6' Co (^. o) (J (o 6) Co (-, '^ (i '^) Co OUR oy\r is> U Y LAMDIlE, a. m. I I t GREENSBOKO, N. C: PUBLISHEJP BY STERLING, CAMPBELL & ALBRIGHT. 1 s crs . <^ ( i i4>-t^t'i^i ; T PERKINS LIBRARY Uuke University Kare Dooks TRINITY COLLEGE LIBRARY DURHAM, N. C. 1903 Gift of Dr. and Mrs. Drcd Peacock J ^ %. O XT H O "W M SCHOOL ARITHMETIC. B Y S. LA-jSTDER, a., m. )0 *•* kO GREENSBORO, N. C: 18 6 3 PUBLISHED BY STERLING, CAMPBELL & ALBRIGHT. Richmond, Va., W. Harorave White. Entered according to Act of Congress, in the year I860, By S. lander, In the Clerk's Office of tlie District Court of the Con- federate States, for the District of Cape Fear, North-Carolina. i 'J 4i P 11 E F A C E . In presenting to the public perhaps the first Arithmetic whose authorship and publication belong exclusively to the Confederate States, I call attention to the following as its leading characteristics. J . The pupil is furnished with a model for each class of operations, by which he may know precisely what kind of explanation is required of him as he recites. 2. The distinction between abstract and concrete num- bers is carefully kept up throughout the whole book. 3. The Tables of Relations of Concrete Numbers are un- usually full and convenient. 4. The problems are designed to call into exercise the pupil's practical common sense, as well as to assist him in acquiring a correct knowledge of Arithmetic. 5. The results of about two-thirds of the problems are given : those of the* remainder are omitted, and a few of those given contain intentional errors, to test the pupil's self-reliance. 6. Progressions, and Mensuration, and the ordinary methods of extracting roots, are excluded entirely, because they lie beyond the province of Arithmetic. 7. No space is wasted by the insertion of questions on the text. A teacher who can not instruct without the help of questions, will succeed but poorly even with them. IV PREFACE. 8. The discouraging contradictions -which are so nume- rous in ''our best Arithmetics/' have beew sedulously avoided. For the neat appearance of the book I am much indebt- ed to my publishers, who have spared neither pains nor ex- pense to bring it out creditably ; and I am under especial obligations to my friend Prof. Theo. F. Wolle, of Edge- worth Female Seminary, without whose constant vigilance in revising the sheet.'?; no approximation to its present ac- curacy could have been attained. I invite my fellow-teachers to try the book by the unly <ure test, the test of the school-room ; and I will thank- fully receive any propositions of improvement which their examinations may suggest. Our Own Primary Arithmetic will follow this as soon a.< possible. ' S. LANDER. LiNCOLNTON, N. C, August Ij 1863, 6 N T E N T S Introductiou, v i Arabic Notation, , -^ Roman Notation , 1 ♦^ Operations, • Addition of Abstract Integers, 17 Subtraction of Abstract Integers, 2r, Multiplication of Abstract Integers. 31 Division of Abstract Integers, 41 Contraction in Addition, 54 Contraction in Subtraction, 5:") Contractions in Multiplication, 5() Contractions in Division,. Gf) Oeneral Principles of Division, 72 , Measures and Multiples, 82 Prime Factors, ■• ■ • • • 99 Involution, 100 Evolution, lOi Grreatest Common Measurv; 103 Least Common Multiple, •-• 10(5 <'ommon Fractions, lOD Reduction of Common Fractions, 12(' Addition of Common Fractions, 131 Subtraction of Common Fractions, 134 Multiplication of Common Fraction.^ 139 Division of Common Fractions — 146 Cancellation, 15r' Decimal Fractions, — Notation,... 154 Addition of Decimal Fractions,.... 159 Subtraction of Decimal Fractions, 161 Multiplication of Decimal Fractions, 162 Division of Decimal Fractions, 164 Contraction in Multiplication, 167 Contraction in Division 168 Tl CONTENTS. Relations of Common and Decimal Fractions, §169 Concrete Numbers, — Relations, 177 Addition of Concrete Numbers, 178 Subtraction of Concrete Num.bers, 180 Multiplication of Concrete Numbers,, c 182 Division of Concrete Numbers, 184 Reductipn of Concrete Numbers, 186 Compound Numbers, , . , 199 Addition of Compound Numbers, 200 Subtraction of Compound Numbers, 201 Multiplication of Compound Numbers, 202 Division of Compound Numbers, ., 208 Aliquot Parts ; or, Practice, 205 Contraction in Multiplication, 208 Contraction in Division, 209 Ratio, 210 Simple Proportion, 2 1 G Compound Proportion, 224 Partitive Proportion ; or. Fellowship, 229 Conjoined Proportion ; or. The Chain Rule, 232 Percentage, i 233 Jnterest,. ..,......; 242 Partial Payments, c 250 Compound Interest,. 251 Discount,. 252 Bank Discount,... , 255 Average, ^61 Alligation Medial, 264 Alligation Alternate,.*. _ 266 liquation of Payments,.. 268 ARITHMETIC. INTRODUCTION. § 1 . Arithmetic is the science of numbers. ^ 2. A unit is any single tiling f as, one, one dollar. § 8. A number is a collection of units ; as, three, two <lollars, four men. § 4. An abstract number is one whose unit is not speci- tied ; as, two, forty, seventy-one, eight. § 5. A concrete number is one whose unit is specified : us, ten dollars, forty men, seventy-one bales, eight books. § 6. Two or more numbers are similar when they have the same unit ; as, two, five, and seventy ; three men and six men. § 7. Two or more numbers are dissimilar when they have different units ; as, two, five dollarst, seventy men, three books. iVb^e.— All abstract numbers are similar. § 8. A compound number is a concrete number expressed in two or more denominations;, as, three dollars, fifty cents; ten hogsheads, forty gallons, three gills ; ten miles, seven furlongs, seventeen rods. §9 ARABIC NOTATION, ARABIC NOTATION. ^ 9. Notation is the luethod of expressing numbers \>y figures. The Ar'abic system, which is the one in common use, is called also the decimal system, partly because it employs ten figures. These figures are : naught or zero, 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine. The figure is used to fill vacant places^ and is omitted in reading. » § 10. A ten is a collueiioAi often units, and is called :i unit of the second order. A hundred is ten tens,»or one hundred units, and is called a unit of the third order. A thousand is ten hundreds, or one thousand units ; and is called a unit of ihQfourtli order. So, ten units of any order jiiake one of the next higher. § 11. A single figure denotes units; as, 3, three units. 5, five units, 7, seven units. § 12. When two figures are written together, the one on the right denotes units, and the rthertens; as, 23, two tcn.^- and three units, that is, twenty-three units ; 34, three t^wi^ and four units, that is, thirty-four units, or, simply, thirty- four. Read 27, 66, 73, 37, 84, 48, 99, ^6, 43, 55, 79, 80, KS. § 13. When three figures are written together, the one on the right denotes units, the next tens, and the other hundreds ; as, 123, one hundred, two tens, and three units, or, one hundred and tw^enty- three ; 321, three hundred and twenty -one ; 132, one hundred and fliirty-two ; 402, four hundred and two. ARABIC NOTATION. §15 Read 647, 864, 420, 301, 753, 587. 357, 735, 608, 740, 047, 306, 700, 609, 069, 009, 290, 391, 001. § 14. When more than three figures are written together, they arc separated into periods of three figures each, begin- ning at the right ; find^ in each..period, the three figures de- note respectively units, tens, and hundreds, of that period. § 15. The names of the periods and their order from right to left arc given in the following 3 a o o ■X3 'rJ Ph H a» <y CO tjj o c -3 r— * </, • cr fl a o »— 1 a o O t! C3 O '3 O 't, rA 1-^ C a a E-i ca EH t^ 123,004,500,060,000,000,700,089,000,897,060,000,543,210. Thfe above number is read, — one hundred and twenty- three duodecillions, four undeeillions, five hundred decil- iious, sixty nonillions, seven hundred scxti-llions, eighty- nine quintillions, eight hundred and ninetj -seven trillions, .-ixty billions, five hundred and forty-three thousand, two hundred and ten. BuLE FOii READING i»UMBERS.—>S'<rjDamfe the jicjiivei^ into periods of three figures each, beginning at the right; then, heginning at the left, read each period as if it stood alone, •md pronounce the 7i(ime of the period after reading it, Kead the following : 12^ 21, 37, 86; 793, 842, 209, 319; 1^346, 7907, 5432, 8642, 4C04, 1861, 1775 ; 24608, 13579, 10724, 40047, 78009; 475213, 570903, 400101, 300003; 1230456,2040608,7901035,8000005 ; 70083790,245000542, 5430102046, 146070080009, 9000800706040, 600005000 r §15 ABSTRACT NUMBERS. Write tbe following numbers in figures : 35. Seventy-four. 36. Four hundred and forty-eight. 37. Five thousand, three hundred and ninety-seven. 38. Sixty thousand, and seventeen. ^ 39.. Seven hundred and forty thousand, eight hundred and forty-one. 40. Eight millions, sev-enty thousand, and seventy-nine. 41. Ninety-four millions, sixteen thousand, four hundred and fourteen. 42. Thre« hundred millions, and three. 43. Two billions, two millions, two thousand, and two. 44. Ten billions, ten millions, ten thousand, and ten. 45. Nine hundred and twenty-five billions, eight thou- sand, and sixteen. 46. Eight trillions, seven billions, sixty millions, five thousand, and four. 47. Seventy trillions, eighty -nine millions, and twenty- one. 48. Six hundred and forty-two trillions, three hundred thousand. 49. Fifty-three quadrillions, eleven billions, and seventy- three. 50. Four hundred and four quintillions, two hundred and two millions. 51. Thirty sextillions, forty quintillions, fifty trillions, six hundred and two. 52. Two octillions, four quadrillions, six hundred and eight thousand. 53. Ten decillions, twelve nonillions, fourteen millions, and ninety-nine. 54. Nine ijonillions, ten* millions, and twenty-seven. 10 ROMAN NOTATION. §16 ROMAN NOTATION. yj) IG. The Koman Notation employs the following seven letters : I one, V five, X ten, L fifty, C one hundred, D five hundred, and M one thousand. All integral numbers may be denoted by combining these letters accordiug to the following rules : 1. Any letter doubled denotes twice its simple value ; tripled denotes three times, and so on. Thus, 11=2, XX=20, CCC = 300. 'J 2. If a letter of less value is placed after one of greater value, tlie less is to be added to the greater. Thus, ¥1=6, XV=15, CGL=250. 3. If a letter of less value is placed he fore one of greater value, the less is to be subtracted from the greater. Thus, iy==4, XC=90, CD = 400. 4. If a letter of less value is placed hciwetni^o of greater value, the less is to be subtracted from the sum of the other two. Thus, XIX= 19, Xiy = U, XCIX=99. 5. A dash placed over a letter multiplies its value by 1000. Thus, "1=50000/0 = 100000. The above rules are sufficiently exemplified in the fal- lowing T.4lI5LiJEC. I--=l XI-=11 XXI=21 C=100 II----2 Xll-^] ■-. XXII=22 CC=200 111=3 xiri=i3 :vXIII=2.3 CD=400 IV=^1 XlVc^li XXX=30 D=500 V=5 XV=15 XL=;:40 DC=600 VI=6 XVI^IG L=50 M=1000 ^ VIIr=7 XVII=17 LX=60 MC=1100 ' vni=8 XVIII=18 LXX=70 MM=2000 IX=9 XIX=19 LXXX=80 11=1000000 X=^10 XX=::20 XC=90 U MDCCCLXIII:^1808 §1' ABSTRACT NUMBERS. OPEKATIONS. There are four operations in Arithmetic ; Addition, Sub- traction, Multiplication, and Division. We will explain these operations in succession, first with reference to ab- stract numbers, and afterwards with reference to concrete numbers. ADDITION OF ABSTRACT NUMBERS. § 17. Addition is the operation of finding one number equal to several other numbers put together. § 18. The result of addition is called the sum of the num- bers added. Thus, 10 is the sum of 6 and 4. § 19. The sign of addition^ -,'-, is read jo/t^s. When pla- ced before a number, it denotes that it is to be added to any other additive number with which it is connected. Thus, ()-f4, 6 plus 4, denotes four added to six. § 20. l^hQ sign of equality^ =, is read i^ equal to. When, placed between two expressions it denotes that they are equal to each other. Thus, 6+4=10 Also, 7+4 + 3 = 8 + 6. 1 aud are 2 3 and are 3 4 and are 4 5 aud are ;' '1 and 1 are 3 3 and 1 are 4 4 and 1 are 5 5 and 1 are G li and 2 are 4 3 and 2 are 5 4 and 2 are 6 5 and 2 are 7 'J and 3 are 5 3 and 3 are 6 4 and 3 are 7 5 and 3 are 8 2 and 4- are 3 and 4 are 7 4 and 4 are 8 5 and 4 are 9 li and 5 are 7 3 and o f.re 8 4 and 5 are 9 5 and 5 are 10 il and 6 are 8 3 and C are 9 J 4 and 6 are 10 5 and 6 are 11 2 and 7 are 9 3 and 7 are 10 1 4 and 7 are 11 5 and 7 are 12 2 and 8 are 10 3 and 8 are 11 4 and 8 are 12 5 and 8 are IS 2 and are 11 3 and are 12 4 and 9 are 13 5 and 9 are 14 12 ADDITION OF INTEGERS §22 G and arc G G and 1 nre 7 6 and 2 nre 8 G and 3 are 9 G and 4 are 10 G and 5 are 11 G and G are 12 G and 7 are 13 G and 8 are 14 <i and 9 are 15 7 and 7 and 7 and 7 and 7 and 7 and 7 and 7 and 7 and 7 and are 7 1 are 8 2 are 9 3 arc 10 4 are 11 5 are 12 G are 13 7 are 14 8 are 15 9 arc 16 8 and 8 and 8 and 8 and 8 and 8 and 8 and 8 and 8 and Sand are ^ T are 9 2 are 10 3 arc 1 1 4 are 12 5 are 13 6 are 14 7 are 15 8 are IG 9 are 17 9 andO 9 a4id 1 9 and 2 9 and 3 9 and 4 9 and 5 9 and 6 9 and 7 9 and 8 9 and 9 are U are 10 are 1 1 are 12 are 13 are 14 are 15 are IG are 17 are 18 Note. — Let the above table bo thoroughly memorized before the pupil advances farther. Ex. 1. Add togetkcr 102741, 42102, and 3050. 102741 ^ 21. Model.— 2 and 1 are 3; 5 and 4 arc 42102 9 ^ I ^j^(2 7 arc 8 ; 3 and 2 are 5, and 2 are 3050 7 4 i_ The sum is 147893. 147893 Note. — Let the teacher see to it that the pupil recites precisely according to the model ttoth here and -wherever a model is given. Explanation. — First, the numbers are arranged with units of the same order in the same column. Then, begin- ning at the right, the numbers in each column are added to- gether, and the sum is placed underneath in the same column . 2. Addtogetlier23456, 10203, and 56030. Sum, 89689. 3. Find the sum of 120242, 334124, and 224612. Note. — Let the pupil first say, -'Add the numbers together/" and then proceed as in the model. 4. What is the sum of 2400, 1505, and 3074 ? Ans. 6979. 5. Add 270, 102, 314, and 301 together. Sum, 987. 6. Add together 94085, 16275, and 3367. 94085 § 22. Model.— 7 and 5 are 12, and 5 1627S are 17, set down 7 ; 1 and 6 are 7, and 7 3367 are 14, and 8 are 22, set down 2 ; 2 and 3 Sum, 113727 are 5, and 2 are 7 ; 3 and 6 are 9, and 4 are 13, set down 3 ; 1 and 1 are 2, and 9 are 11, set down 11. The sum is 113727. 13 §22 ABSTRACT NITMBERS. , Explanation. — After arranging tlie numbers as in § 21, the sum of tlie column of units is found to be 17 units, that is, 1 ten and 7 units ; hence, the 7 is placed under the col- umn of units, and the 1 is afterwards added in with the column of tens. The sum of the column of tens are 22 tens, that is, 2 hundreds and 2 tens ; hence, the right hand 2 is placed under the column of tens, and the other 2 is added in with the column of hundreds. The sum of the column of hundreds is 7 hundreds, and the 7 is placed underneath in that column. The sum of the column of thousands is 13 thousands, that is, 1 ten-thousand and 3 thousands ; hence, the 3 is placed in the column of thousands, and the 1 is added in with the column of ten-thousands. The sum of the column of ten* thousands is 11 ten-thousands, that is, 1 hundred-thousand and 1 ten-thousand ; hence, the right hand 1 is placed in the column of ten-thousands, and the other 1 in the place of hundred-thousands. Rule. — Arrange the numbers icith units of the s<^nie. or- der in the same column. Beginning at the right, find the sum <?/ each column ; if this sum is expressed hy one figure, set it down under the col- umn; but if it is expressed by more than one figure, set th-e right hand figure under this column, and add the remaining figure or figures in with the next column. Set down the whole sum of the last column. Pnoor. — 1, Add as before, but begin at the top of each column. Or, 2. Find the sum of all the numbers but one, and to this sum add the number excepted. Ex. 7. Add together 234, 15G, 987, and 358. Sum, 1735. 8. Add together 1020, 304, 66, and 9. Sum, 1389, 14 ADDITION' OF INTEGERS. §22 9. Add together 2739, 9647, 271, 17, and 2U50. 10. Add together 169078, 270189, and 928608. Sum, 1367875. 11. Add together 27090, 2709, 2^905, 27, 2709050, and 2r0. Sum, 3010051. 12. Find tha sum of 369764, 275863, 10794, 273, 102469, and 1861. 13. Find the sum of 173594, 240680, 10305, 678, and 976531. Sum, 1401788. 14. Find the sum of 97347S25, 89734782, 28973478, 828973478, and 98289734. Sum, 1143319297. 15. Find the sum of 1928374560, 192837456, 1928, 19283745, 1928374, 192837, and 19283. 16. 907050301 4-80604020-1- 123123123-=what ? Ans. 1110777444. 17. 146-f 1375-M3795-f 246820-f 24682=what ? Ans. 2S681b. 18. 2620-h6202-f7593-4-3694-f 1735=what? 19. "What is the sum of 3426, 9120634, 52714, 9987, 1137, and 97579? 20. What is the sum of 26322, 50555, 37684, 898955, and 9024 ? Ans. 1022540. 21. WhatisiAie sum of 41084, 293347, 9139919, and 46552? Ans. 9520902. 22. What is the sum of 245301, 586642, 51407, 1752, 71283, and 42061 ? 23. What is the sum of 10, 105, 1057, 10572, 105723, 1057234, 10572349, 105723496, and 1057234968? Ans. 1174705532. 24. What is the sum of 135792468,246813579,159483726, 372684951, 123456789, 896745321, 896453217, arid 400500746 ? Ans. 3231930797. 15 §23 ABSTaACT NUMBERS. SUBTRACTION OP ABSTRACT NUMBERS. § 23. Subtraction is the operation of finding the differ- ence between two iiurabers, by taking the less from the greater. § 24. The number to he aubtracted is called the subtra- hend. § 25. The number to he diminislied is called the imnn- tmd. § 2%. The residt of subtraction is called the remainder or the difference. § 27. The sign of subtraction, — , is read minus. When placed before a number, it denotes that it is to be subtracted from the number with which it is connected. Thus, (>— 4, 6 minus 4, denotes 4 taken from 6. Also, 7—3=4. § 28. The remainder is not changed by increasing the minuend and the subtrahend equally. Thus, Min. 27 27 + 15=42 274-240=267 27 + 306;=333 Sub. 16 16+15=31 16+240=256 16 + 306=322 Kern. 11 n nn ~Y\ from 1 leaves 2 from 2 leaves from 2 leaves 1 2 from S leaves 1 from 3 leaves 2 2 from 4 leaves 2 from 4 leaves 3 2 from 5 leaves 3 from § leaves 4 2 from 6 leaves 4 from 6 leaves 5 2 from 7 leaves 5 from 7 leaves 6 2 from 8 leaves G from 8 leaves 7 2 from 9 leaves 7 from 9 leaves 8 2 from 10 leaves 8 from 10 leaves 9 2 from 11 leaves 9 from from from from 3 from from from from from from leaves leaves 1 leaves 2 leaves 3 7 leaves 4 8 leaves 5 9 leaves 6 10 leaves 7 11 leaves 8 12 leaves 9 16 SUBTRACTION OF INTEGERS. §29 4 from 4 lenves i 5 from 6 leaves i ^ from 6 leaves 4 from 5 leaves 1 j 5 from 6 leaves 1 i 6 from 7 leavea 1 I from leaves 2 ! 5 from 7 leaves 2 1 ^ from 8 leaves 2 i from 7 leaves 3 5 from 8 leaves 3 i 6 from 9 leaves 8 4 from 8 leaves 4 5 from leaves 4 ' 6 from 10 leaves 4 It from leaves 5 5 froru 10 leaves 5 I 6 from 11 leaves 6 1 from 10 leaves G 5 from 11 leaves 6 ; 6 from 12 leavei G 4 from 11 leaves 7 i» from 12 leaves 7 : 6 from 13 leaves 7 I from 12 leaves 8 from 13 leaves 8 : G from 14 leaves 8 '{ from 18 leaves 5 from 14 leaves 9 G from 15 leaves 9 7 from 7 leavea 8 from 8 leaves ' 9 from 9 leaves 7 from 8 leaves 1 8 from 9 leaves 1 \ 9 from 10 leaves 1 7 from 1> leaves 2 8 from 10 leaves 2 j 9 from 1 1 leaves 2 7 from 10 leave." 3 8 from 11 leavob 3 ' 9 from 12 leaves 3 7 from 1 1 leaves 4 8 from 12 leaves 4 9 from 13 leaves 4 7 from 12 leaves o 8 from 13 leave? 5 i 9 from 14 leaves 5 7 from 13 leaves G 8 from 14 leaves G ; 9 from 15 leaves G 7 from 14 leaves 7 8 from 15 leaves 7 1 9 from IG leaves 7 7 from 15 loaves 8 8 from 16 leaves 8 J 9 from 17 leaves 8 7 from IG leaves 9 8 from 17 leaves 9 ! 9 from 18 leaves Mill. 8ub. Rem. I. From 976348 subtract 35127. § 29. Model.— 7 from 8 leaves 1 ; 2 976348 35127 941221 from 4 leaves 2 ; 1 from 3 leaves 2 ; 5 from leaves 1 ; 3 from 7 leaves 4 ; from 9 leaves 9. The remainder is 941221. Explanation— The subtrahend is placed under the min- uend, with units of the same order in the same column. Then, beginning at the right, each figure of the subtrahend is taken from the corresponding figmre of the minuend, and the remainder is set underneath in the same column. 2. From 127936 subtract 14312. Rem. 113624. 3. From 96898 subtract 13456. 4. Subtract 864231 from 9557654. 5. Subtract 1024370 from 12357799. B 17 Rem. 123423. Rem. ] 1333429. §30 ABSTRACT NUMBERS. 6. Subtract 327739 from 573G47. Min. 573647 §30. Model.— 9 from 17 leaves 8 ; 4 Sub. 327739 from 4 leaves ; 7 from 16 leaves 9 ; 8 Rem. 245908 from 13 leaves 5; 3 from 7 loaves 4: 3 from 5 leaves 2. The remainder is 245908. Explanation. — After arranging the numbers as in § 29, it is required to take 9 units from 7 units : this can not be done ; hence, 1 ten, that is, 10 units, is added to the minu- end, giving 17 units, from which 9 units taken leaves 8 units. Then, because the minuend is increased 10 units or 1 ten, the subtrahend must be increased the same amount (§ 28). This gives 4 tens to be taken from the 4 tons of the minu- end, leaving tens. Again, 7 hundreds can not be taken from 6 hundreds ; hence, 1 thousand, that is, 10 hundreds, is added to the minuend, giving 16 hundred:?, from which 7' hundreds taken leaves 9 hundreds. Then, because the minuend is increased 10 hundreds or 1 thousand, the sub- trahend must be increased the same amount. The same kind of reasoning will explain the rest of the operation. Rule, — Place the siihtraJiend under (lie niinucnd, wilh units of the same order in the same column. Beginning at the rights take each figure of the subtrahend from the corresponding figure of the minuend. If any figure of the minuend is less than the corresponding figure of thc^ subtrahend, add 10 to this minuend figure ^ and add 1 to the subtrahend figure in tlie next column. Proof. — 1. Add the remainder to the subtrahend ; the sum will be equal to the minuend. Or, 2. Subtract the remainder from the minuend ; the difference will be equal t© the subtrahend. ' 18 SUBTEACTION OF INTEGERS. 'M Ex. 7. From S9G take 307. 8. From 1842 take 961. 9 From 2719 take 1827. 10. From 12791 take 9872. 11. From 24598 take 20689. 12. From 978637 take 97863. 13. From 1654278 take 755429. 14. Take 678902 from 896454. 1^. Take 1724937 from 1963869, 16'. Take 23468579 from 60050040. 17. Take 9879789 from 9900000. 18. Take 7890845 from lOOOOOOO. 19. Miiiuend = 1234567, Subtrahend Eem. 589. Eem. 881. Eem. 2919. Rem. 3909. Rem. 898849. Rem. 217552. Rem. 36581461. Rem. 20211. 765432. Rem. 469135. Begin by saying, "Subtract tlie Subtrahend from the Rem. 181765. Rem. 7777782. Ans. 2887879. one million, and Ans. 999901. Note Minuend."" ^ 20. Min.=290178, Sub. = 108405. 21. Sub.=20499, Min.-~1900623. 22. Sub. =987631, Min. =8765413. 23. 12646723-975894^=what? 24. 2468000— 970053 = what? 25. What is the difference between ninety -nine 1 Xoic. — Begin, "Subtract the less number from the greater.'" 26. What is the difference between thirty-seven billions, and eleven ? Ans. 36999999989. 27. What is the difference between nine thousand six hundred and thirteen, and five hundred and forty-two ? 28. What is the difference between eight thousand and tw«nty-six, and eight hundred and twenty-six? Ans. 7200. 29. What is the difference between five thousand four hundred and ninety, and seven hundred and sixty-two 1 19 |3l ABSTRACT NCMBERf. ^MULTlPLICATrON OF ABSTKAOT NUMBER.S. § ol. Multiplication is the operation of finding a num- Vier ■which shall contain one of two given num})ers as many times as there are units in the other. Thus, o times 6 are 18 : here 6 is multiplied by 3, be- '■auae 18 contains 6, 3 times. § 32. The number io he multiplied is called the muUi' f)Hcand. § 33. The inultiplying number is called the juuftiplier, § 84. The result of multiplication is called the product. § 35. Either the multiplicand or the multiplier is called a factor of the product, and thej both are called itfi factors. In general, one number is a factor of any other number which contains it on exact number of timea. Thus, 3 is a factor of 18 ; 4 is a factor of 12, or of 20 ; 5 is a factor of 10, of 15, of 30, or of 45. § 36. The si'^n of multiplication^ X, when placed between two numbers, denotes that one of them is to b« multiplied by the other. It is read times, when placed after the mul- tiplier, and multiplied by, when placed after th« multipli- cand. Thus, to denote that 6 ia to be multiplied by 3, we may Miy, 3x6, 3 times 6, or 6x3, 6 multiplied by 3. To denote the successive multiplication of more than two num- bers, periods are used. Thus, 2.3.5 = 30. 2 times 3 times f)=«30. § 37. The product of any two abstract factors ia the i^ame, no, matter which is used as multiplier. Thus, 3 x6s— «X3==18; 4x5-=5x4=20; 10x8-=8xl0=«80. 20 MULTIPLICATION OF INTEGERS. §o7 IVIUI-iTlJeLlCATIOlV T^VBLE. Oace is Twice are 3 times are It Once \ is 1 Twice 1 are i> 3 times 1 are o Once 2 is 2 Twice o Ml are 4 3 times 2 are 6 Once 3 is 3 Twice 3 are G 3 times 3 are 9 Once 4 is 4 Twice 4 are 8 3 times 4 are 12 ' hice T) is 5 Twice 5 arc 10 3. times 5 are 15 Once « is G Twice G are 12 '"> times G are 18 Once 4 is 7 Twice 7 are 14 8 times 7 are 21 Once 8 IS 8 Twice 8 are IG o times 8 are 24 Ouce 9 is 9 ' Twice 9 are 18 time3 are 27 Once 10 is 10 Twice 10 are 20 8 times 10 are 30 Once 11 is 11 Twice 11 are 22 times 11 are 33 Once 12 t is 1 o Twice 12 are 24 1 3 times 12 are 36 4 times are 5 times are 6 times are 4 times 1 are 4 5 times 1 are 5 6 times "1 are 6 4 times •) are 8 5 times O are 10 6 times 2 are 12 4 timei) 3 arQ 12 times 3 are 15 6 times 3 are 18 4 times 4 are IG 5 times 4 are 20 6 times 4 are 24 4 times 5 are 20 5 times 5 are 25 G times 5 are 3ft 4 timet G jire 24 5 times 6 are 30 times G are 36 4 timts 7 are 28 5 times 7 are 35 6 times 7 are 42 4 times 8 are 32 5 times 8 are 40 G times 8 arc 48 4 times ■ 9 are 36 5 times are 45 6 times 9 are 64 4 times 10 are 40 5 times 10 are 50 6 times 10 are 60 4 times 11 are 44 5 times 11 are 55 6 times 11 are GG 4 times 12 are 48 5 times 12 are 60 6 times 12 are 72 7 time:.> arc 8 times arc i iraes are 7 times 1 are 7 8 times 1 arc 8 9 t imes 1 are 9 y times 2 are 14 8 times are 16 9 t imes 2 are 18 ^ times 8 arc 21 8 times 8 are 24 9 t imes 3 are 27 T times 4 are 28 8 times 4 are 82 9 t imes 4 ar« 36 7 times T) are 35 8 times 5 are 40 9 t imes 5 are 45 7 tim(;s G aio 42 3 times G are 48 9 t imes 6 are 54 7 times 7 are 40 8 times 7 are 53 1 9 t iini>8 7 are {Vd *T times 8 jire oG 8 times 8 are 64 9 t imes 8 are 7a- "7 times are 68 8 times 9 are 72 9 t imes 9 arc 81 »T times 10 are 70 8 times le arc 80 9 t imes 10 are 90 T times 11 are 77 8 times 11 arc 88 9 t imes 11 are 9& 7 tiraos 12 arc »4 8 times 12 arc 9G 9 t imes [2 are 108 21 ABSTRACT NUMBERS. 10 10 10 times 10 times 10 times 10 times times times 10 limes 10 times 10 times 10 times 10 times 10 times 10 times are 1 are 2 are 3 a,re 4 are 5 are G are 7 are 8 are 9 are 10 are 11 are 12 are 11 til 10 n ti 20, 11 til SO 11 ti 40 11 ti 60 11 ti GO 11 ti 70 11 ti 8§ 11 ti 90 11 ti 100 11 ti 110 11 ti 120 11 ti mes are 12 times are mes 1 are 11 12 times 1 are 12 mes 2 are 22 times 2 are 24 mes 3 are 33 12 times 3 are 36 mes 4 are 44 12 times 4 are 48 mes 5 are 55 12 times 6 are 60 imes 6 are 66 12 times 6 are 72 mes 7 are 77 12 times 7 are ■84 imes 8 are 88 12 times 8 are 96 mes 9 are 99 12 times 9 are 108 mes 10 are 110 12 times 10 are 120 mes 11 are 121 1 ^ X.JU times 11 are 132 imes 12 are 132 12 times 12 are 144 Ex. 1. Multiply 24307 by 3. Multiplicand, 24307 § 33. Model.— 3 times 7 are Multiplier, 3 21, set down 1 ; 3 times are 0, Product, 72921 and 2 are 2; 3 times 3 are 9 ; 3 times 4 are 12, set down 2; 3 tfimes 2 are 6, and 1 are 7. The product is 72921. Explanation. — Tke smaller factor is placed under the larger. Then, beginning at the right, each figure of the upper number is taken 3 times, the right hand figure of each product is set down, and the remaining figure, if any,, is added to the next product. 3 times 7 units are 21 units, that is, 2 tens and 1 unit ; hence, 1 unit is set in the units' ]jlace, and 2 tens are added to the product of the tens. 2. Multiply 24307 by 40. Multiplicand, 24307 § 39^ Model.— 4 times' 7 are Multiplier, 40 28, sef^down 8 ; 4 times are 0, Product, 972280 aad 2 are 2 ; 4 times 3 are 12, set down 2 ; 4 times 4 are 16, and 1 are 17, set down 7 ; 4 times 2 are 8, and 1 are 9 : annex 0. The produ<;t is 972280. Explanation. — Since 10 units of any order make one of t4e next order on the left, any number is rflultiplied by 10 22 MULTIPLICATION OF INTEGERS. §40 by merely moving eacli of its figures one place to the left, and putting a in the place of units. Hence, to multiply by 40, each figure of the product by 4 is set one place to the left, and the units' place is filled with a 0. 3. Multiply 24307 by 43. Multiplicand, Multiplier, 1st partial prod. 2nd' partial prod. Product, 24307 43 72921 97228 1045201 § 40. Model. — 3 times 7 are 21, set down 1 ; 3 times are 0, and 2 arc 2 ; 3 times 3 are 9 ; 3 times 4 are 12, set down 2 ; 3 times 2 are 6, and 1 are 7 :— 4 times 7 are 28, set down 8 under 2 ; 4 times are 0, and 2 are 2 ; 4 times 3 are 12, set down 2 : 4 times 4 are IG, and 1 are 17, set down 7 ; 4 times 2 are 8, and 1 are 9. Add the partial products : 1 ; 8 acd 2 are 10, set down ; 1 and 2 are 3, and 9 are 12, set down 2 ; 1 and 2 are 3, and 2 are 5; 7 and 7 are 14, set down 4 ; 1 and 9 arc 10, set down 10. The product is 1045201. Explanation. — The upper nuniLur is multiplied, first by 3, as in § 38, and then by 40, as in § 39, except that the at the right is omitted, as being unnecessary, since the several figures can be placed in their proper columns with- out it. It must be remembered, however, that the second partial product is not 97,228, but 972,280. 4. Multiply 3047 by 246279. 5. Multiply 794378 by 4608. Multiplier, Multiplicand, Product, 246279 ; 3047 T723953 985116 738837 750412113 Multiplicand, Multiplier, ! Product, 794378 4608 6355024 4766268 3177512 3660493824 23 §40 ABSTRACT NUMBERS, Rule. — 1. When either factor contains hut one valuable figure. Set the smaller factor under the lafger. Beginning at the right, multiply each figure of the upper number by the lower number, set doicn the right hand figure of the product, and add the remaining figure, if any, to the next product ; but set doivn the whole of (he last product. 2. When the smaller factor contains more than one valua- ble figure. Set it under the larger ; multiply the upper fac- tor by each figure of the lower, setting the firat figure of each partial product under tin- .'n-i'iijli^ing figure which produced it, and add the partial products together in thai order. Proof. — Multiply the lower factor by tlie upper. Ex. 6. Multiply 3469 by 3. 7. Multiply 4'by 268. ' 8. Multiply 45274 by 5. 9. Multiply 56295 by 6. 10. Multiply 75397 by 7. 11. Multiply 9 by 98765. 12. Multiply 21179 by 27. 13. Multiply 97825 by 34. 14. Multiply 86906 by 45. 15. Multiply 279862 by 52. 16. Multiply 192837 by 67. 17. Multiply 293705 by 75. 18. Multiply 246835 by 83. 19. 1964326 x98=« what? Ans. 192503948. NoU. — Begin, " Maltiply the first number by the second.'" 20. What is the product of 2975x375 ? Aus. 1 1 15625. 21. What is the product of 3047x287 ? 24 Prod. 10407 Prod. 1072. Prod. 226370. Prod. 527779. Prod. §88885. Prod. 3326050. Prod. 3910770 Prod. 12920079. Prod. 22027875. DIVISION OF INTEGERS. §48 22. What isthe product of 40535x403? Ans. 19983755. 23. What isthe product of 4-027x4027? Aus. 16216729. 24. 719x729=whiitr 25. 92730465xl794 = wh.it > Aus. 1G635845421U. 2(5. 81G2035x28r>45=whatr' Au3. 233801492575. DIVISION OF ABSTRACT NUMBERS. § 41. Division is the operation of finding how many times one number is contained in an otlier. Thus, 4 in 20, "> times : her« 20 is divided hy 4, since 4 is contained 5 times in 20. § 42. Or, Division is the operation of separating a num- ber into some number of equal parts. Thus, if 20 is di- vided into 4 equal parts, each or the parts is 5. § 4$. The number to he divided is called the dividend. i^ 44. The dividing number is called the diviHor. § 45. The result of division is called the quotievl. § 46. When tlie division is not complete, the undivided p^rt of the dividend is called the remainder. Thus, Sin 35, 4 times, with 3 over ; here 35 is the dividend, 8 is the divi- sor, 4 is the quotient, and 3 is the remainder. § 47. The sign of division, -T-,is read dividcdhy. When placed between twu numbers, it denotes that the one before it is to be divided l>y tlie one after it. Thus, 20-7-5::rr:4. § 48. Division is sometimes denoted by placing the divi- dend over the divisor with a line between them. Tiius, ?i-=r4. 9r, Z') ^48 ABSTRACT NUMBERS. I3I^V^lS10I«ir TJLBLE. 1 in 0, no time | 2 in 0, no time 3 ] n 0, no time 1 in 1, onee 2 in 2, once o n 13, Onee 3 in 2, twice 2 in 4, twice 3 ] m 6, twice 1 in 3,' 3 times 2 in 6, 3 times 3 ] n 0, 3 times 1 in 4, 4 times 2 in 8, 4 times 3 i n 12, 4 times 1 iu 5, 5 times . 2 in 10, 5 times 3 in 15, 5 times 1 in ^, 6 times 2 in 12, 6 times 3 n 18, 6 times 1 in 7, 7 times 2 in 14, 7 times 3 ] n 21, 7 times I in 8, 8 times 2 in 10. 8 times 3 LU 24, 8 times 1 in 9, 9 times 2 in 18, 9 times 3 m 27, 9 time^ 4 in X no time ^ 5 in 0, no time 6 ] m "^"o, no time 4-in 4, once 5 in 5, once 6 m 6, once 4 in 8. twice 5 in 10, twice 6 m 12, twice 4 in 12, 3 times 5 in 15, 3 times 6 in 18, 3 times 4 in 16, 4 times "^5 in 20, 4 times 6 m 24, 4 times 4 in 20, 5 times 5 iu 25, 5 times 6 n 30, 5 times'. 