there is no waste in matching the figure ? 376 COMPLETE ARITHMETIC. Denominate Numbers. 335. Plastering and Papering. 1. How many square yards of plastering in a room (walls and ceiling) that is 15 ft. by 18 ft. and 12 ft. high, an allow- ance of 1 2 square yards being made for openings ? Note. — In estimating the cost of plastering, allowance is made for " openings " (windows and doors) only when they are very large in proportion to the wall to be covered. Why are plasterers unwilling to deduct the entire area of all the openings ? 2. At 24^ a square yard how much will it cost to plaster a room that is 17 ft. by 20 ft. and 10 feet from the floor to the ceiling, deducting 1 6 square yards for openings ? 3. How many "double rolls" of paper will be required for the walls of a room that is 14 ft. by 16 ft. and 11 ft. high above the baseboards, if an allowance of 1 full "double roll" is made for openings ? Note. — Wall paper is usually 18 inches wide. A "single roll" is 24 ft. long. A " double roll " is 48 ft. long. In papering a room 11 ft. high it would be safe to count on 4 full strips from each " double roll." The remnant would be valueless unless it could be used over windows or doors. Since each strip is 18 inches wide, a " double roll " will cover 72 inches (6 ft.) of wall measured hori- zontally. 4. At 12^ a "single roll," how much will the paper cost for the walls of a room that is 12 ft. by 14 ft. and 7 ft. above the baseboards, if the area of the openings is equivalent to the surface of 2 "single rolls" of paper ? 5. Find the cost, at 25^ a square yard, of plastering the walls of a room that is 48 ft. by 60 ft. and 18 feet high, deducting 30 square yards for openings. PART III. 377 Denominate Numbers. 336. Faem Pkoblems. Find how many acres in — 1. A piece of land 1 rod by 160 rods. 2. A piece of land 7 rods by 160 rods. 3. A piece of land 13 rods by 160 rods. 4. A piece of land 22 feet by 160 rods. 5. A piece of land 8^ yards by 160 rods, (a) Find the sum of the five results. 6. A piece of land 8 rods by 80 rods. 7. A piece of land 17 rods by 80 rods. 8. A piece of land 37^ rods by 80 rods. 9. A piece of land 618f feet by 80 rods. 10. A piece of land 550 yards by 80 rods. (b) Find the sum of the five results. 11. A piece of land 12 rods by 40 rods. 12. A piece of land 27 rods by 40 rods. 13. A piece of land 46 rods by 20 rods. 14. A piece of land 36 rods by 20 rods. 15. A piece of land 264 feet by 20 rods. (c) Find the sum of the five results. 16. A piece of land 1 rod by 1 mile. 17. A piece of land 11 rods by 1 mile. 18. A piece of land 66 feet by 1 mile. 19. A piece of land 99 yards by 1 mile. 20. A piece of land 198 feet by ^ of a mile. (d) Find the sum of the five results. 21. A piece of land |^ of a mile long and as wide as the schoolroom. 378 COMPLETE ARITHMETIC. Denominate Numbers. 337. Faem Problems. 1. A piece of land 1 foot wide and 43560 feet long is how many acres ? 2. Change 43560 feet to miles. 3. A piece of land 1 foot wide must be how many miles in length to contain 1 acre ? 4. Some country roads are 66 feet wide. How many acres in 8^ miles of such road ? 5. How many acres in 1 mile of road that is 4 rods wide ? 6. A farmer walking behind a plow that makes a furrow 1 foot wide will travel how far in plowing 1 acre ? 7. A farmer walking behind a plow that makes a furrow 16 inches wide will travel how far in plowing 1 acre ? 8. If a mowing machine cuts a swath that averages 4 feet in width, how far does it move in cutting 1 acre ? 9. If potatoes are planted in rows that are 3 feet apart, (a) how many miles of row to each acre ? (b) How many rods of row to each acre ? (c) If 4 rods of row on the average yield 1 bushel, what is the yield per acre ? 10. Strawberry plants are set in rows that are 2 feet apart, (a) How many miles of row to the acre ? (b) How many rods of row to the acre ? (c) How many feet of row to the acre ? 11. If corn is planted in rows 3^ feet apart and if the " hills " are 3|- feet apart in the row, how many hills to each acre ? PART III. 379 Geometry. 338. To Find the Area of a Circle. 1. Cut one half of a circular piece of paper as indicated in the diagram. Observe that if the circle is cut into a very large number of parts and opened as shown in the figure, the circumference of the circle becomes, practically, a straight line. Note. — Imagine the circle cut into an infinite number of parts and thus opened and the circumference to be a straight line. Observe that a circle may be regarded as made up of an infinite number of triangles whose united bases equal the circumference and whose altitude equals the radius. Hence to find the area of a circle we have the following : EuLE I. Multiply the circumference hy ^ of the diameter. 2. It has already been stated that if the diameter of a circle is 1, its circumference is 3.141592. Hence the area of a circle whose diameter is 1 is (3.141592 x |) .785398. 3. A circle whose diameter is 2, is 4 times as large as a circle whose diameter is 1 ; a circle whose diameter is 3, 9 times as large, etc. Hence to find the area of a circle we have also the following : EuLE II. Multiply the square of the diameter hy .785398. 4. The approximate area may be found by taking f (or .78) of the square of the diameter. (See Note 9, p. 445.) 380 X)OMPLETE ARITHMETIC. 339. Miscellaneous Problems. 1. Find the approximate area of a circle whose diameter is 20 feet. 2. What is the area of a circle whose diameter is 1 foot ? 1 yard ? 1 rod ? 1 mile ? 3. What is the area of a circle whose diameter is 2 feet ? 2 yards ? 2 rods ? 2 miles ? 4. A horse is so fastened with a rope halter that he can feed over a circle forty ieet in diameter. Does he feed over more or less than 5 square rods ? 5. Find the approximate length (in rods) of the side of a square containing 1 acre. 6. Find the approximate diameter (in rods) of a circle whose circumference is one mile. 7. Find the approximate area of the circle described in problem 6. 8. Find the approximate circumference of a circle whose diameter is 30 rods. 9. The expression "a bicycle geared to 68" means that the machine is so geared that it will move forward at each revolution of the pedal shaft as far as a 68-inch wheel would move forward at one revolution. How far does a bicycle "geared to 68" move forward at each revolution of the pedal shaft ? A bicycle "geared to 70" ? 10. What is the approximate circumference of the largest circle that can be drawn on the floor of a room 40 ft. by 40 ft. if at its nearest points the circumference is 2 feet from the edge of the floor ? DENOMINATE NUMBERS. Volume Measure. 340. The standard unit of volume measure is a cubic yard, which is the equivalent of a 1-yard cube. This unit, like the cubic foot and the cubic inch, is derived from the corresponding unit of linear measure. Cubic Measure. 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.). 27 cubic feet = 1 cubic yard (cu. yd.). Exercise. 1. Show by a drawing that there are 27 cu. ft. in a 1-yaid cube. 2. How many cubic inches in 1 half of a cubic foot ? 3. How many cubic inches in a ^-foot cube ? 4. How many cubic feet in 1 third of a cubic yard ? 5. How many cubic feet in a ^-yard cube ? 6. Estimate in cubic feet the amount of air in the school- room. 7. Estimate in cubic yards the amount of air in the schoolroom. 8. Estimate in cubic inches the capacity of your dinner box. 9. Estimate in cubic feet the capacity of some wagon box. 10. Estimate in cubic inches the volume of the school globe.* * A globe is a little more than ^ of the smallest cube from which it could have been made. See note 10, p. 445. 381 382 COMPLETE ARITHMETIC. Denominate Numbers— Volume Measure. 341. Wood is usually measured by the cord. A cord is a pile 4 feet wide, 4 feet high, and 8 feet long, or its equiva- lent. Hence — 128 cubic feet = 1 cord. Problems. 1. Estimate the number of cords of wood that could be put upon the floor of the school room if the desks were re- moved and the wood piled to the depth of four feet. 2. If 4-foot wood is piled 6 feet high what must be the length of the pile to contain 100 cords? 3. How many cords of wood in a pile 8 feet wide, 8 feet high, and 1 6 feet long ? 4. Compare the amount of wood in the pile described in problem 3, with the amount in a pile one half as wide, one half as high, and one half as long. 5. If I pay $1.10 a cord for sawing wood, cutting each 4-foot stick into 3 pieces, how much ought I to pay for cut- ting each 4-foot stick into 4 pieces ? 6. A pile of wood 4 ft. high, 4 ft. wide, and 192 ft. long contains cords. How many cords in a pile 4 feet high, 192 feet long,^ and 46 inches wide ? 7. A pile of wood is as wide as it is high and 32 feet long. It contains 9 cords. What is the width and height of the pile ? 8. How many cords of 4-foot wood can be piled in a cellar that is 24 feet wide and 32 feet long, provided the pile is 4 feet high and one end of each 4-foot stick touches a wall of the cellar ? PART III. 383 Denominate Numbers— Volume Measure. 342. Rough Stone is usually measured by the cord. A pile 4 feet high, 4 feet wide, and 8 feet long or its equiv- alent, is 1 cord. Note. — One cord of good stone is sufficient for about 100 cubic feet of wall. Hence in estimates it is customary to use the number 100 instead of 128 ; thai is, as many cords of stone will be required for a given wall as 100 cubic feet is contained times in the number of cubic feet in the wall. Problems. 1. Estimate the number of cords of stone necessary for a cellar wall 18 inches thick, the inside dimensions of the cellar being 15 feet by 18 feet and 7 feet deep, no allowance being made for openings in the wall. 2. What are the outside dimensions of the wall of the cellar described in problem 1 ? 3. What length of wall 7 feet high and 18 inches thick is equivalent, so far as amount of stone is concerned, to the cellar wall described in problem 1 ? 4. If ^ of the depth of the cellar described above is to be below the surface of the ground, how many cubic yards of earth must be excavated ? 5. How many per cent less of stone will be required for a 16-inch wall than for an 18-inch wall ? 6. Estimate the stone necessary for a wall 100 yards long, 1 1 feet high, and 2 feet thick. 7. If the specific gravity of stone is 2|^ and each cord is equivalent to 100 solid feet, how much does a cord of stone weigh ? 8. If the specific gravity of a certain stone is 2i, what is the weight of a block 8 feet by 2 feet by 2 feet ? 384 COMPLETE ARITHMETIC. Denominate Numbers — Volume Measure. 343. An ordinary brick is 2 in. by 4 in. by 8 in. and weighs about 4 pounds. Problems. 1. How many bricks are equivalent to 1 cubic foot ? Note. — When bricks are laid in mortar in the usual way, about 22 bricks are required to make a cubic foot of wall. 2. Estimate the number of bricks necessary for a cellar wall 12 inches thick, the inside dimensions of the cellar being 15 feet by 18 feet, and 7 feet deep, no allowance being made for openings in the wall ? 3. What are the outside dimensions of the wall of the cellar described in problem 2 ? 4. What length of wall 7 feet high and 12 inches thick is equivalent, so far as the number of bricks required is concerned, to the cellar wall described in problem 2 ? 5. If f of the depth of the cellar described above is to be below the surface, how many cubic yards of earth must be excavated ? 6. Estimate the number of bricks necessary for a wall 100 yards long, 11 feet high, and 1 foot thick. 7. If a brick is exactly 2 in. by 4 in. by 8 in. and weighs exactly 4^ lbs. what is its specific gravity ? (Note i6, p. 446.) 8. Find the approximate weight (in tons) of a pile of bricks as long as your school-room, 2 feet wide, and 4 feet high. 9. Find the approximate weight of a chimney, outside dimensions, 16 in. by 16 in., and 20 ft. high, the flue being 8 in. by 8 in. PART III. 385 Denominate Numbers — Lumber. 