4 in 24, 6 times 5 iu 30, 6 times 6 m 36, 6 times 4 in 26, 7 times 5 in 35, 7 times 6 n 42, 7 times 4 iu 32, 8 times 5 in 40, 8 times 6 n 48, 8 times 4 iu 36, 9 times 5 in 45, 9 times 6 : LU 54, 9 times 7 in 0, no time 8 in 0, no time 9 : n 0, no time 7 in 7, once 8 in 8, once ,9 in 9, once 7 in 14, twice 8 in 16, twice 9 m 18, twice 7 in 21, 3 times 8 in 24, 3 times 9 m 27, 3 times 7 in 28, 4 times 8 in 32, 4 times 9 U 36, 4 times 7 in 35, 5 times 8 in 40, 5 times 9 m 45, 5 times 7 in 42, 6^ times 8 in, 48, 6 timo-: 9 m 54, 6 times 7 in 4VJ, 7 times 8 in 56, 7 times 9 in 63, 7 times 7 in 66, 8 times 8 in 64, 8 times 9 in 72, 8 times 7 in 63, 9 (times 8 in 72, 9 times 9 in 81, 9 times 10 in 0,^ no time 11 in 0, no time '12" in i. ), no time 10 in 10, once 11 in 11, once 12 in V. I, once 10 in 20, twice 11 in 22 twice 12 in 2- :, twice 10 in 30, 3 times 11 in 83, 3 times 12 n 86 ), 8 times 10 in 40, 4 times 11 in 44 4 times 12 m 4^ \, 4 times 10 in 50, 5 times 11 in 55 5 times 12 in 6( ), 5 times 10 in 60, 6 times 11 in 66, 5 times 12 m 71 J, 6 times 10 in 70, 7 times 11 in 77 7 times 12 Q 8^ t, 7 times 10 in 80, 8 times 11 ia 88 ' 8 times 12 m 9( J, 8 times 10 in 90. 9 tiinos 11 iri 99 9 timp= 12 in 10^ 5, 9 times 26 DIVISION OF INTEGERS- §50 I. SHORT DIVISION. Ex. 1. Divide 3096 by 3. J)W^<^r,r^ 3) 3096, Dividend. ^ 49^ Model.— 3 in 3, 1032, Quotient. once ; 3 in 0, no time ; 3 in 9, 3 times; 3 in 6, twice. The quotient is 1032. Explanation. — The divisor is placed on the left of the liviilotid. Then, beginning at the left, the number in each rder of units is divided by 3, and each quotient figure is et in its proper column. Ex. 2. Divide 806X4 by 2. Quot. 40312. 3. Divide 8048 by 4. Quot. 2012. 4. Divide 90369 by 3. Quot. 30123. 5.. Divide 17120 by 8. 8)17120 g 50 MopEL.— 8 in 17, twice, with I over, 2140 set' down 2 ; 8 in 11, once, with 3 over, set down 1 ; 8 in 32, 4 times ; 8 in 0, no time. The quotient is 2140. Explanation. — 8 is not contained in 1, that is, in 1 ten- thousand, in its present form ; hence, 1 ten-thousand is re- duced to 10 thousands, and added to the 7 thousands, mak- ing 17 thousands. 8 is contained twice in 16 ; so that there i3 1 thousand still undivided. This is reduced to 10 hun- dreds, and added to the 1 hundred, making 11 hundreds. 8 is contained once in 8 ; so that there are 3 hundreds still undivided. These are reduced to 30 tens, and added to the 2 tens, making 32 tens. 8 is contained in 32 just 4 times. The of the dividend is retained in the quotient, to cause the several quotient figures, 2 thousands, 1 hun- dred, and 4 tens, to occupy their proper plares .27 §51 ABSTRACT NUMBERS. Ex. 6. Divide 36374 by 9. 9)36374 § 51^ Model.— 9 in 36, 4 times; 9 in 4041... 5 3, time, with 3 over, set down ; 9 in 37, 4 times', with 1 over, set down 4 ; 9 in 14, once, with 5 over, set down 1 in the quotient, and 5 a? remainder. The quotient is 4041, and the remainder 5. Explanation. — The division of the 5 units might be denoted ^-, as in § 48. Rule. — -Set the divisor on the left of the dividend, unth a line between them, and one under the dividend. Beginning at the left, see how many times the divisor is roiitained in each figure of the dividend, and set the result u/nder the dividend. Whenever there is a remainder, prefix it to the next fgure of the dividend, before dividing. If the divisor is not contained in any figure, except tlw first^ set under such figure, and regard it as a remainder, pjROOF. — Multiply the quotient by the divisor : the prod-' uct, increased by the remainder, if any, will be equal to th© dividend. Ex. 7. Divide 73052 by 2. Quot. 36526. 8. Divide 222345 by 3. Quot. 74115. 9. Divide 123456 by 4. lO! Divide 790530 by 5. Quot. 158106, 11. Divide 78920472 by C. Quot. 13153412. 12. Divide 945 by 7. * 13. Divide 1240128 by 8. . Quot. 155016. 14. Divide 743200173*^ by 9. Quot. 82577797. 15. Divide 4703750 by 10. 16. Divide 9009 by 11. Quot. 819. i7. Divide 721428 bv 12. Quot. 60119. 28 DIVISION OP INTEGERE. §52 U. LONG DIVISION. Ex. 18. Divide 2966232 by 925. Dividend, 29662321925, Divisor. 2775^ 13206, Quotient , ., ^, ^ . ■ ig^Q 29, 3 times ; multiply . _-^ . the divisor by 3 ; 3' ^7"^'^ times 5 are 15, set '^:^j^ down 5 ; 3 times 2 are 682, Remainder. 6, and 1 are 7 ; 3 times 9 are 27, 5et down 27 : ubtract the product from the dividend; 2; 5 from G leaves 1 ; 7 from 16 leaves 9; 8 from 9 leaves 1 : — 9 in 19, twice; multiply the divisor by 2; twice 5 are 10, set down 0; twice 2 arc 4, and 1 are 5; twice 9 are 18, set down 18 : subtract the product from the previous remain- der ; 3 ; U from 2 leaves 2 ; 5 fi^om 11 leaves 6; 9 from 9 leaves 0: — 9 in 6, no time ; annex 2: — 9 in 62, 6 times; Multiply the divisor by 6 ; 6 times 5 are 30, set down ; () times 2 are 12, and 3 arc 15, set down 5; 6 times 9 are 54, and 1 are 55 : subtract the product from the previous remainder ; from 2 leaves 2 ; 5 from 13 leaves 8 ; 6 from 12 leaves 6; 6 from 6 leaves 0. The quotient iso206,nnd the remainder 682. Explanation. — The divisor is placed on the right of the dividend, for convenience in multiplying. The number is used throughout as a trial divhor. As two figures ©f the real divisor arc thus omitted, two figures of each partial dividend must be omitted also. Hence, in the third step, we say 9 in 6, and not 9 in 62, until we have annexed an additional figure. The first quotient figuile stands for 3000 ; hence the first product is really 2775000, and the first remainder 191232; but, as we do not need all these 29 >53 ABSTRACT NUMBERS. figures for tke next step, we begin to subtract only one place to the right of the last valuable figure in the prod- uct- The division of the remainder might be expressed as in § 48. Ex. 19. Divide 6593 by 19. 6593119 57 oT« § 53. Model. — 2 in 6, 3 times; multiply ~7."q~- the divisor by 3 ; 3 times 9 are 27 ; sot nn. down 7 ; o times 1 are 3, and 2 are 5 : sub- tract the product from the dividend ; 9 ; ^^^ 7 from 15 leaves 8; 6 from 6 leaves :-^ — ^'^^ 2 in 8, 4 times ; multiply the divisor by 4 ; 4 times 9 are 36, set down 6 ; 4 times 1 are 4, and 3 are 7 : subtract this product from the previous remainder ; 3 ; 6 from 9 leaves 3 ; 7 from 8 leaves 1 : — 2 in 13, 7 times; multiply the divisor by 7 ; 7 times 9 are 63, set down 3 ; 7 times 1 are 7, and 6 are 13, set down 13 : subtract the product from the previous re- Qiainder; 0. The quotient is 347. Explanation. — If the second figure of the divisor is less than 5, the first figure is the trial divisor ; but, if the second figure is greater than 5, the trial divisor is one more than the first figure. If, on multiplying, a quotient figure be found to be too large or too small, let it be diminished or increased a unit at a time until the right result is attained. Rule. — Set the divisor on the right of the dividend, with a line between ihem, and one under the divisor. Beginning at the left, see how often the divisor is contained in the first part of the dividend : the result will he the first figure of the quotient. Multiply the divisor by this quotient figure, and subtract the product from that part of the divi- dend which loas used, annexing to the remainder the next figure of the dividend. 30 DIVISION OF INTEGERS. §53 Take this remainder as a second partial dividend, and from it obtain the second quotient Jif/ure. Muhiphj (he divi- sor by thi^Jigure, and subtract the product from the previous remainder^ annexing to the second remainder the next figure of the dividend. Continue this process till all the figures of ili'- dividend have been used. If any partial dividend will iiot contain the divisor, set in the quotient^ annex an other figure of the dividend, and divide again. Proof. 1. — The same as in § 51, for short division. Or, 2. Subtract the remainder, if any, from the divi dend ; divide this remainder by the quotient, and the re- sult will be the divisor. Ex. 20 Divide 18950 by 25. « Quot. 758. 21. Divide 17136 by 36. 22. Divide 42581 by 49 Quot. 8t)9. 23. Divideud=lG7'01, Div-i.-ur^57 Quot. 293. P^ Note. — Begin, "Divide the Dividend by tbo Divuor.'' 24. Dividend— 265oG, Divi3or^()2. 25. Dividend^l5076872, Divisor=:72. Quot. 209401. 26. Dividend— 30744, Divisor=:84. Quot. 366. 27. Divisor=97, Dividend==84002. " 28. Divisor=^125, Dividend-=15625. Quot. 125. 29. Divisor=:273, Dividendr=104832. Quot. 384. 30. Divisor=354, Dividend=94l64. 31. Divisor=465, Dividend^2G7375. Quot. 575. 32. Divisor=531, Dividend=340902. Quot. 642. 33. Divisor=^685, Dividend=543205. 34. Divisor:==721, Dividend=r2728264. Quot. 3784. 35. DiTisor=829, Dividend=5697717. 31 .^63 ABSTRACT NUMBERS. 36. r>ivisor=937, Dividendr=981976. Qu»t. 1048. 37. 5754375-4-1125=:whatl Ans. 5116. 38. 4515625-^2125=:what! 39. 48284964-T-3094=what ? Ans. 15606. 40. 24896825~-4105=what ? Ans. 6065. 41. 27206656-T-5216=what? 42. 45782172-^6327=what? Ans. 7236. 43. 313201258-f-7153=what? Ans. 43T86. 44. 293834463995 ~ 8405==what f 45. 572473044-i-9516=what? Ans. 60159. 46. 93939874943-- 10471=what? Ans. 8971433. 47. 151807041— 12321==what? 48. Dividend=l 2741 53376, Divisor=23456. Qwot. 54321. 49. Dividend==18p9739176, I>iTisor=34056. Quot. 54Q2I. 50. Dividend=2642079580, Divisor=40565. 51. Dividend==:2900 124304, Divisorzz=56504. Quot. 513p. 52. Divisorrr=:65405,Dmdend=6677 19645. Quot. 1020». 53. Divisoi— 74316, Dividend=4734969624. 54. DiTisor=81634, Dividend=7571 145330. Quot. 92745. 55. Dmsor=:95703, Dividend=1299551037. Quot. 13579. 56. Divisor=97531, Dividend=2999956029. .57. Divisoi— 36805, l)ividend==800655970. Quot. 21754, 58. Divisor=::234282, Dividend=83596737522. Quot. 356821. 59. Divisor=5276431, DiTidend^7105901 538475. 32 CONTRACTED ADDITION OF INTEGERS. §54 CONTRACTION IN ADDITION. ; Note. — The judicious teacher will omit this and most of the following contractions as his classes proceed through the book the first time. Ex. 1. Add together the following numbers : 469375 237924 472437 853214 975318 242326 §54. Model.— 26 and 10 are 36, and 8 are 44, and 10 are 54, and 4 are 58, and 30 are 88, and 7 are 95, and 20 are 115, and 4 are 119, and 70 are 189, and 5 are 194, set down 94: — 1 and 23 are 24, and 50 are 74, and 3 are 77, and 30 are 107, and 2 are 109, 3250594 and 20 are 129, and 4 arc 133, and 70 are 203, and 9 are 212, and 90 are 302, and 3 are 305, set down 05 : — 3 and 24 are 27, and 90 are 117, and 7 are 124, and 80 are 204, and 5 are 209, and 40 are 249, and 7 are 256, and 20 are 276, and 3 are 279, and 40 are 319, and 6 are 325, set down 325. The sum is 3250594. Explanation. — Beginning at the right, and taking two columns at a time, we take in first the tens and then the units, as we go up the column, and set down the two right hand figures of each sum. Ex. 2. 123456 789012 345678 901234 567890 987654 821098 765432 4801454 o. 1234 5678 9012 3456 7890 1357 9246 8987 46860 4. 235689 124578 135792 468097 531086 420987 654321 555775 5. 14250663 32215941 10340285 92341967 82395786 17084657 40558476 91623378 6. 819349 720258 630167 541076 452985 363894 274703 185612 7. 120341 989052 878163 767274 656385 545496 432107 321098 3126325 380811153 38 §55 ABSTRACT NUMBERS. CONTRACTION IN SUBTRACTION. Ex. 1. From 970347 take the sum of 14375; 226899, 12534, and 369708. ?Z?!?^ § 55. Model.— 8 and 4 are \2, and 9 are 14375 21, and 5 are 26, from 27 leaves 1 ; 2 and 3 226899 are 5, and 9 are 14, and 7 are 21, from 24 12534 leaves 3 ; 2 and 7 are 9, and 5 are 14, and 8 ^69708 are 22, and 3 are 25, from 33 leaves 8 ; 3 and 346831 9 are 12, and 2 are 14, and 6 are 20, and 4 are 24, from 30 leaves 6 ; 3 and 6 are 9, and 1 are 10, and 2 are 12, and 1 are 13, from 17 leaves 4 ; 1 and 3 are 4, and 2 are 6, from 9 leaves 3. The remainder is 346831. Explanation. — As 26, the sum of the subtrahend units, can not be take^ from 7, the units of the minuend, we add 2 tens, that is, 20 units, to the minuend, and afterwards add 2 tens to the subtrahend. (§ 28.) yote. — Let the pupil be required to use this contraction when- ever it can be applied. Ex. 2. From 1000 take 9+98-f-176-f 254-f 289. Rem. 174. 3. From 9125 take 8-f 88 +888 -f 1297+3945. Rem. 2899. 4. From 10275 take 1245 + 373 5 + 298 6 J- 895. Rem. 1414. 5. From 87579 take 1477+2796 + 8972 + 10896. Rem. 63438. 6. From 120225 take 246+1357+97531 + 1358. 7. From 72575 take 575+2575+4575+15575. 8. From 4970 take 250-|-325-|-348-|-2211. 9. From 22907 take 3916.|-2821-|-4302-|-2309. 34 CONTRACTED MULTIPLICATION OF INTEGERS. §57 CONTRACTIONS IN MULTIPLICATION. Ex. 1. Multiply 7325 by 100. 752500 ^ ^^' ^^^^^i" — Annex two naughts to the multiplicand. The product is 732500. Explanation. — We annex to the multiplicand as many ci- phers as there are annexed to the 1 of the multiplier. (§ 39.) Ex. 2. Multiply 1358 by 10. Prod. 13580. 3. Mult 4. Mult 5. Mult 5. Mult 7. Mult ply 2468 by 100. ply 4579 by 1000. ply 86725 by 10000. ply 1020 by 100. ply 32500 by 1000. Prod. 246800. Prod. 4579000. H. Multiply 32500 by 25000. 32500 __25000 1625 650 812500000 § 57. Model.— 5 times 5 are 25, set down 5 ; 5 times 2 are 10, and 2 are 12, set down 2 ; 5 times 3 are 15, and 1 are 16, set down 16: — twice 5 are 10, set down under 2 ; twice 2 are 4, and 1 are 5 ; twice 3 are 6 :— add the partial products : 5 ; 2 ; 5 and 6 are 11, set down 1 ; 1 and 6 are 7, and 1 are 8 :— annex 5 naughts. The product is 81250000o! Explanation. — After finding the product of the valua- ble figures, we annex to it as many naughts as there are in the right of both the factors. Ex. 9. Multiply 27500 by 350. 10. Multiply 1250 by 1500. 11. Multiply 747000 by 250. 12. Multiply 19500 by 1400. 13. Multiply 124750 by 3000. 14. Multiply 2795000 by 2700. 85 Prod. 9625000. Prod. 1875000. Prod. 186750000. Prod. 27300000. §58 ABSTRACT NtJMBERS, 15. Multiply 3759 by 104. 3759x104 I 58^ Model.— 4 times 9 are 36, set _j_^^*^ down 6, two places to the right of 9 ; 4 390936 ' times 5 are 20, and 3 are 23, set down 3 ; 4 times 7 are 28, and 2 are 30, set down ; 4 times 3 are 12, and 3 are 15, set down 15 : — add the partial products : — 6 ; 3 ; 9 ; 5 and 5 are 10, set down ; 1 and i are 2, and 7 are 9 5 3. The product is 390936. Explanation.— If the multiplier has only two valuable figures, the first of which is 1, we multiply by the other valuable figure, and set the first figure of the product as far to the right of the units figure of the multiplicand as this figure is to the right of the 1 . Ex. 16. Multiply 2376 by 12. Prod. 28512. 17. Multiply 47475 by 107. Prod. 5079825. 18. Multiply 57875 by 10080. Prod. 583380000. 19. Multiply 275 by 1009. Prod. 277475. 20. Multiply 4479 by 10006. 21. Multiply 795310 by 10500. 22. Multiply 1025 by 7001. 1025x7001 § 59. Model.— 7 times 5 are 35, 7175 set down 5, three places to the left of 7176025 5 ; 7 times 2 are 14, and 3 are 17, set down 7 ; 7 times are 0, and 1 is 1 5 7 times 1 are 7 : — add the partial products : — 5 ; 2 ; ; 5 and 1 are 6 ; 7 ; 1 ; 7. The product is 7176025. Explanation. — If the multiplier has only two valuable figures, the last of which is 1, we multiply by the other valuable figure, and set the first figure of the product as far to the left of the units figure of the multiplicand as this figure is to the left of the 1. Ex 23. Multiply 7893 by 51. Prod. 402543. 36 CONTRACTED MULTIPLICATION OF INTEGERS. m 24. Multiply 4685 by 601. 25. Multiply 23795 by 7010. 26. Multiply 1375 by 8001. 27. Multiply 20478 by 90010, Prod. 2815685. Prod. 166802950. 28. Multiply 27346 by 99. "'^o-^^fi ^ ^^' ^^^^^T^-— ^^^^ex 2 naughts to the .J*l___ multiplicand : — subtract the multiplicand 2707254 from the result; 6 from 10 leaves 4; 5 from 10 leaves 5 ; 4 from 6 leaves 2 ; 7 from 14 leaves 7 ; 3 from 3 leaves ; from 7 leaves 7 ; from 2 leaves 2. The product is 2707254. Explanation. — Since 9 is 1 less than 10, we may multi- ply any number by 9, by subtracting the number from 10 times itself. If therefore the multiplier consists of 9's alone, we annex to the multiplicand as many naughts as there are nines in the multiplier, and subtract the multi- plicand from the result. Ex. 29. iVIultiply 124795 by 9. 30. Multiply 24735 by 99. 31. Multiply 1469 by 999. lultiply 70095 by 99. 53. Multiply 9999 by 256. (§ 37.) 34. Multiply 1276538 by 999. 35. ^Multiply 8365712 by 99. Prod. 1123155. Prod. 2448765. Prod. 1467531. Prod. 6939405. Prod. 2559744. 36. Multiply 2754 by 54. 27540 on-. ,r r. ^ . §61. Model. — o4=9 times 6. First, mul- tiply by 9 :— (§ 60.) 4 from 10 leaves 6 ; 6 from 14 leaves 8 ; 8 from 15 leaves 7 ; 3 from 7 leaves 4 ; from 2 leaves 2. The product is 24786. Multiply this product by 6 :— 6 times 6 are 36, set down 6 ; 6 times 8 are 48, and 3 are 51, set down 1 ; 6 times 7 are 42, and 5 are 47, 37 2754 24f86 6 148716 m ABSTRACT NUMHKRS. set down 7 ; 6 tiroes 4 are 24, and 4 arc 28, set down 8 ; 6 times 2 are 12, and 2 are 14, set down 14. The product is 148716. Explanation. — If the multiplier is the product of two or more numbers, we may multiply the multiplicand by either of those numbers, and this product by an other, and so on. Ex. 37. Multiply 3725 by 35. 38. Multiply 17075 by 48. 39. Multiply 473729 by 49. 40. Multiply 279o6 by 56. 41. Multiply 124684 by 64. 42. Multiply 247372 by 72. Prod. 130375. Prod. 843G00. Prod. 23212721. Prod. 1564416. 43. Multiply 21857 by 714. 21857 714 149499" 298998 T5248898 §62. Model. — 14 is twice 7. First, multiply by 7 : — 7 times 7 arc 49, set down 9 under 7 of the multiplier ; 7 times o are 35, and 4 are 39, set down 9 ; 7 times 3 are 21, and 3 are 24, set down 4 ; 7 times 1 are 7, and 2 are 9 ; 7 times 2 are 14, set down 14. The product is 149499. Multiply this product by 2 : — twice 9 are 18, set down 8 under 4 of the multi- plier; twice 9 are 18, and 1 are 19, set down 9; twice 4 are 8, and 1 are 9 ; twice 9 arc 18, set down 8 ; twice 4 are 8, and 1 are 9 ; twice 1 are 2. Add the partial pr(-ducts : 8 ; 9 ; 9 and 9 arc 1>^, set down 8 ; 1 and ^^ arc 9, and 9 are 18, set down 8 ; 1 and 9 are 10, and 4 are 14, set down 4 ; 1 and 2 are 3, ami 9 are 12, set down 2 ; 1 and 4 :n\. 5 ; 1. The product is 15248898. Explanation. — If one part of the multiplier is a factor of an other, the work may be contracted as in the model, placing the first figure of each product immediately under 88 CONTRACTED MULTIPLICATION OP INTEGERS. §64 the right hand figure of the corresponding part of the mul- tiplier. Ex. 44. Multiply 12479 by 654. 45. Multiply 24793 by 56248. 46. Multiply 97635 by 53545. 47. Multiply 86436 by 497. 48. Multiply 23047 by 488. 49. Multiply 902756 by 366108, Prod. 8161266. Prod. 1394556664. Prod. 5227866075. Prod. 42958692. 50. Multiply 225 by 25. 4)22500 ^(53^ Model.— Annex 2 naughts to the 5625 multiplicand : — divide the r-esult by 4 : — 4 in 22, 5 times, with 2 over, set down 5 ; 4 in 25, 6 times, with 1 over, set down 6; 4 in 10, twice, with 2 over, set down 2 ; 4 in 20, 5 times. The product is 5625. Explanation. — Annexing 2 naughts multiplies by 100, (§ 56) : hence, since 100=4x25, we divide the product by 4, to get the true product. Ex. 51. Multiply 10275 by 25. Prod. 256875. 52. Multiply 28832 by 25. Prod. 720800. 53. Multiply 72725 by 25. . Prod. 1818125. 54. Multiply 84287 by 25. Prod. 2107175. 55. Multiply 96248 by 25. 5G. Multiply 8324728 by 25. 57. Multiply 274 by 125. 8 )274000 ^ (54 Model.— Annex 3 naughts to the 34250 multiplicand : — divide the result by 8 : — 8 in 27, 3 times, with 3 over, set down 3 ; 8 in 34, 4 times with 2 over, set down 4 ; 8 in 20, twice, with 4 over, set down 2 ; 8 in 40, 5 times ; 8 in 0, no time. The product is 34250. 39 §65 AHBTRACT MMBBRS. Explanation. — Annexing 3 naughts multiplies by 1000, (§ 56): hence, since 1000—8x125, wc divide the product by 8, to get the true product. Kx. 58. Multiply 125 Ly 125. Prod. 15620. 59. Multiply 625 by 125. Prod. 78125. 60. Multiply 1776 by 125. Prod. 222000. 61. Multiply 34070 by 125. Prod. 4259875. 62. Multiply 934478 by 125. 63. Multiply 7840349 by 125. CONTRACTIONS LN Di\ iiilO.X. Ex. 1. Divide l25wl 125 64 ^^ ^^ MoDKL. — Cut off two figures at thf ' right. The quotient is 125, and the remain der 64. ExPL-\NATiON. — We cub off at the right, for remainder, a.s many figures as there are naughts at the right of the I • •f the divisor. The remaining figures on the leftconstitut« the quotient. 2. Divide 34000 by 10. (]uot. 3400. 3. Divide 74500 by 100. Quot. 745. 4. Divide UJ740 by 100. Quot. 107; Hem. 40. 5. Divide 24G000 by lOUO. Quot. 24(;. 0. Divide 147375 by 1000. 7. Divide 24680 by 100. 8. Divide 98630 by 800. 8,00)086,30 ^QQ MoDKL.^Out off the 2naught.s 123 — 230 at the right of the divisor, and 2 fig- ures at the right of the dividend : — then, 8 in 0, once, with 1 over, set down 1 ; 8 in 18, twice, with 2 over, set down 2 ; 8 in 2(.), 3 times, with 2 over. Tho (quotient is 123, and th(? rcmaindo^- '^'^'^ 40 CONTRACTED DIVISION OF INTEGERS. §68 Explanation. — The remainder after dividing is prefixed to the dividend figures cut off, to constitute the true re- mainder. Ex. 9. Divide 127569 by 270. (.^uot. 4724; Rem. 189. 10. Divide 56000 by 700. Quot. 80. 11. Divide 3230000 by 1700. Quot. 190U. 12. Divide 24600 by 2400. Quot. 10 : Hem. 600. 13. Divide 7346790 by 72900. 14. Divide 135073 by 21800. 15. Divide 275 by 5. ^"^[^ ' § 67. Model.— Multiply the dividend by 2 : _^ twice 5 are 10, set down ; tv/ice 7 are 14, and 55,0 1 are 15, set down 5 ; twice 2 are 4, and 1 are 5 : — divide this product by 10. (§ 65.) The (quotient is 55. Explanation. — Since the dividend is already 5 times the required quotient, multiplying it by 2 gives (2x5) 10 times the quotient. The part cut off at the right, by this plan, is twice the true remainder. Ex. 16. Divide 10024 by 5. Quot. 2004; Kern. 4. 17. Divide 2725 by 5. Quot. 545. 18. Divide 49720 by 5. Quot. 9944. 19. Divide 598405 by 5. Quot. 119681. 20. Divide 479324 by 5. 21. Divide 2379156 by 5. 22. Divide 329 by 25. ^2^ i^ 68. Model.— Multiply the dividend by 4 . 4 times 9 = 30, set down 6 ; 4 times 2=8, and 3 i /i- =ll,set down 1; 4times3 = 12, and 1 = 13, set down 13 : — divide this product by 100 (J; 65.) The quotient is 13, and the remainder 4. 41 §69 ABSTRACT NUMBERS. Explanation. — Since the dividend is already 25 times the required quotient, multiplying it by 4 gives (4x25) 100 times the quotient. The part cut off at the right, by this plan, is 4 times the true remainder. Ex. 23. Divide 293235 by 25. Quot. 11729; Rem. 10. 24. Divide 148532 by 25. Quot. 5941 ; Rem. 7. 25. Divide 2475 by 25. Quot. 99. . 26. Divide 193450 by 25. Quot. 7738. 27. Divide 34795 by 25. 28. Divide 107059 by 25. 29. Divide 23725 by 125. 23725 I Qg Model. — Multiply the dividend by ^ 8:8 times 5 are 40, set down ; 8 times 2 189,800 are 16, and 4 are 20, set down ; 8 times 7 are 56, and 2 are 58, set down 8 ; 8 times 3 are 24, and 5 are 29, set down 9 ; 8 times 2 are 16, and 2 are 18, set down 18 : — divide this product by 1000. (§65.) The quotient is 189, and the remainder 100. Explanation. — Since the dividend is already 125 times the required quotient, multiplying it by 8 gives (8x125) 1000 times the quotient. The part cut off at the right, by this plan, is 8 times the true remainder. Ex. 30. Divide 724350 by 125. Quot. 5794; Rem. 100. 31. Divide 111000 by 125. Quot. 888. 32. Divide 246625 by 125. Quot. 1973. 33. Divide 57935 by 125. Quot. 463 ; Rem. 60. 34. Divide 793575 by 125. 35. Divide 125364 by 125. 36. Divide 10202 by 42. 42 CONTRACTED DIVISION OP INTEGERS. §71 2)10202 3)5101 § 70. Model. — 42=2 times 3 times 7. First, divide by 2 : — 2 in 10, 5 times; 7)1700—1 2 in 2, once ; 2 in 0, no time ; 2 in 2, 242 — 6 once : — divide this quotient by 3 : — 3 in 5, once, with 2 over, set down 1 ; 3 in 21, 7 times; 3 in 0, no time ; 3 in 1, no time, with 1 over, set down in the quotient, and 1 as remainder : — divide this quotient by 7 : — 7 in 17, twice, with 3 over, set down 2 ; 7 in 30, 4 times, with 2 over, set down 4 ; 7 in 20, twice, with G over, set down 2 in the quotient, and 6 as remainder. The quotient is 242, and the remainder 38. Explanation. — If the divisor is the product of two or more numbers, we may divide the dividend by either of those numbers, and the quotient by an other, and so on. The true remainder is found by multiplying each remainder by all the divisors previous to the one which produced it, and adding together the several products. Ex. 37. Divide 7346 by 56. Quot. 131 ; Hem. 10. 38. Divide 347934 by 35. Quot. 9940 ; Rem. 34. 39. Divide 92384 by 64. Quot. 1443; Rem. 32. 40. Divide 83495 by 45. Quot. 1855 ; Rem. 20. 41. Divide 745106 by 72. 42. Divide 656215 by 96. 43. Divide 34635 by 285. ^1^^^|^_?5 § 71. Model.— 3 in 3, once :— once 5 is J^r 121 5, from 6 leaves 1; once 8 is 8, from 14 ;*^^ leaves 6 ; once 2 is 2, and 1 are 3, from 3 ^^^ leaves : annex 3 : — 3 in 6, twice : — twice 5 are 10, from 13 leaves 3 ; twice 8 are 16, and 1 are 17, from 21 leaves 4 ; twice 2 are 4, and 2 are 6, from 6 leaves : annex 5 : — 3 in 4, once :— -once 5 is 5, from 5 leaves 0; once 8 is 8, from 13 leaves 5 ; once 2 is 2, and 1 arc 3, from 4 leaves 1. The quotient is 121, and the re- mainder 150. 43 §72 ABSTRACT NUMBERS. Explanation. — The products are not written, but are iui mediately substracted as in § 55. Note. — Let all the exercises in Long Division hereafter be per- formed by this plan. Ex. 44. Divide 136895 by 725. Qiiot. 188 ; Rem. 595. 45. Divide 247986 by 836. Quot. 296 ; Rem. 230. 46. Divide 358097 by 749. Quot. 478 ; Rem. 75. 47. Divide 469108 by 5275. Quot. 88 ; Rem. 4908. 48. Divide 5702195 by 4386. 40. Divide 68132050 by 5295. GENERAL PRINCIPLES OF DIVISION. § 72. If the divisor remain unchanged, and the dividend be multiplied by any number,the quotient will be multiplied by the same number. Thus, 32-i-8=4 : then, 64-^8=8. § 73. If the divisor remain unchanged, and the dividend be divided by any number, the quotient will be divided by the same number. Thus, 32-f-8=4 : then, 16-^8 = 2. § 74. If the dividend remain unchanged, and the divisor be multiplied by any number, the quotient will be divided by the same number. Thus, 32-^8=4 : then, 32-^16 = 2. § 75. If the dividend remain unchanged, and the divisor be divided by any number, the quotient will be multiplied by the .-ame number. Thus, 32^8=4: then, 32-^4=8. § 76. If the dividend and the divisor be both multiplied by the same number, the quotient will remain unchanged. Thus, 32-8.-4: then, 64-^16:zr=4. § 77. If the dividend and the divisdr be both divided by the same number, the quotient will remain unchanged. — Thus, 32-T-8==l: then, lG--4=ri. 44 PROMISCUOUS PROBLEMS. 5J80 PROMISCTTOTTS PROBLEMS. 1. The subtrahend is thirty thoiisaDcl and forty-five ; the remainder is fortj'-six thousand eight hundred and ninety: what is the minuend ? Ans. 769i:j5. § 78. Minuend — Subtrahend=Remainder. Minuend — Remainder=Subtrahend. Subtrahend 4- Remainder.-rrrMinuend. 2. The minuend is three hundred thousand ; the subtra- hend is ninety-nine thousand three hundred and seventy- four : what is the remainder? Ans. 200024. 3. The minuend is seventy thousand and twenty-nine ; the remainder is sixty-five thousand and forty-six : what i.* the subtrahend ? 4. The multiplicand is twenty-seven thousand and four ; the product is seven hundred and twenty-nine millions, two hundred and sixteen thousand, and sixteen : what is the multiplier ? Ans. 27004. § 79. Multiplicand X Multiplier =Product. Product-7-Multiplier= Multiplicand. Product-f- Multiplicand = Multiplier. 5. The multiplicand is four thousand and seventy-two : the multiplier is one thousand one hundred and six : what is the product 1 Ans. 4503632. 6. The product is ninety-three thousand three hundred and sixty-one ; the multiplier is eighty-nine : what is the multiplicand 1 7. The divisor is one thousand and nine ; the quotient is nine hundred and ten : what is the dividend ? Ans. 918190. § 80. Dividend-i-Divisor=Quotient. Divisor xQuotient=Dividend. (Dividend — Remainder) -^Quotient =Divisor. Quotient xDivisor-f- Remainder =Dividend. 45 §)^0 ABSTRACT NUMBERS. 8. The dividend is nineliuudred and forty- five thousand, eight hundred and eighty-eight ; the divisor is two thousand and four : what is the quotient ? Ans. 472. 9. The dividend is one hundred and forty-eight thou- sand; the quotient is three hundred and forty-two; the re- mainder is two hundred and fifty-six : what is the divisor ? 10. The quotient is one thousand and three ; the divisor is one thousand and two : the remainder is one thousand and one : what is the dividend ? Ans. 1006007. 11. Find the sum of two hundred and forty-five thou- sand, nine hundred and seven, seventy-four thousand and seventy-four, one hundred and nine thousand and nine, and three hundred and ninety-seven. Sum, 429387. 12. Find the difference between two hundred thousand, and one hundred and eighty-seven thousand six hundred and fifty-four. 13. Find the product of one million three hundred and seventy-five, and one thousand three hundred and seventy- five. Prod. 1375515625. 14. Find the quotient of three millions divided by six thousand two hundred and seventy-nine. Quot. 477 ; Rem. 4917. 15. What number is that from which if 2407, 4072, 724, and 7240 be subtracted, the remainder will be 7042 ? 16. What number is that to which if 2407, 4072, 724, and 7240 be added, the sum will be 15000 ? Ans. 557. 17. What number is that by which if 2047 be multiplied, the product will be 15151894 1 Ans. 7402. 18. What number is that by which if 2025042 be di- vided, the quotient will be 2021 ? 19. 247-f 1023— 9344-3720— 4142-f245=:what? 20. (247-154)-^3-f(247-f 154)x3=what? Ans. 1234. 46 PROMISCUOUS PROBLEMS, §81 § 81. A parenthesis enclosing two or more numbers shows that their united value is to be subjected to the operation indicated immediately before or after the parenthesis. For example, in the preceding problem, the difference of 247 and 154 is to be divided by 3, and the sum of 247 and 154 is to be multiplied by 3, and the product and the quotient arc to be added together. Two numbers thus connected are called a binomial ^ three numbers are called a trinomial; four, a tetranomial ; five. a pcntanomial ; six, a hexanomial, &c. The 20th problem is read, '^ Binomial 247 minus 154 divided by 3 plus binomial 247 plus 154 multiplied by 3 is equal to what?" 21. 3247 + 247-47 + 7-(247— 474-7)=what? 22. (987— 876-f333)-H(765-543)-f-210-95=what y Ans. 117. 23. 27— 30-^10-f (475 — 399)H-4=:what? Ans. 43. 24. (204-60)-^6-(90-|.10)--5-[-(76-!.12)-T-4=what '^ 25. 204— 60-f-6— 90-l-10-^5-K76-|-12)--4:=what ? Ans. 128. 26. (204— G0)-^6-j-90-l-10-5-|-(76-|-12)^4r=:what ? 27. (204— 60)--6— (90-|-10)-f-5-|-7G— 12-^4r=what ? 28. 204— 60--6— (90-|-10--5.!-76)— 12-^4=rwhat ? Ans. 23. 29. 123-1-41— (123— 41>|-123x41—123^41=what? Ans. 5122. 30. 123-1-41— (123— 4l)-|-(123x41— 123) ^41==what ? 31. J[(742-v-2)-r-53]x27-lJ-^53=::what? Ans. 1. 32. [(199-78)-f-ll-(199-43)--78]x(12-3)=lwhat? Ans. 81. 33. [(117-43) x2]--37-K138-128) x37==^hat 1 47 ABSTRACT NUMBERS. MEASURES AND MULTIPLES. § 82, An eve7i number is one which can be exactly divided by 2. Thus, 12, 4, 36, 58, and 70, are even numbers. Xote. — All even numbers end in either 2, 4, 6, 8, or 0. § So. An odd number is one which can not be exactly di- vided by 2, Thus, 9, 17, 25, 33, and 41, are odd numbers. Kote, — AH odd numbers end in either 1, 3, 5, 7, or 9. § 84. A prime number is one which is not the product of two other numbers. Thus, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83,89, and 97, are all the prime numbers less than 100. X-A^BLK of I*I?,I^i:]K INTUMCBEl^^S ixp to lOOO. o 43 103 1 173 241 317 401 479 571 647 739 827 1 919 3 47 107 1 179 251 331 409 487 577 653 743 829 929 53 109 181 257 337 419 491 687 659 751 839 9,37 . 1 59 113' 191 263 347 421 499 593 661 757 853 941 i lll Gl 127 193 269 349 431 603 599 673 761 857 947! jl8 67 131 197 271 353 433 509 601 677 769 859 963 !l7 71 137 199 277 359 439 521 607 683 773 863 967 119 73 139 211 281 367 443 523 613 691 787 877 971 '23 79 149 223 283 373 449 511 617 7U1 797 881 977 29 83 151 227 293 379 457 547 619 709 809 883 983 31 89 157 229 307 383 461 557 631 719 811 887 991 37 97 163 233 311 389 463 563 641 727 821 907 997 41 101 167 239 313 397 467 569 643 733 823 911 § 85. A composite number is one which is the product of two other numbers. Thus, 4, 6, 9, 15, 21, and 30, are composite numbers, because 2x2=4, 2x3=6, 3x3=9, 3x5=15, 3x7=21, and 5x6=30. Is 20 prime, or composite ? 25 ? 28 ? 31 ? 34? 37 ? 40 '/ 43? 501 57? 64? 71? 78? 85? 92? 99? 106? 217? 328 ? 439 ? 48 MEASURES AND MULTIPLES. §87 §86. Powers. — The first jyo we r of a number is the num- ber itself. Thus, 5 is the Jirst power of 5 ; 7, of 7 ; 10, of 10. The second poicer of a number is the product of the num- ber multiplied by itself. Thus, 8G is the second power of G, because 6x6=^36: 81, of 9, because 9x0—81 : 100, of 10, because 10x10=100. The second power of a number is usually called its square. The third power of a number is the product of the num- ber multiplied by its square. Thus, 8 is the third power oi 2, because 2x4=^8 : 64, of 4, because 4x16=64 : 216, of 6, because 6x36=216: 512, of 8, because 8x64=--:5l2: 1000, of 10, because 10x100 = 1000. The third power of a number is usually called its cube. In like manner, what is the fourth power of a number ? What is the sixth power 't The 7iinth power ? ka. § 87. Roots. — The first root of a number is the number itself. The second root, or the square root, of a number is one of the tico equal factors which produce it. Thus, 5 is the square root of 25, because 5x5=25. 15 has no square root, because its two factors, 3 and 5, are not equal. The third root, or the cube root, of a number is one of the three equal factors which produce it. Thus, 3 is the cube root of 27, because 3.3.3=27. 30 has no cube root, because its three factors, 2, 3, and 5, are not equal. 25 has none, because it has only two equal factors, 5 and 5. 