344. A foot of lumber is a board 1 foot square and 1 inch thick or its equivalent. (Note ii, p. 445.) Note 1. — An exception to the foregoing is made in the measure- ment of boards less than 1 inch in thickness. A square foot of such boards is regarded as a foot of lumber, whatever the thickness. Exercise. Tell the number of feet of lumber in each of the following boards, the thickness in each case being one inch (or less) : 1 in. wide and 12 ft. long. 2 in. wide and 12 ft. long. 3 in. wide and 12 ft. long. 4 in. wide and 12 ft. long. 7 in. wide and 12 ft. long. 13 in. wide and 12 ft. long. 9 in. wide and 12 ft. long. 12 in. wide and 12 ft. long. (a) How many feet (of lumber) in the eight boards ? Problems. 1. How much lumber in 6, 12 -ft., 1-in. boards whose widths are 11 in., 13 in., 9 in., 10 in., 12 in., and 14 in.? 2. How much lumber in 5, 12-ft., |--in. boards whose widths are 10 in., 12 in., 12 in., 11 in., and 14 in.? 3. How much lumber in 7, 12-ft., |^-in. boards whose widths are 9 in., 8 in., 5 in., 7 in., 8 in., 6 in., and 9 in. ? 4. How much lumber in 8, 12-ft., 1-in. boards each of which is 12 inches wide ? 5. How much lumber in 54, 12-ft., 1-in. boards each of which is 6 inches wide ? (b) Find the sum of the five results. 386 COMPLETE ARITHMETIC. Denominate Numbers— Lumber. Problems. Note 2. — A 14-foot board contains ^ more lumber than a 12-ft. board of the same width and thickness. Hence to find the number of feet of lumber in 14-foot boards, find the number of feet in as many 12-foot boards and add to the result ^ of itself.* 1. How much lumber in 5, 14-ft., 1-in. boards whose widths are 11 in., 12 in., 12 in., 15 in., and 10 in. ? 2. How much lumber in a pile of 14-ft. boards whose united width is 8 feet 7 inches ? 3. How much lumber in 56, 14-ft. boards each of which is 6 inches wide ? f 4. How much lumber in 24, 14-ft. boards each of which is 12 inches wide ? (a) Find the sum of the four results. Problems. Note 3. — A 16-foot board contains | more lumber than a 12-ft. board of the same width and thickness. Make a rule for finding the number of feet of lumber in 16-foot boards. 1. How much lumber in 5, 16-ft., 1-in. boards whose widths are 12 in., 10 in., 14 in., 13 in., and 12 in. ? 2. How much lumber in a pile of 16-foot boards whose united width is 9 feet 8 inches ? 3. How much lumber in 48, 16-ft. boards each of which is 6 inches wide ? 4. How much lumber in 34, 16-ft. boards each of which is 12 inches wide ? (b) Find the sum of the four results. * Take the nearest integral number of feet. I How much lumber in one 14-foot board 6 inches wide? PART III. 387 Denominate Numbers— Lumber. Problems. Note 4. — A Ij-incli board contains i more lumber than a 1-inch board of the same width and length. A l|-inch board contains J more lumber than a 1-inch board of the same width and length. 1. How much lumber in 4, 12-foot, IJ-in. boards whose widths are 12 in., 13 in., 14 in., and 13 in. ? 2. How much lumber in 4, 16-foot, 1^-in. boards whose widths are 13 in., 16 in., 12 in., and 13 in. ? 3. How much lumber in 4, 18 -foot, l|^-in. boards, each of which is 12 inches wide? 4. How much lumber in 4, 16-ft., 1^-in. boards, each of which is 6 inches wide ? (a) Find the sum of the four results. Problems. Note 5. — A "2 by 4, 12" is a piece of lumber 2 in. thick, 4 in. wide, and 12 feet long. Find the number of feet of lumber in each of the follow- ing items : 1. 16 pieces 2x4, 12. 2. 18 pieces 4x4, 12. 3. 25 pieces 2x8, 12. 4. 30 pieces 2x6, 12. 5. 20 pieces 4 x 6, 12. (b) Find the sum of the five results. Observe that in a 12-foot piece of lumber there are as many feet as there are square inches in the cross-section. A piece of lumber 1 in. by 1 in. and 12 feet long is 1 foot of lumber; a piece 2 in. by 2 in. is 4 feet of lumber ; a piece 2 in by 3 in. is 6 feet of lumber, etc. 388 COMPLETE ARITHMETIC. Denominate Numbers— Lumber. Pkoblems. Note 6. — In the measurement of timbers of all sizes it is custom- ary to consider each piece as containing the integral number of feet nearest to the actual content. Thus, a piece of 2 x 4, 14, actually contains 9| feet, but in all lumber yards it is counted as 9 feet. A piece of 2 X 4, 16, actually contains 10| feet, but it is counted as 11 feet. Find the number of feet of lumber in each of the following items : 1. 16 pieces 2x4, 14. 2. 24 pieces 4x4, 14. 3. 32 pieces 2x8, 14. 4. 17 pieces 4 x 6, 14. 5. 15 pieces 8 x 8, 16. 6. 12 pieces 4 x 10, 16. 7. 14 pieces 8 x 12, 16. 8. 6 pieces 12 x 12, 24. (a) Find the sum of the eight results. Pkoblems. Note 7. — " Lumber at $15 per M," means that the lumber is sold at the rate of $15 per 1000 feet. Find the cost : 1. 26, 16-foot, 6-in. fence boards @ $15 per M. 2. 34, 14-foot, 12-in. stock boards 3. 20 pieces 2x4, 16, 4. 14 pieces 4 x 6, 18, 5. 25 pieces 4 x 6, 16, 6. 18 pieces 4x4, 14, (b) Find the sum of the six results. $18 per M. $16 per M. $16 per M. $15 per M. $15 per M. PART III. 389 Qeometry. 345. To Find the Solid Content of a cylinder or of a right prism.* Observe that in any cylinder or right prism the number of cubic units in one layer 1 unit high (as indicated in the diagrams) is equal to the number of square units in the area of the base. Thus, if there are 4i square units in the area of the base there are 4|^ cubic units in one layer. The content of the entire solid is as many times the cubic units in one layer, as the solid is linear units in height. Hence the rule as usually given: " Multiply the area of the base hy the altitude." To THE Teacher. — This rule must be carefully interpreted by the pupil. He must not be allowed the misconception that area multiplied by any number can give solid content, except through such interpretation as is suggested in the above observation. (Note 12, p. 446.) Problems. 1. Find the solid content of a square right prism whose base is 6 in. by 6 in., and whose altitude is 8 inches. 2. Find the approximate solid content of a cylinder 6 inches in diameter and 10 inches long. 3. Find the solid content of a triangular prism the area of whose base is 15 sq. in., and whose altitude is 11|- inches. 4. Find the solid content of an hexagonal prism the area of whose base is 18 inches, and whose altitude is 10|- inches- * A right prism is a solid wiiose bases, or ends, are similar, equal, and parallel plane polygons, and whose lateral faces are perpendicular to its bases. 390 COMPLETE ARITHMETIC. 346. Miscellaneous Problems. 1. Find the solid content of an octagonal right prism the area of whose base is 24 square inches, and whose altitude is 15 inches. 2. What is the solid content of a cylinder, or of any right prism, the area of whose base is 30 square inches, and whose altitude is 1 2 inches ? 3. How many cubic feet of earth must be removed to dig a well 6 feet in diameter and 20 feet deep ? * 4. Find the approximate number of feet of H-m. lumber required to make the lining of the sides of a cylindrical silo that is 20 feet in diameter and 30 feet deep. 5. Find the approximate number of cords of rough stone in a cylindrical pile that is 1 6 feet in diameter and six feet deep. 6. Find the approximate number of brick necessary for a solid cylindrical foundation that is 9 feet in diameter and 4 feet high. 7. If the average specific gravity of the brick and mortar used in the foundation described in problem 6, is 1.9, how much does the entire foundation weigh ? 8. Find the weight in kilograms of a column of water 1 decimeter square and 10 meters deep. 9. Find the weight in pounds of 1000 feet of white pine 1-inch boards, the specific gravity being .6. 10. Find the weight of a load (1 cubic yard) of wet sand, the specific gravity being exactly 2. * The exact number of cubic feet cannot be expressed in figures. An approxi- mation that will answer many practical purposes may be obtained by regarding the circle (base) as 3 of its circumscribed square. If an answer more nearly accurate is required use .78 instead of 2. DENOMINATE NUMBEES. Capacity. 347. The standard unit of capacity used in measuring liquids is a gallon. A gallon equals 231 cubic inches. Liquid Measure. 4 giUs (gi.) = 1 pint (pt.). 2 pints = 1 quart (qt.). 4 quarts = 1 gallon (gal.). 31^ gallons =: 1 barrel (bbl.). Observe that 1 cubic foot = nearly 7 1 gallons. Observe that 4.2 cubic feet = nearly 1 barrel. A kerosene barrel contains about 52 gallons. It equals nearly 7 cubic feet. Problems. 1. Find the capacity (approximate or exact), in gallons, of a rectangular tank 3 ft. by 4 ft. by 8. ft. 2. Find the approximate capacity, in gallons, of a cylin- drical tank 4 feet in diameter and 4 feet deep. 3. Find the approximate capacity, in barrels (31|- gal.), of a rectangular tank 2 ft. by 4 ft. by 12 ft. 4. Find the approximate capacity, in barrels (31^ gal.), of a cylindrical cistern 6 ft. in diameter and 6 ft. deep. 5. Find the approximate capacity, in barrels (31^ gal.), of a cylindrical cistern 12 ft. in diameter and 6 ft. deep. 6. Find the approximate capacity, in barrels (31|^ gal.), of a cylindrical cistern 12 ft. in diameter and 12 ft. deep. 391 392 COMPLETE ARITHMETIC. Denominate Numbers— Capacity. 348. The standard unit of capacity used in measuring grain, fruits, vegetables, lime, coal, etc., is a bushel. A bushel equals 2150.4 cubic inches. Note. — In measuring large fruits, vegetables, lime, and coal, the unit is the "heaped busliel." A heaped bushel equals about 11 " stricken bushels." Dry Measure. 2 pints (pt.) = 1 quart (qt.). 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.). A bushel is nearly IJ cubic feet. A " heaped bushel " is about 1 J cubic feet. A "dry gallon" (4 quarts dry measure) equals 268.8 cubic inches. Enough "ear corn" to make, when shelled, one bushel, occupies about 21 cubic feet. If the corn is inferior in quality it will occupy more space than this — sometimes 2^ cubic feet. Problems. 1. Find the capacity in bushels of a wheat bin 8 ft. by 8 ft. by 10 ft.* 2. Give the dimensions of the smallest bin in which 1000 bushels of oats may be stored. 3. In a bin 12 feet square, there is rye to the depth of 7|- feet. How many bushels ? 4. How many bushels of potatoes (without heaping the bin) may be stored in a bin that is 8 ft. by 4 ft. by 6 ft.? 5. If the com is of excellent quality, how many bushels of " shelled corn " may be expected from a crib of ear corn, 8 ft. by 10 ft. by 80 ft.? *For many practical purposes the approximate ratio (li) of the bushel to the cubic foot will give in such problems as these, results sufficiently, accurate. PART III. 393 Denominate Numbers— Weight. 349. The standard unit of weight in common use is a pound Avoirdupois. Avoirdupois Weight. 16 ounces (oz.) = 1 pound (lb.). 2000 pounds = 1 ton (T.). The abbreviation for 1 hundredweight (100 lb.) is cwt. Miscellaneous Weights. 1 gallon of water 1 gallon of milk 1 gallon of kerosene 1 cubic foot of water 1 bushel of wheat 1 bushel of beans 1 bushel of clover seed 1 bushel of potatoes 1 bushel of shelled corn 1 bushel of ear corn 1 bushel of rye 1 bushel of barley 1 bushel of oats 1 barrel of flour 1 barrel of beef or pork = about 8^ lb. = about 8.6 lb. = about 6^ lb. = 62^ lb. = 60 lb. = 60 lb. = 60 lb. = 60 lb. = 56 lb. = 70 lb.* = 56 lb. = 48 lb. = 32 lb. = 196 lb. = 200 lb. Problems. Find the cost — 1. Of 2650 lb. coal at S5.50 per ton. 2. Of 2650 lb. oats at 24^ a bushel. 3. Of 3330 lb. wheat at 80^ a bushel. 4. Of 4650 lb. potatoes at 42^ a busheL (a) Find the sum of the four results. *This means the amount of ear corn required to make 1 bushel of shelled corn. 394 COMPLETE ARITHMETIC. Denominate Numbers — Weight. Problems. Find the cost — 1. Of 2560 lb. hay at $7.50 per ton. 2. Of 1430 lb. straw at 30^ per cwt. 3. Of 2|- tons meal at f of a cent a pound. 