16 has none, because it has four e«^ual factors, 2, 2, 2, and 2. In like manner, what is the fourth root of a number ? What is the seventh root ? The sixteenth root ? &c. D 49 §88 ABSTRACT KUiMIiEKS. •^ bS. Th.Q prime f (I do i's oi n composite nuitiber arc the prime numbers by whose continued multiplication the num- ber is produced. Thus, the prime factora of 9 are 3 and 3/ because 3x3=9 : the prime factors of GO are 2, 2, 3, and 5, because 2.2.3.5=60. § 89. A measure of a number is a number which is con- tiuned in it a number of times without a remainder. Thus, 3 is a measure of 12, because 3 is contained exactly 4 time.s iri 12 : 4 is a measure of 30, because 4 is confeaiiied exacfli/ 9 times in 30. Is 5 a measure of 10 ? 25 ? 37 ? 40 ? 63 ? Q>~j \ 80 ? Is 6 a measure of 7 ? 12? 20? 30? 39? 48? 54? Is 7 a measure of 14? 19? 28? 36? 42? 48? 63? § 90. A multiple of a number is a number which containn it a number of times loithout a remainder. Thus, 12 is a multiple of 3, because 12 contains 3 exactly 4, times : 36 is x multiple of 4, because 36 contains 4 exactly 9 times. Is 40 a multiple of 2 ? 3 ? 4 ?, 5 ? 6? 7? 8? 9? 10? 20? Is 56 a multiple of 2 ? 4 ? 7 ? 8 ? 9 ? 10 ? 14 ? 20 ? 24 ? Is 60 a multiple of 2? 3? 4? 5? 6? 7? 8? 10? 12? 15? §91. (7o?nmo?i means belonging equally to two or more numbers. § 92. One number is a common measure of two or more numbers, if it is a measure of each of them. Thus, 3 is a measure of 9, also of 12, also of 18 ; hence, 3 is a common measure of 9, 12, and 18. Also, 4 is a common measure of 8, 24, 32, ai)d 48. Is 2 a common measure of 4, 6, and 10 ? Is 3 a common measure of 6, 10, and 15 ? Is 4 a common measure of 12, 16, and 20 ? Two or more numbers may have several common meas- 50 iires. Tlui., :, .id 361iave as c.......i.... ......,.^i'cs 2, o, 4, 6, and 1*2. In this case, lU is, of course, the greatest com- DKjn ::!easure of 2-i and 36. § 93. Ohe number is a common multiple. of two or more numbers, if it- is a multiple of each of them. Thus, 40. is a multiple of 5, also of 8, also of 10 ; hence, 40 is a common uiultiple of 5, 8, and 10. Also, 45 i- a o.ovir.i.nu multiple of 3, 5, and 9. Ls 10 a common multiple of 2 and 5 ? Is 15 a common multiple of 3 and 6 ? Is oO a common multiple of 2, 5, and 10 .' Two or more numbers alwajs have several common mul- tiples. Thus, 4, 3, and 6, have as common multiples 12, 34, 3t3, 48, 60, &o. In this case, 12 is, of course, the kcial eommon multiple of 4, 3, and 6. § 94. Two or more numbers are prime to each other, if they have. np common measure. Thus, 81 and 64 are prime to each other. Also, 20, 27, and 77 are prime to each other. § 95. 2 is a measure of every number which ends in ei- ther 2, 4, 6, 8, or 0. (§ 82. Note.) o is a measure of a number, if it is a measure of the sum of the figures which* denote the number. ThuS; 3 is a meas- ure of 246, or 462, or 624, or 612, or 426, or 261, or 2064, or 4602, &c., because 3 is a measure of 6-f 4-f 2, that is. of 12. 4 is a measure of a number, if it is a measure of the num- ber denoted by its two right hand figures. Thus, 4 is a measure of 768, or 1860, or 95372, or 1112316, because 4 is a measure of 68, or 60, or 72, or 16. 5 is a measure of every number which ends in either 5 or 0. Thus, 5 is a measure of 20, or 55, or 100, or 275. 51 J{96 AESTE.ACT NUMBERS. o is a measure of erery even number of which 3 .is a measnre. Thus, 6 is a measure of 462^ or 4512, or 1236 : but not of 471, or 632L 8 is a measure of a number, if it is a measure of the number denoted by its three right hand figures. Thus, 8 is a measure of 34800, or 753064, because 8 is a measnre of 800, or 64. 9 is a measure of a number, if it is a measure of the sum of the figures which denote the number. Thus, 9 is a meas- ure of 891, or 1728, or 253269, because 9 is a meavsure of 18, or 18, or 27. 10 is a measure of every number which ends iii 0. 100 is a measure of every number which ends in 2 naughts. Is 2 a measure of 3040 ? 4047? 28? 1112? 10124? Is 3 a measure of 258? 869 ? 12345678 ? 5169 ? 2571 '( Is 4 a measure of 125784 ? 24680 ? 57932 ? 14760 ? 1 1 12? Is 5 a measure of 245? 12450? 7824? 12570? 3457? Is 6 a measure of 570 ? 378 ? 45S42 ? 123456 ? 12324 ? Is 8 a measure of 5070120 ? 247080? 1479008? 1234? Is 9 a measure of 1234566 ? 68472 ? 1357 ? 1476 ? Is 10 a measure of 240 ? 245 ? 3795 ? 7630 ? 1460 ? § 96. A measure of a number is a measure of any one of its multiples. Thus, 6 is a measure of 18: then it is a measure of 36, or 54, or 72, or 90. § 97. A common measure of two or more numbers is a measure of their sum. Thus, 8 is a common measure of 16, 24, and 40 : then it is a measure of 80. § 98» A. common measure of tAO numbers is a measure of their difference. Thus, 9 is a common measure of 18 and .^4 : then it is a measure of 36. 52 PRIME FACTORS. §99 PulME FACTORS. Ex. 1. Resolve 7'*^00 into its priDie factors. 2)7800 2)3900 ^^^- J^I^r>EL. — Divide the namber by 2. 'MI^Qi^ (§50). Divide tlie quotient by 2. Divide •^-C this cjuotieut by 2. Divide this quotient by '^1?.I5 5. Divide this quotient by 5. Divide this 5)195 quotient hj 3. This quotient is a prime uum- jj^gT^ ber. The prime f-icfnrs of 7800 arc 2, 2, 2, ' 1. '_ 5, 5, 3, and l.S. o Explanation. — It is better to divide first by 2 as often as possiblcj then b}'- 5, and then by the other prime numbers in succession. The several divisors and the last quotient are evidently the prime factors, of the number. Rule. — Divide the given niiviber by one of it^ prime nimsures ; divide the quotient hy one of its prime measures ; continue thus dividing until a ^^ rime number is obtained fo a quotient: the several divisor.^ and the last quotient will be ihe prime factors of the given nu,'7\ber. Proof. — The continued product of the prime factors will he equal to the given number. Ex. 2. Resolve 524 into its prime factors. P. F. 2, 2, and 131. 3. Resolve 460 into its prime factors 4. Resolve 770 into its prime factors. P. F. 2, 5, 7, and 11, 5. Resolve 880 into its prime factors. P. F. 2, 2, 2, 2, 5, and 11. 6. Resolve 999 into its prime factors. 7. Find the prime factors of 1040. P. F. 2, 2, 2, 2, 5, and 13, 53 §100 ABSTRACT NTJMBTJRS. 8. Find the prime factors of 1160. P. F. 2, 2, 5, and 29. 9. Find the prime factors of 1275. iO. What are the prime factors of 1300 t Ans. 2, 2, l), 5, and 13. iL What are the prime factors of 1590 ? An&. 2, 5, 3, and 53. 12. What are the prime factors of 1738? 13. What are the prime factors of 19500 ? Ans. 2, 2, 5, f), 5, 3, and 15. 14. What are the prime factors of 966000 ? Ars 2, 2, 2, 2, 5, 5, 5, 3, 7, and 23. 15 What are the prime factors of 825000 ? 16. What arc the prime factors of 13572001 Ans. 2, 2, 2, 2, 5, 5, 3. 3, 13, and 29. INVOLUTION. § 100. Involution is the process of finding i^ power oi -^ number. From the definitions of the several powers in §86, it is evident that any power of a number is fo raid L_j taking the number as a factor in miiltiplication as nianj times as there are units in the number of the power, Ex. 1. What is the square of 7? Ans 49, 2. What is the cube of 3 ? Ans. 27. 3. Yv^hat is the fourth power of 2 't 4. What is the fifth power of 2 ?' Ans. 32. 5. What is the fourth power of 5 ? Ans 625. 6. What is the cube of 9 ? 7. What is the square of 19? Ans. 361. 8. What is the cube of 15 ? Ans. 3375. 9. What is the fourth power of 20? 54 JiVOLUTJON. §102 EV(^LUT10X. ^\0l EvoLU^i*iON is the process of finding, a >oo^ of ;i given power. The method here explained is applicabli only to such numbers as have precise roots. The method of extracting aj»proxiraate roots of imperfect powers can not be explained without the use of algebraic formulas, and consequently is not given in this treatise. Ex 1. What is the cube root of 21G ? r 2)2 16 2)108 2^54 § 102. MoDEii.-— Resolve the given nuRi- ber into its prime factors. (§99.) It con- po)'-7 tains three twos and three threes. Hence. 3y >= cube root is 2x8=6. Explanation. — Since the cube root of a number is ouv; of the three equal factors which produce it, we separate the prime factors into sets of three equal prime factors each, and selecting one from each setjthe produc^ of those ^elected is evidently the cube roqt of the given nilmber. For ari v other root, wo separate into sets of as many prime factors each a? there ape units in the ri-.mber of the root. 3' he prime factors ')e*eparattd as above, '.ho required rv)ot can not be exactly found, eitlun- by this, or by afly vtfK'r method. Kui. .-■ . ..:/.;. .,.:;:.„ ^ ■...,..- into ila .....,:: fac!or>> : separate the factors into groups of as many equal factor,'. each as there are units in the niimher of the root} select one faetor from each f^reiip, and multiply togetlter (hose selected : their product tcill he the root n quired. Proof — Raise the root to the corresponding power. Tht^ result will bo counl to the given number. 55 §10o ABSTRACT NUMBERS. GREATEST COMMOX MEASURE. Ans. 6. An». 3, Ans. 10. Ans. 4. Ex. 2. What is the square root of 100 ? Ans. 10. 3. What is the cube root of 125 ? 4. What is the fourth root of 1296 ? 5. What is the fifth root of 243 ? 6. What is the sixth root of 64 ? 7. What is the fourth root of 10000 1 8. What is the fifth root of 1024 ? 9. What is the cube root of 3375 ? 10. What is the square root of 12321 ? Ans. 111. 11. What is the square root of 65536 ? A^a^. 256, 12. What is the fourth root of 65536 ? 13. What is the eighili .■ .,t of 65536? 14. What is the sixteenth root of 65536 ? 15. What is the square root of 390625 ? 16. What is the fourth root of 390625?" 17. What is the eighth root of 390625 ? 18. Yfhat is the cube root of 10077696? 19. What is the ninth root <'.f 10077696? 20. What is the cube root oi 42875 ? 21. What is the square root of 122500 ? 22. What is the square root of 7569? Ans. 4. Aus. t.) Ans. 25. Ans. 5. Ans. 6. Ans. 35. Ans. 87„ Ex. 1, and 480. 2)60, . Ein 150, ;d the 480 5)30, 75, 240 3) 6, 15, 48 2, 5, 16 2. ,5.3=oU grep.test common measure of 60, 150, § 103. Model — Divide each of the given numbers by 2. (^- 50). Divide each of these quotients bj 5. Divide each of these quotients bji 3. These quotients are prime to each other. 2.5.3=30. 30 is the greatest com- mon measure of the 2:iven numbers. 56 GREATEST COMMON MEASURE. Explain A'iiuN. — In this operaiiuu it u uul iicoe^haij n>i the divisors to be prime nuiubcTs. Wo might liave divided by 10 and by 3, or by 5 and by 6. lluLE. — Divide each of the given numbers hy any one cj their common meamrcs ; divide each of these qnoiienta by ne of iheir comm07i measures ; continue f hits diriding un- , il the quotientc, become prime to each other : the continued product of the divisors will be the grcated comrnnii. mt'.asurc q f the g iven n k m bers . Ex. li. Fin'"' t'"' 'greatest common measure of '^6, 12'), -16, and 234. 30=2.2.3.3 § 104. Modkl.— Resolve 36 126 = 2. 3.3.7 inj-Q ity pi-iine factors. (§ 99). '?16 = 2.2.3.3. 2.3 36=2.2.3.3. Resolve 126 into 234=2^_3.3. 13 ^^^ primefactors. 126=2.3.3.7. 2.3.3. = 18. Resolve 216 into its prime fac- tors. 216=2.2.3.3.2.3. Re- solve 234 into its prime factors. 234=2.3.3.13. 2.3.3 = 18. 18 is the greatest common measure of the ?iven nnm];ei"». ExPLANATlo.v. — The prime factors are arranged Vr-ith equal factors in the same column, as far as possible. The full columns contain the factors that are common to aU the numbers. The product of these factors is the greatest common mea.sure of tlio numbers. Rule. — Resolve each of the given numbers info il.s pri/iw factors ; select those factors ichich are common to all the numbers: the continued product of these factors will l-r iJtc (jreatest common measure of the given numbers. 57 §105 ABSTRACT NUMBERS. o. Find the greatest common measure of 108 and 261 261:108 216 2 108!45 90!^~- ■^* 105. Model.— Divide 261 by 108. il (S 71). Divide 108 by 45. Divide 45 f^^^ hy'lii. Divide 18 by 9. There is do '-''^ 2 reoiainder, 9 is the greatest common >o7^ — measure of the (nven numbers. _____ Explanation. — is a measure of IS, (§ 89); hence it i.^ a measure of 2x18, or 36, (§ 96) ; hence, of 86-f 9, or 45, (§ 97) ; hence, of 2 x45, or 90, (§ 96) ; hence, of 108, {% hi) : hence, of 216, (§96); hence, of 261, (§ 97);.. hence it is a common measure of 108 and 261. YhULE^—r-DLvldeihe larger ?iumber h;j the smalUr ; then divide the smaller numher by the remainder^ and continue divvUnq the last divinor bv the laM. remainder. until there ?*? no reinuinder : tlie last divisor iriU he fitc r/reatcst co'mrn^ rr. :'■ ■ ,' • of the given numhers. To jind the' r/rentest common 'me<!^ure of mor*'. than two numbers^ find the greatest common measure of two of them, (hen find the greatest co7nnio?i measure of this mmmire and an other of the mnnbos^ and w on : th' last common meas- ure will he Ihe greateU common measwe of alt the numher; . Either of the above methods may be used in the follow- ing exercises. Ex. 4. Find the greatest common measure of 48, 64,iipd H2. ' G. 0. M..d«. 5. Find X\iq greatest common measure of 08, 1 19, and 357. (I. Find the greatest conimon messuro of GO. OjQ, and 10'-. GREATEST COUMON MEASUr..K. >^i()5 7. Vi d the greatest common measure of c : 1 .:;. (1. C. M. 13. i!^ ' ' lie greatest commou mcasjuro of 40, 60, and 200. G. C. M. 2U. 9. Find the greatest common measure of 96, lt?8, and 320, 10. Fiud the greatest common measure o^ 164, '-87> and 451. ^ 0, M. 41. 11. Find the greatest common mea«iirc of 63, 126, i^ 15, and 441. ' G. 0.U, 68. 12. Find the greatest common measure of 150, '376, and (i75. 13. Find the greatest common measure of. 40, CO, 68, and 204. ^^ ^\ M. 4. H. FiM.^ 0- . ' :; r 2, nnfl n:)-.. M..214. I :\ Find the greatest common measure of 63. 189, 315, and }i}. Fuid the greatefat common me.'i^u re of 152, *j80, and 532, ,. ..^-. ' ■ r. r'. jy[, 7^. 17. Fijid the greatest common mcasv. ■■ 1 , 7, and '^^IL ^ G. C. M. 17. i '. d the greatest comuion measure of iOQ,' IJ^O, 210. and 4JV~. 19. Find til i34, 190, and • 1140. G. CM. 38. 20. Find the greate.'ic common measure of 54, 108, 324, nnd378. G C. M. 54. Si. Find the greatest common measure of oQ, 84, 140, and 106. * a. C. M. 28. 22. Find the greatest common Rinr.sHro of 75, 1:25, 375, and 67^. G C. M. 75. Fin<] the greateist common measure of 46, 115, aad 161 . 50 106 ABSTRACT NUMBERS. LEAST COMMON MULTIPLE. 2)40, 60, 150 2) JO, 30, 75 ^3)10, 15, 75 3)2, :J, 15 2. 1. 5 Ex. 1. Find the least common multiple of 40, 60,and 150. § J 00 Model. — Divide each of the numbers by 2. (§ 50). Divide some of the quotients by 2. Divide each of these quotients by 5. Divide some of these quotients by 3. These quotients are prime to each other. __ 2.2.5.3.2.5=600. 600 is the least 2.2.5.3.2.5=600 common multiple of the given num-^ bers. Explanation. — We divide two or more of the given numbers by any prime number that will divide them with- out a remainder ; and two or more of the resulting num- bers by any prime number that will divide them without a remainder : and so on, till the quotients are prime to each other : — remembering to repeat in the line below, such numbers as cannot be divided. By this means, every factor of each number is used, and hence the result is a commov multiple of the numbers; but no factor of either number is used more than once, and hence the result is their least common multiple. p^uij/. — Divide two or more of the given numbers hy any prime romraon measure; take the qiwfients and the undivid- ed numbers for a new set ; divide two or marc of theAii by (my prime common measure; and ^o on, uniil the resulting numbers are prims to each other; the eontiniied product oj the resulting numbers and all the divisors loill he the least common multiple of the gicen numbers. Ex. 2. Find tlie least common multiple of 3C, 120, and 216. 60 LEAST COMMON MULTIPLE. §107 36=2. 2. 3 J^ § l^'^- Model. — iiesolve 06 126=2. 3.3.7 '^^^^^ ^^^ prime factors. (§99.; 2IQ 2.2.3.3 2.0 3Sr=2.2.3.3. Resolve 12G into wwoonoo if^ia its prime factors. 120=-2.3.3.7. ^.Z.6.6.i.4.6~l0l^ Resolve 216 into its prime fac- tors. 216=2.2.3.3.2.3.-^2.2. 3. 3. 7. 2. 8=^1512. 1512 is the least common multiple of the given numbers. Explanation. — The prime factors are arranged as in ^ 104, and one factor is taken from each column, whether full or not. Rule. — jResolve eacliof tJic given numbeni into its prime factors ; multiply together all the factors of the largest nitiTi' bcr, and all the factors of the other numbers that arc not fourid hi the largest number ; the product xcill bn the lea^t common multiple of the given numbers. Either of the above methods may be used in the following exercises. Ex. 3. Find the least common multiple of 5, 6, and 7. 4. Find tlie least common multiple of 2, 4, 6, 8, 12, and 16. L. C. M. 48. 5. Find the least common multiple of 3, 6, 9, 12, and 18. L. C. M. 3G. 6. Find the least common multiple of 5, 10, 12, and 15. 7. Find the least common multiple of 6, 12, 24, and 48. L. C. M. 48. S. Find tlie least common multiple of 8, 24, and 72. L. C. M. 72. 9. Find the least common multiple of 3, 9, 18, and 72. 10. Find the least common multiple of 2, 3, 4,5, 6, 10, 12, 15, and 20. L. C. M. 60. 11. Find the least common multiple of 3, 5, 7, and 11. L. C. M. 1155. 61 >ji07 iT,^TR\CT NUMBER?. l^. I\..:\ -A.^ iy^vfnurm multiple of 2,3,4,6,8.12, and 24. 13. Find the least common multiple of 3, 7, and 18. L. CM. '^73. 11. Find tlie least com,mon multiple of 2, 4, 7, and 14. . \ L. C. M, 28. 15. Find the least common ninltiple of 3, 5, 15, and 30. 10. Find the least common multiple of 2,4,8, 16, and 32. L. C. M. 32. 17. Find the least common multiple of 3, 4, 6, 8, and 9. L. CM. 72. 18. Find. the least common multiple of 2, 3, 6, and 9. 19. Find the least common multiple of 4, 6, 8, 12, 16, and 32. L. C M. 96. 20. Fi id the least common multiple of 2, 4, 5, 10, and '^0. L. C M. 20. PROMISCUOUS PROBLEMS. 1. Read 279301682038040. 2. Read i207§008u40009750. 3 . Write twenty-seven billions, three hundred and three millions, four hundred and seventy-five thousand, and eighty-nine. 4. Write five hundred and five billions, and fifty-five. 5. Add 3 millions 24 thousand and 17, 4 hundred thou- sand 7 hundred and 98, 4 millions 247 thousand and 56, and 724 thousand 8 hundred and 29. Sum, 8396700. 6. Add twenty, 2 hundred and 2, 2 thousand and 27, 20 thousand 278, 202 thousand 7 hundred and 89, and 2 millions 27 thousand 8 hundred and 90. §2 PROMISCUOUS PROBLEMS. §107 7, UnB 9 millions and 9, subtract 5 iiu]Iiaub789 adred and 54. 3216355. 8. From 80 milliaiis 85 thousand and 8, subtract 65 inil- •liops 764 tboueand 3 hundred and "^ ' Kem. 143206o'3> Multiply 4 hundred and 70 i ' "^ ^ "'"■••' • r.d !■ thousand 8 (mndred and 1 . 10. Multiply 90 thousaud 7 hundred aud'S, b}^ 80 tliou- sand Ghundred and 4. Prod. 7::nilS5J^20. 11. Divide 2 billions 59 million' ' ' - -ousand and. 72, 50 thousand 7 hundred and 9. Quot, 40608. 12. ]'>ividc 8 billions 777 millions 887 thousand 5 hui:- dro 31, by 97 thousand 5 hundred and 31. 13. The minuend is 4 hundred thpusand 4 hundred ; the subtrahend is 364 thousand 7 hundred aud 26 : what ks the remainder? - ' Ans. 35674. 14 The minuend is 57 thousand and 57 ; the subtrahend 27 thousand 5 hundred and 79 : what is the remainder 1' An.'i. mis. 15. The minuend ic? 75 thousand and 6^i ; the remainder ■ • 36 thousand and 57 : what is the subtrahend 1 16. The subtrahend is 3 millions and 75 ; the remainder I'j 5 hundred thousand 7 hundred and 5 : what is the minu- end ? Ans. 3500780. 17- The remainder is 777 thousand 7 hundred and 7 ; t-hc subtrahend is 654 thoiis.fnd 3 hundred and 25; what is the minuend ? Aus. 1432032. 18. The multiplicand is 3 millions and 75 ; the multi- plier is 5 hundred thousand 7 hundred and 5 : what is the product ? 19 The multiplier is 3 thousand 3 hundred and 3 ; the multiplicand is 75 thousand 4 hundred and 25: what is the product ? Ans. 249128775. 63 §107 ABSTllACT NUMBERS. 20. The product is 670 millions 592 thousand 745 ; the multiplier is 12 thousand 845: what is the multiplicand T Ans. 54321. 21. The multiplicand is 40 thousand 5 hundred and 6 : •Le product is 413 millions 282 thousand 7 hundred and 18 : what is the multiplier ? 22. The dividend is 1 billion 546 millions 2(')3 thousand 5 hundred and 4; the divisor is 71 thousand 2 hundred and 17 : what is the quotient? Ans. 21712. 23. The dividend is 2 billions 162 millions 6 hundred thousand ; the remainder is 19 thousand 4 hundred and 90 ; the quotient is 24 thousand and 6 : what is the divisor ? Ans. 90085. 24. The divisor is 14 thousand and 20 ; the quotient is 2 thousand 3 hundred and 45 : what is the dividend ? 25. The divisor is 7 thousand and 2 ; the quotient is 2 thousand and 7 ; the remainder is 2 thousand and 7 : what is the dividend ? Ans. 14055021. 20. Resolve 3 thousand and 80 into its prime factors. P. F. 2,2,2, 5, 7, 11. 27. Resolve 5 thousand 4 hundred and GO into its prime factors. 28. Resolve 4 thousand and 4 into its prime factors. P. R 2,2,7, U, 13. 29. Find the greatest common measure of 58, 87, and 2610. G. C. M. 29. 30. Find the greatest common measure of 118, 177, and 295. 31. Find the greatest common measure of 4^', 80, 128, and 176. Q. C. M. 16. 32. Find the least common multiple of 3, 7, 9, 12, and 18. L. C. M. 252. 64 COMMON FRACTIONS. §109 33. Find tbo least common multiple of 2, 5, 8, 11, and 14. 84. Find the least common multiple of 2, 4, 7, 11, 16, and 22. i L. C. M. 1232. 35. What number is that to which if 1231, 8912, 5G78, 45G7, and 9123 be added, the sum will be 47275 1 Ans. 17761. 31. What number is that from which if 1234,8912,5678, 45li7,and 9128 be subtracted, tlie remainder will be 47275? 37. What number is that by which if 9876 be multiplied, the product will be 121919220? Ans. 12345. 38. What number is that by which if 5483886 be divided, the quotient will be 2468 ? Ans. 2222. FRACTIONS. § 108. A FRACTTON is a part of a unit. Thus, one half, three fourtlis, two fifths, five sixths, four sevenths, three eighths, five ninths, and seven tenth-i, are fractions. Fractions are of two kinds, Common and Decimal. COMMON FRACTIONS. § 109. A common fraction, or, simply, a fraction, is de- noted by two terms!, one above, and the other below, a hori- zontal line. The term above is called the numerator ^ the term below is called the denominator. Thus, the above fractions are denoted, i, f , f , |^, f, f, -|, ■^. The numera- tors are 1, 3, 2, 5, 4, 3, 5, and 7 : the denominators are 2, 4, 5, 6, 7, 8, 9, and 10. E 65 §110 ABSTRACT NUMBERS. Point out the numerator and tlie denoujinator of eacli of the following fractions : i, -|, f, ^, f, |, |, -V» -rr» T2» tt> 5 7_ _P_ 3 _R_ _2_ _3_ _4_ _J>_ _fi_ _7_ _8_ _n_ lH 1 I 1^ 14" 15» 10' 17' 18' 19' 20' 2 1' 2 2' 23' 2 4' 2 3> 26» 37'26>3»> iA i± ^5- JL1 <^ JLS a o 3_7 4- "^ 2^JLF 3^' 31' 32' 3 '3' 3 4' 35' 1200' 2000' 376' 9 72* § 110. A fraction is rf-ad by |>ruinnniciiig after the nu- merator the ordinal of the denominator iti the singular or the plural number accorditig as the numerator is «»ne or mora than one. Thus, -t is read, one fifth ; |, two fit's hs; ^, three twenty -firsts ; -g^, four thirty-.-ecoMds ; ^^^17, five two hundred and-ninths; -^jy^^, t^ix two thoui^and-a nd- sec- onds ; -g^y'og^, seven three thoiisand-one-hundred-and .-ixths. But, if the denominator is 2, the fraction is read half or halves, and not second or seconds. Thus, -}, one half; -f, thrre halves. IIj^'kI thp followinfy • ^ ^- * — " -' "_ JL. _+_ _p_ _i_ s , _ 4 ^- _6_ __?_ _8_ _9_ 1 O. _:i_ _4_ _S_ _6_ _7_ _8_ _?i_ J D 1 1 IT' 16' 17' 18' 19> 20' 21' 22' 2 3' 24» 25' 2 6' 2 7» 28»2 9'30> 12 1 3 X± XJi -L<> 11. -lA JL^l ST' 32' 33> 34' 35' 30' 37' 3b' § 111. A fraction \^ prod me d by dividing a unit or a iQunjber into some nuniber of e<|ual f»arts. Thus, \ is pro- duced by dividing the unit into four etjual parts: f, bj dividing 2 into 5 equal parts ; ^, by dividing 5 into 9 e(jual parts. The nun)erator is the divi<lend, the denominator is the divisor, and the value of the fraction is the quotient. See $48. How is I produced ? |-? |? i? -,% ? ^? A? i^? ^? VP ? iJ. ? u ? Xs • 2 4' 35* § 112. Otherwise, a fraction may he produced by divid- ing a unit into some number of equal parts and con.-tder* ing either one or several of thesb parts. Each of these parts is called a ^//ar/ />;/<«/ 7/1* /V ; and a fraction is f^irher One or several fractional units. The denominator shows C6 COMMON FRACTIONS. ^113 into liow many partes the unit is divided, and the imnierator fihows iiow luany parts there arc iti the fraclii)ii. Thus, iit |, one fifth is the fractiotial unit, and the fraction containg three of these units ; in ^, the fractional unit is one niutli^ and the fraction contains seven of them. In this view, how is ^^^ produced ? 3^^? {r ? f? J^ ? y\f 7 ? _S ? j< ? 2 ? _5_ ? _«_? « 4 • a r> • 7 • 9 • 11" 10* § 113. The o t/iifi of a fraction is the quotient of its nu- merator divided by its denominator. Tliis value depends on the value of tjie fractional units?, as well as <u\ the niun- her of tliein. If the fractional units of sevt-ral fraeti.mt are ef|ii:jl, of course the greatest fraction is the one which has the most fractional units. Tiiat is, if the denoui'nutors arc e(|nal, the groate.^'t fraction is the one wisieli has tho greatest numerator. Aga'n, if the nuniher of frictional units in several fraction!* is the same, of ct)urse thegieiresl^ fraction is the one which has the largest fraotiMia! nu'ib^ But, tiie larger the number of parts into which a unit i^ dividt^d, the sn>aller each part must be. Therefore, if tho uumiM-ators of ^several fractions are equal, the greatest frac- tion is tlie one which has the smallest den(Mninator. How do I and * compare in value ? -^- and f ? y and ^t r- and ^ ? ^V »»<1 t*t ? A a"^ t\ ? xV »" ^ t\ ^ U ^'»"J i^« ^ ^ and I- ? -/3 and -,\1 -,% and A? t and ^ ? -* and /j-? \i- a"<l M? U «nd H? li and f^? || and fi? ^and^f From the definition in § 1U8, the number of fractional units in a frae;ion must be less than the number of pjirti into which tlie unit is divided ; that is, the numerator must be less than the denominator. Larger numbers, however, may be expressed in a fractional form ; and such expres- siocs are improperly called fractions also. Ilejce the fol- lowing distinctions: — 67 §114 ABSTRACT NtJMBERS. §114. A proper \rnct\im h owe whose numerator is Icsm than its <lon»»niinator, antl wh/»se value is less ihnn a uirit. Tl'iis I, ly ^, -♦, J., I, A, A, _z^, _«_^ .7^.^ and |.}, arc proper fractious. § Ho. An improper fraction is one wlioFe nunicrator if not /r.v.s- tlun ita (Iciioniiniitor, jind whose value is not ]e$n than a unit. Thus, -?, t, 2, {-, §, ^ V, V, Vj Vu',^*"^ -f^ ^«ir« jinpr(»per fractious. :I.s I a proper or an improper fraction ? * ? ^ ? -| ? -^\T JL; ? nV 11^ 12? 17? 209 20 9 W? ft ? P? »V f»? 7^ * 1 ■ J,;J • ♦ ' & • 'JO' 1 > * 2 5 ' 1 5 • 1 o • w • ¥ • y • w * § lJ(j, A unit is often ouilcd, for distinction, un iTif/yral IBnit ; and a number of integral units is called an integer, .er an inlegral number. § 117. A mixed number is one composed of an integer and a fraction. Thus, 5^, read five and one half, is a mixed number. So al.-o, (i^, 3^, 18^, 31;, GG^. A fraciional unit or a fraction may be divided into anj nuuilMjr of equal parts. Thu'^, if I be divided into o equal part.", Ciich of the parts is J of ^. So, if ^ of I be divided into 4 equal parts, 3 of these parts arc ;^ of ^ of ^. Such cxpiev-ioiJB are called comjiouiid fractions. Jlencf, § 118. A compound fraction is a fraction of a fraetioiL Thu., I of -^ i of vV, I of }, l], of 12X, and ^ of ^ of 33^ are a)njj)ound fractions., § 119. A complex fraction is one which has a fraction for 2 its numerator ojr its denomitator or each. Thus, - , read two divided by five and one half, is a complex fraction, 6i 12i- .} of I \ of 18J *^" "^''' y ' 37i* "33i ' vTf 12.i- 68 DEDUCTION OF COMMON FBACriONR. ^123 re: UCTION OF COMMON FRACTIONS. § 120. T() reduce any mimbcr. cither fracti)nar or inte- gral, i^ to cha'ige its /'ftni of expression wlthmt r/i>n>f/ing{ ifs Vdfiir, Thus, a unit ni>i)' be reduced to *, or to y, or tfll I'o" • — i '"^^^ ^'^ reduced to ®, or to -}-i, or to ^\ : — (j\ nja v be reduced to ^^, or to ^£ : — and y ,iiay be reduced lo 12^. § 121. A fraction is in its lowest terms \vht;n its tcrrafl are pnuie to each other. (^ 04). Thus, -^, \^ |, -y, ;^, -J, and 'J', are each in its lovve-t terms. Ex. 1. Reduce \^ to its Newest term?. 4^;if^l4:i^'9j^|i ^ 122. MuDKL.— Divide botk terms by 4 : 4 in 14 t, 3() times; 4 in .^TG, 144 times: divide both these quofit-nts by 4: 4 in oG, !> times; 4 in 144, oG rime-;: divide both fho.»o <juo- tienrs by 9 ; 9 in 9, once ; 9 in HG, 4 times. 'J'h<^vo (|U0- tients are prime to each other : hence, \ is the given Irao- tion in its lowest terms, ]Cxri,AVAT!<>N,— By eompHririg§^ 77 and 111, it is evi- dent th it the value of a Iracfioii i> not changed »iy dividing, both its ternjs bythesame nnnjber: and by jsucci^s.sive divi- sions, its terms may alwa\s be made piiujc to each oiher. Ex. -. lied nee -^^' !- to its lowest term^. 2v -llf^ll^ § 128. M(n.Ki,. — Find the greatest com- mon measnre of the terms of tln^ fiacfion. (§ 108i. Their greatest con)nn'ri measuie is 27. D.vide. both terms bv 27 : 27 in 5G7, 21 times; 27 in 67;'), 25; times. 21 twenty iitths is the given fraciion iii its iowest terms. Rui-R for reducing a fraction to its lowest terms. 1. Dciilii hotk tf^.rm< hj (till/ conitiiim mnsan; dioidt Loth t/iKSf (jU'ffifnfs })ij ttiii/ cninniou inensnrr • (dh/ .-o itti^ Ultlll ihi' qii'itif.uf>i arc in'Im.f to r<irh nlhr ; (lit: lual (juuhcHfj will tfS the l(jioeil term.< of the ij'ocn fractiun. |124 ABSTRACT NUMBERS. Or, 2. Divide lofh ferrns It/ tlidr f/rrafcst cornnion iiiea*- urf : the qnofiniis will he the loua^t lams, Ex. 3. Reduce -^l^^^ to its lowest termis. 4. Reduce f i^^ to its lowest terms. Value, •^. 5. Reduce ^|-^ to its lowest terms. A'^al. |-, G. Reduce -*-f^- to its lowest terms. 7. Reduce |f^-2- to its lowest terms. A'nl. -jV J^. Reduce \-^^\ to its lowest terms. Yal. \\^ P. Reduce -^—-r to its lowest terms. 18 9 4 10. Reduce f-|{)^ to its lowest terras. Yal. ^, 11. Reduce -|^|-f to its lowest terms. Yal. -/-j^ 12. Heduce ^-^-^-l to its lowest terms. 13. Reduce -ff-^ to its lowest terms. Yal. l^^, 14. Reduce ||-^ to its lowest terms. Yal. -^^^ 15. Reduce ^44^4-^ to its lowest terms. 10. Hedtice ^ffl to its lowe-t terms. Yal. ^^^ 17. Reduce -^Vs ^^ '^^^ lowest terms. Yal. t/qV 18. R<duce 14x1 ^^ ^^^ lowest terms. 19. Reduce |f?-i to its lowest teims, Yal. f?-^^ 20. Reduce -^-«ff^ to its lowest terms. Yal. |^-^ 21. Tn Ytt* ^f'W mapy units i' 8G8'16 § 124. MoDKL.— Divide tlie numer- f58 ,54tt=«'>4-} at(»r })y the deiiomiuafor. (?^ 71). Thb 4, rpiotient is f)-4, and tlie rtniaindor 4, Redufo -.-V to its luwest teruiS. The jiren fraction is equal to 54^. KxriiANATiON. — Since IG sixteenths make a unit, the BTimher of units in SG*"^ sixteenths is ecjual to the number pf limes IG is contained in 8C8. See § U.S. Ri'LK for reducing an improper fraction to a whole or a nixed number. / 70 DEDUCTION OF COMMON FRACTIONS. §125 D'lvlde the numerator bij the ilenominafffr ; the qKotient will he the integral part. Place the denondnator umUr the rem ai inter for the fracti mul part, Ex. 22. [n y* how ininy units? Anf*. 25. 23. Ill Y l^ow many units? Ans. 24.1:. 2\. In -i--^— ^'<J^^ many units ? 25. In y liow many units? Ans. 3-2-. 28. Reduce J-«n. to units. Val. 20. 27. Reduce ^-^.s> to units. 28. Reduce \-Y ^^ units. Val. 14-*-. 29. Reduce V/ t^> ufiits. Val. 8^. 80. Reduce ^.-y to units. 31. Reduce W^ to a njixed number. Val. 134. 32. Reduce Y/ ^0 a mixed number. Val. 12^^. 30. Reduce "Yt^ ^0 a mixed number. 84. Reduce y^^ to a mixed number. Val. 11-^V S5. Reduce y^ to a mixed number. Val. 10-^-. 36. Reduce ^-^^ to a mixed number. 37. Reduce y/ to a mixed number. Val. S^^. 85. Reduce ^^^ to a mixed number. Val. *7~\ S9. Reduce \\/ to a mixed number. 40. Reduce VV ^^ a mixed number. Val. 69V 41. Reduce 10 units to fourths-. § 125. MiM)Kf..— Multiply 4 fourths by 10. 10=*jp The product is 40 fourths : hence 10 uLits— 40 fourths. ■ ExPLAN.VTioN.— Since 4 fourths make a unit, 10 units= 10 times 4 fourths; that is, 40 fourths. Rui.K for reducing a whole number to any fractional de- nominiition. Mif/tlpij/ the nu'^ber of fractional units in a unit by the number of units. 71 §126 ABSTRACT NUMBERS. Ex. 42. In 3 units how many fifths ? 43. In 5 units how many sevenths? An?. ^^. 44. In 6 units how many ninth;^ ? Ans. '^, 45. In 7 units how many elevenths? 46. Ueduce 8 to thirteenths. Val. VV- 47. Reduce 9 to fifteenths. A'al. VV. 48. Reduce 11 to seventeenths. 49. Reduce 12 to twentieths. Val. ^*s, 50. Reduce 13 to twent- ^ nrths. Val. W. 51. Reduce 14 to tweni^-i.iDths. 52. Reduce 15 to thirty-fifths. Yal. •'W' 53. Reduce 10 to a fraction with denominator 40. Val. Vo^ 54. Reduce 17 to a fraction with denominator 4H. 55. Reduce 18 to a fraction with denominator 53. Val ^-^ 56. Reduce i9 to a fraction with denominator 59. Val. ^}-l^, 3 57. Reduce 20 to a fraction with denominator 05. 58. Reduce 21 to a fraction with denominator 71. Val ±±SJL, ' 59. Reduce 22 to a fraction with denominator 77. 60. Reduce 23 to a fraction with detiomiuator 85. 61. Reduce 16^- to an improper fraction, 16i = V + 2 = V § ^-^- MoDKL.— Reduce 16 units to halves. Add o2 halves and 1 half. — The sum is 33, halves: hence 16^ is equal to 33 halves. Rule for reducing a mixed number to an impro[>er frac- tion. Rtduce the infeger to the (JcnomI nation of the fraction ^ add the two numerators tojefher, and under their sum mttthe common denominator. 72 REDUCTION OF COM MOM FRACTIONS. §127 Ex. Ci. Reduce 3| to tliiids. Val. y. G-J. R«^.(iuce 4.^- to fourths. 