4. Of 1|- tons corn husks at 1^ cents a pound. 5. Of 3420 lb. hay at $8.00 per ton. (a) Find the sum of the five results. Find the cost — 6. Of 2140 lb. oats at 24^ a bushel. 7. Of 2140 lb. corn at 2^ a bushel. 8. Of 2140 lb. wheat at 90^ a bushel. 9. Of 2140 lb. barley at 36^ a bushel. 10. Of 2140 lb. rye at 42^ a bushel. (b) Find the sum of the five results. Find the cost — 11. Of 520 lb. clover seed at $6.30 a bushel. 12. Of 520 lb. potatoes at 75^ a bushel. 13. Of 520 lb. beans at $2.15 a bushel. 14. Of 520 lb. corn at 20^ a bushel. 15. Of 520 lb. ear corn at 35^ a bushel. (c) Find the sum of the five results. Find the approximate weight — 16. Of a barrel of kerosene. 17. Of 1 quart of milk. 18. Of the oats that will fill a bin that is 4 ft. by 4 ft. by 9 ft. 19. Of the water that will fill a taxxk that is 2 ft. by 2 ft. by 12 ft. PART III. 395 Denominate Numbers— Weight. 350. Troy weight is used in weighing gold, silver, and jewels. Troy Weight. 24 grains (gr.) = 1 pennyweight (pwt.). 20 pennyweights = 1 ounce (oz.). 12 ounces = 1 pound (lb.). 351. Apothecaries' weight is used in mixing medicines and in selling them at retail. Apothecaries' Weight. 20 grains (gr.) = 1 scruple (3). 3 scruples = 1 dram ( 3 ). 8 drams = 1 ounce ( § ). 12 ounces = 1 pound (Bb). Note 1. — The pound Troy and the pound Apothecary are equal, each weighing 5760 grains. The pound Avoirdupois weighs 7000 Troy or Apothecary grains. Note 2. — The ounce Troy and the ounce Apothecary are each 480 grains; the ounce Avoirdupois is 437 1 grains. Query. — Which is heavier, a pound of feathers or a pound of gold ? An ounce of feathers or an ounce of gold ? Problems. 1. Change 5 lb. Avoirdupois weight to pounds, ounces, etc., Apothecaries' weight. 2. How many 5 gr. powders can be made from one Avoir- dupois ounce of quinine ? 3. One Avoirdupois ton of gold is how many Troy pounds ? 4. Twenty-four Troy pounds equal how many Avoirdupois pounds ? 396 COMPLETE ARITHMETIC. Denominate Numbers — Time. 352. The standard units in the measurement of time are the day and the year. Note 1. — The solar day is the interval between the time when the sun is on a given meridian and the time when it appears on that meridian again. These intervals (solar days) are not perfectly uni- form. The average of these intervals is the mean solar day — the day noted by our watches and our clocks— the day, one twenty-fourth of which is called an hour. Note 2.— The solar year is the time of one revolution of the earth around the sun, or nearly 365 i mean solar days. The calendar year of 365 days is nearly 6 hours less than the solar year. Four years of 365 days each, would lack nearly 24 hours (4 times J da.) of being equal to 4 solar years. Hence 1 day is added to 365 every fourth year, ^ith the exception noted below. Note 3. — The exact length of a solar year is 365 da. 5 hr. 48 min. 48 sec. Since this lacks 11 min. 12 sec. of being 365^ days, it fol- lows that if every fourth year should contain 366 days, in 400 years the calendar years would amount to 3 days (400 times 11 min. 12 sec.) more than the solar years. Hence three of the years that would otherwise contain 366 days, are made to contain 365 days. The years thus changed to 365 days are those ending the centuries (1800, 1900, 2000, 2100, etc.) unless the number denoting the year is exactly divisible by 400. Measure of Time. 60 seconds (sec.) = 1 minute (min.). 60 minutes = 1 hour (hr.). 24 hours — 1 day (da.). 7 days = 1 week (wk.). 365 days = 1 common year. 52 wk. 1 da. = 1 common year. 366 days — 1 leap year. 52 wk. 2 da. =1 leap year. 12 months = 1 year. PART III. 397 Denominate Numbers— Time. Exercises. 1. If Jan. 1 of a common year is Monday, Feb. 1 is ; March 1 is ; April 1 is ; May 1 is ; June 1 is ; July 1 is ; August 1 is ; September 1 is ; October 1 is ; November 1 is ; December 1 is . 2. Jan. 1, 1899, was Sunday. Tell the day of the week for each of the following dates : 1900, January 1 ; February 1 ; February 8. 1901, January 1 ; February 1 ; February 9. 1902, January 1 ; February 1 ; March 1. 1903, January 1 ; February 1 ; March 10. Problems. 1. How many days from Jan. 1, 1900, to Aug. 17, 19001 2. Jan. 1, 1900, falls on Monday. Upon what day of the week does Aug. 17, 1900, fall? 3. What month begins on the same day of the week as January in every common year ? 4. What months begin on the same day of the week as February in every common year ? 5. If January begins on Sunday, (a) how many Sundays in the month ? (b) How many Mondays ? (c) How many Tuesdays ? (d) How many Wednesdays ? 6. If a common year begins on Sunday, (a) how many Sundays in the year ? (b) How many Mondays ? 7. (a) How many days old are you? (b) How many weeks old? (c) Upon what day of the week were you born? 398 COMPLETE ARITHMETIC. Denominate Numbers— Circular Measure. 353. For the purpose of measurement, every circumference is supposed to be divided into 360 equal parts. Each of these parts is called an arc of 1 degree. 354. An angle is measured by regarding its vertex as the center of a circle, its sides being extended until they cut the circumference. The angle is measured by the arc lying between its sides. If the intercepted arc is an arc of 45 degrees, the angle is an angle of 45 degrees ; if the arc is 20 degrees, the angle is 20 degrees, etc. Circular and Angular Measure. 60 seconds (") =: 1 minute ('). 60 minutes = 1 degree (°). 360 degrees = 1 circumference. 355. In geography, a meridian is a north and south line on the surface of the earth, extending from pole to pole. 356. Longitude is distance east or west in degrees (or parts of degrees) from a given meridian. Longitude is usually measured from the meridian that passes through Greenwich. Problems. Find difference in longitude between — 1. Eome, 12° 27' east, and Washington, 70° 2' 48'' west. 2. Washington and Chicago, 87° 37' 30" west. 3. Chicago and Denver, 104° 59' 23" west. 4. Denver and San Francisco, 122° 24' 15" west. 5. San Francisco and Berlin, 13° 23' 53" east. 6. Washington and Honolulu, 157° 50' 36" west. 7. Washington and Manila, 121° east. PART III. 399 Denominate Numbers— Longitude and Time. 357. One degree of longitude corresponds to 4 minutes of time. Explanatory — The sun seems to move over 360 degrees of longitude in 24 hours ; over 1 degree in ^^-^ of 24 hours = 4 minutes. Problems.* 1. When it is noon at Greenwich, what is the time at Washington, 77° 2' 48''? 2. When it is noon at Washington, what is the time at Greenwich ? 3. When it is noon at Portland, Maine, 70° 15' 40", what is the time at San Francisco, 122° 24' 15"? 4. When it is noon at San Francisco, what is the time at Portland, Maine ? 358. Standard Railroad Time. Every railroad train in North America is run on the time of some one of the follow- ing meridians : 60th meridian (passing through Labrador). 75th meridian (passing near Philadelphia). 90th meridian (passing near St. Louis). 105th meridian (passing near Denver). 120th meridian (passing near Carson City). Observe that each of the above numbers after the first is 15 greater than the one preceding it; and that 15 degrees of longitude corre- spond to 1 hour of time. * At first the pupil should give an approximate answer to these problems, consid- ering integral degrees only. Later, if thought advisable, he may take into the account the parts of degrees. One arc minute corresponds to ^ of 4 minutes of time, or 4 seconds of time. One arc second corresponds to gsW of 4 minute^ of time, or b% of one second of time. That is, each arc degree corresponds to 4 minutes of time; each arc minute, to 4 seconds of time; each arc second, to Ss of a second of time. Hence, multiplying by 4 the figures standing for degrees, minutes, and seconds, of longitude Will give the figures standing for minutes, seconds, and 60ths of seconds, of tim«i. 400 COMPLETE ARITHMETIC. Denominate Numbers— Value. 359. The standard unit of value in the United States is the dollar. United States Money. 10 mills (m.) = 1 cent (ct. or ^). 10 cents = 1 dime (d.). 10 dimes = 1 dollar ($). 360. The standard unit of value in Great Britain and Ireland is the pound. Its value, reckoned in United States money, is $4.8 6 6f English, or Sterling, Money. 4 farthings (far.) = 1 penny (d.). 12 pence = 1 shilling (s.). 20 shillings = 1 pound (£). 5 shillings = 1 crown. 21 shillings = 1 guinea. 361. The standard unit of value in France is the franc. Its value, reckoned in United States money, is 19.3^. 362. The standard unit of value in Germany is the mark. Its value, reckoned in United States money, is 23.85^. 363. The standard unit of value in Eussia is the ruble. Its value, reckoned in United States money, is $.772. Problems. 1. Find the value of a guinea in United States money. 2. Find the value of $1000 in English money. 3. Find the value of £5000 in United States money. 4. Find the value of £1000 in Eussian money. 5. Find the value of 4000 marks in English money. 6. Find the value of 8000 francs in United States money. SHOET METHODS. Multiplication and Division. Art. 1. To multiply a number by 50 : Multiply ^ of the number hy 10^. Why? 46 X 50 I- of 46 X 100 ^2300 47 X 50 I of 47 X 100 = 2350 Multiply: 44 by 50 32 by 50 35 by 50 36 by 50 38 by 50 27 by 50 64 by 50 42 by 50 55 by 50 82 by 50 76 by 50 43 by 50 46 by 50 52 by 50 53 by 50 (a) Find the sum of the fifteen products. Art. 2. To multiply a number by 51 : Take 50 times i number, to which add the number itself. Why ? 48 X 50 50 times 48 = 2400 2400 + 48 = 2448 37 X 51 50 times 37 = 1850 1850 + 37 = 1887 Multiply : 26 by 51 34 by 51 35 by 51 46 by 51 32 by 51 29 by 51 24 by 51 42 by 51 43 by 51 66 by 51 84 by 51 33 by 51 36 by 51 38 by 51 39 by 51 (b) Find the sum of the fifteen products. 401 402 COMPLETE ARITHMETIC. Art. 3. To multiply a number by 52 : Take 50 times the number, to which add tivice the number. Why ? 34x52 50 times 34 = 1700 1700 + 68 = 1768 45x52 50 times 45 =r 2250 2250 + 90 = 2340 Multiply : 26 by 52 38 by 52 27 by 52 18 by 52 14 by 52 17 by 52 24 by 52 32 by 52 35 by 52 36 by 52 44 by 52 23 by 52 (c) Find the sum of the twelve products. Art. 4. To multiply a number by 49 : Take 50 times the number, from which subtract the number itself. Why ? 24x49 50 times 24= 1200 1200-24 = 1176 33x49 50 times 33 = 1650 1650-33 = 1617 Multiply : 18 by 49 22 by 49 27 by 49 28 by 49 34 by 49 35 by 49 16 by 49 46 by 49 43 by 49 (d) Find the sum of the nine products. Art. 5. To multiply a number by 33^: Multiply ^ of the number by 100. Why? 36 X 33i I of 36 X 100 = 1200 37 X 33i I of 37 X 100 = 1233^ Multiply: 24 by 331 27 by 33^ 28 by 331- 18 by 33i 21 by 33i 22 by 33^ 30 by 33i 33 by 33^ 35 by 33^ (e) Find the sum of the nine products. PART III. 403 Art. 6. To multiply a number by 34^ : Take 33^ times the member, to ivhich add the numher itself. Why? 24 X 341- 331 times 24 =. 800 800 + 24 = 824 25 X 341 33-1- times 25 = 833i 833i + 25 = 8581 Multiply : 18 by 34-1- 27 by 34i- 24 by 341 21 by 341- 33 by 34^ 39 by 34^ 30 by 341 ^ 36 by 34i 15 by 34i 12 by 34| 16 by 341 17 by 34i (f) Find the sum of the twelve products. Art. 7. To multiply a number by 35i : Take 331 times the numher, to which add twice the member. Why ? 24 X 351 33^ times 24 = 800 800 + 48 = 848 25 X 351 331- times 25 = 8331 8331- + 50 = 8831 Multiply : 21 by 351 30 by 351 27 by 35-1- 18 by 351 66 by 35-i 36 by 35^ 33 by 351 39 by 351 3I by 35^ (g) Find the sum of the nine products. Art. 8. To multiply a number by 32 1: Take 33-1 times the number, from which subtract the number itself. Why? 21 X 321 331 times 21 = 700 700 - 21 = 679 22 X 32^ 331 times 22 = 7331 7331 _ 22 = 7111 Multiply : 15 by 321 21 by 32i- 18 by 321- 24 by 321 30 by 32-1 27 by 32i 33 by 321 39 by 32-1 36 by 32i (h) Find the sum of the nine products. 404 COMPLETE ARITHMETIC. Art. 9. To multiply a L number b} ^ 25 : Multiply ^ of the number by 100. Why? 48 X 25 i of 48 X 100 = 1200 49 x25 i of 49 X Multiply : 100 = 1225 36 by 25 * 32 by 25 33 by 25 40 by 25 28 by 25 29 by 25 24 by 25 16 by 25 19 by 25 52 by 25 48 by 25 50 by 25 (i) Find the sum of the twelve products. Art. 10. To multiply a number by 26: Take 25 times the number, to which add the member itself. Wliy ? 36x26 25 times 36 = 900 900 + 36 = 936 37x26 25 times 37 = 925 925 + 37 = 963 Multiply: 36 by 26 48 by 26 45 by 26 28 by 26 24 by 26 25 by 26 44 by 26 32 by 26 35 by 26 (j) Find the sum of the nine products. Art. 11. To multiply a number by 27: Take 25 times the number, to which add twice the member. Why ? 36 X 27 25 times 36 = 900 900 + 72 = 972 37 X 27 27 25 times 37 = 925 925 + 74:= = 999 Multiply : 48 by 27 52 by 27 37 by 27 32 by 27 16 by 27 17 by 27 28 by 27 24 by 27 26 by 27 (k) Find the sum of the nine products. PART III. 405 Art. 12. To multiply a number by 24: Take 25 times the numhcr, from which subtract the member itself. Why ? 32 X 24 25 times 32 = 800 800 - 32 := 768 33 X 24 25 times 33 = 825 825 - 33 = 792 Multiply : 24 by 24 16 by 24 17 by 24 44 by 24 36 by 24 37 by 24 28 by 24 32 by 24 35 by 24 48 by 24 52 by 24 54 by 24 (1) Find the sum of the twelve products. Art. 13. To multiply a number by 20 : Multiply i of the number by 100. How may a number be multiplied by 21? By 22? By 19? 35x21 20 times 35 = 700 700 + 35 = 735 Multiply: 45 by 21 45 by 22 45 by 19 35 by 21 35 by 22 35 by 19 36 by 21 36 by 22 36 by 19 (m) Find the sum of the nine products. Art. 14. To multiply a number by 1 6| : Multiply J of the number by 100. How may a number be multiplied by 17f? BylSf? By 15-1? 24 X 17| 16f times 24 = 400 400 + 24 = 424 Multiply : 18 by 17f 18 by 18f 18 by 15f 30 by 17f 30 by 18f 30 by 15f 36 by 17| 36 by 18| 36 by 15f (n) Find the sum of the nine products. 13^? By 14^? By 11^? 32 X 13i 12|- times 32 = 400 Multiply : 24 by 131 24 by 141- 16 by 13|- 16 by 14^ 40 by 13|- 40 by 14|^ 406 COMPLETE ARITHMETIC. Art. 15. To multiply a number by 12 J: Multiply i- of the numher hy 100. How may a number be multiplied by 400 + 32 r:. 432 24 by 11| 16 by 14 40 by llj (0) Find the sum of the nine products. Art. 16. To multiply a number by 125 : Multiply ^ of the numher hy 1000. How may a number be multiplied by 126? By 127? By 124? 96 X 125 -1- of 96 X 1000 = 12000 96 X 126 125 times 96 = 12000 12000 + 96 = 12096 Multiply : 120 by 126 120 by 127 120 by 124 320 by 126 320 by 127 320 by 124 240 by 126 240 by 127 240 by 124 (p) Find the sum of the nine products. Art. 17. To multiply a number by 250 : Multiply ^ of the numher hy 1000. How may a number be multiplied by 251? By 252? By 249 ? 48 X 250 i of 48 x 1000 = 12000 48 X 251 250 times 48 = 12000 12000 + 48 = 12048 Multiply : 60 by 251 60 by 252 60 by 249 72 by 251 72 by 252 72 by 249 (q) Find the sum of the six products. PART III. 407 Art. 18. To square 2^, S^, 4^, etc.: Multiply the integer by the integer phis 1, and add \ to the product. 21- X 21- = 2 times 2+2 times |- + ^of2+|-of^ But 2 times -i- + ^ of 2 = 1 time 2 ; and ^ of ^ = \ Hence, 2|- x 2^ = 2^^ + t = ^i HxH = 31^4 + i = 12^ Multiply : H by 4^ H by 5i 6i by 6J n by 7i 8t by 8J 9J by 9^ H by H 21- by 2J 3^ by 3^ (r) Find the sum of the nine products. Art. 19. To square 25, 35, 45, etc.: Multiply the tens' figure^ hy the tens' figure increased hy 1 ; regard the product as hundreds, to which add 25. To explain this rule, think of 25 as 2 tens and ^ of a ten, and apply the explanation given under Art. 18. 25x25 = 2x3 hundred and 25 = 625 35 X 35 = 3 X 4 hundred and 25 = 1225 45 x 45 == 4 X 5 hundred and 25 = 2025 Multiply : 55 by 55 65 by 65 75 by 75 85 by 85 95 by 95 15 by 15 (s) Find the sum of the six products. *The author is aware that the expressions "Multiply the tens^ figure'^ and "the tens' figure increased by 1" are tabooed by the hyi>ercritical. But it is believed that neither obscurity nor misconception will arise from this use of the word figure. The word as here used clearly means the form value of the figure— the number which the figure hy virtue of its shape represent?. 408 COMPLETE ARITHMETIC. Art. 20. To multiply 2| by 2f , 3 J- by 3|-, etc. : Multiply tJi/i integer hy the integer jplus 1, and to the product add the product of the fractions. Observe that this rule will apply only when the integer of the multiplicand and the integer of the multiplier are the same, and the sum of the fractions is 1. H by 2J = 2 times 2 + 2 times ^ + |of2+|of| But 2 times \ + f of 2 = 1 time 2, and f of | = f^ Hence, 2| x 2f = 2x3 + f of i- = ^^% 3| X 3f = 3 X 4 + I of i = 12/^ Multiply : ^hJ^ ^hJ^ ^\^J^ 7iby7f 8fby8i H by 9f (t) Find the sum of the six products. Art. 21. To multiply 24 by 26. 33 by 37, etc.: Multiply the tens' figure hy the tens' figure increased hy 1 ; regard the product as hundreds, to which add the product of the units' figures. Observe that this rule will apply only when the tens' figure of the multiplicand and the tens' figure of the multiplier are alike, and the sum of the units' figures is 10. 22 X 28 = 2 X 3 hundred and 16 = 616 33 X 37 = 3 X 4 hundred and 21 =. 1221 Multiply : 21 by 29 23 by 27 24 by 26 31 by 39 32 by 38 34 by 36 41 by 49 42 by 48 43 by 47 (u) Find the sum of the nine products. PART III. 409 Art. 22. To multiply a number by 15: Multiply the number hy 1 0, and to the product add ^ of the product. 64 X 15 10 times 64 = 640 640 + 320 = 960 45x15 10 times 45 ^ 450 450 + 225 = 675 Multiply : 24 by 15 32 by 15 35 by 15 46 by 15 34 by 15 43 by 15 82 by 15 66 by 15 75 by 15 37 by 15 41 by 15 39 by 15 (v) Find the sum of the twelve products. Art. 23. To multiply a number by 99: Take 100 times the number, from which subtract -the number itself. How may a number be multiplied by 98 ? 36 X 99 100 times 36 = 3600 3600 - 36 = 3564 42 X 98 100 times 42 = 4200 4200 - 84 = 4116 Multiply : 35 by 99 44 by 99 35 by 98 ' 27 by 99 54 by 99 46 by 98 62 by 99 75 by 99 28 by 98 (w) Find the sum of the nine products. Art. 24. To multiply a number by 75 : Multiply ^ of the number by 100. How may a number be multiplied by 66-|? By 62|-? By87i? Multiply : 64 by 75' 24 by 66| 64 by 874- 48 by 75 36 by 66| 48 by 871- 52 by 75 63 by 66f 56 by 87|- 37 by 75 37 by 66| 32 by 87| (x) Find the sum of the twelve products. 410 COMPLETE ARITHMETIC. Art. 25. To divide a number by 25; by 33i; by 12^; by 16| ; by 20 ; by 50. ^ (See pp. 212, 213, and 214, of this book.) 850 - -25 -331 -m -16f -20 -50 = 8 times 4-^2 = 34 933i- = 9 times 3 -f- 1 = 28 637I-- = 6 times 8 -f 3 = 51 750 - = 7 times 6-^3 = 45 960 - = 9 times 5 + 3 = 48 450 - = 4 times 2 + 1 = 9 Divide : 1275 by 25 1166|- by 33i- 762|- by 12i- 950 by 16f 880 by 20 950 by 50 (y) Find the sum of the six quotients. Note. — Without a pencil, tell the integral quotient and the re- mainder resulting from the incomplete division of 1584 by 25. Art. 26. To divide a number by 125; by 250: Observe that 125 is contained in each thousand of a nnmher, 8 times; that 250 is contained in each thousand of a number, 4 times. 7125 -^ 125 = 7 times 8 + 1 = 57 8500 ^ 250 = 8 times 4 + 2 = 34 Divide : 13250 by 250 13500 by 250 13750 by 250 18000 by 250 18750 by 250 18500 by 250 12000 by 125 12500 by 125 12625 by 125 9125 by 125 9375 by 125 9875 by 125 (z) Find the sum of the twelve quotients. Note. — Without a pencil, tell the integral quotient and the re- mainder resulting from the incomplete division of 15450 by 250. PART HI. 411 Art. 27. When the same factor occurs in a dividend and in its divisor, it may be omitted from both without changing their ratio. Hence all the factors that are common to a div- idend and its divisor may he stricken out (canceled) and the quotient (ratio) he unchanged. Divide 180 by 42. Operation No. 1. Operation No. 2. 42)180(4-1 180 ^ ^X2x3x3x5 ^ 30 ^ ^ 2 1^ 42 ^X3x7 7 7 12 2 =r — Observe that the striking out of the 4-^ ' factors 2 and 3 from the dividend and its divisor does not change their ratio — the quotient. II. Divide 420 by 35. Operation No. 1. Operation No. 2. 35)420(12 420 ^ 2x2x3x0x;?^ = — = 12 35 35 ^xli 1 70 70 Observe that if all the factors of one of the numbers are canceled, the number be- comes 1 and not 0. The factor 5 is 5 times 1 : the factor 7, 7 times 1. Hence in the above problem there really remain in the divisor, after the cancellation, the factors 1 and 1 = 1x1=1- III. Divide 48 x 8 x 4 = 1536 by 8 x 4 x 4 = 128. Operation No. 1. Operation No. 2. 12S);g6(12 ^xixi_12_^, $ X ^ X 4 1 Observe that it is not necessary to obtain the prime factors of a dividend and its divisor to employ cancellation in finding the quotient. In the above the composite factor 8 is stricken out of the divisor and out of the 48 of the dividend. 256 256 412 COMPLETE ARITHMETIC. IV. Divide 56 x 35 = 1960 by 15 x 8 = 120. Operation No. 1. Operation No. 2. 120)1960(16. M_xi^_49_^^l Ux^ 3 3 760 '20 Observe that in the above the factor 5 4Q \ is stricken out of 15 and 35, and the factor 8 is stricken out of the divisor and out of the 56 of the dividend. 120 3 Miscellaneous Problems. Note. — Employ " Short Methods " in the solution of the follow- ing problems. How many cords of wood — 1. In a pile 32 feet by 8 feet by 4 feet ? * 2. In a pile 40 feet by 1 6 feet by 6 feet ? 3. In a pile 32 feet by 30 feet by 10 feet ? (aa) Find the sum of the three results. How many acres of land — 4. In a piece 180 rods by 28 rods?f 5. In a piece 64 rods by 96 rods? 6. In a piece 136 rods by 32 rods? (bb) Find the sum of the three results. 7. Multiply 64 by 96 and divide the product by 16 x 24 X 2. 8. Multiply 250 by 72 and divide the product by 16 1 x 3 X 24. (cc) Find the sum of the two results. * Think of a cord as 8 feet by 4 feet by 4 feet. f Think of an acre as 40 rods by 4 rods. PART III. 413 Find the cost — 9. Of 346 acres of land at $50 per acre. 10. Of 346 acres of land at $51 per acre. 11. Of 346 acres of land at $52 per acre. 12. Of 346 acres of land at $49 per acre. 13. Of 254 acres of land at $51 per acre, (dd) Find the sum of the five results. 14. Of 243 ft. iron pipe at 33-i^ a foot. 15. Of 243 ft. iron pipe at 3414 a foot. 16. Of 243 ft. iron pipe at 35^^ a foot. 17. Of 243 ft. iron pipe at 32^^ a foot. 18. Of 156 ft. iron pipe at 35^^ a foot, (ee) Find the sum of the five results. 19. Of 260 lb. butter at 25^ a pound. 20. Of 260 lb. butter at 26^ a pound. 21. Of 260 lb. butter at 27^ a pound. 22. Of 260 lb. butter at 24^ a pound. 23. Of 184 lb. butter at 27^- a pound, (ff) Find the sum of the five results. 24. Of 350 lb. coffee at 12 J^ a pound. 25. Of 350 lb. coffee at 13|^^ a pound. 26. Of 350 lb. coffee at 14J^ a pound. 27. Of 350 lb. coffee at 11^^ a pound. 28. Of 330 lb. coffee at 16|^ a pound. 29. Of 330 lb. coffee at 17f^ a pound. 30. Of 330 lb. coffee at 15f^ a pound, 31. Of 240 lb. coffee at 25^ a pound. 32. Of 240 lb. coffee at 26^ a pound. 33. Of 240 lb. coffee at 27^ a pound, (gg) Find the sum of the ten results. 414 COMPLETE AKITHMETIC. Find the cost — 34. Of 2i tons coal at $2^ per ton. 35. Of 3|- tons coal at $3|- per ton. 36. Of 4 J tons coal at $4^ per ton. (hli) Find the sum of the three results. 37. Of 25 tons of meal at $25 per ton. 38. Of 35 acres of land at $35 per acre. 39. Of 45 M. ft. of lumber at $45 per M (ii) Find the sum of the three results. 40. Of 23 yd. cloth at 27^ a yard. 41. Of 36 yd. cloth at 34^ a yard. 42. Of 42 yd. cloth at 48^ a yard, (jj) Find the sum of the three results. 43. Of 3240 ft. lumber at $15 per M. 44. Of 2460 ft. lumber at $15 per M. 45. Of 1620 ft. lumber at $16 per M. (kk) Find the sum of the three results. 46. Of 99 lb. butter at 23^ a pound. 47. Of 99 lb. butter at 28^ a pound. 48. Of 98 lb. butter at 24^ a pound. (11) Find the sum of the three results. 