64. Re<luce 61 to fifrlis. V;il. \K 65. Ri-duce, 8j to sixihv. • Val. *^*. 66. Rod wee lOy to sevenths. 67. Ke(]uoe 12.^ to eighths. Val. JlOJ.. 68. Reduce 14.} to ninths. Yal. -L^-i 60. Reduce 16 tV to tenth.*. 70. In 17^1 ^i"^v many elevenths? An^. yy. 71. In ISfV hovf many twelfths? 7-j. In 19yV '^*^^ many fourteenths? 7'1. In 2)i\ how many sixreenths? 74-. In 21/^ how many eighteenths? 75. In 22,/ ^- how many tA^eiity firsts? 70. Reduce 23^:^- to an improper fraction. 77. Reduce 24.}"- to an im[>roj)er fraction. 7H. lleduce 25/^ to an improper fraction, 79. Rfdiice 523^3 to an improper fraction. 8u'. Reduce 65 y^- to an improper fraction. 81. R-jduce -•'- to twentieths. •J! ^=y^; ^1-7. Model. — 5 twentieths make one fourth, ^lultiply both terms hy 5: 5 times 3 arc 15; 5 times 4 are 20. a js equal to 15 twentieths. Exp.tiANATioN. — By comparing §.!^ 76 and 111, it is evi- dent that the value of a fraction is not changed hy multi- plying both its terms by the same number. We divide the re(]uired denominator by the given one, and multiply both terms of the fraction by the (juotient. RiiLK for reducing a fraction (o a larger denominator. AInitipf(j infh ferma hj (lie qwitient of the rKjuiitd dc' nomijKtfor divided bj/ fhe ijlccn one. 73 Ans. 2 2 3 "1 u • Ana. 3 2 5 1 6 * Ans. 3 8 S 16 * Yal. R n n a * ♦ Yal. 6 7 8 U 7 * Yal. 1 718 3 3 • Yal. 4 "281 6 » ' • §128 ABSTRACT NUMBERS. Ex. ^2. Hednce | to tenths. Val. yV 80. Heduce A to eiglitecntlis. Yal. i-5,^ 84. Eednce -f to tbirty-fifilis. 85. R.duce |- to forrieths, Val. J^|^ 8(>. T^educo |- to sixty-thirds. Val. -|-|v 87. Reduce —o ^^ ninetieths. 88. R«*diice ^ to ninety-ninths. Val. ^. 81). Heduce ^^ to sixtieths. Val. f^. 90. Reduce -^3 to sixty-fifths. 9i. Reduce yoi; to twentieths. § 128. Model. — Divide both terms by 5c 5]_^.'^_j_^ 5 in 2o, 5 limes ; 5 in 100, 20 timccs. 25 one bundredths=:5 twentieths. For explanation, see § 122. Rule for reducing a fraction to a lower denominator. Divi'le both terms by the (jiiotlent of the given denoinina- tor dii't led btj the required one. Ex. 92. Reduce -/zr to fourteenths. x Val. ,^-. 9o. Reduce -i§ to tifteenths. 94. Reduce f^- to sixteenths. Val. -/g~> 95. Reduce -^^-^ to eighteenths. Val. -^^. 9(). In -^^-^ how many twentieths? 97. Ill f-^- how many twenty fiists ? Ans. ^^w 98. In -ifi how many twenty-thirds? Ans. ^^. 99. In -~V ^^^^ many twenty-fourths? 100. In 22o- ^^'^^v "^^^"y thirtieths? Ans. -?-§.. 101. Reduce -,}, ^:, and |, to a common denominator. (§91). ^ , , § 129. iMoDKL. — Multiply both termn ''^ * ''^ of the liist fiuciion by Ji2 ; 32 times 1 •ff ^% oi ^'"^ '^-; ^- times 2 are 04: multiply both terms of th»'. second fraction h) 16; 16 times are 48 ; ]G times 4 are 64 : multiply both termi 74 PPBUCTION OF COMMON FRACTIONS. §129 of tlio Oiird fr.-iotion b}' 8 ; 8 times 5 nre 40 ; 8 tiiues 8 are 64. The j?ivpn fractions are respectively cijual to 3J, 48, and 40 sixty- fuurtb.". ExrrANATiON. — The values of the fractions are not chart|j(H], because both terms of each fraction are multiplied by the f-anie iiuiuber: and the denominators are alike, be- cause each one is produced by mulfiplyinc together all the given denojiiinators. The n)ulti[)lier 82 fur the first frac- tion is 4 X 8, the product of the other two denominators. And so for the other?. RiJLK tor reducing fractions to a common denominator. Mnlnplij both ferms of euch f taction It/ the product of th* other dinnminotors. Ex. 102. Heduce -|, -^, and -^, to a common denojiiinator. 103. lleduce -^, -^-, and J, to a common deooniinator. Vi* I 1 "^ 10 8 * *"• 3 0' To' 311- 104. lleduce J, 2, and |, to a common denominator. V-i I 2 4 5 a 4 105. Ilt>duee -^, |, and -^, to a common denominaror. 100. Reduce *, ^, and |-, to a common detioniinator. ' "' • 2 10' a 1 » a 1 6^ 107. Reduce -\^ f, and y, to a common denominator. Vi 1 -'■ « _f fi_ JL?J» ' ' '• 3 3 (>' a 3 6' 330* lOH. Re<lnce i, ^, and -y, to a common denoniibutor. 109. Reiluce ^5 I, and -j'\j, to a commoti denominator. V-il ^(^ 1 «<' 2JL0 * «J ' . -7 [. o 1 Tii " ^7 a 0* 110. Reduce -^, -^^, -^\^ and gV, to a common deiionjina* tof V'il _2>oo. _8 !_« Q_ _finoo_ 7 s 6 O ^ '"•2 16do'»216OO'2 1C0O*21G0O" 11 '. Ri'iluee I, -j^j, and -j^g-, to a common dtMjominator. 111.'. Roduee -), 1^, and y'j-, to a common denominator. ' " • « 1> 3 » G J 3 ' a 9 3 * lis. Reduce -j^, ^, and -fo-».^^ ^ common deitoniinaioi-. 114. Reduce ^, fj '^^d *. to a cunimon deuumiiiittor. 75 §130 ABSTRACT NUMBERS. 115. Ileduee \, ~, and ^-, to a common dononiinator. Vj, I 5 t 3 6 4.» * '* • "?!¥' 2T6> TT9' 116. lieduce ^, |, and y, to a common denoiniiuiror. V-il -iL'i- j^3^ iOO * '" • 14 0' 1 4 » 1 ¥!?• 117. lieduee ^, |-, and |-, to a common deuominator. 118. Reduce i, f, and f, to a common denomiitator. V-il -3Jl^ ^7J! _40 '"'• ibO> ISO' 1 sU* 119. Reduce ^-, -f-, and — , to a commovi dennmiii i or. Vi 1 7 SO 1 * '"' "2 8 0' ^so* a 120. Reduce ^, f, and y\, to a common deuominator. L A A 2 4 8 ILM. Reduce |, -|, and |^, to their least common denomi- nator. §130. MoDFL. — Find tliejenst common multiple of the denominators. (§ 1(J(>). 8 is -| I 1^ their leist Ci>ujmon njulti|tle. MuUiplj both terms of the first friicrion by 4 : 4 times 1 are 4; 4 times 2 are 8 : multiply Ixirh terni>i of the • seco'id frdc'i )n by 2: twice 3 are 6; twice 4 are 8; th« third fraction is already of tlie required dcuoiiiiiiurion. — Tlie given fractions are respectiveiy equal to 4, 6, and 5 eighrh.^'. Exp t. A NATION. — To find the proper multiplier for the terms of either f^yiction, we divide the least conimon mul- tiple by its denominator. See § 127, RuLK for reducing fractious to their least common de- noniinafor. Fmd /he hast common multiple of the denominalorii, and rtducc each fraction to the diiiomlii'ilioii expressed hij thU mulfipf''. E'ich fraction iniist first he. iiL its hnct^t terms, Ex. 122. Pteduce f, -4,, and |, to their least con m m de- \ n o I n i ) i a t o r . Ya 1 . y\ » fV? -]rh 123. Reduce I, |, and ^, to their least commou duuomi- iiatur. 7G \ 1?ED€CT10N OF COMMON FRACTIONS §180 124. Il(3duce J, -f-, and |, to tlieir least common dcnoiiii- Udior. vai. ies, ins^ lo.s* 125. lleduce ^, I, and y^, to their least comnKHi dcMiomi- nator. Val. -,V ^\„ H^. 126. lleduce ^., ^, and -^, to tlieir least comniuii dciiomi- h, tor. 127. lleduce -^, |, and ^"^,^5 to their least common denomi- nator. Va!. -/-, /^, I-*-. 128. Reduce .}, ^, ^, and i, to their least common de- nuujiiiator. Yal. -j\-, ^\) ■^•\, -^-^, 129. Reduce -^, i, ^, ^, and -j\, to their Uaat common de- nominator. 180. Reduce |, I, |, -|, and |;^, to their least common de- iioiuinator. ^ Yal. aa, -^I, f^, -|^.> |->. 181. Reduce IJ, 2, >ij To> ^'^'^ "sfj *^ their least common deiu.minator. " Val. if, f*-, |f, -^, if, 132. Reduce l, ^, J-^, and H, to their least common de- nominator. 138. Reduce -^, -^\, ^Vj ^^^^ 2^5 to their least common de* nominator. Val. |, -*, |, 1- 184. Rtuliice I, I, -j^^, and -\l, to their least cumuxm de- nominator. Val. -j'V, -ft-, -i«3^, -j^a^. 135. Reduce -^\, -^\-, -^-^^ and -^\, to their lea.st common deiiomiuator. 181). Reduce I, -|, |-, f, and ^, to their least common de- i^^i"i"ator. Val. |§, >^, -*^, -*C,-6-C-. 187. Reduce -J, f, /^, ^f, and |-^, to their least common denominator. Val. -J-^, 1% -i*, |-o, if. 138. Reduce i, a, *, |, 3^, ^'*^, and l^, to their least com- mon denominator. *139. Reduce i, f, -*, y^^^, and i^, to their least common dt^nominator. Val. H-, -||, |f, -*i, H* 14U. Reduce J, «, -^y /g, and 5%, to their least common denominator. Val. |^, -1%, -^, /„> »V- 77 §131 ABSTRACT NUMBERS. ADDITION OF COMMON FilziCTIONS. Ex. I. Add^-, A,|., and|. § 181. Model. — 1 and 3 are ¥ + ¥ + ¥ + { = ¥ = 2 4,JH.d 5 are 9, ai.-l 7 are K).— lo eii^htlis is eijual to 2. The Bum 18 2. ExPLAMAT'loN. — Since all the fraction-! have the samo fractional uni^, their ntKuerators are ad led \'oi' the tiunier- ttor of tlie sum, and the comiuoii deuominator is taken as its defioiuiiiator. Ex. 2. Add I, I, and -|-. 3 § 132. MoDFL. — Reduce the given ^ * '^ fractioLs to 'heir least c Mintioa de- ♦ ^fi-|_L=ij~2* nominator. (§ l:-^0). 4and<)arel0, arid 7 a?e 17. 17 eighths i.s e([aal to 2^,. The f^um is 2^.. ExPi.AVATroN. — It is evidently impossible to add the given fr-ictions wiihout reduction. 3 fourths and 7 eighths make Tieither 10 fourth's nor 10 eighths. It is not essential to reduce to tlie Irasf common denominator; but this gen- erally re<|nires less labor than to reduce simply to a com- mon denominator. Ex. 3. Add 241:. 351|, 179|-, and 187. 24.V .l:+i-i-|- §133. MooKL.— Re- '^ 2 I 4 . i_ ij(_iji duce the fracti.ins to 179^ "b'ii"t*tt H H their least comm.tn de- 1q^ nominator. (§ 130). — 742 f 2 and 4 are (5, and 7 are 13. 13 eighths is equal to 1 and 5 eighths, set down ^; 1 and 7 are H. and 9 are 17, and I are 18, and 4 are 22, -^et down 2 : 2 and 8 are 10, ant) 7 are 17, and 5 are 22, and 2 are 24, set down 4; t and 1 are 3, and 1 are 4, and 3 are 7. The sum is 742f, 78 ADDITION OF C 'M3I0N FRACTIONS. §133 Ruri*^, — Reduce the fractinns to their least cotnnto.i de- nomlnutor : a Id the numerators, and under their .smn set the coni'noa deiomlnttor. Reduce the result to iis lowest terms or to a mixed number ^ as the case mat/ be. Ex. 1. Ada I, I, and f. Sum, 1^-, 5. Add 1,1, A, and ± Sum, 2. G. Add i,f.A,/;,a„d^. 7. Add f, f, f, and f. Sum, 2|-. 8. Add-^, I, A, and |. Sum, 1a 9. Add 1-, A, A, ^, and ^ 10. Add -L, f, A, and >-. Sun.,2j-5. 11. Add^, ^, A, and A Sliu., 2^^^. *■-• ^^'J'l a» 4> 5> To> anu -j^j. l;i. Add i 4-, A, and yV Suiii, l2-|. 14. Add i, i. A, 3Z_, and -i|. Stun, 23V 15. Add A, ^, 8, ,A, a„j i,. 16. Add A, ^, I, X, |i., and a^. S.hu, 4}|, 17. Add A, I, 3, -^-, ^^_., and 3^^. Smn, 2ai. 18. Add A, I-, A, and «. 19. Add A. i, L, and -Jj-. Sun., ^«-V^. 20. Add 2^, 3f, 4A, and 5*. Sim, 16aa. 21. Add 4, 8A, 9^, and llf. 22. Find ihe suiu of IQa, 21a. 32^-, 43^*^-, and 54/^. Sum, 162^*0. 23. Find the sura of 19, 23a, 16a, and 27a. Su.n, 83». 24. Find die sum of At, 2\'^, 3», 25a, and 33a 25. Find the sum of 12^, 18^, 33a, 87a, and 93^ Sum, 245A 26. What is the sum of », 6a, 3|, 2a, and 98 ? Ans. 111». 27. What is the sum of 1, 2a, 3a, 6a, and 9a? 79 S134 ABSTRACT NUMBERS. 28. AVhat is the sum of 4,^, 5-«, 17|, and 18y^ ? An?. 46 1^. 29. What is the sum of 2i, 251, 125^, and 325tV? Ans^ 478/j-. 30. What is the sum of If, 4|, 7«, 10-H-, and 13-}-^? SUBTRACTION OF COMMON FRACTIONS. Ex. 1. From -} take f. 7_.3_4_i §134. M->DEL.— B from 7 leaves 4. 4 eighths is equal to -}. The reiuaiade? b B 2 IS^, Ex. 2. From i take i 1 § 135. Model. — Keduce the fractions to ^ their least conimon denominator. (>? loO). -^— |^=i 2 from 3 lenves 1, that is, 1 sixch. The re- mainder is -}.. a' Ex. 3. From 32?- take ISf. 32|- x_A ^ 1S(). Model.— Reduce the •^Q? y g_^ fractions to their least common 14i- "B'~s — 8 denomi»)ator. (§130). G from 7 leavt's 1, set down |-; 8 from 12 lenves 4 ; 2 from 3 leaves 1. The rcmaiuder is 14}. ExiLANATiON. — Any numher of fractional units may evidently be subtracted from a larger number of fractional units of the same denomination, just as one number of sim- ple units is subtracted from an other. If the given frac- tions liave different denominators, they must first be re- duced to a common denominator : ^ — ^ = neither | nor ^ just as 7 dollars — 3 cents=neither 4 dollars nor 4 cents. Ex. 4. From 27 take 19}. 27 § 137. MoDKL. — 1 from 8 leaves 7, set down ■L^i -^r 10 from 17 leaves 7; 2 from 2 leaves 0. 7l- The remainder is 7|. 80 8" SUBTRACTION OF COMMON FRACTIONS. §138 Ex. 5. From 9^ take 6h ^8 A —i § 138. Moi^EL. — Reduce the fractions "2 to their least common denominator. — 2-5. 8~-¥ (§ 130). 4 from 9 leaves 5, set down ^ 7 from 9 leaves 2. The remainder is 2| Explanation. — When the fraction in the minuend is less than that in the subtrahend, we add an integral unit to the minuend fraction, subtract the subtrahend fraction from this sum, and then add 1 to the unit^ of the subtra- hend before subtracting from the units of the minuend. Rule. — Reduce the fractions to their least common de- nominator ; sid>tract the numerator of the stibtraheQid from the numerator of the minuend; and under the remainder set the common denominator. Jf 111 suhtracting one mixed number from an other, the suhtrcdicnd fraction should he larger than the one in the min- uend, reduce an integral unit to the common denomination of the fractions, add if to the minuend fraction, subtract the subtrahend fraction from this sum, and add one to the sub- trahend in the column of units, Ex. 6. Subtract f from y^ 7. Subtract I from ^. Rem. -^, 8. Subtract -{j^ from -^^. Rem. i. 9. Subtract f from -Li. 10. Subtract -f- from |. Rem. -i-f . 11. From \-l take «. Rem. ^. 1 2. From -}/^ take |. 13. From 5§- take f. Rem. 5^- 14. From 7^ take 4^. Rem. S^^-. 15. From 8f- take 7f , 16. Minuend=l7-iV; Subtrahend=6i Rem. lOff. F 81 §139 ABSTRACT NUMBERS. 17. Minuend:=2003^; Subtrahend=105^. Rem. 94| 18. Minuend =:42i; Subtrabend = 27yV 19. Minuends 721; Subtrahend =24fV- Rem. 47| 20. Minuend=l75; Subtrahend =:83|-. Rem. 91-1; 21. Subtrahend =661-; Minuend=106^ 22. Subtrahend = 171; Minuend = 27i Rem. 10/, 23. Subtrahend=lf ; Minuend=4.f. Rem. 3| 24. Subtrahends 7-«-; Minuend=8y%. 25. Subtrahend 11-}^ ; Minuend=20f5-. Rem. 8f|f 26. What is the difference between 12|i and 21ii? Ans. 8-}^ 27. What is the difference between 16-^-|^ and 10-JvV? 28. What is the difference between 100 and 33^? Ans. 66| 29. What is the difference betwcc;. : 9} and 20yV? Ans. I-I- 30. What is the difference between 75 and 68^-1 MULTIPLIOATION OF COMMON FRA^CTIONS. Ex. 1. Multiply I by 7. § 139. Model. — 7 times 3 are 21 : Ax 7= V=2| 21 eighths is equal to 2|, The prod- uct is 2-^r. Explanation. — Comparing §§ 72 and 111, we see that the value of a fraction is multiplied by a whole number by multiplying its numerator by the number. Ex. 2. Multiply |- by 3. § 140. Model. — 3 in 9, 3 times : 5 Ax3=f=l|- thirds is equal to If. The product is 1^ 82 MULTI1>LIGATI0N OF COMMON FRACTIONS. §143 - Explanation, — Comparing §§ 75 and 111^ we see that the value of a fraction is multiplied by a whole number by dividing its denominator by the number. When the mul- tiplier is a measure of the denominator, this method if preferable to the other. Ex.3. Multiply 47|- by 9. ^'^4: § 141. Model.— 9 times 3 are 27 : #27 fourths ^ is equal to 6-2-, set down a • 9 times 7 are 63, and 429f G are 69, set down 9 ;. 9 times 4 are 36, and 6 ■ ^ are 42. Tbe product is 429-2. Explanation. — As in whole numbers, we begin with the lowest denomination, and reduce each partial product to the next higher denomination, setting down the remaining units of the denomination in question, and reserving the units of the next denomination to be added to the nest product. Ex. 4. Multiply ;] by f . 3 7_2_i §142. Model.— 7 times 3 are 21: S 4. X s — 3-z times 4 are 32.. The product is fi. Explanation. — To multiply by |- is the same as to mul- tiply by 7 and divide the product by 8. 7 times 3 fourths =:21 fourths, and 21 fourths-^8 = 21 thirty-seconds: since a fraction (or a quotient) is divided by multiplying the de- nominator (or the divisor). (§§ 74, 111). Ex. 5. Multiply 30,^ by l. op.1 ^ §143. Model.— Reduce 30i to ^^^" ^ *' fourths.(§l26). It is equal to 121 i|-i X i= VV =7-5%^ fourths. Once 121 is 121 : 4 times 4 are 16. 121 sixteenths is equal to 7^. The product is 7^\. 83 U44 ABSTRACT NUMBERS. Ex. 6. Multiply 30^ by 5^ 30i X 6i- ^ ^'^^' ^^^^^^- — I'^educe the * ^ mixed numbers to improper i|i X V = i-^/J = 166f fraction?. (§ 126). 11 times 121 are 1831 : twice 4 are 8, 1331 eighths is equal to 166^. The product is 166|. Explanation. — It is often easier to reduce a mixed iiumber to'Hin improper fraction before multiplying, if the other factor is not a whole number. Rule. — To multiply a simple fraction by a whole number; Divide the denominator of the fraction, or else multiply lis numerator^ by the whole number. To multiply a fraction by a fraction. Multiply each term of the one fraction by the correspond- ing term of the other. A mixed number may be reduced to an improper frac- tion, or its parts may be multiplied separately. Ex. 7. Reduce | of 4 to a simple fraction. „ . \ § 145. Model. — Twice 4 are 8 : 3 times '3 5" i'5 5 Q^pQ 1^ fj^i^Q given fraction is equal to -^^g. Explanation. — One third of 1 fifth is evidently 1 fif- teenth ; 1 third of 4 fifths is 4 times 1 fifteenth, that is, 4 fifteenths ; and 2 thirds of 4 fifths is twice 4 fifteenths, that is, 8 fifteenths. Rule for reducing a compound fraction to a simple one. Multiply together the several fractions ivhich compose it. Ex. 8. Multiply a by 4. Prod. 1^. 9. Multiply A by 7. 10. Multiply -l by 8. Prod. 7. 11. Multiply I by 12. Prod. 10. 12. Multiply yV by 15. MULTIPLICATION OF COMMON TRACTIONS. §145 13. Multiply -j^o by 5. Prod. ^. 14. Multiply 2i by 7. Prou. 17l. 15. Multiply 8-1 by 8. 16. Multiply 16f by 15. Prod. 250. 17. Multiply 19^. by 20. Prod. 397^. 18. Multiply 207-;} by 13. 19. What is the product of 315-^ and 19 ? Prod. 5995|. 20. What is the product of -'■- and -•} 1 'Prod. f-. 21. What is the product of | and a? 22. AYhat is the product of J^ and ^5- 1 Prod. -}. 23. Reduce ^ of §- to a simple fraction. Val. -*. 24. Keduce jj of ;^ to a simple fraction. 25. Reduce -^- of -f to a simple fraction. Val. -f-l. 26. Reduce -} of ^l of ^ to a simple fraction. Val. ■^. '11 . Reduce -i- of -f of -^^ to a simple fraction. 28. Reduce -^ of 7-^- to a simple fraction. Val. 2-}. 29. Reduce \- of \ of 7^ to a simple fraction. Val. l^-. 30. Reduce -"; of -^ of 8^ to a simple fraction. ol. Find the product of f of f and \: of 12? . Prod. 1-^^. 32. Find the product of f of f and 83^-. Prod. 2^\, 83. Find the product of -^- of 66f and -I of 100. 34. Find the product of f of 250 and -f of 21. Prod. 900. 35. Find the product of -\ of ^ of 210 and \ of 83|. Prod. 670^. SO. What is the product of 16| and 16^? 37. What is the product of 30^ and 60|- ? Ans. 183a^. 38. What is the product of 111-,^^ and 20-^ ? Ans. 2277^^. 39. What is the product of 275 and "^ of a of 36? 40. What is the product of 303 and |-of 20 ? Ans. 173^-. 41. What is the product of 3|- and 4^4? 42. What is the product of ^ of f and f of 3f ? §146 ABSTRACT NUMBERS. DIVISION OF COMMON FRACfTIONS. Ex.1. Divide i-^. by 3. jL^_i.3— _5_ § ^"^^^ Model.— 3 in 15, 5 times. The ^'^ ' ^° quotient is 1 6' Explanation. — Comparing §§ 73 and 111, we see that the value of a fraction is divided by a whole numbet* by dividing its numerator by the number. Ex. 2. Divide f by 5. K 3 ■^— To § 147. Model.— 5 times 4 are 20. The quotient is 3 twentieths. Explanation. — Comparing §§ 74 and 111, we see that the value of a fraction is divided by a whole number by multiplying its denominator by the number. Ex. 3. Divide f^ by |. §148. Model. — 3 in 15, 5 times: -ff-f-f =:;f=li 4 in 16, 4 times. 5 fourths is equal to 1|:. The quotient is l-i. Explanation. — 15 sixteenths-^3=i5 sixteenths (§146): but the divisor 3 fourths is only one fourth of 3 ; hence the quotient is 4 times 5 sixteenths, that is, 5 fourths. (§ 140). Again, since division is the reverse of multiplication, the process for division should be the reverse of that for multi- plication : and since -|-x-f-=:-j^-g^, it is evident that -^^-^2^=-^. Ex. 4. Divide | by |. § 149. Model.— 8 times 3 are 24 : 7 .l-hf ?=ff =-f times 4 are 28. 24 twenty-eighths ia equal to f. The quotient is ^. ExPIiANATiON. — 3 fourths-^7=:3 twenty-eighths (§147): but the divisor 7 eighths is only one eighth of 7 ; hence, by §75, the quotient is 8 times 3 twenty-eighths, that is, 24 twenty- eighths. (§ 139). 86 DIVISION OF COMMON FRACTIONS. §151 Again, Multiplying both terms of the dividend by 56, we have a||-^|^= I4=t- ^^> Multiplying both terms by 14, we have -;|^f -^|-=-f, the same result as before. Ex. 5. Divide 2731- by 5. § 150. Model. — 5 in 27, 5 times with 2 l?_i over, set down 5 ; 5 in 23, 4 times with 3 54f over, set down 4; 5 in 10, twice, set down f. The quotient is 54f. Explanation. — We divide the integer as usual, and reduce the 3 remaining units to thirds, making 9 thirds, which added to the given 1 third makes 10 thirds, and this divided by 5 gives 2 thirds. If the numerator of \£ had not been divisible by 5, we would have multiplied its de- nominator by the divisor, as in § 147. Ex. G. Divide 3^- by 12f . ^^ .193 §151- Model. — Reduce the 3"" *-'3 given mixed numbers to improp- i_o_^.3.s__3^__5_ QY fractions. (§126). 3 times 10 are 30 : 38 times 3 are 114, 30 one-hundred-and-fourteenths is equal to 5 nineteenths. The quotient is 5 nineteenths, KuLE.— -To divide a simple fraction by a whole number. Divide the numerator of the fraction^ or else multiply its denominator J by the tchole number. To divide a fraction by a fraction. Divide each term of the dividend by the correspo/iding term of the divisor. Or, Multiply each term of the divi- dend by the other term of the divisor. To divide a whole number by a fraction. Divide the dividend by the denominator of the divisor, (t.nd multiply the quotient by the numerator . A mixed number will mostly better be reduced to an im- proper fraction. 87 §162 ABSTRACT NUMBERS. Ex. 7. Reduce 2i of to a simple fraction. 2i-^iof A 40. 6 ' § 152. Model. — Reduce the terms to simple fractions. Divide |- hy ^. 8 times 5 are 40 : 3 times 2 are 6. — 40 sixths is equal to 6f . The fraction is equal to 6|. Rule for reducing a complex fraction to a simple one. Divide its numerator by its denominator. Ex. 8. Divide f by 5. 9. Divide -^ by 8. 10. Divide if by 3 11. Divide if by 6 12. Divide 40 by f 13. Divide 200 by 14. Divide 175 by 15. Divide \% by | 10. Divide |-^- by f. 17. Divide i| by f . 18. Dividend=-^' 19. Dividend^! 20. Dividend=f 21. Dividend=:y^ 22. Divisor =f 23. Divisor^l 24. Divisor={r 25. Divisors ^ 26. Divisorz^i off: 27. Divisors I of f : 28. Divisor=:f of f : 29. Divisor=f of l : 30. Divisor=i of 12^ divisor =-|. 4 : divisor=|-. divisor=:f. divisor=i. dividend =-^. dividend=^|. dividend ={'. dividend =.^. dividend = |. dividend=-f-. dividend = *- of ^. dividend=f of ^0-.. : dividend = 1 of y\ 88 given Quot. ^V Quot. -V Quot. -^-^. Quot. 466|-. Quot. 49. Quot. ia. Quot. |. Quot. 2^Q, Quot. i'A. Quot. li Quot. ^^, Quot. ,«. Quot. Iff. Quot. ^. Quot; 1\%. DIVIvSION or COMMON FRACTIONS. §152 :;l. Dividend=:12A-: clivisor=4. Quot. 3^. :i2. Dividend:rz207i : divisor^G. Quot. S^-^V :;3. DivideDd=i45f : divisor^lS^. 34. Dividends 70^: divisor=68^. Quot. l-^-f^. 35. Dividends 27-^: divisor=:55i ' Quot. |-f|. 36. Dividend^:^ of 28^ : divisori=:| of 43^}. 37. Dividend=:|- of a ; divisor=:i of 275. Quot. ■^^^. :]8. Dividends 2 of i of I- : divisor=| of 17-^-. Quot. ■^. 39. Dividend=| of 27.V: divisor=^ of 38^. 40. Pteduce — "- to a, simple fraction. Val. 3-|. 4." 41. Reduce -^ to a simple fraction. Val. --^y\. 4r2. Reduce — ^ to a simple fraction. 41 ^r of ? 43. Reduce " " to a simple fraction. Yal. ■^\. 2, of 4^ 44. Reduce ~ — — -^ to a simple fraction. Yal. i|-. -} ot 19 '}. of ?- 45. Reduce --—7-"- to a simple fraction.. I oi 7i ^ 27 46. Reduce — -.— — - to a simple fraction, Val. 2tV- ■I of 30 ^ ' "" 1- of 20 47. Reduce — — — - - to a simple fraction. Val. -;;%• -|ofl7-i- ^ ? of 2^- 48. Reduce to a : iinpio fraction, 3461 2i- 40. Reduce ~ to a simple fraction. Val. 1- 1 of 27j 50. Reduce - — rr~ to a simple fraction. Val. WA\« f-of72f " '''^^' 89 ^153 ABSTRACT NUMBEHS. CANCELLATION. § 153. In multiplication of fractions, and in some other similar operations, the labor may be often diminished by canceling all the factors common to the numerators and the denominators, and afterwards multiplying together the re- maining factors of each. This is simply reducing the re- sult to its lowest terms in advance. It is customary to draw a line through a number that has "been canceled. Ex. 1. Multiply U by Ti^. ^ 7__^ Model. — 45 in 45, once; 45 in 90, f^' pfp"~14 twice: 7 in 7, once; 7 in 49, 7 times :- 7 2 the numerator is 1 ; the denominator is 7x2 = 14. The product is ^L. Ex. 2. Divide I- of j- of f by f of if of ^V- i of f of i---f of i| of ^\ Model.— 3 ^ in 3, once ; 3 - of 2 of '? X ^ of ?^ of — =.. «J- = 10-^- ^^ ^' ^^^^® • ^ 27^^^/^ 1 « ^ in7, once;7 2 2 in 21, 3 times: 9 in 9, once ; 9 in 9, once : 5 in 5, once ; 5 in 10, twice : the numerator is 3 X 27=81 ; the denominator is 2.2.2=8. The quotient isV = 10i. Ex. 3. Divide the product of 77 and 96 by the product of 22 and 24. 2 Model.— 11 in 77, 7 times ; J 1 in 22^ ^ /¥ twice : 2 in 2, once ; 2 in 96, 48 times • Ilj^.^14, 24 in 24, once; 24 in 48, twice. The ^^ X ^4' ^ quotient is 7 x 2= 14. ? 90 PROMISCUOUS PP.OBLEMS. SI 53 Ex. 4. Divide 11 x 21 x 26 by 3 x 13 x 14. ^^^Jy^^li Model.— 3 in 3, once; 3 in 21, 7 11 X // ^xjg times : 7 in 7, once ; 7 in 14, twice : 13 ^x/^x/^ in 26, twice; 13 in 13, once: 2 in 2, ^ once ; 2 in 2, once. The quotient is 11 Es. 5. Multiply A of i-f- by -J^ of f ^^ of if. Prod. ^ 6. Multiply -V of I- of if by I of -y-. 7. Multiply f of V by a of if. " Prod. ^'^ 8. Divide ^ of AA by V of V of -^V- Quot. i 9. Divide f-t of if by -Jl. of ^. 10. Divide A of A of I by"A of aa. Quot. 2i 11. Divide the product of 22 and 56 by the product of 44, 28, and 16. Quot. -,„ 12. Divide the product of 72 and 96 by the product of 60 and 64. 13. Divide the product of 27, 2S, and 29 by the product of 35, 86, and 37. Quot. -/a.. 14. Divide 10 X 11 x 12 by 22 x 24 x 30. Quot.' J-^. 15. Divide 25 x 27 x 32 x 36 by 15 x 18 x 24 x 28. PROMISCUOUS PROBLEMS. 1. What is the sum of 275, 386, 497, and 608 ? 2. What is the difference between 275386 and 497608 ? Ans. 222222. 3. What is the product of 275386497 and 608 ? 4. What is the quotient of 275386497 by 608 ? Ans. 452938J-AA. -J. Add the difference between 395 and 4:22 to the sum of 39, 54, and 202. . Sum, 3 91 §163 ABSTRACT NUMBERS. (>. Subtract the sum of 25 and 19 from their product. 7. Multiply ibe difference of 25 and 19 by tlieir sum. Prod. 264.. 8. Divide the product of 36 and 45 by their difference. Quot. 180. 9. Resolve 7050 into its prime factors. 10. What is the greatest common measure of 25, 250, and 375 ? " Ans. 25. il. What is the least common multiple of 5, G, 10, and 12 ? Ans. 60. 1-.'. Reduce -rV-^V to its lowest terms. i z o u 13. In "^"^ how many units ? Ans. 10-^^. 14. In 19 units how many nineteenths ? Ans. ^^. 15. In 15|- how many fifths ? 16. In f how many forty-fifths ? Ans. -|-|.. 17. In -^q\ how many twenty-fifths ? Ans. -^V- 18. Reduce -^, -^j and {- to a common denominator. 19. Reduce ^, |, and {^ to their least common denominator. Vol 2 7 .TO 2_5. 20. Add -5^, Jj,-, and i}. Sum, 1. 21. What is the sum of } of *- and ^ of i ? 2^ 22. What is the sum of f of 10-^ and — -^— , ? Ans. 7/,%. •" "* i of 17 23. What is the difference between 19^- and 26^ ? Ans. 6-1-^. 24. What is the difference between a of 27 and -^- of 24? 25. What is the product of 27i.and -v of 77 ? Ans. 706|-. 26. What is the product of '^-—^rr- — - and — --- '/ Ans. 1. ^ 22^ .} or 11 27. What is the quotient of I of 47 by 25^ ? 28. What is tlie product of 5- of 47 and ^ of 25 ? Ans. 43-1-*-. 93 I'LlOiVllSCUO'tJS PROBLEMS. §l5o -I- of 27.^ ^ of 19 , 29. Wlhit h the quotient of ^—xz — - by -— ^ ■ ? Aids. 3,W.V SO. Add the product of -^ of 27 and ~l of ^ to their difference. 31. Subtract the quotient of -f- of 45 by -} of 24 from their sum. Eem. lO^'j. A of 5- 32. Multiply the sum of ^" " and ? of 7'Sl by their dit- ference. Prod. 262-jyVc- 33. Divide the product of 25^- and 17^,- by their sum. 34. What number is that to which if 3-§- , 5^^-, 6f, and 10-/^, be added, the sum will be 30-]j- ? Ans. 3-^-. 35. What number is that tVoni which if of, of, 6^^, and 10^%, be subtracted, the remainder will be 30-^ ? Ans. 57-|-. 36. What number is that by which if the sum of 3f, 5|^ 6{?, and 10y%, be multiplied, the product will be 30-|-? 37. What number is that by which if the sum of 3f, 5f, 6|, and lOy'^;-, be divided, the quotient will be 30-1- ? Ans -^-i- 88. What is the sum of -}, ,|, 13, and IS^V ? 39. What is the difference between -} and -^\1 Ans. y%. 40. What is the product of 5^^ and ^ ? 41. What is the quotient of -ff ff by 19 ? 42. 3^-5-7-^5 + 16-^7-15'^6=what? Ans. 6|-^. 43. (3 + 5-7)~5 + 16-^7-15-=-6=what? 44. 3 + (5 + 7)~5-fl6-^7-15-J-6=what? Ans. 5-^-§.. 45. 3 + 5-7-^6-fl6 + 7-15-^6=what? Ans. 27yV- 46. (3 + 5-7)-^-5 + (16 + 15-7)-f-6=what? 47. 3 + (5 + 7)-^5 + 16+(15-7)-^6=:what? Ans. 22H-. 48. 3 + (7-5)-^(5 + 16) + (15-7)-^6=what? Ans. 4f 49. 34.7_5^(5_l_16 + 15)-7-f-6=what? 93 §154 ABSTRACT NUMBERS. DECIMAL FRACTIONS. § 154. A decimal fraction is one whose denominator is some poivcr of ten and is not expressed in writing. § 155. In the Arabic or decimal system of notation (§10), we observed that, in passing from the units' place to the left, a unit of any order is ten times a unit of the preceding order ; or that, in passing from left to right, a unit of any order is 07ie tenth of a unit of the preceding order. If this law be extended to the right of units, the next order will be tenths, the next hundredths, the next thousandths, &c., as in the following ■ . ^ ti -a ^ » I t2 1 S r 2 S S ^ i ^ ^^ 23 45.234576595875 As 100 is 3-V of 1000, 10 is 3-V of 100, and 1 is J^ of 10, so one tenth is -^ of 1, one hundredth is -=^^ of 1 tenth, one thousandth is J^ of one hundredth, &c. § 156. To write any number of tenths, then, we simply put the proper figure one place to the right of units ; for hundredths, we put the figure two places to the right, &c. To determine the position of units and the relative posi- tions of the fractional orders, we place a period, called the 94 NOTATION OF DECIMAL FRACTIONS. §158 units^ point, between units and tenths ; or to the left of tenths, if the expression is entirely fractional. Thus, 2.3, two and three tenths ; 3.02, three and two hundredths ; 5.32, five and three tenths and two hundredths ; .005, five thousandths ; .0006, six ten-thousandths ; .00004, four hundred-thousandths; .000008, eight milliontht-. In integral numbers, this point, being unnecessary, is never written : but in fractional or mixed expressions, it must never be omitted. § 157. It will be observed that the number of places oc- cupied by the numerator of a decimal fraction is equal to the number of naughts in its denominator. If the ordinary exprei?sion cf the numerator does not require so many places, each place intervening between the units' point and the left hand figure of the numerator must be filled with a naught. Thus, .002, 2 thousandths; .023, 23 thousandths ; ,0203, 203 ten-thousandths ; .0023, 23 ten-thousandths ; .0004, 4 ten-thousandths ; .002034, 2034 millionths. § 158. A decimal fraction is read, like a common frac- tion, by pronouncing after the numerator the ordinal of the denominator. Sometimes, in reading a mixed number, to prevent ambiguity, it is necessary to pronounce the word "units" after the integer. Thus, three hundred and fif- teen thousandths is written .315 ; but 300.015 is three hun- dred units and fifteen thousandths : so, 7000.0275 is read seven thousand units and two hundred and seventy-five ten- thousandths. Read the following decimal fractions: — .1, .3, .5, .7, .8 • .01, .05, .09, .11, .25, .34, .47, .51, .63, .75, .87, .99; .001^ .005, .015, .025, .075, .125, .219, .375, .487, .567, .605, .777, .808, .999; .0001, .0012, .0125, .1275, .3525, .6225, .7203, 95 §158 ABSTRACT mTMBERS. .8007, .9883, .9999 ; .00001, .00014, .00225, .03275, .33125, .42075, .53003, .70007, .87078, .99999 ; .000001, .000017, .000175, .003175, .063175, .475327, .796305, .634008, .320075, .200017, .200325; .0000001, .0000025, .0000275, .0020705, .0357675, .7500786; .00027625, .02700625, .23450275, .00073513, .23570025, .125346798, .000000125, .0007600025, .27340709025, .70030005345, .000257025702. Read the following mixed numbers : — 3.3, 70.5, 35.7, 2.02, 3.25, 75.75, 24.05, 7.07, 30.003, 400.025, 25.125, 375.375, 1.001, 2.325, 2.0275, 300.0025, 17.0017, 1.0005, 2000.0002, 21.2125, 325.03725, 9180.20025, 1000.02207, 7025.00025, 6278.374375, 2000.0002325, 3375.00000765, 27.0000027, 3200.000000075, 2500,0000036975. Write tlie following in figures : 110. Seventy-four hundredths. 111. Four hundred and forty-eight thousandths. 112. Five hundred units and three hundredths^. 113. Seventy-five thousandths. 114. Five hundred and three thousandths. 115. Five hundred units and three thousandths. 116. Three hundred and twenty-seven ten-thousandths. 117. Three hundred units and twenty-seven ten-thousandths. 118. Seventeen and seventeen hundred-thousandths. 119. One thousand units and two thousand two hundred and seven hundred-thousandths. 120. Three thousand two hundred units and seventy-five millionths. 121. Six hundred and three ten-thousandths. 122. Two thousand four hundred and sixty-one and three hundred and nineteen millionths. 96 ADDITION OF DECIMAL FRACTIONS. §160 ADDITION OF DECIMAL FRACTIONS. Ex. 1. Add .3, .23, .175, and .025. 3 *23 § 159. Model. — 5 and 5 are ^0 ; 1 and 2 175 ^'^^ ^' ^^^ ^ ^^^ ^^' ^^^ ^ ^^^ ^^' ^^^ down 3 ; 025 ^ ^"^ ^ ^^® "' ^^'^ ^ ^^'^ ^> ^^^ '^ ^^^ ^' — '-— Point before 7. The sum is .73. Explanation. — Beginning at the right, we find the sum of the first column to be 10 thousandths, equal to 1 hun- dredth exactly. We do not set down the naught here, be- cause a naught at the right of a decimal fraction does not issist in determining the orders of the other figures. The 1 hundredth is added in with the column of hundredths, which amounts to 13 hundredths, equal to 1 tenth and 3 hundredths. Setting o under the column of hundredths, we add the 1 tenth in with the column of tenths. We then place the units' point at the left of the tenths. (§ 156). Ex. 2. Add .3, 3.5, 3.15, 35.25, and 171.275. § 160. Model.— 5 ; 7 and 5 are 12, and •'^ 5 are 17, set down 7 ; 1 and 2 are 3, and 2 'z'^ are 5, and 1 are 6, and 5 are 11, and 3 are oi'ot -^-^J ^^^ down 4 ; 1 and 1 are 2, and 5 are 7, 171 07f^ and 3 ar6 10, and 3 are 13, set down 3 ; 1 ^'^'^'^ and 7 are 8, and 3 are 11, set down 1 ; 1 213.475 and 1 are 2. Point before 4. The sum is 213.475. Explanation. — The sum of the column of tenths being 14, that is, 1 unit and 4 tenths, we set 4 under the column of tenths, and add 1 to the column of units. We place the units' point between the units and the tenths. (§ 156). a 97 §160 ABSTRACT NUMBERS. Rule. — Arrange the numbers with units of the same order in the same column ; and add as in luholc numbers. (§ 22). Place the units^ point on the left of the tenths figure in the sum. Proof. — The same as in whole numbers, (§ 22). Ex. 3. Add 1.2, 3.56, 45.67, and 56.789. 4. Add 1.3, 5.79, 24.68, and 90.275. Sum, 122.045. 5. Add 27.72, 365.9, 125.008, and 236.115. Sum, 754.743. 6. Add 135.709, 246.008, 145.008, and 236.709. 7. Add 1.35795, 135.795, and 13579.5. 8. Find the sum of 2.465, 25.G09, 100.206, and 146.27. Sum, 333.950. 9. Find the sum of 100.0001, L- 1.4012, 412.5124, and 421.5214. 10. Find the sum of 1234.58,78.9012,3456.789, 10.234567, and 890.13575. Sum, 5670.020517. 11. Find the sum of 907.0503, 890.7054, 785.4321, and 25.457. Sum, 2608.6448. 12. Find the sum of 12.012575, 120.125725, 1201.257725, and .270825. 13. Find the sum of .760027, .000176, .012012, and .027945. Sum, .800160. 14. What is the sum of .230495, .341507, .452618, and .563729 ? Ans. 1.588349. 15. What is the sum of 2.30495, 34.1507, 452.618, and 5637.029 ? 16. What is the sum of 12.000012, 250.0025, 75.075, and 175.0175 1 Ans. 512.095012. 17. 175 + 6.115 + 123.1341 + 172.21275 + 5637.175:^ what? Ans. 6113.63685. 98 SUBTRACTION OF DECIMAL FRACTIONS. §161 18. 52.8G72 + 549.72-f927.365 + 57.10715 + 13.575=wliat? 19. 79.105 + 131.187 + 19.4201 +2643.13 + 34.8H6-t = whai? Ans. 2907.6785. 50. 3844.04 + .444584 + 6.14644 + 6847.34 + 77.9899= what ? Ans. 10775.960924. SUBTRACTION OP DECIMAL FRACTIONS. Ex. 1. From 275.075 take 87.1275. ^P'^I-- § 1^1- Model.— 5 from 10 leaves 5 ; 8 "'• ^-'^^ from 15 leaves 7 ; 3 from 7 leaves 4 ; 1 187.9175 from 10 leaves 9 ; 8 from 15 leaves 7; 9t from 17 leaves 8 ; 1 from 2 leaves 1. Poiut before 9. The remainder is 187.9475. Explanation. — After placing the subtrahend under the minuend with units of the same order in the same column, we find 5 ten-thousandths in the subtrahend and no ten- thousandths in the minuend. Adding 1 thousandth, that is, 10 ten-thousandths, to the minuend, we subtract from this the 5 ten-thousandths of the subtrahend. Then, be- cause the minuen'd is increased 10 ten-thousandths or 1 thousandth, the subtrahend must be increased the same amount. (§28). The same kind of reasoning will explain the rest of the operation. We place the units' point be- tween the units and the tenths. (§ 156). Rule. — I^lace ike subtrahend under (he minuend, with *units of the same order in the same column, and subtract as In whole numbers. (§30). Place the unita^ point on the left of the tenths figure in the remainder. (|^ 156). Proof. — The same as in whole numbers. (§ 30). 99 S161 ABSTRACT JSUMBER.^. Ex. 2. From 8.96 take 8.07. Rem. 5.89. 3. From 2.719 4:ake 1.827. 4. From 97.8637 take 9.7863. Rem. 88.0774. 5. Take 67.8902 from 896.454. Hem. 828.5638. 6. Take 17.24937 from 1963.869. 7. Take 234.68579 from 6005.004. Rem. 5770.31821, 8. Take 98.79789 from 99.000099. Rem. .202209. 9. Minuend is 1284.567 ; Subtrahend is .76542„ 10. Minuend is 29017.05 ; Subtrahend is 10.8405. Rem. 29006.2095. 11. Minuend is 2098.76 ; Subtrahend is 454.698. Rem. 1644.062. 12. Minuend is 1201.257725 ; Subtrahend is 120.125575. 13. Subtrahend is .012095; Minuend is .027945. Rem. .01585. 14. Subtrahend is 2.30495 ; Minuend is 34.1507. Rem. 31.84575. 15. Subtrahend is 12.000012 ; Minuend is 250.0025. 16. Subtrahend is 75.075 ; Minuend is 175.0175. Rem. 99.9425. 17. 5637.175-I72.2l275=:what? Ans. 5464.96225, 18. 927.305-57.190715=what? ' 19. What is the difference between one millionth, and ninety-nine thousandths ? Ans. .098999. 20. What is the difference between thirty-seven billionths, and one hundred and eleven thousandths ? Ans. .110999963. 21. What is the difference between six billionths, and nine hundred and ninety-nine thousandths ? 22. What is the difference between three millionths, and three hundred and six thousandths ? Ans. .305997. 100 MULTIPLICATION OF DECIMAL FRACTIONS. §16S MULTIPLICATION OF DECIMAL FRACTIONS. Ex. 1. Multiply 5.3 by 6.25. 6 25 CO § 162. Model. — 3 times 5 are 15, set down — -— 5 ; 3 times 2 are 6, and 1 are 7 ; 3 times 6 ^^1^ are 18: — 5 times 5 are 25, set down 5 under ^^^^ 7 ; 5 times 2 are 10, and 2 are 12, set down 2 ; o3.125 5 times 6 are 30, and 1 are 31: — add the par- tial products : 5 ; 5 and 7 are 12, set down 2; 1 and 2 are 3, and 8 are 11, set down 1 ; 1 and 1 are 2, and 1 are 8 ; 3. Point before 1. The product is 33.125. ExrLANATiON. — Reducing both factors to improper frac- tions, and multiplying as in § 142, we have -^-f^ x {-^= 'VoW? and this product reduced to a mixed number becomes S3. 125. as in the model. If any decimal mixed number be reduced to an improper fraction, the numerator will con- sist of the same figures as the given mixed number. Hence we multiply as in whole numbers. The location of the units' point in the product is found by observing that the number of naughts in the denominator of either factor is the same as the number of figures in the numerator, aijd that the product of any two powers of ten is obtained by annexing to 1 as many naughts as there are in both factors togetiier. There are, therefore, as many fractional figures in the product as in both factors together. Ex. 2. Multiply ,15 by ,3, •^^ § 163. Model. — 3 times 5 are 15, set down 5 ; 3 times 1 are 3, and 1 are 4. Prefix one naught. .015 Point before 0, The product is .045. Explanation. — When the product does not contain onough figures to express its proper denomination, we pre- fix one or more naughts to supply this deficiency. 101 >164 ABSTRACT NUMBERS. Rule. — Midt^i-ily as in loJioIe numhers, and ^Doint off ai> many fractional Jigiires in the product an there arc in hotU the facfon^y prc/?x?*wr7 navghts lohen necessary to mahe up the number. Proof. — The same as in whole numbers, (§ 40). Ex. 3. Multiply 12.42 by 3.2. 4. Multiply 25.25 by 2.5. ProJ. 63.125. 5. Multiply .25 by .25. Prod. .0625. 6. Multiply 5.5 by 5.5. 7. Multiply 211.79 by 2.7. ' Prod. 571.833. 8. Multiply 97.825 by .34. Prod. 33.2605. 9. Multiply 275.005 by 5.005, 10. Multiply 869.06 by .045. Prod. 39.1077. 11. Multiply 27.9362 by .0052. Prod. .14526824. 12. 192.837x6.7== what? 13. 293.705 X .075=what ? Ans. 22.027875. 14. 3.047 x2.87=what? Ans. 8.74489. 15. 2.975 x.375=what? 16. 4.027 X 402.7== what ? Ans. 1621.6729- 17. What is the product of 247.742 and 10.035 ? 18. What is the product of 307.0005 and .000375? 19. What is the product of 175.025 and 25.0175 ? Ans. 4378.6879375.* .20, What is the product of 1200.375 and 162.625 ? DIVISION OF DECIMAL FRACTIONS. Ex, 1. Divide 2.25 by .3. .3)2^ § 164. Model.— 3 in 22, 7 'times, with 1 7.5 over, set down 7 ; 3 in 15, 5 times. Point before 5. The quotient is 7.5. 102 DIVISION OF DECIMAL FRACTIONS. §166 Explanation. — Since the divisor and the quotient arc factors of the dividend, there must be as many fractional figures in the dividend as there are in both the factors. (§ 162). Hence, to find the number of fractional figures in the quotient, we subtract the number in the divisor from the number in the dividend. 26.40 2475 3.2 1650 1650 Ex. 2. Divide 26.4 by 8.25. on-' §165. Model. — Annex one naught to the dividend : 8 in 26, 3 times ; multiply the divisor by 3 ; 15, 7, 24 ; subtract the product from the dividend ; 5, 6, 1 ; annex : 8 in 16, twice ; multiply the divisor by 2 ; 10, 5, 16 ; subtract the product from the previous remainder ; 0. Point before 2. The quotient is 3.2. Explanation. — As the number of fractional figures in the divisor exceeds the number in the dividend, we annex a naught to the dividend to make them equal. We after- wards find it necessary to annex an other naught to com- plete the division. This makes 3 fractional figures in the dividend ; and, as there are 2 in the divisor, there must be one in the quotient. Ex. 3. Divide 4 by 15. 4.00 30 15 .266 + 100 90 100 _90 10 § 166. Model. — Annex 2 naughts to the dividend ; 15 in 40, twice ; multiply the divisor by 2 ; 10, 3 ; subtract the product from the dividend ; 0, 0, 1 ; 15 in 100, 6 times; multiply the divisor by 6 ; 30, 9 ; subtract the product from the previous remainder; 0, 1; annex 1: 15 in 100, 6 times ; &c. Point before 2, — The quotient is .266 + . 103 §166 ABSTRACT NUMBERS. Explanation. — Since the dividend can be extended only by annexing naughts, it is evident that, if the same remain- der should occur twice in succession, the same quotient figure will occur and will give rise to the same remainder again ; so that the same circuit of operations will occur per- petually. In such cases the quotient can not be obtained exactly, but we can always make an approximation suffi- ciently near for any practical purpose. Rule. — Divide as in ?'-7'o^/^ numbers, and point off as tnamj fractional figures in 6-.... (quotient as the numher in the dividend exceeds the numher in the divisor, ijrefixtng naughts when necessary to make up the numher. If the nmnher of fractioiial figures in the divisor exceeds the numher in the dividend, annex to the dividend as many naughts as may he necessary to make the numher in the divi- dend at least equal to the number in the divisor. Note. — When the division can not be exactly performed, we put the sign + at the right of the quotient. Proof. — The same as in whole numbers. (§ 53). Ex. 4. Divide 1728 by .12.' 5. Divide 1728 by 1.2. 6. Divide 172.8 by 12. 7. Divide 17.28 by 12. 8. Divide 13 by 245. 9. Divide 2.7 by 900. 10. Divide 189.75 by .759. Quot. 250. 11. Divide 84.099 by .097. Quot. 867. 12. Dividend is 4435.2, divisor is .84. 13. Dividend is .8928, divisor is 1.24. Quot. .72. 14. Dividend is 7049.754, divisor is 8.7034. Quot. 810. 15. Dividend is 2.4416, divisor is 43.6. 104 CONTRACTED DIVISION OF DECIMAL FRACTIONS. §168 16. Divisor is 47, divideDd is 22.09. 17. Divisor is 18.07, dividend i^ .12649. 18. Divisor is 180.7, dividend is .012649. 19. Divisor is .125, dividend is 2.25. L'O. Divisor is 18, dividend is 19. Quot. .47. Quot. .007. Quot. 18. CONTRACTION IN MULTIPLICATION. Ex. 1. Multiply 23.25 by 10. • § 167. Model. — Remove the 23.25 X 10rr232.5 point one place to the right. The product is 232.5. Explanation. — To multiply by any power of ten, we simply remove the units' point as many places to the right as there are naughts in the multiplier, annexing naughts when necessary. See § 155. Ex. 2. Multiply 232.5 by 100. 3. Multiply 10.25 by 1000. 4. Multiply 246.25 by 100. 5. Multiply 875.275 by 10. 6. Multiply 96.0025 by 10000. 7. Multiply .0025 by 1000. 8. Multiply .0007 by 100000. 9. Multiply .05 by 1000000. 10. Multiply .0065 by 10000. Prod. 23250. Prod. 24625. Prod. 8752.75. Prod. 2.5. Prod. 70. Prod. ()0' CONTIUOTION IN DIVISION. ^■ Ex. 1. Divide 23.25 by 10. §168. Model. — liemove the 23.25 -f-lOmz 2.325 point one place to the left. The (juotient is 2.325. 105 ^169 ABSTRACT NUMBERS. Explanation.^— To divide by any power of ten, we sim- ply remove the units' point as many places to the left as there are naughts in the divisor, prefixing naughts when necessary. See §155. Ex. 2. Divide 2.325 by 100. Quot. .02325. 3. Divide 10.25 by 1000. 4. Divide 246.25 by 100. Quot. 2.4625. 5. Divide 875.275 by 10. Quot. 87.5275. 6. Divide 9G.0025 by 10000. 7. Divide 2500 by 1000. Quot. 2.5. 8. Divide 7000 by 100000. Quot. .07. 9. Divide .05 by 1000000. 10. Divide .0065 bv 10000. Quot. .00000065. ■<©>- RELATIONS OF COMMON AND DECIMAL FRACTIONS. § 169. Every decimal fraction may be expressed in the form of a common fraction by simply removing the units' point, writing the denominator under the numerator, and re- ducing, if necessary, to its lowest terms. Thus, .5=-!%-=:-^. Also, .25=-^\\=l:. Ex. 1. Reduce .375 to a common fraction. Val. |. 2. Reduce .625 to a common fraction. Val. ^. 3. Reduce .1875 to a common fraction. 4. Reduce .3125 to a common fraction. Val. -j^, 5. Reduce .05 to a common fraction. Val. -^^. 6. Reduce .0015 to a common fraction. 7. Reduce 00025 to a common fraction; Val. ^o^. 106 COMMON AND DECIMAL FRACTIONS. . §171 8. Reduco .004375 to a common fraction. Val. xoVo- 9. Reduce ,08125 to a common fraction. 10. Heduce .0175 to a common fraction, Yal. ^^-^^j. §170. If the denominator of a common fraction has no other prime factor than 2 or 5, it may be reduced to a dec- imal form by multiplyiog both its terms by such a number as will make the denominator a power of ten, removing the denominator, and putting the units' point at its proper place in the numerator. Thus, multiplying both terms of -I by 25, we have -jY^j which may be written, .25. Ex. 11. Heduce | to a decimal fraction. Yal. .4. 12. Reduce {: to a decimal fraction, 18. Reduce f to a decimal fraction. Val. .625. 14. Reduce ./^ to a decimal fraction, Val. .35. 15. Reduce ^}| to a decimal fraction. 16. Reduce l^ to a decimal fraction. Val. .475. 17. Reduce ,.^^y to a decimal fraction. Val. .0375, 18. Reduce '^-Jj to a decimal fraction. 19. Reduce -^\ to a decimal fraction. Val. .5625. 20. Reduce -^^ to a decimal fraction. Val. .09375. § 171. If the denominator of a common fraction has nei- ther 2 nor 5 as a prime factor, it cannot be reduced to a decimal form. We can make an approximation, however, sufficiently near for all practical purposes, by the following plan. Taking the example of last section, if we multiply both terms of l by 100, we have i^, and then dividing both terms by 4, we have -^^^q, that is, .25. In other words, A common fraction is reduced to a decimal form hi/ dividing its numerator hy its denominator, (§ 166). This is the gen- eral rule, and is but a repetition of what we learned in § 113. But let us attempt to apply this rule to the frac- 107 §172 ABSTRACT NUMBERS. tlon |-. I>ividing, we have 3 in 20, 6 times, with 2 over ; again, annexing an other naught, we have o in 20, 6 times, with 2 over ; and so on, evidently forever. Again, reduce -i\ to a decimal form. Dividing, we have, ll^n 20, once, with 9 over ; 11 in 90, 8 times, with 2 11)2.0000 over; 11 in 20, once, with 9 over, .1818+ again ; and 11 in 90, 8 times, with 2 over, again; and so, evidently, these two quotient figures might be repeated to the end of time. § 172. Such expressions as these are called ^wre repctends, and they are denoted by placing a dot over the repeating figure when there is but one, or by placing dots over the first and last repeating figures when there are several. Thus, 1^.6; xV-.iS; fff=.275. Ex. 21. Reduce ^ to a repetend. 22. Reduce f to a repetend. Yal. .285714. 23. Reduce y\- to a repetend. Yal. .2*7. 24. Reduce -^-^ to a repetend. 25. Reduce xV ^o ^ repetend. Val. .2941176470588235. § 173. If the denominator of a common fraction has ei- ther 2 or 5 or both, and other prime factors, the quotient of its numerator by its denominator will be partly a deci- mal fraction and partly a repetend. Thuo', |=:.8333 + , or .83. Also, ^^j=A1666 + , or .416. Also, -,\-=.2083 ; and -^^=.104<16. These expressions are called Tnixed rcpetends. Ex. 26. Reduce -^^ to a mixed repetend. Val. .585. 27. Reduce ^ to a mixed repetend. 28. Reduce -^^ to a mixed repetend. yal.'.2i4285t. 29. Reduce -^-^ to a mixed repetend. Val. .46. 80. Reduce |J- to a mixed repetend. 108 COMMON AND DECIMAL J-RACTIOJSS. §176 §174. To reduce a jJiif'c^ rrpetcnd to a common fraction, toe remove the units' point, vjrite for denomhiator as m'jny nines as there are repeating figures, and reduce the result to lis lowest terms. Again, ,,V=-Oi. -^=.05, U=AO, ^ = .25, f^-.50: Also, -,J^:= .001, WV=.010, ^=.075, ^^i]=:.275, &c. § 175. From these facts we learn that a pure rcpeteiid is read by pronouncing after its numerator the ordinal of the number formed of as many nines as there are figures in the repetend. Thus, .1=1-, .5^'7=|-^, &c. Ex. 31. Reduce .27 to a common fraction. ^. 82. Reduce .72 to a common fraction. . -=& IT* 33. Reduce .36 to a common fraction. 34. Reduce .185 to a common fraction. -^. 35. Reduce .270 to a common fraction. -^j-, 36. Reduce .7^2 to a common fraction. 37. Reduce .801 to a common fraction. -^^-^^-^ 38. Reduce .9001 to a common fraction. uf-p* 39. Reduce .8877 to a common fraction. 40. Reduce .9765 to a common fraction. i-i-fi- § 176. A mixed repetend is a complex fraction, having for its denominator some power of ten, and for its numera- tor a mixed number: the fractional part of the mixed num- ber having for its denominator a saries of nines. 83 41 « '2083- Thus,.85is--;.416is-^;.2083is^». To reduce a mixed repetend to a simple common fraction, we must first reduce the numerator to an improper fraction. This makes it necessary to multiply the integral part by 9 or by a series of nines ; and this multiplication can be most 109 §176 ABSTRACT NU31BERS. readily accomplished by §60. Take tLe second of tbe above examples, for instauce. Annexing one naught to 41, and subtracting 41 from 410 369 ^^100=^"^^ *^^ result, we have 369 _il __^ ~0~ ' ~"900 as the product of the in- 369 375 tegral partby the denom- 9 inator. To this product adding the numerator 6, we have 375 as the numerator of the improper fraction. Dividing 2.I-A by 100, we find -|-^, which should then be re- duced to its lowest terms. This result could be more easily obtained by f^uhtracting the decimal ])art from, the whole repctend for the numerator , and hij taking for the denominator as many nines as there are repeating' fgiircs, followed hy as many naughts as. there are decimal figures. Thus, .83r=i^=:f ; .2083=(2083-208)=i§^^=:^ Ex. 41. Pteduca .123 to a common fraction. t^- 42. Keduce .50t5 to a common fraction. 43. Reduce .779t to a common fraction. -^-|-|. 44. Reduce .1*76 to a common fraction. -^^. 45. Keduce .4554 to a common fraction. PROMISCUOUS PROBLEMS. 1. What is the sum of 247 millionths, 26 ten-thousandths, 163 hundred-thousandths, 3 thousandths, and 19 hun- dredths ? Ans. .197477. 2. What is the difi'erence between 19 units and 19 mil- lionths? Ans. 18.999981. 110 PROMISCUOUS PROBLEMS. §176 3. What is the product of 273 thousandths and 11*7 ten- thousandths? 4. What is the quotient of 17 ten-thousandths by 16 hun- dredths ? Ans. .10625. 5. What is the Bum of the product of 5 tenths and 5 hun- dredths, and the quotient of 5 tenths bj 5 hundredths ? Ans. 10.025. G, What is the difference between the sum of C hundredths and 6 units, and the product of 6 hundredths and 6 units? 7. What is the product of the sum of 12 thousandths and 84 hundredths, and their difference? Ans. .115456. 8. What is the quotient of the product of 506 thousandths and 78 hundredths by their sum? Ans. .306905 + . 9. Add 27 hundredths, 538 thousandths, C4 ten-thou- sandths, and 9768 ni'llionths. 10. Subtract the product of 39 hundredths and 54 thou- sandths from their sum. E-em. .42294. 11. Multiply the quotient of 36 hundredths by 45 ten- thousandths by their difference. Prod. 28.44. 12. Divide the sum of 497 thousandths and 608 ten-thou- sandihs by their difference. 13. What number is that to which if 13 hundredths, 13 thousandths, 13 ten-thousandths, and 13 millionthsbe added, the sum will be 13 units? Ans. 12.855687. 14. What number is that from which if 11 hundredths, 12 thousandths, 13 ten-thousandths, and 14 hundred-thou- sandths be subtracted, the remainder will be 15 mil- lionths? Ans. 123455. 15. What number is that by which if 79 thousandths be multiplied, the product will be 54115 billionths ? Ill ^176 ABSTRACT NUMBERS. IG. What number is that bj whiich if 6375 millionths be divided, the quotient will be 5 thousandths ? Ans. 1.275. 17. The subtrahend is 25 ten-thousandths, the minuend is 2 tenths; what is the remainder ? Ans. .1975. 18. The subtrahend is 25 thousandths, the remainder is 2 hundredths ; what is the minuend ? 19. The remainder is 13 millionths, the minuend is 13 thousandths;" what is the subtrahend ? Ans. .012987. 20. The multiplicand is 75 thousandths; the multiplier is 25 ten-thousandths ; what is the product ? Ans. .0001875. 21. The multiplier is 18 thousandths, the product is 369 millionths; what is the multiplicand ? 22. The product is 1482 ten-millionths, the multiplicand is 95 hundredths; what is the multiplier? Ans. .000156. 23. The divisor is 19 huitdredths, the quotient is 21 thou- sandths ; what is the dividend ? Ans. .00399. 21. The dividend is 65 and 12 hundredths, the divisor is 17 and 6 tenths ; what is the quotient ? 25. The quotient is 14 hundredths, the dividend is 322 thousandths ; what is the divisor t Ans. 2.3. 26. What are the prime factors of 3500 ] Ans. 2, 2, 5, 5, 5, and 7. 27. What are the prime factors of 756 ? 28. What different prime numbers will exactly divide 700 '^ Ans. 2, 5, and 7. 29. What different prime numbers will exactly divide 850 ? Ans. 2, 5, and 17. 30. What is the least common multiple of 7, 8, 10, and 14? 112 PROMISCUOUS PROBLEMS. §176 31. What is the smallest number that may be exactly di- vided by either 9, 10, 12, or 15 ? Ans. 180. 32. What is the smallest number that may be exactly di- vided by either 21, 36, 48, or 72 ? Ans. 141. 33. What is the greatest common measure of 45, 54, and 108? 34. What is the largest number that will exactly divide either 75, 100, or 150 ? Ans. 25. 35. What is the largest number that will exactly .divide either 96, 192, or 240 ? Ans. 48. 36. What is the sum of 1-, l, -^, |, and 3%? 37. What is the difference between f and |^? Ans. ^\. 38. What is the product of | and j-^- 1 Ans. |-|. 39. What is the quotient of ^ divided by -^? 7i 40. What is the value of -^ ? Ans. |^. 15i 41. What is the value of f of -f- of 7^- ? Ans. If^. 42. What is the value of .^25 + . 025 + .715 + .225 ? 43. What is the value of .0237-.002375 ? Ans. .021325. 44. What is the value of .027 x .0027 ? Ans. .0000729. 45. What is the value of .0144-^3.6 ? 46. What is the sum of -^,-i\, yV? and ^ 1 Ans. l-i-|. 47. What is the difference between -^\ and -^ ? 48. What is the sum of 216 thousandths, 37 hundredths, 15 ten-thousandths, and 10 units? Ans. 10.5875. 49. What is the difference between 206 ten-thousandths, and 27 millionths ? 50. What is the value of .211 + 3.07 + 29.6 + .0735 ? Ans. 32.9545. 51. What is the value of .6501 x .736089 ? 52. What is the value of .4396^9.3 ? n 113 §177 CONCRETE NUMBERS. CONCRETE NUMBERS. § 177. The relations of the concrete numbers in most common use are set forth in the following I. United States Money. 1 mill. (m.) — 1 — lO of a cent; 10 mills = • 1 cent, (ct.) — 1 10 <i a dime ; 10 cents = 1 dime. (d.) — 1 — 10 a ic dollar 10 dimes =z 1 dollar, m — 1 — 1 o U il eagle ; 10 dollars = 1 eagle, (E.) m. ct. d. $ E. II II II II rH O O O rH O O 1 1 o 1 10 100 = 1 10 1 10 1 10 ^^ 1000 1 — 1 oo — 1 10 = 1 = 1 1 oooo 1 1000 1 100 1 1 10000 = 1000 = 100 = 10 — 1. The denominations dime and eagle are very little used in calculation. In stead of 14E. 5$, 7d, 5ct., we usually write $145.75. II. £n^lisli Currency; or. Sterling Mouej 1 farthing, (qr.) - 4 farthings =: 1 penny, (d.) = 12 pence = 1 shilling, (s.) - 20 shillings = 1 pound, {£). 114 i of a penny; i-V " " shilling; ^ " " pound ; RELATIONS. §17' qr. d. s. £ 1 = 1 4.- — 1 48 * = 9 60 4 1 1 , 1 — 12 ~^ — 2 40 48 z=: VA :rr 1 = 1 20 960 — 240 =: 20 — 1 The pound sterling is represented by a gold coin, called a sovereign, valued at $4.84, U. S. currency. Farthings are usually written as fractions of a penny. III. FreificU Currency. 1 centime, (cent.) = J- of a decimej 10 centimes = 1 decime, (dec.) = /^ '^ " franc ; 10 decimes = 1 franc, (fr.)- cent. dec. fr. ^ 10 — 100 10 = 1 = iV 100 = 10 = 1 Accounts are kept in francs and centimes. The franc is valued at 18ct. 6m., U. S. currency. --«yp~ IV. Troy IRTeiglit. USED FOR WEIGHING GOLD, SILVER, JEWELS, <fec. 1 grain, (gr.) = ^l of a pennyweight ^ 24 grains = 1 pennyweight, (dwt.)= -^ of an ounce.; 20 penny weights^ 1 ounce, (oz.)= -^^ of a pound .; 12 ounces = 1 pound, (lb.) gr. dwt. oz. lb. 1 = -1- = -1- = 1 24 480 5760 24 — 1 _ _i_ — 1 ^^ ^ 20 240 480 ::=: 20 = 1 = JL. 5760 = 240 = 12 = 1 115 §177 CONCRETE NUMBEHS. ¥. Apotliecaries' W' elglit. USED IN MlJrjJVO MEDICINES. 1 grain, (gr.) - ^of a scruple 20 grains ~ 1 scruple, (so.) or 9 = i ^' " dram ; 3 scruples =r 1 dram, (dr.) or o 6 ^' ounce • 8 drams = 1 ounce, (oz.) or 5 1 <6 12 " pound ^ 12 ounces = 1 pound, (lb.) or lb g^' sc. dr. oz. lb. 1 ^^ "2 0" "6"0 = 24 = 8 = 1 — 4 S 1 — 2 4 ,1 = 1 -20 60 480 5 7 6 1 2 8 & 1 e 1 T2" :5760 = 288 =96 = 12 = 1 The poun d Apothecaries' is the same as the pound Tro}^ ¥1. Avoirdupois 'Weaglat. USED FOE, WEIGHING ALL AllTICLES EXCEPT THOSE MENTIONED ABOVE. • 16 drams 16 ounces 25 pounds = 4 quarters = 20 hundredweight: dr. 1 16 256 6400 25600 512000 1 dram, (dr..)= -^ of an ounce ; = 1 ounce, (oz.)= J^- of a pound ; = 1 pound, (lb.)= J-5 of a quarter ; = 1 quarter,(qr,)= ^of ahundredweight' = 1 hundred weight,(cwt,) = -^V of a ton; oz. 1 1 6 1 16 400 1600 32000 lion, (T.). lb. qr. 1 . 2 5 6' Ji._ . 1 6 ■ 1 : 25 : 100 : 2000 : 110 1 6 40 1 4 00 1 2 5 cwt. 2 5 6 00 1 16 1 100 T. 4 80 4 — 1 = 20 = 5 12 000 1 3 2 00 1 2 _1_ 8 1 2 O 1 RELATIONS. §177 144 pounds Avoirdupois = 175 pounds Troy or Apothe- : 7000 gr. Troy ; 1 oz. Avoir. = 437.5 gr. cariess 1 lb. Avoir. Troy. The following denominations also belong here : 28 pounds 112 " 2240 " 14 " 21^ stone 8 pigs 50 pounds of salt 56 60 56 100 ]96 200 1 long quarter ; 1 long hundredweight ; 1 long ton ; 1 stone; 1 pig; 1 fother. " corn '^ wheat '' butter " salt fish " flour " beef, pork, or fish; 1 bushel, 1 bushel. 1 bushel. 1 firkin. 1 quintal, 1 barrel. 1 barrel. Til. Frencli Wciglits. 1 milligramme = -^-^ of a centigramme; 10 milligrammes=l centigramme 10 centigrammes=l decigramme = iV 10 decigrammes =1 gramme == -^ 10 grammes =1 decagramme = -^-^ 10 decagrammes =1 hectogramme^= -^ 10 hectogrammes =1 kilogramme = y^^ 100 kilogrammes =1 quintal = y^^ " " millier; 10 quintals =1 millier, or 1 ton of sea water. yV " " decigramme; ^ '' gramme ; ^ " decagramme ; ^ " hectogramme; ^ " kilogramme ; i a quin tal 1 gramme = 15.433 grains Troy. 117 §177 CONCRETE NUMBERS. Till, ftiong- Measure ^ or, I^inear Measure. USED IN MEASURING LINES, OR DISTANCES. 1 inch, (in.) = = yV 0^ ^ ^00* J 12 inches — 1 foot, (ft.) = = i " " yard; 3 feet = 1 yard, (yd.) - - ^\ " " rod ; 5i yards 1 rod, (rd.) = = ^^ " " furlong ; 40 rods — 1 furlong , (fur.) = = i " " mile; 8 furlongs — 1 mile, (mi.) in, ft. yJ- rd. fur. mi. 1 — '^ -- — 1 1 — 1 — 1 -^ 12 3 6 19 6 7 9 20 63360 12 = 1 : ] 2 . 1 1 3 3 3 6 6 5 2 8 36 = 8 = = 1 2 1 1 1 1 2 2 1 7 60 198 =r 16i= H = 1 1 1 4 ■3T0 7920 — 660 : = 220 = 40 1 i 63360 = 5280 : = 1760 = 320 8—1 The following denominations are sometimes used : 3 barley corns = 1 inch ; 6 points — 1 line ; 12 lines 1 inch ; 4 inches = 1 hand ; 9 " — 1 span ; 18 " :zjL 1 cubit; 21.9" 1 sacred cubit ; 3 feet : — : 1 pace; 6 feet z= 1 fathom ; 1 69i miles 1 1 degree < of latitude. IX. Swrveyor's I<oiig Measure. 7.92 inches 1 link. 00- 2V of a rod ; 25 links = 1 rod. (rd.) = i " " chain ; 4 rods = 1 chain, (ch.) = -jV" "furlong; 10 chains = 1 furlong, (fur.) - i « " mile ; 8 furlongs = 1 mile, (mi.) 118 RELATIONS. §177 in. 7.92 — 198 — 792 = 7920 = 63360 = 1. 1 — 25 — 100 = 1000 = 8000 = rd. ch. "2 5" TcTo 1 _ i ^ 4—1 40—10 — 320 — 80 — fur. mi. 1 — 1 1000 SOOO 1 — 1 40 3 2 1 1 10 »o 1 = i 8—1 X. Square Measure. USED FOR MEASURING SURFACES OF LAND, PAINTING, PLASTERING. PAVING, &c. 1 squareinchj (sq.in.)=y^-^ofasquarefoot; 144 square inches=l " foot, (sq.ft. )= I '^ " " yard; 9 « feet =1 '' yard,(sq.yd.)=j-^y" '^ perch; 30-i « yards =1 perch, (P.) = ^V " " ^^^^ ; 40 perches =1 rood, (R.) = |- " " acre ; 4 roods =1 acre, (A.) =^^0" "square mile 640 acres =1 square mile, (sq.mi.). sq.in.sq.ft.sq.yd. P. R. A. sq.mi. ^ ^~~ 1 4 -i ~~ Ta ITe"" 3 8 2 o ¥ "~~ 1 5 6 8 "i~6 o" — "e "aT 2~6 4 0" To 1 4 4 8 9 eoU 144 — 1 — 1 — 4_ — \ — 1 — 1- ^^^ -^ V 10«9 10890 43560 '>. 7 k 7 s l9Qfi — q _ 1 _ t__ — 1 — !„ 39204 = 272:1 = 30^ z= 1 = -,V = 1568160 = 10890 = 1210 = 40 =.: 6272640 = 43560 =: 4840 =. 160 :.= 4 = . _ ^-^^ 4014489600 = 27878400 =z 3097600 = 102400=2560=640=1 XI. Cubic Measure. USED FOR MEASURING THE CONTENTS OF SOLIDS. 1 cubic inch, (cu.in.)=yyi^-g-of acubicft.:; 1728 cubic inches=l " foot, (cu.ft.) = ^V *' " " J^-'^ 27 '' feet =1 '' yard,(cu.yd.). cu. in. cu. ft. cu. yd. 1 — _3_. — L ■^ 1728 4 6050 1728 =z 1 = ^\ 46656 =27 = 1 119 §177 CONCRETE NUMBERS. Also, 40 cubic feet of round timber = 1 ton ; 50 " " '' hewn timber = 1 ton ; 42 " " '• shipping = 1 ton ; 16 " '' " wood = 1 cord foot; 128 " " ^' wood = 1 cord. Also, 231 cubic inches=l gallon Liquid or Wine Measure; 268|- " " =1 '^ Dry Measure ; 282 « " =1 " x\le Measure, (out of use.) 537-?- " " =1 ponk; 2150i " " =1 Lu.liel. XII. £iiq3£ld Measure 5 or, Wisae Measiia'e. USED IN MEASURING LIQUIDS ; AS, MOLASSES, SPIRITS. WINE, WATER, &c. 1 gill, (gi.) = I: of a pint ; 4 gills = 1 pint, (pt.) — 1- " " quart ; 2 pints = 1 quart, (qt.) = -^ " " gallon ; 4 quarts =z 1 gallon, (gal.) = -^-3 " " barrel ; 31-|- gallons = 1 barrel, (bbl.) =: -} " " hogshead; 2 barrels — 1 hogshead(hhd.) == ^ " " 2 hogsheads = 1 pipe, (pi.) = ^ pipe « " tunt 2 pipes r= 1 tun, (tun). gi. pt. qt. gal. bbl. hhd. pi. tun. 1 - i 4 = 1 — Y — Z 126 Z 1 — 1 — 1 — 32 lOOb 3016 s ^^ "2T5'"2 ^^ "sinr 4. 126 2 52 •*- 3 G 3 3U-= 1 = i = 1 — 1 4 3 2 a 6 i 1 1 8 = 2 32 = 8 1008 = 252 1 s 2 1 g: 1 — 1 504 100 tt- _1 — 1 12 6 2 5 2 2016 == 504 = 252 = 63^= 2 = i = 1 1 2 4 4032 :=1008 = 504 = 126 = 4 = 2 = 1 = i 8064 =2016 = 1008 = 252 = 8 = 4 = 2 = 1 Also, 42 gallons = 1 tierce ; 2 tierces = 1 puncheon. 120 RELATIONS. §177 XIII. Ale Measaire. FORMERLY USED FOil MEASURING MALT LIQUORS AND ?IILK, WHTCli NOW. HOWEVER, ARE GENERALLY MEASURED BY LIQUID MEASURE. 1 pint, (pt.) = ^ of a quart ; 2 pints = 1 quart, (qt.) = -} " " gallon; 4 quarts = 1 gallon, (gal.) = ^fo " " barrel; :>6 gallons =: 1 barrel, (bbl.) - | " " hogshead ; li barrels =: 1 hogshead, (hhd.) Also, 9 gallons = 1 firkin, 2 firkins = 1 kilderkin. (qt.) (gal.) (pk.) - 1 i 1. 1 ii *' gallon ; " peek ; " bushel; , (bu.) gal. pk. bu. 1 — 1 1 6 1 — 1 XIV. itry Measure. USED FOR MEASURING GRAIN, FRUITS, VEGETABLES, SALT, Ic. 1 pint, (pt.) =: \- of a quart ; 2 pints = 1 quart, 4 quarts — 1 gallon, 2 gallons — 1 peck, 4 pecks = 1 bushel, pt. qt. 1 - i = 8 =r. 4 ^ i 16:=.Sz::.2rrrl^A Also, 5 bushels zzi 1 barrel, of corn ; 8 bushels r::: 1 quarter ; 36 bushels — 1 chaldron. In the Confederate States, corn is usually bought and sold by the barrel. A barrel of corn should contain 280 pounds. 121 y, 3 2 1 — 1 a' — V §177 CONCRETE NUMBERS. V ^ — XV. Time. 1 second. (sec.) 1 GO of a minute ; 60 seconds — 1 minute, (min.) T 6 an L hour ; 60 minutes = 1 hour, (hr.) 1 •J. 4 a day; 24 hours z=z 1 day, (da.) 4. 146 1 a year ; 365^^ days = 1 year, (yr.) 1 1 a decade; 10 years = 1 decade. (dec.) 1 10 ii century ; 10 decades = 1 century, (cent.) sec. min. hr. da. yr. dec. cent. I— 1 _ 1 _ _ 1 1 1 0"^ _ 1 •^ 6 3 e 00 S64;06 3155760O 3155760 "yi 5 5 7 6 OOOO 60 = 1 = -i-. 1 1 1 1 1 14 40 " 525960 5 2 5 6 00 52596000 3600= 60 : == 1 = ^v - 1 1 8 7 6 6 — 1 S 7 6 6 S 7 6 6 00 86400 = 1440 = 24 = 1 146 1 2 1 7 30 5 3 6 5 2 5 31557600 = 525960 = 8766 = 365^ = 1 ^ = 1 1 10 100 315576000 = = 5259600= 87660 = 3652^ = = 10 = = 1 = tV 3155760000 = 52596000 = 876600 = 36525 = 100 = 10 = 1 x\lso, 7 days = 1 week, (wk.) ; 30 or 31 days = 1 month, (mo.) ; 12 months = 1 year. According to the table, 3651 days make a year. To ob- viate the difficulty arising from the fraction, we reckon three years of 385 days each, and one of 866 days. This long year is called leap year. The leap years are those whose numbers are exactly divisible by 4 ; except that the centennial years are not leap years unless their numbers are exactly divisible by 400. Thus, 1860 and 1848 were leap years ; but 1900 will not be leap year, because it is not divisible by 400. The year is also divided into four seasons; Spring, Sum- mer, Autumn, and Winter. These consist of the following months : 122 RELATIONS, §17' r 3. March, Spring, < 4. April, I 5. May, 6. June, 7. July, 8. Aucust, Summer, (Mar.) lias 31 clays. (Apr.) " 30 " (May) " 31 (Jun.) " 30 (Jul.) " 31 '' .^..., (Aug.) " 31 " ( 9. September,(Sept.) " 30 " Autumn, MO. October, (Oct.) " 31 '^ ( 11. November,(Nov.) " 30 " r 12. December, (Dec.) " 31 " Winter, < 1. January, (Jan.) '^ 31 '^ { 2. February, (Feb.) " 28 " leap year, 29. In most business transactions 30 days are considered a month. XVI. Circular Measure. USED IN SURTEYING, GEOGRAPHY, AND ASTRONOMY. 1 second, ('' or sec.)=g^^ of a minute; 60 seconds =1 minute, ('or min.)^:-^^ " " degree; 60 minutes =1 degree, (° or deg.)= ^^j " "• sign ; 30 degrees =1 sign, (S.)=y\- " " circumference; 12 signs =1 circiimference,(C.) fr 1 = 60 = 1 3600 = 60 108000 = 1800 1^96000 = 21600 Also, 60 degrees =: 1 sextant =-1- of a circumference ; And, 90 " =1 quadrant =a. « « " 123 6 O o S. c. 1 = 1 = 1 36 OO 10 8 000 12 9 6000 1 = 1 1 S 00 = 1 GO 2 16 1 = 1 30 = 1 360 30 = 1 z=z I 1 2 360 =z 12 :;:;::: 1 §178 CONCRETE NUME"£:RS. 24 sheets 20 quires 2 reams 5 bundles XYII. Faper. 