49. Paid $15.50 for ribbon at 16|^ a yard. How many yards did I buy ? 60. Paid $24.75 for ribbon at 12|^ a yard. How many yards did I buy ? (mm) Find the sum of the two results. PRACTICAL APPROXIMATIONS. So far as practicable, solve the following problems without the aid of a pencil. At least, exercise the judgment on every problem before making any figures^. 1. The specific gravity* of iron being about 1\, how much does a cubic foot of it weigh ? How much does a cubic inch of iron weigh ? 2. A 4-inch iron ball weighs about pounds. A 2- inch iron hall weighs about pounds. 3. An iron rod, 1 inch in diameter and 12 feet long, weighs about pounds. An iron rod 2 inches in diam- eter and 12 feet long weighs about pounds. 4. A sheet of boiler iron, 8 feet square and f of an inch thick, weighs about pounds. 5. What is the weight of the water that will fill a tank 2 feet wide, 2 feet deep, and 1 feet long ? 6. The specific gravity of limestone is about 2^. What is the weight of a piece of limestone that is 4 feet square and 3 inches thick ? 7. The specific gravity of seasoned white pine is about .5 ; that is, a piece of white pine weighs about 5 tenths as much as the same bulk of water weighs. How much does a pine board 1 foot wide, 1 inch thick, and 12 feet long, weigh? 8. What is the weight of a stick of timber 12 inches by 12 inches and 20 feet long, if its specific gravity is .7 ? 9. What is the weight of 1000 feet of green lumber if its specific gravity is .9 ? *When we say that the specific gravity of iron is about 1\, we mean that it weighs about 71 times as much as water, the same bulk being considered. 415 416 COMPLETE ARITHMETIC. 10. The specific gravity of sand is about 2. How much does a load (27 cu. ft.) of it weigh ? 11. If the specific gravity of granite is 2.7, how much does a cubic yard of it weigh ? 12. The specific gravity of brick and mortar is nearly 2. What is the weight of a cubic foot of brick wall ? 13. If a brick, 8 in. by 4 in. by 2 in., weighs 4J pounds, is its specific gravity more or less than 2 ? That is, does 1 cubic foot of bricks weigh more or less than exactly twice as much as 1 cubic foot of water ? 14. If the specific gravity of Athens (Illinois) limestone is 2.4, and if a cord of it is equal to 100 soM feet, how much does a cord of this stone weigh ? 15. If bricks, 8 in. long, 4 in. wide, and 2 in. thick, are laid on their largest face, how many bricks will be required for a walk 6 feet wide and 100 feet long, making some allowance for imperfect bricks and breakage in handling? 16. What is the capacity in gallons of a tank 5 feet long, 2 feet wide, and 2 feet deep ? 17. Give the dimensions of a tank the capacity of which is 225 gallons. 18. A certain cylindrical tank is 8 feet in diameter. Each foot in depth will contain how many gallons ? 19. One inch of rain-fall will give how many pounds of water on a horizontal surface 30 feet by 40 feet ? 20. Two inches of rain-fall will give how many barrels (31 1^ gal.) of water on a horizontal surface 40 feet by 60 feet? 21. A one inch rain-fall will give how many tons of water to the acre ? 22. Your school-house lot is what part of an acre ? PART III. 417 .23. A one inch rain-fall will give how many barrels (31|- gal.) of water on the school-house lot ? 24. Your school-room floor is what part of an acre ? 25. The distance around your school-room is what part of a mile ? 26. How many bushels of oats would be required to till your school-room to the depth of 3 feet ? 27. A shed as large as your school-room would hold how many cords of wood ? 28. Give the dimensions of a crib that will hold 1000 bushels of corn. 29. Give the dimensions of a pile of wood that contains 40 cords. 30. Apiece of land 100 feet long and 5 rods wide is what part of an acre ? 31. How many cubic inches in a cylinder 8 inches in diameter and 10 inches long? 32. How many cubic inches in an 8-inch sphere ? 33. If 468.25 be multiplied by .5106 will the product be more or less than 234 ? * 34. If 484,079 be multiplied by .251, will the product be more or less than 121 ? 35. If 2480 be multiplied by .2479 will the product be more or less than 620 ? 36. If 6400 be multiplied by .74, the product will be how many less than ^ of 6400 ? 37. If 4800 be multiplied by 1.6, the product will be how many more than 1|- times 4800 ? 38. If 366.06 be divided by |, will the quotient be more or less than 366 + J- of 366 ? * Observe that 234 is J of 468. 418 COMPLETE ARITHMETIC. 39. If 25.2314 be divided by J, will the quotient be more or less than 51 ? 40. If 250 be divided by .26, will the quotient be more or less than four times 250 ? 41. If cheese is worth 17 cents a pound, how much should be paid for (a) 2 lb. 3 oz.? (b) 3 lb. 5 oz.? (c) 1 lb. 7 oz.? (d) 4 lb. 9 oz. (e) 2 lb. 11 oz.? 42. If meat is worth 15 cents a pound, how much should be paid for (a) 1 lb. 4 oz.? (b) 2 lb. 6 oz.? (c) 1 lb. 7 oz.? (d) 2 lb. 9 oz.? (e) 3 lb. 13 oz.? 43. If cheese is worth 12|^ cents a pound, how much should be paid for (a) 2 lb. 8 oz.? (b) 2 lb. 4 oz.? (c) 3 lb. 1 oz.? (d) 4 lb. 9 oz.? (e) 4 lb. 15 oz.? 44. A wagon box 10 feet by 3 feet by 16 inches will contain how many bushels of shelled corn? 45. The surface of a sphere is equal to 4 times the area of a circle having the same diameter as the sphere. How many square inches in the surface of a 10-inch sphere? 46. How many square inches in the surface of a 20-inch sphere ? 47. If from a cylinder of wood the largest possible cone be cut, exactly -| of the wood will be cut away. The solid content of a cone is therefore exactly ^ of a cylinder having the same base and the same altitude (length). How many cubic inches in a cone the diameter of whose base is 8 inches and whose altitude is 12 inches? 48. How many bushels of grain in a conical pile whose diameter is 6 feet and whose altitude is 4 feet ? 49. At 10 cents a square yard, what is the cost of paint- ing the outer surface of a cylindrical standpipe whose diameter is 15 feet and whose altitude is 36 feet? PAET III. 419 50. At 1 cents a square foot, what is the cost of lining a cylindrical tank (curved surface and bottom) whose diameter is 10 feet and whose depth is 12 feet? 51. A sphere is exactly ^ and a cone exactly -i- of a cyl- inder of the same dimensions. Find the solid content and compare the following : (a) A 6 -inch cube. (b) A cylinder 6 in. in diameter and 6 in. long. (c) A 6 -inch sphere. (d) A cone ; base 6 in. in diameter, altitude 6 in. 52. A cubic foot of steel weighs 490 lb. (a) What is the weight of a cylinder of steel 1 foot in diameter and 1 foot long ? (b) Of a sphere of steel 1 foot in diameter ? (c) Of a cone of steel, base 1 foot in diameter, altitude 1 foot ? 53. A circular piece of land 20 rods in diameter contains more or less than 2 acres ? 54. At sight, give approximate answers to the following : (a) Interest of $450.25 for 1 yr. 3 da. at 6% ? (b) Interest of $4000 for 29 da. at 6% ? (c) Interest of $250 for 1 yr. 8 mo. 5 da. at 6% ? (d) Interest of $500 for 1 yr. 11 mo. 16 da. at 6% ? (e) Interest of $751.27 for 2 yr. 6 mo. 1 da. at 4% ? (f) Interest of $149.75 for 1 yr. 7 mo. 29 da. at 6% ? (g) Interest of $298.97 for 1 yr. 6 mo. 4 da. at 8% ? (h) Interest of $495 for 15 da. at 6% ? (i) Interest of $1200 for 6 da. at 8% ? (j) Interest of $600 for 20 da. at 4% ? (k) Interest of $397.28 for 2 yr. 6 mo. at 6% ? (1) Interest of $5000 for 5 mo. 29 da. at 8% ? (m) Interest of $3000 for 2 mo. 29 da. at 7% ? (n) Interest of $4000 for 3 mo. 29 da. at 6% ? 420 COMPLETE ARITHMETIC. 55. At sight, give answers to the following, that are true to dollars; then, with the aid of a pencil, if necessary, obtain answers that are true to cents. (a) Cost of 2970 lb. coal at $4.50 per ton V (b) Cost of 3520 lb. hay at $11.50 per ton?'-^ (c) Cost of 1490 lb. straw at $4.25 per ton ? (d) Cost of 2460 lb. bran at $10.00 per ton?^ (e) Cost of 2310 lb. oil meal at $21 per ton ? (f) Cost of 2240 lb. beef at $6.10 per cwt.? - (g) Cost of 1560 lb. pork at $4.50 per cwt.? (h) Cost of 2150 lb. flour at $3.05 per cwt.? (i) Cost of 1200 lb. lard at $5.90 per cwt.? (j) Cost of 1400 lb. tallow at $3.55 per cwt.? (k) Cost of 1000 lb. nails at $3.05 per cwt.? (1) Cost of 2240 ft. lumber at $15 per M.?' (m) Cost of 4500 lath at $2.50 per M.? (n) Cost of 6740 brick at $6.00 per M.?' (o) Cost of 1997 ft. lumber at $27.50 per M.? (p) Cost of 198 lb. butter at 27|- cts. per lb.?' (q) Cost of 20 3|- lb. cheese at 16 cents per lb.? (r) Cost of 2440 lb. oats at 24 1- cents per bu.?' (s) Cost of 1680 lb. oats at 23^ cents per bu.? > 2970 lb. is nearly IJ tons. 2 What is the cost of 3500 at $12 per ton ? 3 At $10 a ton, how much does 1 lb. cost? * At $15 i)er M., how much is 1 foot worth? " 6740 brick are nearly 62 M. « 200 lb. butter at 27J cents is worth how much ? ' At 24^ a bushel, 1 lb. of oats is worth how much ? MISCELLANEOUS PEOBLEMS. Note. — The following problems are selected mainly from sets of examination questions supplied to the author for this purpose by one hundred school principals and superintendents. 1. Each edge of a cube is diminished by -^^ of its length. (a) By what fraction of itself is the volume diminished ? (b) By what fraction of itself is the surface diminished ? 2. How many cubical blocks, each edge of which is ^ ft., are equivalent to a block 8 ft. long, 4 ft. wide, and 2 ft. thick ? 3. A ladder 78 ft. long stands perpendicularly against a building. How far must it be pulled out at the foot that the top may be lowered 6 ft. ? 4. A merchant sold | of a quantity of cloth at a gain of 20% and the remainder at cost. (a) His gain was what per cent of the cost ? (b) If he gained $7.29 what was the cost of the goods ? 5. What must I pay for 4% stock to get 5% on the investment ? 6. The cubical content of one cube is eight times that of another : (a) How does an edge of the first compare with an edge of the second ? (b)How does the surface of the first compare with the surface of the second ? 7. A creditor receives $1.50 for every $4.00 that is due him and thereby loses $301.05. (a) What was the sum due him? (b) What per cent of the debt did he lose ? 421 422 COMPLETE ARITHMETIC. 8. At $20 per M., board measure, what is the cost of the following: A stick of timber 30 feet long and 14 inches square, and a plank 18 feet long, 8 inches wide, and 2^ inches thick ? 9. A and B hire a pasture for $85 ; A puts in -8 cows and B puts in 12 cows. How much should each pay ? 10. Simplify the following : ^ of J- of li. • 11. Seven times John's property plus $32200 equals 21 times his property. How much is he worth ? 12. Two men engage in business with a joint capital of $5000. The first year's gain was $1760, of which one received $1056. How much capital did each furnish? 13. Thirty-five per cent of the men in a regiment being sick, only 637 men were able to enter battle. How many men were there in the regiment ? 14. A lawyer collected 80% of a debt of $2360 and charged 5% commission on the sum collected. How much did the creditor receive ? 15. Write a negotiable note for $500, making yourself the payee and James J. Kogers the maker. Interest at the legal rate. 16. A speculator bought stock at 25% below par and sold it at 20% above par. He gained $1035. How much did he invest ? 17. What is the rate per cent per annum if $712 gains $142.40 in 3 yr. 4 mo.? 18. A person asked for a lot of land, 40 % more than it cost him, but finally reduced his price 15% of his asking price and sold it, making $9.50. (a) What per cent did he make ? (b) How much did the land cost him ? (c) How much did he receive for it ? PART III. 423 19. Purchased stock at a premium of 8 per cent. What rate of interest do I receive on the investment if it pays an annual dividend of 6 % ? 