1 sheet, (sh.) 1 quire, (q^-) 1 ream, (rm.) 1 bundle, (bdie.) 1 bale. ^13: of a quire ; 2V ** " ream; i " " bundle ; i " " bale: -•♦^ 1 unit, = tV of a dozen ; 12 units =z 1 dozen, (doz.) = ^V " " gi'oss; 12 dozen = 1 gross, (gr.) — -,V " " g^^^* S^°^^ 5 12 gross := 1 great gross. Also, 20 units = 1 score. OPERATIONS ON CONCRETE NUMBERS. The numerical processes are the same for concrete num- bers as for abstract. In this place^ therefore, we are to dis- cuss only the denominations of the several results. ADDITION OF CONCRETE NUMBERS. §178. Dissimilar numbers can not be added together. Thus, 3 dollars and 5 cents malre neither 8 dollars nor 8 cents. § 179. The sum of several similar numbers is similar to the numbers added. Thus, 3 dollars and 5 dollars make S dollars ; 3 cents and 5 cents make 8 cents. Ex. 1. Add $1075, $2157, $3779, and $4209, 124 ADDITION'. !^i79 2. Add ^47, £bd, £29, nnd £.63. Sum, £192. ". Add 27fr., 3Gfr., 297fr., and 365fr. 4. Add 291b., "STlb., 491b., and 581b. Sum, 1731b. 5. Add 45sc., 2Si?c., 143sc., and 2S7sc. Sum, 5l.)3se. G. Add iOOcwt., 205cwt., 177cwt,, and 329c wt. 7. Add 2479 grammes, 147 grammes, and 986 grummeis. Sum, 3GI2graui. 8. Add 276yd., 299jd., 4G9yd., and 357yd. Sum, 140] yd. 9. Add 79mi., 227mi., 37mi., and 475mi. 10. Add 306cu. ft., 279cu. ft., and 520cu. ft. Sum, 1105cu.ft. 11. Add 575A., 209A., 105A., and 258A. Sum, 1147A. 12. Add 27gal., 72ga]., 298gal., and 143gal. 13. Add 15bbl., 28bbl., 19bbl., 247bbl., and SGbbl. Sum, 395bbl. 14. Add 47bu., 475bu., 407bu., and 4750bii. Sum, 5679bu. 15. Add 27da., 38da., 52cla., and 93da. 16. Add 12°, 26°, 37°, and 45°. Sum, 120°. 17. Add lOrui., 14rm., 7rm., and 22rm. Sum, 53rm. 18. Add 6doz., 27doz., 14doz., and 97doz. 19. Add 12^-lb., 33Ub., 37ilb., and 83-^lb. Sum, 166pb. 20. Add 3|mi., I6|mi., 18:^ mi., 62irai., and 42|mi. Sum, 143i{:mi. 21. Add 19^qt., 20^qt., 7^.qt., and 28^qt. 22. Add 3.251ir., 6.5hr., .275hr., and 700.075hr. Sum, 710.1hr. 23. Add 47.5pt., 57.75pt., .375pt., and .0625pt. Sum, 105.6875pt, 24. Add 9.73pk., lO.Olpk., 17.75pk., and .1775pk. 25. Add lO.llpk., 7.369pk., and 1.002pk. 125 §180 CONCRETE NUMBERS. SUBTRACTION OF CONCRETE NUMBERS. §180. Subtraction can not be performed upon dissimilar numbers. Thus, 3 cents from 5 dollars leaves neither 2 cents nor 2 dollars. § 181. The difference of two similar numbers is similar to those numbers. Thus, 3 dollars from 5 dollars leaves 2 dollars ; 3 cents from 5 cents leaves 2 cents. Ex. 1. From £245 take .£196. Rem. £49. 2. From 25cwt. take 6c wt. Kem. 19cwt. 3. From 793rd. take 546rd. 4. From 17246sq. ft. take 8472sq. ft. Rem. 8774sq. ft. 5. From 635cu. yd. take 473cu. yd. Rem. 162cu. yd. (5. From 47 decigrammes take 29 decigrammes. 7. From 479bhd. take 3981ihd. Rem. 81hhd. 8. From 272pt. take 199pt. Rem. 73pt. 9. From 365da. take 175da. 10. From 360° take 275° Rem. 85°. 11. From 27fs. take 19is. Rem. 8is. 12. From $75^ take $59i. 13. From $19.75 take $ .99. Rem. $18.76. 14. From 270^fr. take 197|fr. Rem. 72ifr. 15. From 77in. take 17.75in. 16. From 3706sq. yd. take 897isq.yd. Rem. 2808^sq.yd. 17. From ^ of 246cu. ft. take -\ of 317cu. ft. 18. From 525iqt. take 252^qt. 19. From 27bu. take I7.25bu. Rem. 9.75bu. 20. From SS^cu. in. take 31icu. in. Rem. 2y\cu. in. 21. From 725dwt. take 339.17dwt. 22. From .2468d. take .08642d. Rem. .16038d. 23. From .1751b. take .0171b. Rem. .1581b. 126 MULTIPLICATION. §183 MULTIPLICATION OF CONCRETE NUMBERS. § 182. Every muUipUer must be an abstract number. — Thus, if we wish to find the cost of o yards at 25 cents a yard, it is evidently absurd to say, "3 yards times 25 cents," or, '' 25 cents multiplied by 3 yards." "We multi- ply 25 cents by 3, because 3 yards cost 3 times the price of 1 yard, that is, 3 times 25 cents. § 183. The product is always similar to the multiplicand. Thus, 3 times 25 cents are evidently 75 cents ; 6 x 7 ab- stract units=:i42 abstract units ; 4 x $10=$40 ; 5 x 6-yards =30 yards. Ex. 1. Multiply $3179 by 27. 2. Multiply 2764bu. by 4G. 3. Multiply 3S5da. by 19. 4. Multiply 347oz. by 83. 5. Multiply 2047cwt. by 109. 6. Multiply 347fr. by 201. 7. Multiply 467A. by 5297. 8. Multiply 6386pi. by 578. 9. Multiply 7475pk. by 689. 10. Multiply £69 by 4234. 11. Multiply 224 by 4759. 12. Multiply 8564wk. by 790. 13. Multiply 9563 by 801. 14. Multiply 10742doz. by 912. 15. Multiply 20s. by 16750. 16. Multiply 5}yd. by 746. 17. Multiply 16.5ft. by 165. 18. Multiply 30.25sq. yd. by 3.025. 19. Multiply 7.92in. by 198. 127 Prod. $85833. Prod. 127144bu. Prod. 28801OZ. Prod. 223123cwt. Prod. 2473699A. Prod. 3691108pi. Prod. .£292146. Prod. 1066016. Prod. 7659963. Prod, 9769704doz. Prod. 4103yd. Prod. 2722.5ft. Prod. 1568.16in. i ^184 CONCRETE NUxMBERS. 20. Multiply 31igal. by 1008. Prod. 31752gal. 21. Multiply SGo^da. by 365^. 22. Multiply $29.75 by 29.75. Prod. $885.0625. 23. Multiply $100,375 by 37.5. Prod. $3764.0625. 24. Multiply 279.5 by 27.95. DIVISION OF CONCLIETE xXUMBERS. {M84. Division ia the reverse of multiplication. In multiplication, the two factors are given, to find tbe prod- uct ; in division, the product and one of its factors are given, to find the other factor. The dividend corresponds to the product; the divisor may correspond to either the multiplicand or the multiplier, and the quotient corre- sponds to the other. ' Thus, 6 X 25gal.=150gal. Conversely, 150gal.-^6=:25gal. Or, 150gal.-^25gal.=G. § 185. Either the divisor or the quotient must be ab- stract, and the other must be similar to the dividend. In other words, if the dividend and the divisor are simi- lar, the quotient is abstract : if the divisor is abstract, the (juotient is similar to the dividend. The remainder is always similar to the dividend. (§46). Ex. 1. Dividend=:45ct., divisor=^-3. Quot. 15ct. 2. I)ividend==:$750, divisor=$25. Quot, 30. 3. Dividend=::=1000bu., divisor=40. 4. Dividend- 2451b., divisor=5. Quot. 49^b. 5. Dividend=3003, divisor=ll. Quot. 273. 6. Dividendz=]728cu. in., dmsor=48cu. in. 7. Dividend=:7007yd., divisor=13yd. Quot. 539. 128 REDUCTION. §188 8. Divisor=l7mi., dividend=2S9mi. Quot. 17. 9. Divisor=25, dividends 1175gi. 10. Divisor=:27, dividend=:2971ir. Quot. llbr. 11. Divisor=109, dividend =:2398qt. Quot. 22qt. 12. Divisors 245cu. ft., dividend = 5880cu. ft. 13. Divide 642780 dozen by 36 dozen. Quot. 17855. 14. Divide 79008oz. by 96. Quot. 823o2. 15. Divide 847665qr. by 345qr. 16. Divide 3475cwt. by 296. Quot. 11.739 + cwt. 17. Divide 1001s. by 27s. Quot. 37.074. 18. Divide SS^^ft. by 172ft. 19. Divide 372.25sq. yd. by 250sq. yd. Quot. 1.489. 20. Divide iA. by 13 1-. Quot. .03tA. 21. Divide 243flb. by 19f. 22. Divide 799.6T. by 87.'5T. Quot. 9.104 + • 23. Divide 34.1.5 grammes by 19.25 grammes. 24. Divide 177pt. by 771. Quot. .2295 + pt. KEDUCTION OF CONCRETE NUMBERS. § 186. A compound number may be reduced to a simple one, or a simple concrete number to a-compound one by tbe application of the following rule according to tbe circum- stances of the case. §187. Rule. — Find from the proper table the value of one of the given units in terms of the required denomination ; and multiply this value by the number of the given units. Ex. 1. Reduce 6 gallons to pints. §188. Model. 6gal.=r6x4qt. = 24qt. 24qt. = 24x2pt.=48pt. Hence, 6gal.=48pt. I 129 §189 CONCRETE NUMBERS. Explanation. —Since 4qt.=:lgal.,6gal.j tliat is, 6 times lgal.=6 times 4(it. And since 2pt. = lqt., 24qt., or 24 times lqt.=24 times 2pt. Observe that in each instance the product is similar to the multiplicand. (§183.) Otherwise^ 6gal. = 6 x 8pt.=:48pt. Ex. 2. Reduce 6gal, 3qt. Ipt. to pt. §189. Model. 6gal. = 6x4qt.r=24qt. 24 + 3^:27 27qt.r=27 x 2pt. = 54pt. 54 + 1=55 Hence, 6gal. 3qt. lpt. = 55pt. Explanation. — After reducing the 6gal. to qt., the giv- en 3qt. may be added to the result. (§ 179.) And after reducing the 27qt. to pt., the giv s Ipt. may be added to this result. OtlieriDise, 6gal.=6 x 4qt. = 24qt.=24 x 2pt.=:48pt. 3 '^' = 3x2^^= 6 " Hence, 6gal. 3qt. lpt.=:55pt. Otherwise, 6gal.=6 x 8pt.r::48pt. 3qt. =3x2" = 6" Ipt. = 1_^ Hence, 6gal. 3qt. lpt. = 55pt. Evidently the final result is not affected by the order in which the several reductions are performed. Ex. 3. Eeduce i^lb. to oz., dwt., &c. § 190. Model. TVlb.=yV of 12oz.=l-^oz. ^^oz.=|: of 20dwt. = 4dwt. Hence, -Jylb. = loz. 4dwt. Explanation. — This example differs from the first only in the fact that here each multiplier is a fraction. 130 REDUCTION. §193 Ex. 4. Reduce 3795P. to A., R., &c. §191. Model. 3795P. = 3795x-J,tR.=H^R. = 94R, 35P. 94R.=: 94 X 1:A.= V A. = 23A. 2R. Henoe, 3795P. = 23A. 2R. 35P. Explanation. — This example differs from the preceding only in the fact that here each multiplicand is a fraction. Ex. 5. Reduce 6gal. 3qt. Ipt. 3gi. to hhd. §192. Model. 3gi. =3 x ipt. = a pt, 3^qt. =3^- X igal. = U gal. 6||gal. = 6|ixJ3hhd.=^^hhd. Explanation." — Here both factors are fractional. Ex. 6. Reduce 30bu. Ipk. 3qt. Ipt. to pk. § 193. Model. 30bu. =r 30 x 4pk. = 120pk. Ipk. = 1 « 3qt.= 3x ipk.= .375 lpt.=: t\V^'= •0625_ Hence, 30bu. Ipk. 3qt. Ipt."^21^43f5pk. Explanation. — This example is but a combination of two of the preceding ones, and seems to require no addi- tional explanation. Ex. 7. In $14, how many mills ? Ans. 14000m. 8. In jS15, how many pence ? Ans. 3600d. 9. In 19fr., how many centimes 1 10. In 221b., Troy, how many dwt. ? Ans. 5280dwt. 11. In 251b., Apothecaries', how many scruples ? Ans. 7200sc. 12. In 261b., Avoirdupois, how many drams ? 13. In 31 hectogrammes, how many decigrammes ? Ans. SlOOOdec. 14. In 45 miles, how many feet ? Ans. 237600ft. 131 S193 CONCRETE NUMBERS. Ans. 2048qt. Ans. 1728000sec. Ans. 8160sh. Ans. 720doz. 15. In 49fur., ho^ many chains? 16. In 50 A., how many square yards? Ans. 242000sq. yd. 17. In 65cu. yd., how many cu. in. ? Ans. 2566080cii. in. 18. In 72gal., how many gi. ? 19. In 64bu., how many qt. ? 20. In 20da., how many sec. 1 21. In 29°j how many seconds ? 22. In 17rm., how many sheets ? 23. In 5gr. gross, how many doz.? 24. In M, 3s. 2d., how many qr. ? 25. In 5fr. 7dec. Scent., how many centimes ? Ans. 578cent. 26. In 61b. 5oz. 3dwt., how many gr. ? Ans. 37032gr. 27. In 31b. 6oz. 5dr. 2sc., how many sc. ? 28. In 28T. lOcwt. 3qr., how many lb. ? Ans. 570751b. 29. In 1 millier, 5 quintals, how many grammes ? Ans. 1500000gr. 30. In 2rd. 3yd. 2ft., how many in. ? 31. In 10 chains, 1 rod, how many links ? Ans. 10251k. 32. In 4sq. yd. 6sq. ft., how many sq. in. ? Ans. 6048sq.in. 33. In lOcu. ft. 400cu. in., how many cu. in.? 34. In 2 tuns, Ipi. Ihhd., how many gal. ? Ans. 693fral. 35. In 5bu. 2pk. Igal. 3qt., how many pt. ? 36. In Icent. 6dec. 5yr., how many yr. ? 37. In 2S. 25° 45', how many seconds ? 38. In 2rm. lOqr. 12sh., how many sh. ? 39. In 3gr. 4doz,, how many units? 40. Reduce -}£ to s. and d. 41.- Reduce ffr. to decimes. 42. Reduce fib. Troy, to oz., dwt., &c. 43. Reduce ^ lb. Apothecaries', to oz., &c. Yal. 2oz. 3dr. 12ffr. 132 Ans. 366pt. Ans. 308700^ Ans. 1212sh. Val. 6s. 8d. Val. 6fdec. REDUCTION. §193 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. Reduce Reduce Reduce Reduce Reduce Reduce Reduce R.educe Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Val. 3cwt. Iqr. S^lb. Val. 22fP. ^T. to cwt,, qr., &c. ^-mi. to fur., &c. f A. to R., P., &c. ;^rcu. yd. to cu. ft., &c. Val. 4cu. ft. 864cu. in. f-gal. to qt., pt.j and gi. ^jjhu. to pk., gal., &c. J da. to lir., min., &c. Yal.2pk. Igal. 2|qt. Val. 4lir. 48mm. Val. 2qr. 14.4sh. Val. Idoz. 9.6un. Val. lOoz. 8dwt. 8gr. Val. lib. 4dr. .375° to min. and sec. .I3rra. to qr. and sh. .15gr. to doz. and units 975qr. to £. oOOOgr. to lb. 300sc. to lb. Apothecaries' 600000dr. to T. llOOOin. to mi. Val. Ifur. 15rd. 3yd. 2in. 600001k. to mi. Val. 7mi. 40ch 4000000sq. yd. to sq. mi. 60000cu. in. to cu.yd. Val. leu. yd. 7cu. ft. 1248cu. in, lOOOOgi. to tuns. Val. 1 tun, 60gal. 2qt lOOOpt. to lihd., A\q Measure. 250pt. to bu. Val. 3bu. 3pk. 5qt, eOOOOOmin. to yr. Val. lyr. 51da. 16hr 2000000" to circumferences. 27000sh. to rra. Val. 56rm. 5qr 19000 units to gr. gross. Val. lOgr. gr. llgr. lldoz. 4 units lid. 3qr. to £. 9oz. 9dwt. 9gr. to lb. Val. .789 -fib 6dr. 2sc. 15gr. to lb. Val. .07204 + lb, Iqr. 151b. to T. S19o CONCRETE NUMBERS. 73. Reduce 20rd. Syd. to fur. Val. .52!>'7fur. 74. Reduce 2rd. 201k. to ch. Val. .7ch. 75. Reduce IR. lOP. to A. 76. Reduce leu. ft. lOcu. in. to cu. yd. Val. .03725 + cu. yd. 77. Reduce Ipt. Igi. to gal. Val. .15625gal. 78. Reduce Ihhd. Ibbl. to tuns. 79. Reduce 3pk. Igal. 3qt. to bu. Val. .96875bu. 80. Reduce lOhr. 15min. BOsec. to da. Val. .4274 + da. 81. Reduce 1*^ 10' 30" to S. 82. Reduce 2qr. 12sli. to rm. Val. .125rm. 83. Reduce Igr. lOdoz. 10 units, to gr. gross. Val. .1585648igr. gr. 84. Reduce J12, 10s. 6d. 3qr. to s. 85. Reduce iOlb. 9oz. 9dv/t. 9gr. to oz. Val. 129.46875oz. 86. Reduce 31b. 5oz. 5dr. Isc. lOgr. to dr. Val. 333. 5dr. 87. Reduce IT. lOcwt. Iqr. 20ib. to cwt. 88. Reduce Irai. 7fur. 20rd. 3yd. to rd. Val. 620.o4rd. 89. Reduce 3ch. 2rd. 101k. to rd. Val. 14.4rd. 90. Reduce lOA. 3R. 20P. to R. 91. Reduce 2cu. yd. 6cu. ft. 75cu. in. to cu. ft. Val. 60.0434 + CU. ft. '92. Reduce lOgal. Iqt. Ipt. 3gi. to pt. Val. 83.75pt. .93. Reduce 2bu. Ipk. 3qt. to pk. 94. Reduce Ida. Ihr. Imin. Isec. to min. Val. 1501.0l6min. 95. Reduce 1*= 10^ 30'' to minutes. Val. 70^'. 96. Reduce 2rm. 3qr. 5sb. to qr. 97. Reduce IT. Icwt. Iqr. lib. loz. to lb. Val. 2126.06251b. 98. Reduce Isq. yd. Isq. ft. Isq. in. to sq. ft. Val. 10.00694sq. ft. *- 134 PROMISCUOUS PROBLEMS. §195 PROMISCUOUS PROBLEMS. 1. Bought a dress for $12, a cloak for $15, a bonnet for $7, and a pair of gloves for $1 : what did they all cost ? ' 15 § ^^^' Model. — The whole cost is the sum of y the several prices : hence, add $12, $15, $7, and I $1. (§ 179). The sum is $35 : hence, they all cost $35. 2. A owns $10475 in real estate, $3850 in slaves, $4095 in good notes, and $1415 in cash ; what is the value of his whole estate ? A. $19835. « 3. Three men form a partnership : A invests $2445 ; B, $2890 ; G, $1959 : what is the whole investment ? . 4. A miller bought from one man 147 bushels of wheat, from an other 98 bushels, and from a third 273 bush- els ; how mueh wheat did he buy from the three ? 5. A farmer raised on one farm 415 bushels of wheat, 548" bushels of corn, 827 hundred weight of hay ; on the other, 293 bushels of wheat, 487 bushels of corn, 2SQ hundred weight of hay : how much did he raise on both farms ? A. 708bu. wheat, 1035bu. corn, 613cwt. hay. (). Bought a farm for $2875, and sold it for $3225 ; what did I gain t ^""225 ^ ^^^' ^^^^^'^- — ^^^^ g^^^^ '^^ *^® difference 9R7^ between what I gave and what I received : -fiir hence, subtract $2875 from $3225. (§181.) $350 The difference is $350 : hence, I gained $350. 7. A farmer owning 725 acres, sells 375 acres ; how much land has he remaining? A. 350A. 185 §190 CONCRETE NUMBERS. 8. A man divides $3000 among three sons^ giving A ^985, and B $1235 : how much does he give C ? A. $780. 9. Burnt a kiln of 100000 bricks ; sold at different times 3475, 2800, 40150, and 35000 ; how many are still unsold ? 10. The distance from Charlotte to Groldshoro', via High Point, is 223mi., from Charlotte to High Point is 79mi.; how far is High Point fj;om Groldshoro' ? A. I4lmi. 11. What cost 2471h. of hn^ -n. at 19ct. per lb.? 9A7 IQ + §186. xiioDEL. — 2471b. cost 247 times ooL^ the cost of lib.: hence, multiply 19ct.,by -^ 247. (§183.) The product is 4693ct.: 4693ct. hence, the bacon cost 4693ct. 12. How many cents are in 25 dollars ? 13. How many gallons in 14hhcl.? A. 882gal. 14. What will 94bbl. of flour cost at $8 per bbl.? $752. 15. How many pages in 475 volumes of 296 pages each ? IG. A father divides $5460 equally among his 4 sons; what does each son receive ? § 197. Model. — Each son's share is one 4)$5460 fourth of the whole : hence, divide $5460 $1365 by 4. (§185.) The quotient is $1365: hence, each son's share is $1365. 17. If 755A. of land cost $12835, what will one acre cost? A. $17. 18. If 125 slaves sell for $75125, what is their average value ? 19. If 85 bales of cotton weigh 386751b., what does each bale weigh '/ A. 4551b. 20. On 475A. of land I raised 15675bu. of wheat ; how much per acre ? A. o3bu. 21. In 478241b. flour, how many bbl. 136 PROMISCUOUS PROBLEMS. §198 ^woci^ii, 1(^n^^. ^198. MoDEL.—As 1961b. Blake a 478241b. 1961b. i ^ , xi i r i,vi • i x QP„ bbl., the number oi bbl. is equal to '^r'0 4 '^^^' the number of times 1961b. are con- f.r.r. tained in 478241b. : hence, divide ^^^ 478241b. by 1961b. (§185.) The quotient is 244 : hence, there are 244bbl. 22. How many cu. yd. iu 13122cu. ft.? A. 486cu. y^. 23. In 11823s., how many G.I A. 563G-. 24. How many xVcres can be bought for $5658 at $23 per Acre ? 25. If a vessel make 376mi. per day, how long will she be in making 7144mi.? A. 19da. 26. Find the sum of two thousand and forty-seven, three thousand six hundred and fifty, sixty-three thousand and five, ten thousand four hundred and three, and four hundred and seven. Sum, 79512. 27. Find the diftereuce between ten thousand and forty- two, and eight thousand seven hundred and ninety- nine. 28. What is the product of seven thousand three hundred and seventy-five, and one hundred and twenty-five? A. 921875. 29. IVhat is the quotient of eight thousand six hundred and twenty-five, by one hundred and twenty-five ? A. 69- 80. How many days in 4wk.? 31. How many hours in 28da.? A. 672hr. 32. Hovv^ many minutes in 672hr.? A, 40320min. 33. How many seconds in 40320min.?" 34. The minuend is 91 thousand 8 hundred and 75, the sub- trahend 8 thousand 9 hundi-ed and G9 ; what is the remainder ? A. 82906. 35. The subtrahend is 4 thousand 2 hundred and 96, the 137 §198 CONCRETE NUMBERS. remainder 6 thousand 2 hundred and 84 : what is the minuend ? * A. 10580. 26. The remainder is 7 hundred thousand and 94, the minu- end 2 millions 3 thousand; what is the subtrahend? 37. How many hours in 40320min.'? A. 672hr 38. How many days in 672hr.? A. 2Sda. 39. How many weeks in 28da.? 40. How many min. in 2419200sec.? A. 40320min. 41. The multiplicand is 37 millions 43 thousand and 25, the multiplier 8 thousand and 64 ; what is the produef? A. 298714953600. 42. The multiplicand is 7 hundred and 25, the product 593 thousand 7 hundred and 75 ; what is the multiplier ? 43. The multiplier is 4 thousand 9 hundred and 7, the prod- uct 42 millions 813 thousand 575 ; what is the mul- tiplicand ? A. 8725. 44. What cost 347yd. of rope at 9ct. per foot 1 A. $93.69. 45. How many qt. in 7gal. 2qt.? 46. How many qt. in 8gal. Iqt.? 47. How many pt. in 8gal, Iqt. Ipt.? 48. How many gi. in 7gal. 3qt. Ipt. Sgi.l 49. The dividend is 11 millions 210 thousand 202, the di- visor 7 thousand and 2 ; what is the quotient ? A. 161. 50. The divisor is 8 thousand and 4, the quotient 5 thou- sand and 90 ; what is the dividend ? A. 40740360. 51. The quotient is 1 million 2 thousand and 3, the divi- dend 1 trillion 4 billions 10 millions 12 thousand and 9 ; what is the divisor 1 52. How many sq. mi. in 228S88P.? A. 2Rq. mi. 150A. 211. 8P. 53. How many R. in 1728P.? A. 43R. 8P. 138 PROrvilSCUOUS PROBLEMS. §198 54. How many sq. mi. in 1895A.? 55. How many A, in 1806P.? A. IIA. IK. 6P. 56. What is the sum of 7 thousand, 83 thousand and 40, 9 hundred and 70, and 17 times 5 hundred and 79 1 A. 100853. 57. What is the difference between the product of 85 and 307, and the quotient of 999875 by 125 ? 58. How many lb. in 7qr.? A. 1751b. 59. How mary oz. in 251b. Avoir.? A, 400oz. 60. The distance from High Point to Greensboro' is 15mi., from Greensboro' to Shops 22mi., from Shops to Fta- 4 leigh 53mi.; how far is it from High Point to Raleigh, via Greensboro' and Shops ? 61. The distance from Charlotte to High Point is 79mi., from High Point to Ilalcigh 95mi., from Kaleigh to Goldsboro' 49mi.; hov/ far is it from Charlotte to Goldsboro', via High Point and Raleigh ? A. 223mi. 62. Bought a pair of horses for $375, a set of harness for $55, and a buggy for $187 ; what did the whole cost ? A. $617. 83. Paid $789 for a lot of tobacco, and sold it for $910 ; gained how much ? 64. How many units in 14doz. and 7 ? A. 175. 65. How many units in 3 score and 10 ? A. 70. 66. How many doz. in 12 gross ? 67. How many units in 10 great gross ? A. 17280. 68. Bought 3 stone of potatoes at 2ct. per lb.; what did they cost ? A. 84ct. 69. Bought 10001b. of fish at $9 per quintal ; what did I pay ? 70. What cost 6161b. of butter at $15 a firkin ? A. $175. 71. What cost 247bbl. of flour at $5 per bbl.? A. $1235. 139 §198 CONCRETE NUMBERS. 72. How far will a traiu of cars go in 3 dajs, at 16 miles per hour ? 73. Bought 16yd. of calico at 15cfc., 7yd. of gingham at 25ct., 9yd. of flannel at GSct., and 25yd. of domestics at lOct.; paid 16bu. of corn at GSct.; how much is still due ? A. $1.89. 74. If a hook of 155 pages has 29 lines on each page, and 39 letters in each line, how many letters are in the book ? A. 175305 letters. 75. I deposited in bank $10050 : having drawn out $15, $175, $237, $375, $4165, $394, and $3968, how much have I still on deposit ? ^ 76. The Bible contains 31173 verses ; how many verses must I read each day, to finish it in one year 1 A. 85 verses a day, and 148 verses over. 77. How many sheets of paper in 20 quires ? A. 480sh. 78. How many sheets in 14 reams ? 79. How many reams in 180 quires ? A. 9rm. 80. How many quires in 19 reams ? A. 380qr. 81. A stock-dealer bought 4'7 cows at $19, 29 horses at $135,53 mules at $97, and 155 sheep at $3: he received for them 347 acres of land at $26^ and $4125 in money; how much did he gain ? 82. What will 574bbl. of pork cost at $13 per bbl.'? A. $7462. 83. How far will a man travel in 6da. at 29mi. per da.? A. 174mi. 84. A planter who worked 57 hands, raised 399 bales of cot- ton : how m.any bales did he raise to the hand ? 85. In $45, how many ct.? A. 4500ct. 86. In M, 5s. 6d., how many d.? A. 1026d. 140 PROMISCUOUS PROBLEMS. ijlDb 87. In 240dwt., boyr raaDj oz.? 88. In 39sc.j liow many dr.? A. I3dr. 89. In 3T. 3qr. 201b. 12oz., how many oz.? A. 97532oz. 90. In 7920in., how many yd.? 91. In 4mi., how many ch.? A. 320ch. 92. In 1568160sq. in., how many sq. yd.? A. 1210sq. yd. 93. In 4sq. mi., how many A.? 94. In 4cu.yd. 12cn.ft., how many cu.in.? A. 207360cu.in. 95. In 3025gi., how many hhd.? ' A. Ihhd. 31gal. 2qt. Igi. 96. In Sbu.j how many pt.? 97. In 3da. lOhr. 15mija., how many sec? A. 296l00scc. 98. In 3S. 3° 3' 3'^, how many seconds ? A. 334983''. 99. In 2gr. gr, 3gr. 4doz. and 5, how many units ? 100. In 6rm. 7qr. 8sh., how many sh.? A. 3056sh. 101. In 3gal. 3qt. 8gi., how many qt,? A. 15.375qt. 102. In lObu'. Ipk. Igal. Ipt., how many pk.? 103. In ^6, 6s. 6d. 3qr., how many s.? A. 126.5625s. 104. Add 4:1b. 5 -^-oz., -idwt., and -Vgr., in gr. Sum, 1688.2gr. 105. Add 3.5hr.5 7.75min., and .15sec., in min. 106. Add iA., A.R., and -^j^V., in P. . Sum, 48.1P. 107. Add .25cu. yd., .375cu. ft., and .625cu. in., in cu. ft. Sum, 7.12536 + CU. ft. 108. From .9cwt., take .251b., in oz. 109. From .751b., take .5dwt., in oz. Rem. 8.975oz. 110. From 10.875s., take 9.15^., in qr. Rem. 485.4qr. 111. From 5.5da., take 5.5min., in min. 112. Multiply .75gal. by 7.5, in pt. Prod. 45pt. 118. Multiply 2.25A. by .125, in P. * Prod. 45P. 114. Divide 4.5mi. by 5.4, in rd. 115. Divide 1.55s. by 2.3, in d. Quot, 8.08695d. 141 §199 COMPOUND NUMBERS. OPERATIONS ON COMPOUND NUMBERS. § 199. The operations on compound numbers are analo- jxous to the corresDondino; ones on abstract -iiumbers. ADDITION OF COMPOUND NUMBERS. Ex. 1. Add together 4hhd. 25gal. 3qt., 5hhd. 20gal. 2qt., 7hhd. 17ga]. 2qt. § 200. MoDEL.~2 and 2 are 4, 4hhd. 25gal. oqt. ^nd 3 are 7, 7qt., equal to Igal. 5 '' 20 " 2 '^ 3(^1^^^ gg. (lo^jj 3 . 1 and 17 are 18, 7 " 17 '' 2 '- ^^^ 20 are 38, and 25 are 63, 63 17 " " 3 " gal., equal to Ihhd., set down 1 and 7 are 8, and 5 are 13, and 4 are 17, i7hhd. The sum is 17hhd. 3qt. Explanation. — In simple numbers ten units of any de- nomination make one of the next higher., In compound numbers this uniformity of relation does not exist. Thus in the example above, 4qt. make Igal., but 63gal. make 1 hhd. With this exception, the explanation in § 22 will suffice for this case. Ex. 2. Add ^10, 14s. 9d. 3qr., J^5, 16s. 6d. 2qr.,^7,10d. Iqr., .212, 9s. 9d. 3qr. 3. Add ^4, 10s. lid., £J, 8s. 9d. 3qr., ^8, 10d.,and 16s. 3qr. Sum, =£20, 16s. 7d. 2qr. 4. Add 101b. lOoz. lOdwt. lOgr., 121b. 9oz. 6dwt. 3gr., 91b. lloz. I3dwt. 15gr., and 24ib. Boz. 15dwt. 20gr. Sum, 581b. 4oz. 6dwt. 5. Add 31b. 6oz. 9dwt. 12gr., 61b. 8oz. lOdwt. 12gr., 81b. lloz. 14dwt. 17gr., and 141b. lloz. 8dwt. 5gr. 142 SUBTRACTION. §201 6. Add 101b. 9ok.7dr.2sc.l5gr.,10oz.6dr. Isc. lOgr:, 15 lb. lloz. 7dr. 2gc. 19gr., and 31b. 4oz. 5dr. 6gr. Sum, 3 lib. loz. 3dr. Isc. lOgr. 7. Add lOT. lOcwt,. lOlb. lOoz. lOdr., 14T. 15cwt. 3qf. 151b. 13oz. 15dr., and 25T. 7cwt. Iqr. 201b. 8oz. €dr. Sam, SOT. IScwt. Iqr. 221b. loz. Idr. 8. Add Ssq.mi. 300A. 2Fi.. 25P., 7sq.mi. 525A. 311. lOR, 19sq.mi. 285A. 3R. 19P., and 250A. 25P. Sum, 31sq.mi. 82A. IR. 39P. 9. Add 19ou.yd. 19cu.ft. 19cu.in., 25cu.yd. 25cu.ft. 250cu. in., and lOOcu.jd. IScu.ft. 1555cu.in. 10. Add 4bhd. 40gal. 2qt. Ipt. 3gi., lOhbd. lOgal. Iqt. Ipt. Igi., and 20bhd. 43gal. 3qt. Ipt. 3gi. IBum, 35hhd. 32gal. 3gi. 11. Add lObu. Spk. 7qt. Ipt., 9bu. 2pk. 6qt. Ipt., 16bu. 3pk. Gqt., and 15bu. Ipk. 5qt. Ipt. Sum, 53bu. Iqt. Ipt. 12. Add 30da. ;LOhr. 30min. SOsec, 15da. 15hr, 15min. 15 sec, and lOda. 20hr. 45rain. 15s8c. 13. Add 25<^ 15' 25'', 75° 24' 50", and 15° 50' 45'^ Sum, 116° Sr. 14. Add 2rm. lOqr. 12sh., 4rm. 15qr. ISsh., and 3rm. 9qr. lOsh. 15. Add 2gr. gross, 10 gross, 7doz. 5 units, 4gr. gross, 8 gross, 6doz. 7 units, and 5 gross, 8doz. 6 units. SUBTRACTION OF COMPOUND NUMBERS. Ex. 1. From o-ei7, 5s. 6d. 3qr., take ^8, 10s. 9d. 2qr. o-i" - PA ^ §201. Model. — 2 from 3 leaves q' io^; o a Sir 1 ; ^ ^^'01^ 18 ^e^ves 9 ; 11 from 25 . .^ilLj__l_ leaves 14 ; 9 from 17 leaves 8. The 8, 14" 9 " 1 « remainder is ^8, 14s. 9d. Iqr. 143 §201 COMPOUND NUMBERS. Explanation. — As 9d. can not "be taken from 6d., we add Is., that is 12d., to the minuend, and subtract 9d. from 18d. We then add Is. to the subtrahend, and proceed. See ^§28, 30. Ex. 2. From 501b. 6oz. 15dwt. 19gr., take 101b. 17dwt. Rem. 401b. 5oz. ISdwt. 19gr. 3. From 151b. logr., take 121b. 9oz. lOdwt. 12gr. 4. From lOT. lOcwt. lOoz., take 5T. 15cwt. 20ib. 12oz. lOdr. Rem. 4T. 14cwt. 3qr. 41b. 13oz. 6dr. 5. From 6sq.mi. 2R., take 375A. 25P. Rem. Ssq.mi. 265A. IR. 15P. 6. From 250cu.yd. 20cu.ft. 875cu.in.j take 79cu.yd. 25cu. ft. G95cu.in. 7. From 15T. 15cwt. 3qr. 151b., take lOT. 19cwt. 3qr. 191b. Rem. 4T. 15cwt. Sqr. 211b. 8. From lOhhd. lOgal. Iqt. Ipt., take 9hhd. 33gal. 3qt. ogi. Rem. 39gal. 2qt, ]gi. 9. From 4 tuns, Ipi. Ihhd. 5gal. 2qt. 3gi., take 2 tuns, 60 gal. oqt. 3gi. 10. From 175bu. Ipk. 3qt. Ipt., take 54bu. 3pk. 2qt. . Rem. 120bu. 2pk. Iqt. Ipt. 11. From 27bu. 2pk. Ipt., take ISbu. 5qt. Rem. 14bu. Ipk. oqt. Ipt. 12. From 30da. lOhr. 15min., take 17da. 15hr. 15sec. 13. From 180°, take 74° 14' 45". Rem. 105° 45' 15". 14. From 90°, take 35° 41' 15". Rem. 55° 18' 45". 15. From 100° 17' 30", take 90° 25' 45". 16. From 22T. Scwt. 2qr. 201b., take 12T. 18cwt. 221b. Rem. 9T. lOcwt. Iqr. 231b. 17. From 16hhd. 24gal. 3qt. 2pt., take I4hhd. 37gal. 3qt. 18. From 236bu. 2pk. 5qt. Ipt., take 17bu. 2pk. 7qt. 2pt. 144 MULTIPLICATION. §202 MULTIPLICATION OF COMPOUND NUMBERS. Ex. 1. Multiply £11, 5s. 6d. 3qr. by 15. i^l7, 5s. 6d 3qr. §202. Model.— 15 x 3=45. 45qr.= ^^ lid. Iqr.; sefc down 1: 15x6=90, 259, 3 " 5 " 1 " nnd 11 = 101. 101d.=8s.5d.; set down 5: 15x5=75, and 8 = 83. 88s. = ie4, OS. ; set down 3 : 15 x 17=255, and 4=259. The product is £2bd, 3s. 5d. Iqr. Explanation.— See §§ 38, 200. Ex. 2. Multiply £bS, 10s. 9d. 2qr. by 4. Prod. ^214, 3s. 2d. 3. Multiply 13° 15' 45" by 7. 4. Multiply 25° 30' 45" by 10. Prod. 255° T 30". 5. Multiply 501b. 6oz. 15dwt. 19gr. by 13. Prod. 6571b. 4oz. 5dwt. 7gr. 6. Multiply 121b. 9oz. lOdwt. 12gr. by 16. 7. Multiply 5T. 15cwt. 201b. l2oz. lOdr. by 19. Prod. 109T. 8cwt. 3qr. 171b. 9oz. 14dr. 8. Multiply 2sq. mi. 200A. 2R. 20P. by 22. Prod. 50sq. mi. 573A. 3R. 9. Multiply 3cu. yd. 25cu. ft. 750cu. in. by 25. Prod. 98cu.yd. 14c u. ft. 1470cu. in. 10. Multiply 9hlid. 33gal. 3qt. 3gi. by 21. 11. Multiply 2 tuns, 60gal. 3qt. 3gi. by 24. Prod. 53tijps, Ipi. Ihhd. llgal. Iqt. 12. Multiply 25bu. 3pk. Iqt. Ipt. by 29. 13. Multiply lObu. Ipk. 4qt. by 35. Prod. 363bu. 4qt. 14. Multiply lOda. lOhr. lOmin. lOsec. by 41. Prod. lyr. 62da. 8hr. 56min. 50sec. 15. Multiply 17da. 15min. 15sec. by 50. J 145 §203 COMPOUND NUMBERS. DIVISION OF COMPOUND NUMBERS. Ex. 1. Divide 15° 15' 50'' by 10. 10) 15° 15' 50" §203. Model.— 10 in 15, once, 1° 31' 35" with 5 over, set down 1 ; 10 in 315, 31 times, with 5 over, set down 31 : 10 in 350, 35 times. The quotient is 1° 31' 35". Explanation. — 10 is contained once in 10; so that there are 5° undivided. These 5° are reduced to 300', and added to the 15', making 315'. In like manner, the 5' undivided are reduced to 300", and added to the 50", making 350". Ex. 2. Divide £^0, 16s. 2d. Iqr. by 3. 3. Divide £Q0, Is. 5d. by 4. 4. Divide 291b. 2oz. 2dwt. 2gr. b;, :». Quot. 51b. lOoz. lOgr. 5. Divide 2421b. 5oz. lldwt. 16gr. by 8. Quot. 301b. 3oz. 13dwt. 23gr. 6. Divide 4481b. lOoz. I4dr. by 11. 7. Divide 32T. 2qr. by 15. Quot. 2T. 2cwt. 2qr. 201b. 8. Divide 52sq. yd. 5sq. ft. 128sq. in. by 20. Quot. 2sq. yd. 5sq. ft. lOOsq. in. 9. Divide 97cu. yd. 22cu. ft. SOcu. in. by 26. 10. Divide 91gal. Iqt. Ipt. by 34. Quot. 2gal. 2qt. Ipt. 2gi. 11. Divide 79 tuns, Iqt. Igi. by 45. Quot. Ftun, Ipi. Ihhd. Igal. Iqt. Ipt. Igi. 12. Divide 600bu. 3pk. 6qt. by 60. 13. Divide 8wk. 3da. 71ir. 43min. 20sec. by 70. Quot. 20hr. 20min. 20sec. 14. Divide 1150° 31' 15" by 75. Quot. 15° 20' 25". 15. Divide 1° 41' 40" by 100. 146 I'ROMISCUOUS PROBLEMS. §201 J 6'. Divide 57T. lOcwt. Iqr. 171b. 14oz. by 9cwt. Iqr.lTlb. lOoz. 57T. lOcwt. Iqr. 171b. 14oz. = 1855086oz. Ocwfc. Iqr. 171b. 10oz. = 15082oz. . 1855086oz.-^15082oz. = 123. § 204. MoDEL.—Recluce the dividend to oz. (§ 189). Ke- duce the divisor to oz. (§ 189). Divide the dividend by the divisor. (§ 185). The quotient is 123. Ex. 17. Divide 294lbii. by 45bu. 3pk. 6qt. Ipt. Quot. G4. 18. Divide 97T. llcwt. 3qr. 111b. lOoz. by IT. 6cwt. 2qr. 261b. lOoz. 19. Divide 17bu. Ipk. 6qt. by 2bu. Spk. 5qt. Quot. 6. 20. Divide 51A. IR. IIP. by lA. IP. Quot. 51. 21. Divide 10 tuns, 2hhd. ITgal. 2pt. by 39gal. 6pt. 22. Divide ^£27, 2s. 6d. by 15s. Gd. Quot. 35, PROMISCUOUS PROBLEMS. 1. What is the least common multiple of 15, 24, and 27 ? 2. What is the least common multiple of 9, 25, and 45 ? L. C. M. 225. 3. What is the greatest common measure of 505, 1111, and 3434? 4. What is the greatest common measure of 1015, 1260, and 1330? G. C. M. 35. 5. What are the prime factors of 6105 1 6. What are the prime factors of 4060 ? 7. Divide ^£113, 13s. 4d. by 31. Quot. £S, 13s. 4d. 8. Divide 10 tuns, Ipi. 17gal. 2pt. by 67. 9. Divide 50T. 4cwt. 2qr. I4lb. by 23cwt. 3qr. 171b. 147 §204 PROMISCUOUS PROBLEMS. 10. Divide 1572yd. by 32yd. 3qr. Quot. 48. 11. Multiply 25oz. 8dwt. 17gr. by 100. Prod. 2111b. lloz. lOdwt. 20gr. 12. Multiply 21da. 181ir. 42niin. by 75. 13. Subtract 40A. 3R. 25P. from 79A. 15P. Rem. 38A. 30P. 14. Subtract 4 tuns, Ipi. llihd. 25gal. 3qt. from 5 tuns, Iqt. 15. Add 60mi., 40mi. 7fur. 39rd., and 19mi. Ird. 16. Add 13° 14' 15", 16° 17' 25", 25° 19' 47", and 3° 15". 17. Divide $1521808938 by 234. Quot. $6503457. 18. Divide I42651ihd. by 45hhd. 19. Multiply 4327bu. by 102. Prod. 44l354bu. 20. Multiply 47935gal. by 275. Prod, 13182l25gal. 21. Subtract 2598328fur. from 3002575fur. 22. Subtract 187564329gi. from 923465781gi. ' Rem. 73590l452gi. 23. Add 2479A., 3580A., 1358A., and 9745A. 24. Add ^613575, £2Udb, £9475, and £31525. 25. Divide 82960332 by 84. Quot. 987265 ; Rem. 72. 26. Divide 82071 by 99. Quot. 829. 27. Multiply 24068 by 13579. 28. Multiply 1020908 by 8979091. Prod. 9166825834628. 29. Subtract 3987456002 from 4567398745. 30. Subtract 246 + 357 + 1298 + 982 from 3120. 31. Add 20030405, 910285, 5821090, and 9706845. Sum, 36468625. 32. Add 123, 1234, 12345, 123456, 1234567, 12345678, and 123456789. Sum, 137174192. 33. A. raised 125 bales of cotton, 517bu. corn, 629bu. wheat, and 119bu. rye ; B., 217 bales of cotton, 865bu. corn, 798bu. wheat, and 143bu. rye ; C, 94 bales of cotton, 424bu. corn, 517bu. wheat, and 77bu. rye; and D., 148 PROMISCUOUS PROBLEMS. §204 111 bales of cotton, 512bu. corn, 558bu. wheat, and 98bu. rye. How much of each article did they all raise ? 84. A farmer went to town with $100, and spent $9 for molasses, $13 for sugar, ^11 for coffee, $8 for rice^ $17 for dry goods, and $25 for leather. Horv much money had he left ? Ans. $17^ ii5. Bought 47 acres of land at $19, 5 horses at $125, 10 head of cattle at $21, 14 sheep at $3, and a two-horse wagon for $65 ; what did they all cost ? Ans. $1835. 36. Sold 75 firkins of butter for $1350 ; how much was that a firkin ? 37. What number is that, to which if 245, 379, 124, 212, and 399 be added, the sum will be 1525 ? Ans. 166. 38. What number is that, from which if the sum of 245, 379, 124, 212, and 399 be subtracted, the remainder will be 1525? Ans. 2884. 39. AVhat number is that, by which if twice 19 be multi- plied, the product will be the difference between 4127 and 2759 ? 40. What number is that, by which if 4235 bo divided, the quotient will be 77? Ans. 55. 41. A miller has 5 bins, one of which holds 43bu. 3pk. 5qt.; the second, 39bu. Ipk. 3qt. ; the third, 45bu. Iqt. Ipt.; the fourth, 53bu. 2pk. ; the fifth, 34bu. 3pk. Ipt, i what is their united capacity? Ans. 216bu. 2pk, 2qt. 42. How much time elapsed between Jan. 20th, 1833, and May 25th, 1861 ? 43. Bought 4 lots of land, containing 3B. 27P, each ; how many A. did I buy ? Ans. 3A. 2B. 28P. 44. A wine merchant has 269gal. 2gi. of wine in 30 equal vessels; how much wine is there in each vessel ? 149 P04 PROMISCUOUS PROBLEMS. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 50. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce Reduce ,Reduce Reduce Reduce Reduce Reduce Reduce R.educe Reduce Reduce Reduce Reduce Reduce Val. 69119gr. £40, 19s. lid. 3qr. to qr. 111b. lloz. 19dwt. 23gr. to gr. aib. II5. 75. 29. lOgr. to gr. Val. 2B039gr, 2T. 19cwt. 3qr. 24lb. 15oz. 15dr. to dr. 4L. 2mi. 7fur. 35rd. to rd. Val. 4795rd. 12yd. 2ft. lliu. to in. Val. 467in, 2sq. mi. 600A. 3R. 35P. to P. 25sq. yd. 8sq. ft. lOOsq. in. to sq. in. 21sq. mi. 250A. 2R. to R. Val. 54762R 5cu. in. 20cu. ft. 1600cu. in. to cu. in. 3 tuns, Ipi. llibd. to bhd. ohbd. 60gal. 3qt. Ipt. to pt 2gal. Iqt. Ipt. 3gi. to gi. 5bu. 3pk. 7qt. Ipt. to pt. 30. 75yr. 300da. to da. 4da. lObr. 25min. to sec. 25° 10' J 5" to seconds. 2rm. 15qr. ]2sb. to sh. 4 score and 5 to units. lOOOqr. to £. 6000gr. to lb. Troy. eOOOOOdr. to T. lOOOrd. to mi. 2000sq. in. to sq. yd. 200000sq. rd. to sq. mi. 60000cu. in. to cu. yd. lOOgi. to gal. 500qt. to bhd. lOOpt. to bu. 4000sec. to hr. 200br. to wk. 4000" to degrees. 150 Val. 15bbd. Val. 1999pt, Val. 383pt. Val. 137175da. Val. 90615". Val. 1332sh. Val. £1, lOd. Val. lib. lOdwt. Val. om. Ifur. Val. 3gal. Ipt. Val. Ibu. 2pk. 2qt. Val. Ibr. 6min. 40sec. Val. 1° 6' 40". PROMISCUOUS PROBLEMS. §204 77. Bought 8 firkins of butter at 20ct. per lb., 20qt. mo- lasses at $1 per gal., 3 stone of potatoes at 3ct. per lb., and 9801b. flour at $S per bbl.; what did they all cost ? 78. Bought 5doz. Arithmetics at 30ct. apiece ; sold them all for $27 : how much apiece did I gain or lose ? 79. A owes B for 5001b. of salt fish at $8 a quintal ; B owes A for 8bbl. flour a^ $7 a bbl., lObu. corn at 70ct. per bu., and 5bu. rye at 80ct. per bu. : how does their account stand ? A owes B $18. 80. From 500 subtract the sum of 225, 120, and 75 ; divide the remainder by the diff"erence between 1000 and 960 ; multiply the quotient by 17 ; and add 16 to the product. Sum, 50. 81. Find the sum of the product of 88 and 11, and the quo- tient of 88 by 8. 82. A man has 1184bu. of wheat and 468bu. of corn, which he wishes to pack in equal bags as large as possible. How many bushels will each bag hold ; and how many bags will be required ? Ans. 4bu., and 413 bags. 83. What is the value of 85 + 77-64 + 6 x 19-132-^-4 ? 84. What is the value of 15498-^54 + 41 x 63-27 x 55 ? 85. Find the least common multiple of 4, 44, 132, and 792. 86. Find the greatest common measure of 4, 44, 132, and 792. G. C. M. 4. 87. Two men travel in the same direction from the same place, one 40mi. a day, the other 33mi. a day ; how far apart are they in 7 days ? 88. Two men travel in contrary directions from the same place, one 40mi. per day, the other 33mi. per day ; how far apart are they in 7 days? Ans. 511mi. 89. If 10 persons use a barrel of flour in 57 days, how long will a barrel last one person ? Ans. 570da. 151 §204 PROMISCUOUS PROBLEMS. 90. "What ig the sum of 3 numbers, of which the first is 28, the second 8 times the first, and the third one seventh of the second ? 01. The difference is one hundred thousand four hundred and seventy-six, the minuend is one million ; what is the subtrahend ? Ans. 899524. 92. The minuend is one hundred thousand, the subtrahend is sixty-seven thousand seven hundred and forty-four; what is the remainder ? Ans. 32256. 93. The subtrahend is f:. ■> i Iiundred thousand and forty- nine, the remainder is ninety-nine thousand two hun- dred and seventy-eight ; what is the minuend? 94. The multiplicand is thirty-six thousand seven hundred and seven, the multiplier is eighty thousand and one ; what is the product? Ans. 2936596707. 95. The multiplier is eight h-T^idred and four, the product is sixty-one thousand nine hundred and eight ; what is the multiplicand ? Ans. 77. 96. The product is eighteen billions two hundred and twen- ty thousand, the multiplicnnd is two thousand two hundred ; what is the multiplier? 97. The divisor is one hundred and twenty-five, the divi- dend is nine hundred and eighty-seven thousand six hundred and twenty-five; what is the quotient? Ans. 7901. 98. The dividend is thirty-four thousand eight hundred and forty-eight, the quotient is one hundred and thir- ty-two ; wliat is the divisor ? Ans. 264. 99. The quotient is thirty thousand and seventy, the divi- sor is seven hundred and eight ; what is the dividend? 100. What cost ISrm. of paper at $4, 3doz. Arithmetics at $6 a dozen, and 24 Algebras at 812 a dozen ? 152 ALIQUOT PARTS. §20G ALIQUOT PARTS. § 205. An aliquot fraction is a simple fraction whose nu- merator is 1. Thus, ^V, jf, -}, To> -^Vj ^^® aliquot fractionj^!. An aliquot part of a number is a part denoted by an ali- (|uot fraction. Thus, ?> is an aliquot part of 12, 10s. is an aliquot part of ^61, 20da. is an aliquot part of 2mo. Ex. 1. What is the cost of 5A. 3R. 25P. of land at $45. 50 per A.? 2) $45.50=1A. , §206. M0DEL-5A cost ^ - 5 times as much as lA. ; — hence, multiply the cost of $227.50r:r5A. YK ^ 5 (§183). 2R. is ^) ""*"•/ ^ "tS" ^ Slf- ^^^ ^^^ *^^' \k.\ hence, di- 2) 11.375=rlR. S ' vide the cost of lA. by 2. 4) 5.687=:20P. ^ o^p (§185). IR. is one half of _J^421=r 5P. j • 2E.; hence, divide the cost $268,733 of 2R. by 2. 20P. is one half of III. ; hence, divide the cost of IR. by 2. 5P. is odc fourth of 20P.; hence, di- vide the cost of 20P. by 4. Add the several costs together. The sum is $268,733 : hence, 5A. 3R. 25P. cost $268,733. Ex. 2. What is the cost of 17T. 15cwt. 3qr. 101b. of iron at $36 per T.? Ans. $040.53. 3. What is the cost of lObu. 3pk. 4qt. Ipt. of grass seed at $8 per bu.? 4. What is the yield of 45A. 3R. 15P. of wheat land at 20bu. per A.? Ans. 916.875bu. 5. What is the value of 21b. 9oz. lOdwt. 6gr, of plate at $15 per lb.? Anso'?4l.88. 6. What is the value of 5T. lOcwt. Iqr. 51b. of hay at $25 per T.? 153 §207 ALIQUOT TARTS, 7. What is the cost of 15bu. Ipk. 5qt. of dried peaches at 80 per bu.? Ans. 692.4375. 8. What is the cost of 4gal. Iqt. Ipt. of wine at $4 per gal.? Ans. 817.50. 9. What is the co.st of 37rd. 2^jd. of fencing at $3.50 per rd.? 10. What is the cost of 7yd. 2ft. Gin. of cloth at $7.25 per yd.? Ans. s856.791 + . 1 1. AVhat is the cost of SOsq.yd. 4sq.ft. 72sq.iu. of painting at $.75 per sq.yd.? ' Ans. $22,875. 12. What is the cost of lOcd. SOcu.ft. of wood at $2.50 per cd.? 13. What will a man earn in 9mo. lOda. at $25 per mo.? Ans. $233.a3-J. 14. What will lOcd. of wood cost at $2.62^ per cd.? 2)$10 =cost of lOcd. at $1.00 per cd. $20 — a a a 2.00 (( i( 4) 5 — a u a .50 i( iC 1. '2d= .12.^ 2.02^ a $20. 25= (( § 207. Model,— 10 cords at $1 would cost $10. The cost at $2 is twice the cost at $1 ; hence, multiply the cost at $1 by 2. The cost at 50ct. is one half of the cost at $1; hence, divide the cost at $1 by 2. The costatl2ict. is one fourth of the cost at 50ct.; hence, divide the cost at 50ct. by 4. Add the several costs together. The sum is $20.25: hence, 10 cords at $2.G2| cost $20.25. 15. What is the cost of 3G0bu. of wheat at $1.37-^- per bu.? 16. What is the cost of 15yd. of cloth at £1, 4s. 9d. per yd.? Ans. ^18, lis. 3d. 154 CONTRACTED MULTIPLICATION AND DIVISION. §209 CONTRACTION IN MULTIPLICATION. Ex. 1. Multiply 279 by 33^. q^97qnn ^^^^' ^-foDEL.— Annex 2 nauahts to the o;_wJuu multiplicand :— divide the result'by o The 9300 product is 9300. Explanation.— See §63. When the multiplier is an ali- quot part of any power of ten, we may abridge the work by multiplying by this power of ten (§56) and dividing the re- sult by the denominator of the aliquot fraction. Ex. 2. Multiply 72 by 12i Prod. 900. .3. Multiply 72 by 16|-. 4. Multiply 77 by 14f . Prod. 1100. 6. Multiply as by li Protl. 70. 6. Multiply 684 by 166§. 7. Multiply 273 by 333-}. Prod. 91000. CONTRACTION IN DIVISION. Ex. 1. Divide 2000 by 142^. 2000 §209. Model.— Multiply the dividend by 7 7 :— divide the product by 1000. (§ 65.) The 14 000 quotient is 14. Explanation.— See § 68. When the divisor is an aliquot part of any power of ten, we may abridge the work by mul- tiplying the dividend by the denominator of the aliquot fraction and dividing the product by the power of ten. {^65.) Ex. 2. Divide 150 by 33^. Quot. 4.5. 3. Divide 250 by 14|-. 4. Divide 1500 by 166|-. Quot. 9. 155 ^209 rROMiscuous phoblems. 5. Divide 245 by 12 i. gaot. 19.G. C. Divide 1375 by 11^-. 7. Divide 2468 by 333^. Quct. 7.404. PROMISCUOUS PPtOBLEMS. 1. Find 246 + 2468 + 24680 + 20468 + 24068 + 24608. Sura, 96538. 2. Find the sum of one half, three fourths, two fifths, and one sixth. Sura, 1|^. 3. Add 315, 31.57, 3157, 3.157, and .3157. 4. A farraer raised $357 worth of corn, $475.50 worth of wheat, $123.75 worth of rye, and $446.37-^ worth of other products ; what was the total value of the prod- ucts of his farm ? Ans. $1402.62-v. 5. A's gold mine yielded -j^lb. in one week, B's yielded 7 oz., and C's 125.5dwt.; how many ounces did the three mines yield ? Ans. 19.275oz. 6. A tobacco planter raised from one field 7cwt. Iqr. 20 lb., from an other lOcwt. oqr. 101b., and from an other 15cwt. 151b.; what amount did ho raise in all? 7. A man sold 14 horses at $125.75 a head, and 25 head of cattle at $19.87^ a head ; what did he receive for them all '/ " Ans. $2257.375. 8. Find the value of 135-1357 + 13570-3571-3157. Yal. 5620. 9. What is the difference between thirteen fourteenths and fourteen fifteenths? 10. From 500.05 take 65.556. Rem. 434.494. 11. A dealer sold a lot of bacon for $2425.10^, which cost him $2177.4';^ ; wliat did he gain by the trade ? 156 PROMISCUOUS PROBLEMS. S200 12. Find the difference in K. betvreeii ^A, and 27.5P. 18. Wasliiugton was born Feb. 22, 1732, and died Dec. 14, 1799 ; what was his age? Ans. OTyr. 9mo. 22da. 14. A gave B 2501b. of beef at 8ct. for o?Ah. of leather at 60ct.; how does their account stand '/ Ans. B owes A 20ct. 15. What is the product of 7903 x 3907 ? 16. Multiply three eighths by two and two thirds. Prod. 1. 17. Find the value of 19.275 x 21.125. Yal. 407.184375. 18. A planter who works 47 hands raises to each hand ten bales of cotton averaging 4461b.; how much does his cotton jaeld him at 9ct. per lb.? 19. A farmer owns 4 farms containing each 47A. 25P.; what do the four contain ? ^ Ans. 188A. 2R. 20P. 20. A jeweller has 25 gold rings weighing S^-dwt. each; how many oz. do they all weigh ? Ans. 4|oz. 21. Bought 22gal. of molasses at 75ct., 2471b. of sugar at 16|ct., 1751b. of rice at 6}ct., and 57-^-lb. of coffee at 18 ^ct.; what was my bill ? 22. Find the value of 25000-4-125 + 1475^25. Yal. 259. 23. What is the quotient of three fourths by seventeen thirty-thirds? Ans. l|-t. 24. What is the value of 17.375-^2.5-9.63-^3.3 ? 25. In 392981b. of iiour, how many bbl.? Ans. 200-^bbl. 26. A father divided 778A. 3R. 2 IP. equally among his seven children ; what was each child's share ? Ans. IIIA. IR. 3P. 27. What is the cost of 273bu. of wheat at $1.66f per bn.? 28. What is the cost of 29rra. of paper at $2.75 per rm.? Ans. $79.75. 29. What part of lOda. is 7hr. 15min.? Ans. ■.%%. 157 §209 PROMISCUOUS PROBLEM ij. 30. What part of 5gal. Iqt. is 3qt. Sgi.l 31. Bought I of 17T. 3qr. of iron at 5ct. per lb.; what did I pay ? Ans. $1363. 32. In 31b. Avoirdupois, how many oz. Troy ? Ans. 43.75oz. 33. What part of lib. Avoir, is lib. Troy? 34. How many cubic inches in 40qt. Wine Measure ? Ans. 2310cu.in. 35. What cost 3doz. Arithmetics at $2.75 per doz.jl7 slates at I4ct., 5 gross of steel pens at 93^}ct. per gross, and 300 slate pencils at 31|ct. per hundred ? 36. Bought 156bbl. of flour for $936, and sold the same at ^8.45 per bbl.; what did I gain ? 37. In Ibu., how many qt. Wine Measure ? Ans. 37.236qt. 38. How much heavier is a pound of feathers than a pound of gold ? Ans. 1240gr. 39. Virginia contains 6 I352sq.mi.; North Carolina, 55500; South Carolina, 28000; Georgia, 58000; Florida, 59268; Alabama, 50722 ; Mississippi,47151; Louisiana, 41346; Texas, 325520; Arkansas, 52198; Missouri, 65037; Tennessee, 44000; and Kentucky, 37680. What is the area of the Confederate States of America ? 40. In lib. Troy, how many oz. Avoir.? Ans. 13.0281 -foz. 41. What will 33J-yd. of cloth cost at $4.75 per yd.? Ans. $159,125. 42. What will 17bbl. of flour cost at $7,875 per bbl.? 43. What will 60bu. of wheat cost at $1,125 per bu.? 44. A man borrowed $189.75, and paid at one time $37,375, at an other $23,625, and at an other $19.4375 ; how much does he still owe ? Ans. $109.3125. 158 PROMISCUOUS PROBLEMS. §209 45. A lady bought a silk dress for $21,875, a lace mantle for $15.50, a pair of cloth gaiters for $3.25, and a bonnet for $9,625 ; what did they all cost ? 4Q. If 25yd. of cloth cost $85.50, what does 1yd. cost ? * Ans. $3.42. 47. What will 3651b. of flour cost at $4 per hundred ? Ans. $14.60. 48. How many working days are there in a common year ? 49. If a man receives $2000 a year, how much is that a day? Ans. $5,479. 50. Bought 5bu. at $1.37-^- per bu., and sold them at 5ct. per qt. Wine Measure ; how much did I gain ? Ans. $2,434 + . 51. How many steps of 28in. each, does a soldier take in marching 5 miles? o2. How many bottles containing Iqt. igi. each, can be filled from a hogshead of French Brandy ? Ans. 237.295 bottles. 53. If a familj^ use 15bb]. of flour in a year, how much is that a day ? Ans. 8.051b. 54. If a man travel 29mi. Tfur. I5rd. per day, how far will he travel in 5wk. if he rest on Sunday ? 55. A lady went shopping with £>^, and spent -} of 14s.; how much had she left 1 Ans. 6|s. 50. How many days are there from Jan. 17 to April 6 ? Ans. 79da. 57. Sold one load of hay weighing I.IT., an other weighing 1|^T., andathird weighing 17.3c wt.; what did the three loads weigh ? 58. Bought } of ^ of an acre in one lot, 49P. in an other, and -} of lOR. in an other ; what did the three lots cost at $69.6875 per A.? Ans. $91,029 + . 159 ^209 I'RO.MlbCUOUS PEOBLEMli. 5*>. ^V iiat wiil ^o.ailb. of corn cost at 40ct. per bu.ir Ads. $18.21. GO. What will 10001b. of wheat cost at $1.37 J- per bu.? <il. If Ibu. of wheat will make 451b. of flour, how many> bl>l. will 15001b. make ? Ans. 5.74bbl. C>'2. How many secouds were there in the winter of 1850—60? Ans. 7S62400sec. G3. How many minutes were there in the summer of 1860 ? G4. How many acres of land at $1 per sq.yd. can be bought for S15000 ? Ans. 3.099A. ()5. What will 200mi. of Tclegrapii wire cost at lOct. per yd.? An.s. $35200. 66. How many pounds of flour in 75bbl.? 67. What is the diiference in height between a man 5ft. llin. high, and a horse 16 hands high ? Ans. 7in. Os. Bought 101b. of rhubarb at $6.50 per lb. Avoir., and sold it at 50ct. pnr oz. Troy ; what did I gain ? Acs. 87.916. 69. What cost 2127ft. of lumber at $ .8375 per hundred ? 70. What cost 37560 bricks at 1^7.75 per thousand ? Ans. $291.09. 71. What cotit 17 firkins of butter at 18^ct. per lb.? Ans. $178.50. 72. What cost 5.5 stone of potatoes at 1.5ct. per lb.? 73. What decimal fraction is equal to 02^-^129 ? " Ann. .484496 + . 74. A man dying left $27000 to be divided so that his widow should have one third of it, each one of 4 sons one seventh of the remainder, and each one of 5 daughters one fifth of what was left ; what was each daughter's share ? Ans. $1542.857. 160 PROMISCUOUS PROBLEMS. §209 75. A druggist having bought 60gal. of oil for $97.50, lost 6.25gal. by leakage, and sold the remainder at $2,125 per gal.: what did he gain? 76. A merchant bought two bales of domestic, containing each 20 bolts of 38yd. at 13.25ct. per yd.: what did he pay ? Ans. $174.90. 77. Bought SOObbl. flour at $6.75 per bbl.; sold one third of it at $7,375 per bbl., one half of the remainder at $7.9375 per bbl., and the rest at $8.50 per bbl. : how much did I gain ? Ans. $356.25. 78. What will 2250bu. corn cost at |- of a dollar per bu.? 79. A gentleman's house cost him four times as much as his furniture, and both together cost $4435 ; what did his furniture cost ? Ans. $887. 80. A grocer had 7cwt. 3qr. of sugar, and sold at different times 3icwt., 3iqr., and 1271b.: how many lb. has he remaining ? Ans. 23l^lb. 81. A planter sold 15 bales of cotton averaging 445.51b. at 9ct. per lb., and with the proceeds bought land at $21.25 per A.: how much did he buy ? 82. If 4f yd. of cloth cost $12|, what will 1yd. cost ? 83. What cost 29A. IR. 18P. of land at $45,625 per A.? Ans. $1339.664 -f. 84. What cost 9T. IGcwt. 151b. of iron at $37.75 per T.? 85. A man left -} of his estate to his wife, -} of the remain- der to his son, and the remaining $2500 to his daugh- ter : what was his estate ? Ans. $7500. 86. If 16^ days' work cost $19.75, what will 3:^ days' work cost'/ Ans. $3.89 + . S7. What must I pay for 6:2lb. of butter at 35ct. per lb., , 12vdoz. eggs at 15ct. per doz., 10 chickens at 18fct. apiece, and 30 cucumbers at lOct. per doz.? K IGl §209 pROMiscuoiJS problems'. SS. If Ibbl. of tar cost $3,875, what will l7bM. cost ? Ans. $65,875. 89. What cost 2471b. dried blackberries, at 15ct. per lb.? Ans. $37.05.- 90. What will 1001b. of coffee cost at 61b. to the dollar? 91. How many dollars will pay for 15 pieces of French cal- ico^ each containing 27yd., at 1.2fr. per yd.? Ans. $90,396. '92. "Howm&ny dollars will pay for 75 gross of G-illott's pens at ^So 6d. per gross? Ans. $63,526, 93. What will 45bu. corn cost at 5}^ dimes per bu.? 94. What will 7271b. salt cost at $1.25 per bu.? Ans. $18,175. 95. What will Sbbl. flour cost at $.05 per lb.? Ans. $29.40. 96. How many bu. of corn can be hauled by a team which can haul just 60bu. of wheat t .97. John's height is 3 cubits and a span ; his pony is 14 hands high ; what is the difference of their heights ? Ans. 7in. 98. How many ft. of water is drawn by a vessel which can not sail in less than 3 fathoms 2 feet ? Ans. 20ft. 99. What wiiri280cu.ft. of wood cost at $1.75 per cord? 100. What should be paid for 570bu. of corn, at $2.50 per bbl.? Ans. $285. 101. A merchant bought 21 pieces of cloth, each containing 41 yards, for which he paid $1260 ; he sold the cloth at $1.75 per yd.: did he gain or lose by the bargain ? Ans. He gained $246.75. 102. A man receives } of his income, and finds it equal to $8724.16 : how much is his whole income ? 103. If 322 books cost $371.91, what will 248 books cost at the same rate 1 Ans. $286.44 162 KATIO. §215 EATIO. § 210. The ratio of one number to an other of the same denomination is the quotient of the second divided hy the first. Thus, the ratio of 3 to 12 is 4 ; the ratio of 5ft. to 15ft. is 3 ; the ratio of $17 to $8 is -V- § 211. Since the two numbers compared are necessarily of the same denomination, every ratio is an abstract num- ber. (§ 185.) § 212. Of two numbers compared, the first is called the antecedent, the second is called the consequent, and both to- fcether are called the terms of the ratio. Thus, in the first ratio above, 3 is the antecedent, 12 is the consequent, and 3 and 12 are the terms. §213. A ratio is usually denoted by a colon placed be- tween the two terms. Thus, 3 : 12 is the ratio of 3 to 12 ; so also, 5ft. : 15ft.=:3 ; $17 : $8=-jV- § 214, The ratio of two numbers ef the same nature but of different denominations may be found by first reducing them to the same denomination. Thus, 3ft. : 5yd. = 5; Oct. : $1 = 20. § 215. The ratio of two numbers of different natures can not be found. Thus, 3ft. has no ratio to Sgal. Ex. 1. What is the ratio of 3 to 6 ? Ans. 2. 2. What is the ratio of 10 to 75 ? Ans. 7.5. 3. What is the ratio of 27 to 9 ? 4. What is the ratio of 446 to 1338 ? Ans. 3. 5. What is the ratio of $97 to $485 ? Ans. 5. 6. What is the ratio of 27qt. to 9qt.? 7. What is the ratio of 3qt. to 5gal.? Ans. 6.6. 163 §216 PROPORTION. 8. What is the ratio of 7fur. to llmi.? Ans. 12.57142. 9. What is the ratio of ^£.5 to 15s.? 10. What is the ratio of -} to {-1 Ans. 1.75. 11. What is the ratio of 3.75 to 11.25 ? Ans. 3. 12. What is the ratio of 5:^ to 17^ ? 13. What is the ratio of 3ioz. to 1-Vlb. Aroir.? Ans. 6-^. 14. What is the ratio of 45min. to ihr.? Ans. -i^. 15. What is the ratio pi 1.25cu.ft. to 2.5cu.in.? 16. What is the ratio of f A. to 15P.? Ans. .125. 17. What is the ratio of Ihhd. to 25gal.? Ans. .3668 + . 18. What is the ratio of 1.5 cubits to 65 inches ? SIMPLE PEOPORTION. §216. A 2'>t'oportion is an equality of two ratios. Thus, the two ratios, 5 : 10, and Sin. : 6in., form a proportion. § 217. A proportion is denoted by a double colon between the two ratios, or by a sign of equality between them. — Thus, 5 : 10 :: 3in. : 6in., read, 5 is to 10 as Sin. is to 6in. Or, 5 : 10z=:3in. : 6in., read, the ratio of 5 to 10 is equal to the ratio of Sin. to 6in. § 218. The first two terms of a proportion are called the first couplet^ the last two are called the second couplet: — the first and third terms are called the antecedents; the second and fourth are called the consequents : — the first and fourth terms are called the extremes ; the second and third are called the means. Also, the fourth term is called a fourth proportional to the other three : and, when the means are equal to each other, either mean is called a mean proportional between the two extremes. 164 SIMPLE PROPORTION. §220 § 219. In every proportion, the product of the extremes is equal to the product of the means. For, if 3 : 6 :: 7 : 14, 6-3 = 14-7, or 2= V, multiply- ing both terms of the first fraction by 7, and both terms of the second by 3, we have ~~^ ^^, or 6 x 7=3 x 14. § 220. This property enables us to find any term of a proportion when the other terms are given. If the two means and either extreme are given, to find the other extreme, we divide the product of the means by the given extreme. If the two extremes and either mean are given, to find the other mean, we divide the product of the extremes by the given mean. Ex. 1. 1st Term : 6 :: 5 : 15, what is the first term '/ Ans. 2. 2. 7 : 2nd Term :: 14 : 70, what is the second term ? Ans. 35. o. 5-^- : 22 :: 3rd Term : 40, what is the third term ? ■ 4. 8 : 1.6 :: 50 : 4th Term, what is the fourth term? Ans. 10. 5. 15 : 1.875 :: 3rd Term : 5, what is the third term ? Ans. 40. 6. 2i : 2nd Term :: 7-} : 12 1-, what is the second term ? 7. 1st Term : 1a :: -| of a : 17, what is the first term ? Ans. ^. b. 29 : 2nd Term :: 17 : 49, what is the second term? Ans. 83-}-^. ^. lo : 18 :: 3rd Term : 24, what is the third term? 10. 2i : 31 :: 4^ : 4th Term, what is the fourth term ? Ans, 6.3. 165 §221 PROPORTION. § 221. Whichever term is required, however, the given terms may always be arranged in the first, second, and third places, so that the work shall consist in finding a fourth proportional to the three given terms. Thus, The first question would become, 15 : 5 :: G : 4th Term; The second, 14 : 70 :: 7 : 4th Term ; The third, 12 : 5^ :: 40 : 4th Term. In finding a fourth proportional to three concrete num- bers, the first two terms must be of the same denomination, and this common denomination must be canceled before the operation is performed! In speaking, hereafter, of the first qr the second term, we will always mean the number of ab- stract units in such term. Ex. 11. Find a fourth proportional to 3in., 7in., and $12. 3 : 7 :: $12 : p^2. Model.— Multiply the '- third term by the second. (§183). 3)$84 Divide the product by the first $28 term. (^185). The fourth term is $28. Explanation. — The necessity of considering the first two terms abstract, is evident from the fact that $12 can not be multiplied by Tin., neither can $84 be divided by Bin. liULE. — MuIfqyJj/ the third term hy the second, and divide the 2^'^oduct hy the first. Or, Multiply the third term hy the ratio of the first to the second. Ex. 12. Find a fourth proportional to 15yd., 25yd., and lOda. 13. Find a fourth proportional to $5, $75, and $40. 14. Find a fourth proportional to 7da., 15da., and £2, 63. 9d. 4th Term, £5, 2}d. 16G SIMPLE PROPORTION, §22o 15. Find a fourth proportional to 5|-qt., 14qt., and $17.75. 16. Find a fourtla proportional to 3pk., 2bu., and $7.50. 4th Term, S20. 17. Find a fourth proportional to 1,1^ 43, and 14s. 4th Term, 7s. 9d. S-^^qr, 18. Find a fourth proportional to 3f, 7f, and lOgal. Sqt, 19. Find a fourth proportional to lOgal., 3qt., and 5.5da. 4th Term, .4125da. 20. Find a fourth proportional to 3s., lOda., and 5yd. 4th Term, 1.3$yd. 21. If 15bu. of corn cost $10, what will 27bu. cost? 15 : 27 :: $10 : $18 27 §223. Model.— 110 is the %^1^ 15 third term. Since the cost of 27bu. is 2;reater than that of 120 -l$18 i5bu.^ 15 is the first term, and 27 the second. Multiply the third term by the second. (§ 183,) Divide the product by the first term. (§ 185.) Hence, 27bu. cost $18. Explanation. — The ratio of 15bu. to 27bu. is evidently the same as the ratio of the value of 15bu. to the value of 27bu., that is, the ratio of $10 to the required amount. Hence, the propriety of the proportion. The first and sec- ond terms are considered abstract from the necessity of the case. KuLE. — Take for the third term the given number ivhich is of the same nature with the required term : and, if the re- quired term is evidently greater than this third term, take the greater of the remaining terms for the second and the less for the first ; hut, if the required term' is less than the third term, take the less of the remaining terms for the second and the grtater for the first. Find a fourth proportional to the three terms thus arranged, 167 $223 PROPORTION. Ex. 22. If 101b. of sugar cost $1.25, what will 17.51b. cost? Ans. $2.1875. 23. If 27bbl. of flour cost $150.75, what will 94.5bbl. cost 'i Ans. $527,625. 24. What cost llgal. of molasses, if 49gal. cost $34.47? 25. What cost 3.75A. of land, if 16.375A. cost $400 ? Ans. $91.60 + . 26. If 12 horses eat a load of hay in 10 days, how long would a load last 5 horses ? Ans. 24da. 27. If 12 horses eat a load in 10 days, how many horses would eat it in 5 days ? 28. If 91b. of tea cost $10, what will 111b. cost ? 29. If a 6 penny loaf of bread weigh 5.5oz. when flour is $4.50 per bbl., what should it weigh when flour is $7 per bbl.? Ans. 3.535oz. 30. If -| of a yard of cloth cost $2.75, what will -^-^ of a yard cost? 31. What cost 1751b. of cofi'ee, at 5-i lb. for a dollar ? Ans. $31.81 + . 32. What cost 45 pr. of shoes, if 14 pr. cost $35.50 ? Ans. $114,107 + . 33. If $100 gain $6 in lyr., in how many years will it gain $100? 34. If 12 men build 20rd. of masonry in a week, how many men could build 75rd. in the same time ? Ans. 45 men. 35. If lObbl. of flour will last a company of soldiers ISda., how long will lOOOIb. last them ? Ans. 7.653da. 36. If I4lb. of butter cost $4.25, what will 1.75 firkins cost ? 37. If a boat travels 75mi. in 6hr., how far does it go in 25min.? Ans. 5mi. Ifur. 26.6rd. 38. If 87.51b. of coflfee cost M, 12g. 6d., what will 7.51b. cost? Ans. 7s. lid. 168 COMPOUND PROPORTION. §224 39. If a man can walk lOmi. in 3hr., how far can he walk in 5da. of 8hr. each ? 40. If 5.51b. of sugar cost $1.00, what will lib. cost? Ans. $ .18. 41. If 5ibu. of wheat make Ibbl. of flour, how much flour will 25001b. of wheat make ? Ans. 7.5'7bbl. 42. If l^gal. of molasses cost $1.29, what will l-^hhd. cost? 43. If lOijd. of cloth cost $11,625, what will 16|-yd. cost? Ans. $18,452 + . 44. Iff of a ship cost ^500, 7s. 3d., what will -j^^- of her co«t? Ans. £260, 12s. 1.3125d. 45. If 3 reams of paper cost $7.75, how many reams can be bought for $17.75 ? 46. How many lb. Avoir, are equal to 500 lb. Troy? 47. A grocer was detected in using as a gallon measure a vessel containing 3qt. Ipt. 2^gi. : how many true gal- lons were in 47.5 of his false gallons? A. 45.273 + gal. 48. The same man used for his purchases a vessel contain- ing 4qt. 2gi. : how many true gallons were in 47.5 of these false gallons ? 49. How many of his selling gallons were in 47.5 of his buying gallons? Ans. 52.95 + . 50. If 1001b. of gunpowder require 751b. of saltpetre, how much saltpetre will 22.251b. of gunpowder require ? COMPOUND PROPOKTION. § 224. A compound ratio is the product of two or more simple ratios. 3 ' 5 ) Thus, . '. ^, > :: 12 : 35, is a compound ratio. 169 §225 PROPORTION. § 225. A compound proportion is an expression of equal- ity between a compound and a simple ratio. § 226. A compound ratio is reduced to a simple one, by multiplying together its corresponding terms. Thus, in the above instance, the antecedent 12 is the product of the an- tecedents 3 and 4, and the consequent 35 is the product of the consequents 5 and 7. § 227. If the first three terms of a compound proportion are given, the fourth term may be found by multiplying the third term by the product of the second terms, and dividing this product by the product of the first terms : the first and second terms being reduced to the same denomination in each simple ratio, and then considered abstract. Ex. 1. If 5 hands can hoe 24A. of cotton in 4da., how many Acres can 17 hands hoe in llda.? ^ • ^^^ •• 24A • 224^ \ ^^^^- Model.— 24A. 4 : 11 ^ " * * "* 5- • is the third term. Since 20 137 17 hands can hoc more 2^^^ than 5 hands, 5 is a first 748 374 term, and- 17 a second. Since more land can be hoed in llda. than in 4 2,-0 )448,8A . (la^^ 4 jg ^ first term and 224|-A. 11 a second. 4 times 5 are 20. 11 times 17 are 187. Multiply the third term by 187. (§ 183.) Divide the product by 20. (§185.) Hence the required number is 224|A. Explanation.— As in simple proportion, we take for the third term that which is of the same nature with the re- quired term. The remaining numbers are in pairs of sim- ilar terms, and each pair is arranged ^s if the question de- pended upon it alone. 170 COMPOUND PROPORTION. §^8 Rule. — Take for the third term the given number which is of the same natvre with the required term. Take the re- maining numbers in pairs of the same nature, and arrange each pair as in simple proportion. Multiply the third term by the prodiict of the second termSy and divide this pj'oduct by the product of the first terms. Ex. 2. If 60 men can do a piece of work in 40 days of 8 hours each, how many men can do three times the work in 90 days of 10 hours each % Ans. 04 men. 3. If 10 horses eat 88bu. of corn in 45 days, how many horses will eat 120bu. in 50 days? 4. If I can travel 75 miles in 2ida. of 7-J-hr. each, how many da. of 10 lir. each would it take me to travel 225mi.? Ans. 5|da. 5. If 5 white men can do as much work as 7 negroes, how many days of lOhr. each will be required for 25 negroes to do a piece of work which 30 white men can do in 10 days of 9hr. each r* Ans. 15,12da: 6. If by traveling 7hr. per da. at 4nii. per hr. I go 280mi. in lOda., how far will I go in 12da. by traveling 8hr. per da. afc 4^mi. per hr.? 7. If 12 men build 9 rods of wall in 10 days, how many men can build 27 rods in 5 days ? Ans. 72 men. 8. If $100 gain SfjO in 12mo., what will ^375 gain in 9mo.? Alls. $16,875. 9. If 1001b. be carried lOOini. for 35ct., what will be the freight on 100001b. carried 75mi.? 10. If 9 men, in 10 days, can build a v.all 25rd. long, 3yd. higl), and 5ft. thick, in what time can 20 men build a wall 30rd. long, 4yd. high, and 4ft. thick ? A. 5.76da. 11. If 100 men, in 5da of lOlir. each, can dig a ditch 150 yd. long, .3yd. wide, and 5ft, dec;', how many men, in 171 ,^2^9 PROPORTION. 8da. of 9hr, each, can dig a ditch 200yd. long, S^-yd. wide, and 2yd. deep? A. omitting fraction, 123 men. 12. If $100 gain $7 in 12ino., in how many months will • $700 gain $100? 13. If $100 gain $8 in 12mo., what sum of money will gain $160 in ISmo. ? Ans. 1333.33i 14. If $900 gain $135 in 18mo., what will $100 gain in 12mo. ? Ans. $10. 15. If 6 school-girls spend $72 pocket-money in 4wk., what will 10 girls spend in 6wk. ? 16. If 900 soldiers eat 70bbl. of flour in 20da., how many days will 200bbl. last 3000 soldiers at half rations ? Ans. 34-2-da. PARTITIVE PROPORTION ; OR, FELLOWSHIP. § 229. Partitive Proportion is the division of a num- ber into two or more parts which shall have to each other a given ratio. The terms of the ratio are called the proportional terms, and in the operation they must be regarded abstract. Ex. 1. Divide 450 into three parts, which shall be to each other as 2, 3, and 4. 9 : 2 :: 450 : 100 § 230. Model.--2 and 3 are 5, 9:3:: 450 : 150 and 4 are 9. 9 is to 2 as 450 is 9 : 4 :: 450 : 200 to the first part. Multiply 450 by 2. Divide the product by 9. The first part is 100. 9 is to 3 as 450 is to the second part. Multiply 450 by 3. Divide the product by 9. The second part is 150. 9 is to 4 as 450 is to the third part. Multiply 450 by 4. Divide the product by 9. The third part is 200. Hence, 100, 150, and 200, are the three parts required. 172 TARTITIYE rROPORTION. §230 Explanation. — One half of the first part is evidently equal to one third of the second or one fourth of the third : so that if the whole number be divided into 9 equal parts, the first will contain 2, the second 3, and the third 4, of those parts. Hence the truth of the proportions. Rule. — As the sum of the proportional terms is to cither term, so is the whole number to be divided to the part corre- sponding to that term. Proof. — Add the several parts together : their sum is equal to the whole number divided. Ex. 2. Divide $1000 into three parts which shall be to each other as, 6, 1, and 3. $600, $100, and $300- 3. Divide 141A. into three parts in the proportion of 1^, 3i, and 4}. 4. Divide 226gal. into four parts in the proportion of 3-^., 2-|-, 2i,and 2-^-. 42gal., 56gal., GOgal., and 68gal. 5. Divide 992.95 into four parts in the proportion of 1.25, 3.2, 4.73, and 5.005. 87.5, 224, 331.1, 350.35. 6. Divide $43.20 into four parts in the proportion of 1, 3, 5, and 7. 7. Two men, A and B, engage in business with a joint capital of $6000, of which A furnishes $2500 and B the remainder. What is each one's share of a gain of $1200 ? Ans. $500, $700. 