20. Find the vokime of a cube the area of whose surface is 100.86 square inches. 21. How many apples must a boy buy and sell to make a profit of S9.30, if he buys at the rate of 5 for 3^ and sells at the rate of 4 for 3^ ? 22. Find the cost of 1875 lb. hay at $6.50 per ton. 23. What is the interest on $1200 from Sept. 21, 1898, to May 5, 1899, at 7% per annum ? 24. The area of a square field is 10 acres. What is the distance diagonally across the field ? 25. A "drummer" earns $2500 a year. One thousand dollars of this sum is a guaranteed salary. The remainder is his commission of 5% on his sales. What is the amount of his annual sales ? 26. The area of a triangle is 325 square inches. Its base is 25 inches. What is its altitude ? 27. Gave 6|- lb. butter, worth 36^' a pound, for 3^ gal. oil. What was the cost of the oil per gallon ? 28. A can build a certain wall in 10 days ; B can build it in 12 days, and C in 15 days. In how many days can they build the wall working together ? 29. A horse and a carriage together cost $550. The horse cost I as much as the carriage. Find the cost of each ? 30. A man willed J of his property to his wife, ^ of the remainder to his daughter, and the rest to his son. The difference between the wife's portion and the son's portion was $12480.331 How much was the man worth? 424 COMPLETE AKITHMETIC. 31. (a) How many loads, each containing a cubic yard, will be required to fill a street 150 feet long, 50 feet wide, and 2^ feet deep? (b) How much will it cost at 18^ per cubic yard ? 32. A right triangle has two equal sides. Its hypothenuse is 100 rods long, (a) Find one of its two equal sides, (b) Find its area. 33. What is the ratio of the area of a circle to the area of its circumscribed square ? 34. What is the ratio of the square of the radius to the square of the diameter of the same circle ? 35. A man's tax is $37.50. The rate of tax is li%. Property is assessed at 30% of its value. What is the man's property worth ? 36. A field containing 160 acres is 40 rods wide. At 45^ a rod, how much less would it cost to fence a square field containing the same number of acres? 37. My agent in Baltimore having sold a consignment of grain, after taking out his commission at 3 % and paying a freight bill of $1,125.00, sent me a draft for the amount due me — $19,536.00. For how much was the grain sold ? 38. How many pickets 4 inches wide, placed 3 inches apart, are required to fence a garden 21 rods long and 14 rods wide? 39. At $6.30 a cord, what is the value of wood that can be piled under a shed 50 ft. long, 25 ft. wide, and 12 ft. high? 40. Find the curved surface of a cylinder 6 ft. in diameter and 12 ft. long. 41. (a) What is the bank discount and (b) what are the proceeds on a note for $125 payable in 90 days, the rate of discount being 8 % ? PART III. 425 42. If sugar that cost 5^ a pound is sold at 18 lb. for a dollar, what is the gain per cent ? 43. What is the area of a circle 30 inches in diameter ? 44. What is the volume of a 12-inch globe ? 45. What is the circumference of a circle that is 40 rods in diameter ? 46. The " number belonging " in a certain school was 74. Five were absent in the forenoon and seven in the afternoon. Wliat was the per cent of attendance for the day ? 47. A rectangular piece of land 41 rods by 24 rods is how many acres ? 48. Find the value of x in the following proportion: 17.5 : 25 ::^: 40. 49. A circular ^-mile race track encloses how many acres ? 50. If the same number be added to the numerator and to the denominator of a proper fraction, will it make the fraction greater or less ? 51. A certain roof is 40 feet long and, measured horizon- tally, 24 feet in width. A 2-inch rain-fall should give how many inches in depth in a cistern that receives the water from this roof, the cistern being 6 feet long and 4 feet wide? 62. A can do a piece of work in i of a day. B can do the same amount of work in \ of a day. In what part of a day can both working together do the piece of work ? 53. C can do a certain piece of work in 3 days. D can do the same amount of work in 4 days. In how long a time can both working together do the piece of work ? 54. If the 6-foot drive wheel of a locomotive makes 840 revolutions in moving a certain distance, how many revolu- tions will a 7-foot wheel make in moving the same distance? 426 COMPLETE ARITHMETIC. 55. The circumference of one of my carriage wheels is 12 feet. The circumference of another wheel on the same carriage is 14 feet. How far has the carriage run when the smaller wheel has made exactly 300 more revolutions than the larger wheel ? 56. Is the capacity of a cylindrical pail 6 inches in diam- eter and 5 inches deep more or less than ^ of a gallon ? . 57. At what rate per cent must I invest $800 that in 1 year 6 months it will amount to $854 ? 58. If exactly | of a stick of timber floating in the water is submerged, and if the timber is 12 inches by 12 inches and 30 feet long, how many pounds does it weigh ? Note. — If f of the timber is submerged, it weighs f as much as its own bulk of water. 59. How many pickets are required to inclose a square 2|-acre lot if the pickets are 3 inches wide and 3 inches apart ? 60. The boundary of a certain field is described as follows: Beginning at the northeast corner of section 14 ; thence south, 24 rods; thence west, 20 rods; thence south, 15 rods; thence west, 40 rods; thence south, 41 rods; thence west, 20 rods; thence north, 80 rods; thence east, 80 rods, to the place of beginning. How many acres in the field? 61. At 90 cents a yard, find the cost of carpeting a room that is 15 feet wide and 18 feet long, the cajpet to run lengthwise of the room, there being a w^aste of 1 foot on each breadth, except, the first, for matching; carpet 1 yd. wide. 62. The perimeter of a rectangular field is 144 rods and its length is twice its breadth. Find its area. PART III. 427 63. Change f of a mile to a compound number made up of rods, feet, and inches. 64. Sold 1^ of a barrel of sugar for what ^ of it cost. What was the per cent of loss ? 65. Sold |- of a barrel of sugar for what | of it cost. What was the per cent of gain ? 66. If a merchant sells goods at a uniform profit of 20%, and his sales on a certain day amount to $60, his gain is how many dollars ? 67. Find the cost of 600 ft. of gas pipe, list 28^ a foot, at "55 and 3 lO's off."* 68. Bought for " 60 off " and sold for " 50 off." What was the per cent of profit ? 69. From |^ of a certain number subtract f of it and 27 remains. What is the number ? 70. Divide the number 495 into two parts, the ratio of the parts being as 2 to 3. 71. Divide the number 187 into two parts, the ratio of the parts being as -| to f . 72. The product of a certain number multiphed by If is 352. What is the number ? 73. Find the cost at $16 per M. of 2-in. plank for a floor 24 feet by 42 feet. 74. The perimeter of an oblong is 192 ft., and its length is twice its breadth. Find its area. 75. At $45 per M., find the cost of a board 16 feet long, 18 inches wide, and 14- inches thick. 76. A man bought a horse and a carriage for $315 ; he paid 2i times as much for the carriage as for the horse. Find the cost of each. * " 55 and 3 lO's oflf," means " 55 and 10 and 10 and 10 off.' 428 COMPLETE ARITHMETIC. 77. A house and lot cost $5000. For how much per month must it rent to pay the owner a sum equal to 5 % of its cost and $230 for insurance, taxes, and repairs? 78. If 4 J- lb. of butter can be made from 100 lb. of milk, how much butter per week can be made in a creamery that is receiving 15000 lb. of milk a day? 79. The area of a rectangular piece of land 36 rods long is 900 square rods. How many rods of fence required to enclose the field? 80. A shingle is 4 inches wide.^ If shingles are laid 4|- inches to the weather, each shingle practically covers square inches. Then shingles will cover 1 square foot. On account of waste and short measurements, it is necessary to purchase 9 shingles for every square foot to be covered, if they are to be laid 4^ inches to the weather. Shingles are put up in bunches 20 inches wide and containing 50 courses. Hence each bunch contains 250 shingles. At the lumber yards parts of bunches are not offered for sale. How many bunches of shingles must I purchase for a double roof 35 feet long, rafters 16 feet long, the shingles to be laid 4^ inches to the weather ? 81. The square of a certain number is 576. What is its cube? 82. The area of one face of a cube is 64 square inches. What is the solid content of the cube ? 83. In plowing an acre with a twelve-inch plow the man walking behind it travels (43560 -^ 5280) 8^ miles. What part of 8^ miles will that man travel who plows an acre with a 14-inch plow? With a 16 -inch plow? * Shingles are not of uniform width ; but in counting them at the lumber yards, every 4 inches in width is called 1 shingle. PART III. 429 84. A can do a piece of work in 12 days. A and B can do an equal amount of work in 8 days. In how long a time can B do the work? 85. The edge of one cube is 2|^ times as long as the edge of another cube, (a) The surface of the first cube is how many times the surface of the second cube ? (b) The solid content of the first cube is how many times the solid con- tent of the second cube? 86. At " 50 and 10 off" the net cost was $29.34. Find the list price. 87. How many 160-acre farms in a township 6 miles long and 6 miles wide ? 88. If there is a 4-rod road on every section line* of a township 6 miles square, (a) how many acres of the town- ship in its roads ? (b) How many acres of each square 160- acre farm are taken for roads? 89. What single discount is equivalent to "40 and 20 and 10 off"? 90. Estimate the weight of a 4-inch sod from an acre of ground. 91. A room 16 feet by 22 feet has a floor made of 4-inch tile, (a) How many tiles in the floor? (b) How many tiles in the border of four rows ? 92. If I buy at 20% below list price and sell at 20% above list price, what is my per cent of gain ? 93. Add two hundred and seven thousandths, and two hundred seven thousandths. 94. From nine hundred and eight ten-thousandths, sub- tract nine hundred eight ten-thousandths. * A section is 1 mile square, and half of the width of the road is on each side of every section line. 430 COMPLETE ARITHMETIC. 95. Add two hundred seventy-five tenths, three hundred twenty-four hundredths, and five hundred thirty-six thou- sandths. 96. Multiply six hundred twenty-seven and forty-five thousandths by two and six tenths. 97. Divide one hundred forty-four by twelve hundredths. 98. If the hills of corn are 3|- feet apart each way, (a) how many hills to the acre ? (b) If the corn is cut and shocked, putting " 8 hills square " in a shock, how many shocks to the acre ? (c) If there are "16 hills square " in each shock, how many shocks to the acre ? 99. If it is worth $1.00 a cord to' cut "4-foot wood" into 16-inch pieces, how much is it worth to cut "8-foot wood " into pieces of the same length ? 100. An agent sold 1460 lb. butter at 23J^ a pound. If his commission for selling is 5% and he paid charges amoimting to $8.96, how much should he remit to the, owner of the butter ? 101. A lot 50 feet wide and 120 feet "deep" (long) was sold for $450. This is equivalent to what price per acre ? 102. Weight of wagon and hay, 4750 lb.; weight of wagon 1620 lb. How much is the hay worth at $12.50 per ton? 103. How many acres in 5|- miles of 4-rod road ? 104. The specific gravity of ice is .92. (a) How much does a cubic foot of ice weigh ? (b) How many tons of ice, if packed solid, can be stored in a building 12 feet square, the ice to be 8 feet deep ? 105. How many tons in an acre of ice 1 5 inches thick ? 106. On the first day of May the water-meter at the Illi- nois Institution for the Blind stood at 375,400 (cu. ft.); on PART III. 431 the first day of June, the reading was 477,700. Eegarding each cubic foot as 7i gallons, (a) how many gallons of water were used in May? (b) What was the amount of the bill for water if the price was 12^ per thousand gallons, with a discount of 1 6| per cent ? 107. At $1000 an acre, find the value of a strip of land 4 feet by 125 feet. 108. If the specific gravity of iron is 7^, how many cubic feet in 1 ton of iron ? 109. A merchant marked goods 25% above the cost; he sold them at 25% below the marked price. What per cent did he lose? 110. The cube of a number divided by the number equals 1764. What is the number? 111. What is the edge of a cube whose entire surface is 6144 sq. inches? 112. Seven and one half feet are what part of a rod? 113. Peter has 12|^% more money than Paul; together they have $6.97. How much money has each? 114. Change the following to a common fraction in its lowest terms: •27y3y. 115. Answer the following at sight: (a) Interest of $48 for 2 mo. at 6%. (b) Interest of $375 for 2 mo. at 6%. (c) Interest of $240 for 4 mo. at 6%. (d) Interest of $330 for 4 mo.- at 6%. 116. Answer the following at sight: (a) Divide 125 by .5. (b) Divide 125 by .05. (c) Divide 12.5 by .5. . (d) Divide .125 by 5. 432 COMPLETE ARITHMETIC. SET I. COOK COUNTY, ILLINOIS. Eighth Grade Examination for County Superintendent's Diploma. June, 1897. O. T. Bright, Supt. Time, 9:30 to 12. 1. What is the ratio of (a) .2% to 2% ? (b) 5 -f- .5 = ? (c).05-^5-? (d).5-^.05 = ? (e).005-v-.5 = ? (f) .05 -4- .005 = ? 2. A girl spelled 95% of 60 words. How many words did she miss ? 3. When a vessel sails 160 miles a day, she completes her voyage in 14 days. In what time would she complete it if she sailed 196 miles a day ? 4. A three-inch cube was painted on all sides. It was then cut into inch cubes, (a) How many of the inch cubes were painted on three sides ? (b) How many on two sides ? (c) How many on one side ? (d). How many were not painted at all ? 5. Fill the blanks in the following : A boy had a fish pole 15 feet long. A piece equal to 20% of its length was broken off while catching fish. (a) The part remaining was % of the whole pole. (b) The part broken off was % of the part remaining. (c) The part remaining was % of the part broken off. 6. The N. E. \ of the N. W. i of a certain section of land was fenced off into four equal fields, (a) What is the shortest length of fence necessary ? (b) How much land in each field ? 7. A straight pole 72 feet high is broken 20 feet from the ground, but is not detached. How far from the foot will the top reach ? PART III. 433 SET 11. NEW HAVEN (CONN.) PUBLIC SCHOOLS. Entrance Examination to the High Schools. Fall, 1897. C. N. Kendall, Supt. (Answer 10 entire questions.) 1. (a) 13 oz. is what per cent, of 5 lb. avoirdupois ? (b) A man added 18 cows to his herd, thereby increasing the number 25 per cent. How many cows has he now ? 2. If lead pencils that cost 3 cents each are sold for 5 cents each, what is the per cent of profit ? 3. Find the interest on a note for $330 at 6 per cent, given August 3, last year, and due to-day. 4. Write a negotiable, interest-bearing, promissory note. 5. (a) Divide | of | of 7| by 3f . (b) Subtract 8^?^ from the sum of 5 J, 2|, 4:^^. 6. A rectangular field is 86|^ rods long and 46.875 rods wide. How much wheat will it produce at the rate of 20 bushels per acre ? 7. A rectangular park, the sides of which are respectively 45 rods and 60 rods long, has a walk crossing it from corner to corner. How long is the walk ? 8. If f of 9 bushels of wheat cost $13^, what will | of a bushel cost ? 9. If hay sells for $14 a ton at a loss of 12 J per cent, what must it sell for to gain 1 5 per cent ? 10. How many pounds of cotton at 7-^ cents a pound can a broker buy for $9,225, and retain his commission of 2^ per cent ? 11. Find the proceeds of a 3 months' note for $500 dis- counted at a bank at 6 per cent. 12. If a building 20 feet high casts a shadow of 6 feet, what length of shadow will a church spire 114 feet high cast 1 434 COMPLETE ARITHMETIC. SET III. COOK COUNTY, ILLINOIS. Applicants for Teachers' Second Grade Certificates. June, 1898. O. T. Bright, Supt. " Mental Arithmetic." Time, 20 minutes. Fill the blanks : 1. By a sale of goods I lost 12i%. The cost was % of the selling price. 2. The circumference of a 3-in. circle is % of its radius. 3. A five-inch square is % greater than a four-inch square. 4. I buy apples four for three cents and sell them three for four cents. I gain per cent. 5. A and B are 78 miles apart and walk toward each other ; A walks 3 miles an hour and B 3^ miles an hour. When they meet B will have walked miles. 6. A stick of timber 15 inches square and 32 feet long contains board feet. 7. A horse trots 23f miles in 2 J hours. His rate per hour is miles. 8. $154 was divided among A, B, C, and D, in the pro- portion of ^, ^, ^, ^. C got dollars. 9. A school" contains 50 pupils: Monday, 3 were absent in the forenoon, Tuesday, 2 were absent all day, Wednesday 3 absent forenoon and 2 afternoon, Thursday, 4 absent all day, Friday, all present The per cent of attendance for the week was . 10. 64 gal. of wine and 1 6 gal. of water were mixed. One pint of the mixture contained of a gal. of water. PART III. 435 SET IV. COOK COUNTY, ILLINOIS. Applicants for Teachers' Second Grade Certificate. August, 1895. O. T. Bright, Supt. Time, 60 minutes. 1. If a quarter section of land has fenced within it the largest possible circular lot, how many acres of the quarter section will remain outside of the circle ? 2. Find the value of the following lumber at $15 per M.: 20 pieces 2x4, 18 ft. long. 20 pieces 4x4, 12 ft. long. 20 pieces 3x10, 16 ft. long. 45 16 ft. stock boards, 15 inches wide. 3. (a) What sum invested in 8% bonds at 33^% premium will yield an annual income of $1200 ? (b) What if the bonds were 33^ discount? 4. Find the value of a piece of land 20 ft. x40 rods at $1000 per acre. 5. What is the ratio of 3|- to | ? Answer in per cent. SET V. Examination Department of the University of the State of New York, January, 1898. 100 credits. Necessary to pass, 75. Time, 9:15 a. m. to 12:15 p. m. Answer the first five questions and five of the others, but no more. If more than five of the others are answered only the first five answers will be considered. Give all operations (except mental ones) necessary to find results. Reduce each result to its simplest form and mark it ^ns. Each complete answer will receive 10 credits. 1. Define numerator, denominator, divisor, factor, pro- portion. 2. Find the weight in kilograms of a stone 1 meter square 436 COMPLETE ARITHMETIC. and .4 of a meter thick, assuming that the stone is 2| times as heavy as water. o c- -,•. 3 X 4 X 4.2 3. Simphfy ,^ 4. Find the interest on $375 at 4^% from July 1, last year, to the present time. 5. Multiply 65.15 by 3.14159 and divide the result by 57.296, finding a result correct to three decimal places. 6. Find the cost at $50 an acre of a rectangular field 1650 feet long and 825 feet wide. 7. Find the time required to fill a cistern 8 feet square and 5 feet deep by a pipe which admits water at the rate of 1 quart a second. 8. Make a receipted bill of the following : J. L. Eobbins & Co. sold this day to Samuel Jones 8 yards of cloth at 37|- cents, 24 yards of calico at 8^ cents, 1 dozen handkerchiefs at 12|- cents each, and 3 dozen towels at $2.50 a dozen. 9. Find the cost of four sticks of timber, each 8 inches by 10 inches and 30 feet long, at $15 a 1000 feet board measure. 10. Find the least common multiple of 153, 204, and 510. 11. If 4% bonds to the amount of $8000 face value are bought at 92^%, find the cost of the bonds, and the rate of income on the investment. 12. If 3 men can do a piece of work in 8 days of 10 hours each, how many men will be required to do the same work in 6 days of 8 hours each ? (Solve by proportion.) 13. By selling a horse for $144, a profit of 60 per cent is made; find the cost of the horse. 14. The diameter of a bicycle wheel is 28 inches; find the number of revolutions it makes in going 1 mile. 15. Find the square root of 7, correct to three decimal places. PAKT III. 437 SET VI. Examination for Admission to State High Schools, Minnesota, 1898. Time two hours. Answer any six — no more. If more are attempted and the student does not designate which six he wishes to be graded upon, the first six answers will be taken. 1. a (2) .What is the ratio of 2 to .90 ? b (2) What is the ratio of | to ^V ? c (2) What is the ratio of ^\ to | ? d (2) What is the ratio of 90% to .09 ? e (2) What is the ratio of 70% to 50% ? 2. a (5) What is the cubical content of a cellar 15 ft. wide, 20 ft. long, and 10 ft. deep? (In the solution express all operations in the form of equations.) b (1) What unit (or units) of measure did you use in the example ? c (4) Describe the unit of measure used in measuring boards. 3. (10) A merchant sells an overcoat for S22 ; a suit of clothes for $2 3 ; a hat for S5. On the overcoat he makes 1 % of the cost ; on the suit 15 %, and on the hat 25 %. What per cent of the cost of the goods does he make on the entire sale ? 4. a (5) A man bought a watch and a chain for $70. One-half of the cost of the watch equals f of the cost of the chain. What was the cost of each ? b (5) Analyze. 5. (10) The rates at which A, B, and C work are to each other as 2, 3, and 4. What integers wdll indicate the time it will take each to do a certain piece of work ? 6. (10) How long a rope must a horse have in order that he may graze over an acre of land, if he be tied to a stake in the center of a field ? 438 COMPLETE ARITHMETIC. 7. (10) B buys bank stock at 78 and sells it at 84. C buys railroad stock at 70 and sells at 75. Each buys the same number of shares, and B makes $1000 more than C. How much money did B invest? 8. (10) Make (5) and solve (5) a problem in the solution of which it will be necessary to extract the square root. 9. a (5) What is the interest on $700 for 1 yr. 5 mo. and 10 da. at 7% per annum? b (5) Analyze. 10. (10) A servant is engaged for a year for $280 and a suit of clothes. He leaves at the end of six months and receives $130 and the suit. What is the value of the suit? (An algebraic solution is allowed for this problem.) 11. a (5) How long will it take $1560 at 5% simple interest to gain $426.83^? b (5) Analyze. SET VII. Examination for State Certificates. lUinois, 1898. Time, two hours. 1. (a) Every fraction is a ratio. Explain. (b) Every integral number is a ratio. Explain. 2. In the report of the Committee of Ten it is recom- mended that " the course in arithmetic be at the same time abridged and enriched." (a) Tell what abridgment you regard as important. (b) Tell what enrichment you consider essential. 3. (a) Tell what sense-magnitudes you prefer to use in presenting to third grade pupils the subject of fractions, (b) At wliat stage of the work do you think sense-magni- tudes should give place to imaginative magnitudes ? PART III. 439 4. When and to what extent should pupils in the grades be required to memorize definitions of mathematical terms ? (b) When and to what extent should pupils be required to memorize directions for performing operations ? 5. Mention all the standard linear units with which you are familiar, and give the ratio of each (either exact or approximate) to some other linear unit. 6. What is the weight of 1000 feet of white pine boards (1 inch in thickness) if the specific gravity of the boards is .6? 7. The foundation of my house is 32 feet square on the outside. The house is 20 feet high to the plates and the roof has the usual eave-projections. Give approximately the number of barrels (31i gal.) of water that will fall upon this roof in one year, if the rain-fall is 34 J inches. 8. Give approximately the following ratios: (a) Of the circumference to the diameter of a circle. (b) Of the diagonal to the side of a square. (c) Of the area of a circle to the area of its circumscribed square. (d) Of the area of a circle to the area of its inscribed square. 9. What single discount is equal to a discount of 45 per cent and " 5 lO's," i. e., to "45 and 10 and 10 and 10 and 10 and 10. off," from the list price ? 10. If money is worth 6 per cent annual interest now and prospectively, what is the actual cash value of a note of SI 000 running two years and drawing 5 per cent interest, payable annually ? 440 COMPLETE ARITHMETIC. VIII. The Bank Test.* To THE Tkachp:k. — Below are figures representing 51 sums of money. Procure 51 blank checks and cause them to be filled, using the sums here given. Draw one check from the 51 checks and give the remaining 50 to a pupil to transcribe the sums and find their amount. When the pu^Dil obtains a result the teacher can quickly determine whether it is correct by comparing it with the sum of the 51 checks, less the sum named on the check drawn out. Before the checks are given to the second pupil, the check removed should be replaced and another withdrawn. Thus, although each pupil should obtain a result differing from that obtained by the pupil preceding him, its accuracy can be quickly tested by the teacher. To THE Pupil. — Can you, on first trial, transcribe the sums named on 50 checks and find the amount accurately in 30 minutes ? $324.56 $565.60 $123.20 $75.00 $234.50 $525.40 $312.95 $190.35 $46.45 $112.00 $86.50 $250.00 $325.00 $86.74 $91.23 $50.00 $302.26 $59.29 $12.65 $8.25 $7.75 $875.00 $1.50 $431.05 $201.45 $34.36 $85.40 $90.00 $130.25 $212.24 $230.94 $642.45 $71.20 $708.30 $60.00 $75.00 $1250.25 $6.50 $500.00 $2324.45 $9.10 $101.50 $36.09 $275.00 $150.00 $2.50 $1008.60 $140.65 $256.74 $987.84 $50.00 *In a leading bank in Chicago, it is customary to test applicants for positions as accountants by placing before them 150 checks, requiring each applicant to copy the sums named on the checks and find their amount. The author of this book is informed that the average inexperienced applicant does this in about 30 minutes, with some errors, however, both in transcribing and in footing. An expert accountant can do this amount of work accurately in 6 minutes. PART III. 441 IX. Curious Comparisons. 1. If a pig whose girth is 2 feet weighs 50 lb., what is the weight of a similarly proportioned pig whose girth is 4 feet? 2. If a disk of dough 15 inches in diameter is sufficient for 20 doughnuts, how many such doughnuts can be made from a disk 30 inches in diameter ? 3. The bore of a 1 0-inch gun is how many times as large as the bore of a 2 -inch gun ? 4. The ball of a 10-inch gun is how many times as large as the ball of a 2 -inch gun ? 5. A square, a pentagon, a hexagon, an octagon, and a circle have equal perimeters, (a) Which has the greatest area ? (b) Which has the least area ? 6. The capacity of a cistern 6 feet in diameter and 6 feet deep is about 40 barrels. What is the capacity of a cistern 12 feet in diameter and 12 feet deep ? 7. A 2|-inch pipe is how many times as large as a 1-inch pipe ? 8. If a man 6 feet tall w^eighs 190 lb., how much would a similarly proportioned giant 12 feet tall weigh ? 9. In a certain orchard the trees are 15 feet apart each way and there are 800 trees. How many trees in an orchard of equal size, the trees being 30 feet apart each way ? 10. A ball of yarn 3 inches in diameter is sufficient for one mitten. How many mittens can be made from a ball 6 inches in diameter ? 11. A grindstone was originally 30 inches in diameter. It has been worn until it is but 15 inches in diameter. What part of the stone has been worn away ? 12. A square and an oblong have equal areas. Which has the greater perimeter ? 442 COMPLETE ARITHMETIC. X. Puzzling Problems. 1. If a person traveling as expeditiously as possible from Boston to San Francisco, should mail a letter to his friend in Boston every day at noon, how often would the letters be received in Boston ? 2. If a man and a boy, the boy doing exactly one-half as much work as the man, can hoe one and one-half acres of corn in one and one-half days, how many acres can 6 men hoe in 6 days ? 3. John and James sold apples together. The first day they sold 60 apples at the rate of 5 apples for 2 cents, and re- ceived 24 cents. The second day they divided the apples. John took 30 of the larger apples and sold them at the rate of 2 for 1 cent. James took the remaining 30 apples and sold them at the rate of 3 for 1 cent. They received 25 cents. Why did they receive one cent more the second day than the first ? 4. A pile of four-foot wood stands upon a hill-side. The pile is 8 feet long (measured on the ground), and 4 feet high (measured vertically). Does the pile contain one cord ? 5. A man had shingles enough to cover his house if he laid them 4 inches to the weather. He laid them 4i inches to the weather. What part of the shingles provided remained ? Explain. 6. If on a line of railroad connecting Chicago and San Francisco one passenger train leaves Chicago daily at 6 o'clock a. m., and makes the journey to San Francisco in exactly five days, and one train leaves San Francisco daily at 6 o'clock p. m., and makes the journey to Chicago in exactly five days, (a) a person taking the train at Chicago will meet how many passenger trains while going to San Francisco ? (b) How many trains of passenger cars required to equip the road ? EXPLANATOEY NOTES. Note 1 . The forty-five primary facts of addition are as follows : 1 2 2 3 3 4 3 4 5 4 5 6 4 5 6 1 1 2 1 2 1 3 o 1 3 2 1 4 3 2 2 3 4 4 5 5 6 6 G 7 7 7 8 8 8 7 o 6 7 8 5 6 7 8 9 6 7 8 9 6 1 4 3 2 1 5 4 3 2 1 5 4 3 2 6 8 9 9 9 9 10 10 10 10 10 11 11 11 11 12 7 8 9 7 8 9 7 8 9 8 9 8 9 9 9 5 4 3 6 5 4 7 6 5 7 6 8 7 8 9 The nine facts in full-faced type should receive special attention. Pupils seldom fail to memorize the other thirty-six facts. Note 2. There are eighty-one primary facts of subtraction; that is, two for every primary fact of addition except the 1st, 3rd, 7th, 13th, 21st, 30th, 37th, 42nd, and 45th. The facts of subtraction should be learned while learning the facts of addition. If a pupil really knows that 8 and 9 equal 17, he knows also that 17 less 8 = 9, and 17 less 9 = 8. Note 3. When the sign of multiplication is followed by a frac- tion, it indicates that a certain part of the number preceding the sign is to be repeated as many times as there are units in the numerator of the fraction following the sign; thus, 12 x I, means, that 1 fourth of 12 is to be repeated 3 times; 50 X -5, means, that 1 tenth of 50 is to be repeated 5 times. Note 4. This sign, X, is sometimes so used that it means times, thus, 3 X <156, must be read, three times six dollars. 3x6, may be read, three multiplied by six or three times six. As employed in this 443 444 COMPLETE ARITHMETIC. book, the sign never means times. Instead of 3 x $6, the author pre- fers $6 X 3. It is believed that the restriction of this sign to one use and to one meaning, at least in the first years of arithmetical study, will promote clearness of thought and accuracy in expression. Note 5. Without danger of ambiguity, the sign, X, is sometimes used in this book and elsewhere in place of the word by; thus, 1 pc. of 2 X 4, 12 (to be read, 1 pc. of 2 hy 4, 12) means, a piece of lum- ber 2 inches thick, 4 inches wide, and 12 feet long. Note 6. Besides those problems in which either the multipli- cand or the multiplier is 1, and which require no eifort on the part of the pupil beyond learning to count, there are sixty-four primary facts of multiplication that must be perfectly memorized before the pupil can acquire facility in the process. They are as follows : 2 times 2=4 3 times 2=6 4 times 2=8 5 times 2 = 10 2 « 3=6 3 3=9 4 3 =12 5 3 = 15 2 " 4=8 3 4 = 12 4 4 = 16 5 4=20 2 " .5 = 10 3 5 = 15 4 5 = 20 5 5 = 25 2 " 6 = 12 3 6 = 18 4 6 = 24 5 6 =30 2 " 7 = 14 3 7 = 21 4 7 =28 5 7 = 35 2 " 8 = 16 3 8 = 24 4 8 = 32 5 8 = 40 2 " 9 = 18 3 9 ==27 4 9 =36 5 9 =45 6 times 2 = 12 7 times 2 = 14 8 times 2 = 16 9 times 2 =18 6 " 3 = 18 7 3 = 21 8 3 = 24 9 3 = 27 6 " 4 = 24 7 4 = 28 8 4 = 32 9 4 = 36 6 " 5 = 30 7 5 = 35 8 5 = 40 9 5 = 45 6 " 6 = 36 7 6 =42 8 6 = 48 9 6 = 54 6 ^' 7 = 42 7 7 =49 8 7 = 56 9 7 = 63 6 " 8 = 48 7 8 = 56 8 8 = 64 9 8 = 72 6 " 9 = 54 7 9 = 63 8 9 = 72 9 9 = 81 Although a knowledge of the "elevens" and "twelves" of the table as it is usually given is convenient and helpful, it will be observed that it is not a necessity in the process of multiplication. The facts given above include all that are essentially fundamental. PART III. 445 Note 7. The sign -7-, which is read divided by, has two meanings in concrete problems, which correspond to the two cases in division. In one case it means, find how many times the divisor is contained in the dividend; in the other case it mea^ns, find one of a certain number of equal parts into which the dividend is supposed to be divided. In each case there is division into equal parts. In the first case, the quotient tells the number of parts. In the second case, the quotient tells the size of one part. $18 -T- $2, means, ,y?n(Z how many times $2 are contained in $18. $18 -f- 2, means, find 1 half of $18. (See foot-note, p. 192.) Note 8. There are, in a sense, 128 primary facts of division, — two for each one of the sixty-four facts of multiplication. These facts are so closely related to the facts of multiplication that they should be learned in connection with the multiplication table. If a child really perceives that five fours ( : : : : : : : : : : ) are 20, he will also Know that 4 is contained lu 20 five times, and that 1 fifth of 20 is 4. Note 9. If from a square piece of paper, the largest possible circle be cut, a little less than \ of the paper will be cut away. Hence a circle is a little more than | (.78+) of its circumscribed square. Observe that the diameter of a circle is equal to the side of its circumscribed square. Note 10. If from a cube of wood, the largest possible sphere be cut, a little less than \ of the wood will be cut away. Hence a sphere is a little more than \ (.52 +) of its circumscribed cube. Observe that the diameter of a sphere is equal to the edge of its circumscribed cube. Note 11. A piece of board 1 inch wide, 1 inch thick, and 12 feet long, is 1 ft. of lumber. Hence the number of feet of lumber in any 12-foot stick, is equal to the number of square inches in its cross-section. Note 12. If from a square right prism of wood the largest possi- ble cylinder be cut, a little less than \ of the wood will be cut away. 446 COMPLETE ARITHMETIC. Hence a cylinder is a little more than | (.78 + ) of its circumscribed square right prism. Note 13. If from a cylinder of wood whose diameter equals its altitude the largest possible sphere be cut, exactly ^ of the wood will be cut away. Hence a sphere is exactly | of a cylinder whose diameter and altitude are each equal to the diameter of the sphere. Note 14. If from a cylinder of wood the largest possible cone be cut, exactly | of the wood will be cut away. Hence a cone is exactly ^ of a cylinder of equal diameter and altitude. Note 15. If from a square right prism of wood the largest possible pyramid be cut, exactly | of the wood will be cut away. Hence a square pyramid is exactly -^- of a square right prism of equal base and altitude. Note 16. The specific gravity of a liquid or solid is the ratio of its weight to the weight of an equal bulk of pure water. PROTRACTOR. Carefully paste this sheet upon card-board; then cut out the protractor with a sharp knife and preserve it for use in measuring and in constructing angles. See pages 239, 249, 259, etc. 447 VB 35822 541^20 UNIVERSITY OF CAUFORNIA LIBRARY