8. Two men, C and I), gain $1275 on a capital of which C's share is double D's ; what is each one's share of this gain? Ans. $850, $425. 9. If E invests $2375, and F $3225, and the firm loses $700, how much must each partner lose ? 10. A invests $3000, and B 83500, in a certain business, in which the first year they lose $325. After paying 178 ^230 PROPORTION. this loss from the funds of the firm, they take in C us a partner with a capital of $4000, and the second year the new firm gains ^2035. Hov/ much of this gain is due to each partner ? Ans. A, $570; B, $665; C, 1800. 11. D and E form a partnership for two years, I) contribut- ing $5000, and E $1750. The first year they gain $1350. D spends his share of the gain, but E leaves his share among the funds of the firm. The second year they gain $2130. What is each partner's share of this last gain '/ Ans. D, $1500 ; E, $630. 12. Messrs. Jones, Smith, and Brown gained $5000; what is each man's share of this gain, if Smith owns twice as much of the capital as Brown, and Jones as much as Smitli* and Brown together ? 13. In a certain firm, A owns 1^^, times as much stock as B, C owns 1^- times as much as A, and D owns 1-^ tii^ea as much as C: A's gain on a year's transactions is 8500 ; what is the gain of each of the other partners ? Ans. B, $400 ; C, $600; D, $700. 14. A merchant owes one creditor $2000, and an other $3500 : having failed, he can pay them both only $4015 : how much should* each creditor receive 1 Ans. $2555, and $1460. 15. A man dying wills to one son $2000, to an other son ' $1500, and to his daughter $1250 ; but after paying his debts his executor has in his hands only $3000. How much should he pay to each legatee ? 16. Three partners, A, B, and C, invest as follows : — A in- vests $500 for 2 months ; B, $400 for 3 months ; and C, $300 for 5 months. They gain $740. What ought each partner to receive ? 174 PARTITIVE PROPORTION. §;:;U 500 X 2 == 1000 8700 : 1000 400 X 3 rr: 1200 8700 : 1200 $740 : $200 $740 : $240 $740 ; $300 .300 X 5^:1500 8700 : 1500 3700 §231. Model.— Twice 500 are 1000. 3 times 400 nrc 1200. 5 times 300 are 1500. The sum of tbese propor- tional parts is 3700. [Proceed as in § 230.] Hence A ought to receiye $200, B, $240, and C, ^300. Explanation. — A's investment of $500 for 2 months is equal to an investment of twice $500, or $1000, for 1 month ; so B's $400 for 3 months is equal to 3 times $400, or $1200, for 1 month ; and C's ^300 for 5 months is equal to 5 times $300, or $1500, for 1 month. The several investments, being thus referred to the same unit of time, evidently fur- nish equitjxble proportional terms. This is an example of what is called Compound Fellow- SHIV. Ex. 17. A firm of two partners gained $1750 : what was each partner's share of the gain, if A contributed $3000 for 10 months, and B $2500 for 1 year ? Ans. A's, $875 ; B's, $875. 18. A, B, C, and D, rented a pasture for $100. A kept 20 head of cattle in it 6 months; B kept 25 head 5 months; C kept 30 head 5^- months ; and D kept 50 head 3 months. What part of the rent ought each man to pay ? 19. In a certain partnership A contributed $3000 Jan. 1st; B contributed $2500 Feb. 1st; and C contributed $4000 May 1st. On the 1st of August, they lost by fire $4000. What part of the loss did each partner sustain 1 Ans. A, $1750 ; B, $1250 ; C, $1000. 20. Three partners in trade gained $3008 after 15 months' business. A put in $1000 at first, and $2000 3 months 175 §232 PROMISCDOUS PROBLEMS. afterwards ; B put in at first $4000, but took out $2000 6 months afterwards ; and C put in $2000 at the end of 5 months, and $2000 5 months afterwards. What was each partner's share of the gain ? Ans. A's, $1092 ; B's, $1176; C's, $740. ■ « »> PROMISCUOUS PPtOBLEMS. 8bu. 5 4 45 J. If 8bu. of wheat are worth as much as 15bu. of corn, and 5bu. of corn as much as 2cwt. of hay, and 4cwt. of hay are. worth $6; how many bu, of wheat can be bought for S45 ? 15 45.4.5.8bu.=7200bu. ^ 232., Model.— 2 6.2.15r=180 Set 8bu. on the left, 6 7200bu.^l80 = 40bu. and 15 on the right; 5 on the left, and 2 on the right ; 4 on the left, and 6 on the right, and 45 on the left. Multiply together the terms on the left. Multiply together the terms on the right. Divide 7200bu. by 180. The quo- tient is 40bu. Hence $45 will pay for 40bu. of wheat, Explanation. — This question, commonly referred to a distinct head, called Conjoined Proportion, or the Chain Rule, is merely a complicated case of simple proportion, as will be seen by stating it thus : — 1. If 4cwt. hay cost $6, how many cwt. will cost $45 1 6 : 45 :: 4cwt. : 30cwt. Hence, SOcwt. hay =$45. 2. If 2cwt. hay=z5bu. corn, how many bu. corn = 80cwt. hay? 2 : 30 :: 5bu. : 75bu. Hence, 75bu. corn=$45. 3. If Sbu. wheat=15bu. corn, how many bu. wheat=75bu. corn? 15 : 75 :: Sbu. : 40bu. Hence, 40bu. wheat=$45. 176 CONJOINED PROPORTION. §232 Comparing this work with the model, we see that the me<ans in the proportions are 45, 4, (30,) 5, (75,) and 8 ; and the extremes, except the last, are 6, (30,) 2, (75,) and 15 ; and that, omitting the two terms, 30 and 75, common to both, we have left in the one case the terms on the left in the model, and in the other the terms on the right. And since the product of the means ia equal to the product of the extremes, the product of the terms on the left divided by the product of the terms on the right will give the re- «^uired term. The term similar to the required term is called the odd terra, and the one equivalent to the required term is called the term of demand. Both of these must be placed on the left of the vertical line, and the other terms must be arrang- ed so that equivalents shall be opposite each other, and no two similar terms on the same side. In the operation, all but the odd^term must be regarded abstract. t. If 2bbl. of flour are worth as much as 26fbu. of corn, and 3bu, of corn as much as 7-^-lb. of bacon, how many lb. of bacon are equivalent to 3bbl. of flour? Ans. 1001b. S. If £93 are cqurJ to 2420fr., and 166|fr. are equal to $31, and 87 are C(]ual to 4bu. of wheat, how many bu. of wheat are equal to j£15 ? 4. If A can do as much work in 5 days as B can do in 6, B can do as much in 7 days as C can do in 8, and C can do as much in 9 days as D can do in 10 ; in how many .days can A do as much as D can do in 15 ? Ans. 9.84da. 5. If 10 Ells Flemish are equal to 6 Ells English, and 4 Ells English to 5 yards, and 12 yards to 8 Ells French ; L 177 §282 PROMISCUOUS PROBLEMS. how many Ells French are equal to 16 Ells Flemish? Ans. 8 E. Fr. 6. If 191b. of butter are worth 301b. of cheese, and 191b. of cheese are worth $3 ; how raacy lb. of butter are worth S7.50 ? 7. If a train of cars travel a mile in 2.5min., how long will it be in going 45 miles ? Ans. Ihr. 52.5min. $. If 8 men can mow a meadow in 10 days of I3hr. each, in how many days of llhr. each can 12 men mow it? Ans. 17.'72da. 9. A, B, and C formed a partnership for two years : th« 'first year, they lost $500 ; the second year, they gained $750 ; how much is each partner entitled to at the end of the second year, if A contributed $4000, and B and C $3000 each to the funds of the firm ? 10. A and B formed a partnership for two years from Jan. 1, 1860. On that day, A contributed $1000, and B $500 : July I, 1860, A added $500 to his investment, and Oct. 1, 1860, B added $500 to his : Jan. 1, 1861, A withdrew $250 from the funds of the firm, and Mar. 1, 1861, B contributed $500 more. They gain $1090. How much of this ought each partner to receive ? Ans. A, $600 ; B, $490. 11. If 5bbl. of cider are worth 8bu. of wheat, and 11 bu. of wheat are worth 2T. of coal, and 3T. of coal are worth 501b. of tea, and 41b. of tea are worth 5oz. of quinine, and 7oz. of quinine are worth $6.50; how many dol- lars are lObbl. of cider worth ? Ans. $56,277. 12. If the transportation of 1001b. lOOmi. cost $2.15, what will it cost to transport 25001b. 25mi.? 13. What is the smallest numbeu. that can be exactly di- vided by either .12, 13, or 14 I Ans. 1092. 178 PROMISCUOUS PROBLEMS. §232 14. What is the largest number that will exactly divide either 240, 720, or 840 1 Ans. 120. 1 '). What is the total cost of 4yd. of silk at $1,875 per yd., Syd.'of berege at ^.625 per yd., 3doz. buttons at $.75 per doz., and 7.5yd. of calico at $.25 per yd.? 16. What is the total cost of 51b. of tea at 5s. 6d., 7bu. of corn at 4s. 4d., 8bu. of wheat at lis. 9d., and llgal. of molasses at 7s. 3d.? Ans. j^ll, Us. 7d. 17. What interval elapsed between Dec. 5, 1813, and Mar. 17, 1842 ? Ans. 28yr. 3mo. 12da. 18. What interval elapsed between Jan. 30, 1833, and Sept. 3, 1862 ? 19. John Jones was born Mar. 9, 1827, and was married when he was 22yr. 6mo. lOda, old ; when was he mar- ried ? Ans. Sept. 28, 1849. 20. If 5 men can plough 47-^1 acres in 7|- days, in how many days can G men plough 31 acres ? Ans. 4 days. 21. If o^cwt. of hemp cost ^27.50, how much hemp will cost $33.33i ? 22. A merchant bought SOOObu. of salt : after having sold to A lOO.Sbu., to B 477.75bu., to C 329.8375bu., and to D 1200. 25bu., bow much has he left ? Ans. 891.6625bu. 23. What is the produce of 15.375A. of corn, at 8bbl. 3bu. 3pk. to the acre ? Ans. 134bbl. 2bu. 2pk. 5qt. 24. A field of 25 acres produced 637.5bu. of wheat ; how much was that per acre ? 25. A sum of money divided equally among 17 men gives to each ^17.765 ; if divided equally among 11 men, how much would each get? Ans. S27.455. 26. If 20 men in 35da. earn $320, how many men will earn $480 in 70da.? Ans. 15 men. 179 §233 PROPORTION. 27. If 18 horses eat lObu. of oats in 20(]a., how many horses will eat 60bu. in 36cla.? 28. What is the greatest common measure of 75, 825, and 1575 ? Ans. 75. 29. What is the least common multiple of 46, 230, and 115? Ans. 230. 30. What are the different prime factors of 24400 ? PERCENTAGE. §233. Percentage includes :''• cases of proportion in whichthe first term is one liund. ■■'. The phrase, »er centum, that is, ptr hundred, is usually written, and often pronounced, ^er cent. Thus, in stead of "six dollars per hundred," we usually say "6 per cent." Ex. 1. A lawyer collected $3725 ; what is his commis- sion at 3 per cent.? 100 : 3725 :: $3 : $112.75 This proportion is evi- dently correct : and all similar problems may be solved in the same manner. But, inasmuch as the three given terms have always the same unit, the same result will be obtained by regarding the sec- ond term concrete and the first and third abstract, by di- viding the third by the first and multiplying the second by this quotient. This method, being a little less troublesome, is the one usually adopted. To explain it more fully, we must give the following definitions. §234. The price or amount per hundred is called the rate per cent. Thus, in the above example, 3 is the rate per cent. 180 PERCENTAGE. §238 § 235. If the rate per cent, be divided by 100, the quo- tient is called the rate per unit. In all operations, this is regarded as an ab.'^tract number. Thus, .03 is the rate per unit in the example above. What is the rate per unit for 6 per cent.? for 10 per cent.? for 50 per cent.? for 75 per cent.? for li per cent.] for -J- per cent.? for 100 per cent.? for 33^^ per cent.? for \2\ per cent.? for 18 ^^ per cent.? for \ per cent ? for -^ per cent.? § 236. The number on which percentage is calculated, is called the basis of percentage. Thus, above, $3725 is the basis. § 237. The result of the operation is called the percent' age. Thus, above, $112.75 is the percentage. Ex. 2. Find 5 per cent, of $5750. $5750 § 238. Model.— Multiply the basis by the 1^ rate per unit. (§ 183.) The product is ,^287.- 8287.50 50. Hence the percentage is ^287.50. Explanation. — 5 per cent, of any number is evidently 5 hundredths of that number, and this is found by multi- plying the number by .05. Observe that the rate per unit is simply one hundredth of the rate per cent., and is most conveniently expressed as a decimal fraction. KuLE. — Multiply the basis by the rate per unit. The product vnll be the percentage, Ex. 3. What is 1 per cent, of 7500 ? 4. What 5. What 6. What 7. What 8. What 9. What s 2 per cent, of 250 ? Ans. 5. s 3 per cent, of 275 ? Ans. 8.25. s 4 per cent, of 775 ? s 6 per cent, of $325 ? Ans. $19.50. s 7 per cent, of 89250 ? Ans. $647.50. s 8 per cent, of 725 men? 181 §239 PROPORTION. 10. What is 9 per cent, of 1700 men ? Ans. 153 men. 11. What is -,V per cent, of $1000 ? Ans. 61. 12. What is 1 J- per cent, of 8175 ? 13. What is 2^^ per cent, of 827.75 ? Ans. S.7284375. 14. What is 31 per cent, of 8630 ? Ans. 821. 15. What is 4f per cent, of 795 ? 16. What is 7.} per cent, of 2775.25 ? Ans. 208.14375. 17. What is 9a per cent, of 473.75 ? Ans. 46.190625. 18. What is lO-j-^ per cent, of 275 ? 19. What is 16f per ce^r. of 1500 ? Ans. 250. 20. What is 66f per cent, of 8750 ? Ans. $500. 21. What per cent, of 690 is 115 ? § 239. Model. — Divide the percent- age by the basis. (§ 52.) The quotient is . 16|. IMultiply this quotient by 100. The product is 16|. -Hence, 115 is 23o' I- 1C| per cent, of 690. l690!3 ExPLA-NATiON. — Since the percentage is equal to the basis multiplied by the rate per unit, conversely the rate per unit is equal to the percentage divided by the basis. And, since the rate per unit is one hundredth of the rate per cent., con- versely the rate per cent, is found by multiplying the rate per unit by 100. KuLE, — Divide the percentage by the basis. Tlie quo- tient will be the rate per writ. Midtiplij the rate per unit by 100. The product ivill be the rate per cent. Ex. 22. What'per cent, of 700 is 70 ? Ans. 10 per cent. 23. What per cent, of 375 is 125!;' Ans. 33} per cent. 24. What per cent, of 1000 is 125 ? 25. What per cent, of 550 is 110? Ans. 20 per cent. 182 115.00 690 690 .16-1 4600 100 4140 16|- I>EaC£NTACJE. §239 26. What per cent, of S675 is $:^37.59 ? Ans. 50 per cent. 27. What per cent, of SIOOO is ^875 ? 28. What per cent, of $5000 is ^250 ? Ans. 5 per cent. 29. What per cent, of $10000 is S50 ? Ans. I per cent. 30. What per cent, of $150 is $300 ? 31. What per cent, of 3 JOO is 4000 ? Ans. 133a- per cent. 32. What per cent, of 275 is 302.5 ? x\.ns. 110 per cent, 33. What per cent, of 245 is 735 ? 34. What per cent, of 200 is 500? Ans. 250 per cent. 35. What per cent, of 325 is 2925 ? Ans. 900 per cent. 36. What per cent, of 81.25 is $1.50 ? 37. What per cent, of $1.00 is $.375 ? Ans. 37^ per cent. 38. What per cent, of $.875 is $.50 ? Ans. 57^ per cent. S9. AVhat per cent, of $.66f is $.22| ? 40. What per cent, of $.125 is $.0625 ? Ans. 50 per cent. 41. A commission merchant purchases articles amounting to $247.75 ; what is his commission, at 2^- per cent.? 42. What is the commission on $312, at 12 per cent.? 43. A merchant insured a vessel and cargo, valued at $75000, at 7f per cent. ; what did he pay ? 44. What premium must I pay for the insurance of hlj lifo5*the policy being $5000, and the rate 2.o5 per cent.? Ans. $117.50. 45. What is the premium for insuring $9450, atf per cent.? 46. What is the insurance on a dwelling and furniture val- ued at $25550, at 1,} per cent.? Ans $319,375-. 47. What is the duty, at 40 per cent., on French broafll- cloths valued at $15375 : Ans. $6150. 48. What is the duty, at 20 per cent., on $6250 worth of Italian silk ? 4.9. At 7,} per cent., what is the duty on an invoice of Ge- neva watches, valued at $7475? Ans. $560,635. 183 §239 PROPORTION. 50. At 50 per cent., what is the duty on a ease of Leghorn hats, worth $1500 ? Ans. S750. 51. What tax should be paid on S17725 worth of real es- tate, at i per cent.? 52. What is the tax on $261000, at .15 per cent.? Anp. $391.50. 53. What is the tax on $17150, at 60ct. on $100? Ans. $102.90. 54. What is the amount of a dividend of 3 per cent., on $4200 of bank stock ? 55. The North Carolina Bailroad company declared a div- idend of 2|- per cent. : what did I receive on 14 shares of $100 each ? Ans. $35. 56. A merchant bought broadcloth at $3.50 per yard; at what price must he sell it, to gain 40 per cent.? Ans. $4.90. The percentage must he added to the basis. 57. A grocer bought candles at 25ct. per lb.: how must he sell them, to gain 30 per' cent.? 58. If broadcloth cost $4.00 per yd., how much will it bring at a loss of 35 per cent.? Ans. $2.60. The percentage must he suhtracted from the basis. 59. A dealer bought 50bbl. of flour at $12 per bbl., but.was forced to sell it at a decline of 20 per cent. : what did he get for it all ? Ans. $480. 60. A speculator bought $35000 worth of cotton, and sold it at a loss of 15 per cent. : what did he receive for it ? 61. A man pays $406.25 for the insurance of his dwelling, valued at $32500 : what is the rate per cent.? Ans. 1-^: per cent. 62. A vessel worth -SI 5400 was insured for $539 ; what was the rate per cent.? Ans. 3-^- per cent. 184 PERCENTAGE. §239 63. At what rate per cent, will the insurance on $11500 cost $172.50 ? 64. A man bad his life insured for $277.50': what was the rate per cent., the policy being $10000 ? Ans. 2.775 per cent. 65. If the duty on ^3457 worth of goods is $1037.10, what is the rate per cent.? Ans. oO per cent. 66. What is the rate of duty, when $12657 worth of cloth- ing pays $6828.50 ? 67. If $15000 worth of property pays a tax of $229.50, what is the tax on $100 ? Ans. $ .51. 66. I paid a hroker $21,125 for investing $8450 in Govern- ment stocks; what was his rate of brokerage? Ans. \: per cent. 69. My attorney charged me $260.73f for collecting $3476. 50 : what was his rate of commission ? 70. I bought a farm for $4000, and sold it for $5000 ; what did I gain per cent.? Ans. 25 per cent. The Jirst cost subtracted /roin the selling price leaves the total gain. 71. I bought a farm for $5000, and sold it for $4000 ; what did I lose per cent.? Ans. 20 per cent. The selling 'price subtracted from the jirst cost leaves the total loss. Observe that in each case the first cost is the basis. Hence the difference in the ansiucrs of the last two questions. 72. If I buy calico at lOct., and sell it at 12^-ct., what do I gain per cent.? 73. If I buy calico at 12^ct., and sell it at 15ct., what do I gain per cent.? Ans. 20 per cent. 74. A man bought a house for $7625, and sold it for $8387. 50; what did he gain per cent.? x\ns. 10 per cent. 1S5 5; 240 PR ^PORTION. 75. A man having paid S7625 for his house, was compelled to !^eH it for $6862.50 : how much per cent, did he lose ? 70. By selling an article for ^1300, I gain 30 per cent, on it : what did it cost me ? $1300.00j 1.30 ^,240. Model.— Divide the sell- ■^^^ i SlOOO ing price by 1 + the rate per unit. OUOU (§ 165.) . The quotient §1000 is the first cost. Explanation. — It is evident that l + thc gain per unit : 1 :: the first cost + the whole gain : the first cost. But the selling price, §1800j is evidently the first cost + the whole gain. Then since the second term of the proportion is always 1, it is easy to see the truth of the Rule. — To find the first cost, lohen the selling price and the rate 'per cent, o/ gain are given. Divide the selling price hy 1 -j- the rate per unit. The quotient loill be the first cost. Ex. 77. By selling a piece of muslin for ^50, I gain 100 per cent. ; what did I give for it ? Ans. $25. 78. What did I pay for eggs, if I gain 33^^ per cent, by sell- ing them for IGct. per doz.? 79. A grocer sold a lot of sugar for ^1058, gaining thereby 15 per cent. : what did the sugar cost him ? A. §920. 80. A merchant sells some flour for $924, and gains 12 per cent, on it : what did he pay for it? Ans. $825. 81. The selling price is S1800, the gain 20 per cent.: what is the first cost ? 82. A merchant sold a quantity of cloth for S1410, and thus sustained a loss of 6 per cent. : what did the cloth cost him ? 186 PERCENTAGE. §'241 94 '^'50(3 §241. Model. — Divide the selling 470 price by 1 — the rate per unit. (§165.) ^>7Q The quotient $1500 is the first cost. ~"~m) Explanation. — Evidently, 1 — the loss per unit : 1 :: tiic first cost — the whole loss : the first cost. But the first cost — the whole loss is evidently the selling price, ^1410. The second term of this proportion is always 1, and hence the following EuLE. — To Jin J the Ji rat coiif, when the sellhig price and the. rrtfe j)er cent, of loss are given. Divide the selling pric^ hj/ \—thc rate per unit. The quotient will he the fir.st co?t. Ex. 83. The selling price is ?$8000 ; the loss 20 percent.: what is the first cost ? Ans. $10000. '<4. By selling flour at $12.25 per bbl.,I lost 12^ per cent, of wliat it cost me ; what did it cost? ST). I remitted ^3150 to my commission merchants to lay out in groceries after retaining 5 per cent, of what he spent: how much did he spend for me? Ans. ^3000. This is precisely similar in principle to the foregoing. 86. How much sugar at lOct. per lb. can I get by remit- ting $864,871- to a merchant who cliarges 5 per cent. commission 1' Ans. 34751b. S7. How much stock at ^5 per cent, advance can I buy for S1265 ? .S8. How much stock at 15 per cent, below par can I buy farC935? Ans. §1100. .Si). A father settled his sou with property worth SIOOOO : the first year he lost 20 per cent, of it, and the second year he gained 25 per cent, of what he had left ; how much had he then 't Ans. ^lOOOO. 187 ^242 PB.OPORTION — PERCENTAGE. 90. A merchant sold some sugar for 31402.50, and lost thereby 15 per cent. Wh.it did it cost him? 91. How much stock at :i discount of 3^- per cent, caii I. bought for f-5790 ? Aus. S6000. SIMPLE INTEREST. §242. Interest is the prict paid bj the borrower for the use of money loaned. § 243. The sum of money on which interest is calculated is called the principal. § 244. The sum of the principal and interest is called the amount. §245. The price paid for the use of one hundred doUarpi one year is called the rale per cent, per annum. Ex. 1. What is the interest of ^875, for 2yr. lOmo. 20da.. at 7 per cent, per annum ? ^375 2yr. lOmo. 20da. 7n. c. .07 ' § 246. Model.— i26^=lyr. Multiply^ the priu. 2 ...... iU.wK.. cinai by tlie rale per unit. (§!«3.) The ^52.50=2yr. product, S26.25, is 13.125=:6mo. -j tj,e interest for i 6.562=3mo. -lOmo. y^,,^^ Multiply the 2.187= Imo. j interest for lyr. by •^■o^']"'^^^- \ 20da. 2. The product i^ .364= 5da.. | !$52.50, the interest $75.831 = 2yr. lOmo. 20dn. for 2yr. 6mo. is one half of lyr. Divide the interest for lyr. by 2. The quotient is $13,125, the interest for 6mo. 3mo. is one half of 6mo. Divide the in- terest for 6mo. by 2. The quotient li 36,562, the iuterest 188 SIMPLE INTEREST. §246 for oiiiO. Imo. is one third of 3ino. Divide the iuterest for Smo. by 3. The quotient is 82.187, the interest for 1 mo. loda. is one half of Imo. Divide the interest for 1 mo. by 2. The quotient is SI. 098, the interest f\;r 15da. 5da. is one third of 15da. Divide the iuterest for 15da. by o. The quotient is S .oG4, the inleresi for 5da. Add the partial interests together. The sum is S75.831, the inter- im est for the whole time. ExPLA-NATiOM. — Since the rate is 7 percent, per annum, the interest of the given principal for 1 year is found by multiplying the principal by the rate per unit. Thus far the work is simple percentage. Far longer or shorter pe- riods of time the interest is proportional to the time : hence we take such aliquot parts of the interest for 1 year, &c., as the periods in question severally require. In the calculation of interest, a month is considered equal ■ 30 days, and a year to 360 days. , Rule. — Multij^ly the principal hi/ the rate per unit. The product will he the inf.erefi.t for 1 year. Alidtiplij the 'interest for 1 year hy ihe number of years , and take aliquot parts for periods of time less than a year. To ftriii the amount, add the interest to the principal. Ex. 2. What is the interest of 850 for 2yr. at 6 per cent. per annum ? Ans. ^G. 3. What is the interest of S75 for 6mo. at 7 per cent, per annum 't •i. What is the amount of SlOO for 9mo. at 8 per cent, peu annum ? Ans. ^106. 5. What is the interest of 3125 for 3yr. at 9 per cent, per annum '/ Ans. $33.75. G. What is the iuterest of %22b for 2yr. Gmo. at 10 per cent, per annum ? 189 ^24G riioroiiTiON — rELCEM..^ui:. 7. What IS tlie iiitere.st of ^150!75 for lyr. omo. at 5 per cent, per annum 't Ans. $10.67. 8. What is the amount of 3176.50 for 2yr. 9mo. at G per cent, per annum ? Ans. S204.45. 9. What is the interest of s^305.50 for %r. 5mo. 15<3a. at 6 per cent, per :iiinnni? 10. What is the interest of S574.05 for -^jr. 7iiio. 25da. at 5 per cent, per annum? Ans, 81oS.75. 11. What is the intere^.t of 8615.49 for fiyr. llmo. 22da. at 7 per cent, per annum ? Ans. $257.47. 12. Find the amount of ?-777.75 for !>yr. 2mo. 20da. at 8 per cent., per annum. 13. Find the interest of ^'1225 for ojr. 5n)0. oda. at 5 per cent, per annum. Ans. ?332.G2. 14. Find the intere.s: of S1525.25 for lyr. 2mo. at 8 per cent, per annum. Ans. S 142.356. 15. Find the interest of 62790 for 23'r. 7mo. at 9 per ccrit. per annum. 16. Find the amount of $1724.25 for 12yr. Omo. at 8 per cent, per annum. Ans. ^'3448.50. 17. Find the interest of $3500 for 7yr. omo. IGda. at 7 per cent, per annum. Ans. 81783. 0^'5. 18. Find the interest of -^4275 for 16yr. Smo. at 6 per cent, per annum. 19. Find the interest of S5550 for 15yi'. llmo. 27da. at per cent, per annum. Ana. v?79S7.83{,. 20. Find the amount of $2c;95 for tS) r. 4mo. at 12 per cent. per annum. x\ns. $5990. 21. Find the interest of $3827 for 17yr. 3mo. 15da. at 10 per cent, per annum. Ans. $6617.52. 190 SIMPLE INTi-UFST. §247 CONCISE METHOD FOR 6 PER CJ^NT. PER AKNU.\L Ex. 22. What is the interest of SMT.OO fur 2yr. Cmc^ IS da. at G per cent, per annum ? 2yr. 6mo. 18da. = 30.Grao. {^ 247. M(*DKr .— Rcdnre 200)30.6 ^247.50 ^^^ ^'^^" ^^"^^ ^'' nsonth^,. ~YFb 1 ^S (§1^^.) Divide tho number * h.'okI «^^ months by 20U. (^ 165. ) lOQ^'^A Multiply iho principal by l^d/^0 this quotient, (i^ 162.) The J_Z^iL product is ?P>7.8(1A, the in- $37.86750 terest required. Explanation. — Since the rate is 6 per cent, per annum, or for 12 mouths, one half of the number of montlis is the rate per cent, for any length of time : and this rate per cent, divided hy 100, gives the corresponding rat(3 per unit, by which the principal must be multiplied, to find the interest. For any other rate, we may find the interest at G per cent., and increase or diminish it, as the case may require. For instance, for 7 per cent., add to the interest found by this method, one sixth of its?lf : for 5 per cent., from the interest thus found subtract one sixth of itself; &c. Or, generally, find the interest at 6 per cent., divide it by 6, and multiply the quotient by the given rate. Rule. — Divide the number of monthly in the given time hy 200, and multiplij the jprincipal hy the quotient. The product ivill he the interest. Or, 3Iidtiply the number of years hy 6, and divide the product hy 100 ; Divide the number of months hy 2, and divide the quotient hy 100 .- Divide the number of days by 191 §248 PROPOllTION — PERCENTAGE. 6, and diclile (he quotient hy 1000 ; Add these three results fo<jctherj and multiply the principal hy their sum. After Jill din f/ the interest at 6 per cent., as above, to find the interest at any other rate ; Divide the interest at 6 per rent, hy G, and multiply the quotient hy the required rate. SECOND METHOD FOR 6 PER CENT. Ex. 23. Find the interest of 8275.75 for Syr. lOmo. 21 (la. at 6 per cent, per annum. Syr. lCmo.=46mo. §248, Model. — Reduce the 3)21 v^275.75 years and months to months. — — 7 .467 Divide the number of days by T AOAoT 3. Annex the quotient to the Ifi'S^^O number 01 months. Divide this ^^,^oAa icsult bv 1000. Multiply the principal by this quotient. Di- 2)812877525 ^.iae this product by 2. The $64.3876 quotient is sj61.38£, the inter-. est required. Explanation, — This method is evidently the same in principle as the preceding, and is preferable to the other only on account of its greater freedom from liability to fractions. Of course, the multiplier iu each of these lueth- •ods must be considered abstract. Rule. — To th6 number of months annex one third of the humber of day^s. Divide the number thus produced by 1000. Multiply one half of the principal hy this quotient. Or, Multiply the whole principal by this quotient, and divide the product by 2. The interest for any other rate may be found as in § 247. 192 SIMPLE INTEREST. §249 CONCISE METHOD FOR A^Y S1.4TE PER CENT. Ex. 24. Find the interest of ^^330. 60 for 6mo. 15da. at S per cent, per annum. 12,00)^ 3,60.60 "~$ .3005 g 5 § 249. jModel. — Divide the princi- pal by 1200. Multiply this quotient by 6.5. M'lUiply this product by 8. The product h $15.62|-, the required 15025 18030 ^l,95b25 interest. ^■15.62600 ExPLVNATLON. — The principal -^100=:the interest for 1 year at 1 per cent. This interest-^- I2=the interest for 1 month at 1 per cent. This last x 6. 5 = the interest for 6.5 months at 1 percent. And thtsx8 = the interest for 6.5 months at 8 per cent. Rule. — Divide the principal by 1200. Multiply the quo- tient by the wumber of months in fh'i given time, and this product by the rate per cent. This last product will be the interest.- Either of the above methods may be used in any case. *Ex. 2"'. Find the interest of i?l 3 19.50 for 9 days, at 6 per cent. |rr annum. 26. Find ilie interest of $36")8.75 for 17 days at 6 per cent. per nnaum. " Int. SI 0.366. 27. Find the interest of $5739.2,3 for 2mo. 24da. at 6 per cent, per annum. Int. $80,349. 28. Find the amount of $3738^.375 for 2mo. 6da. at 6 per cent, per annum. Amt. $38096.88. M 193 §249 , PROPORTION — PERCENTAGE. 29. Find the amount of ^1665.25 for lyr. llmo. 9da. at C per cent, per annum. Amt. S1859.25. SO. Find the interest of ^4336.30 for 4yr. 8mo. 13da. at 6 per cent, per annum. ?>1.' Find the interest of ^2758.50 from July 3, 1846, to -May 19; 1855, at 6 per cent, per annum. Int. $i469.36. To find the interval of time, the earliest date must be -iQrr r to J subtractcd from the latcst. In Ibooyr. omo, lyda, 1846 " 7 " 3 " • ^^^^ subtraction, the number ^ u 10 «' IQ <' ^^ emc]! month in the calendar is used, and each month is toaken as equal to 30 days. Ex. 32. Find the amount of $8140.75 from Dec. 9, 1847, to Apr. 27, 1855, at 6 per cent, per annum. Amt. §11747.10. 33. Find the interest of $3^219.15, from Apr. 8, 1850, to June 15, 1855, at 7 per cent, per annum. S4. Find the interest of $6813.45 from Mar. 5, 1855, to Oct. 8, 1862, at 8 per cent, per annum. Int. $4138.035. 35. Find the interest of $856.85 for 6yr. 8mo. 9da. at 8 per cent, per annum. Int. $458,699. 36. Find the amount 6f $742.40 from June 24, 1854, to Mar. 13, 1860, at 7 per cent, per annum. 37. Find the interest of $171.80 from July 29, 1857, to Sept. 1, 1861, at 10 per cent, per annum. Int. $70.24. 38. Find the interest of $670.70 from Apr. 7, 1859, to Oct. 13, 1862, at 9 per cent, per annum. Int. $212,276. ^9. Find the interest of $976.18 from Mar. 1, 1861, to Feb. 10, 1862, at 8i per cent, per annum. 194 SIMPLE INTEREST. §249 40. Find the interest of ^375.85 from Jan. 19, 1860, to Jan. 1, 18G2, at 11 per cent, per annum. Int. $80,619. 41. Find the amount of $6.89 from June 11, 1860, to June 1, 1862, at 9 per cent, per annum. Amt. ^8.11. 42. What is the interest of S89.96 for 2yr. 3mo. 16da.at 8 per cent, per annum ? 43. What is the interest of $325 for 6jr. 7mo. 27da. at 7-^ per cent, per annum ? Ans. $156.88. 44. What is the amount of $1728 from Dec. 29, 1859, to Oct. 9, 1852, at 10 per cent, per annum ? Ans. 152208. 45. W^hat is the interest of $160.08 from May 1, 1851, to Sept. 9, 1854, at 7 per cent, per annum? 46. What is the interest of $18.62 for 3yr. 18da. at 5 per cent, per annum ? Ans. $2,839. 47. What is the interest of i£17, 6s. 9d. for 18mo. at 6 per cent, per annum ? £17, 6s. 9d.=£17.3375 The principal must first be ^^ reduced to pounds, and then ^1.560375 the interest may be found by c£l. 56=^1, lis. 2^d. any one of the preceding methods. ]i]x. 48. What is the interest of ^6427, 18s. 9d. for 2 years at 5:} per cent, per annum ? 49. What is the amount of ,£1096, 15s. 6d. for 4 years at 6-1- per cent, per annum ? ^ Ans. ^61381, 18s. 8d. 50. What is the amount of £120, 10s. for 2yr. 6mo. at 4a per cent, per annum ? Ans. j£134, 36s. l|,d. 51. What is the interest of £270, 10s. >0d. for lyr. 4mo. 20 da. at 7 per cent, per annum ? 52. What is the interest of 1775fr. 75cent. for 3yr. 6mo. at 6 per cent, per annum 1 Ans. 372fr. 90cent. 195 §250 PROPORTION — PERCENTAGE. 53. What is the interest of 2070fr. 65cent. for 2yr. 8mo. 20da. at 7 per cent, per annum ? Ans. 394fr. STcent. 54. What is the amount of o29Tfr. 15cent. for oyr. 15da. at 8 per cent, per annum ? 55. What is the interest of 10720fr. 25cent. for 5yr. 7mo. lOda. at 5 per cent, per annum ? Ans. 3007. G2fr. 56. What is the amount of 20625fr. SOcent. for (5yr. 6mo. 6da. at 6 per cent, per annum 1 Ans. 28689. 79fr, PARTIAL PAYMENTS. The method here given is the one enjoined by the Su- preme Court of North Carolina;, and used iu most, if not all, the States of the Confederacy. § 250. Rule. — Find the amount of tlie given principal to tJie titne of the first p)ciymtnt^ arid if this pa 7/ men f is greater than the interest then due, subtract the payment from the a- mount. Consider the remainder as a second pr in cipaly and find the amount of it from the time of the first payment to the time of the second, and if the second payment is greater fhan the interest last found, subtract the second' payment from the second amount, and consider the remainder as a third principal: and so on. But if any payment is less than its corresponding inter- est, find the amount of the same principal to the time of tJie next payment, and if the sum of these two payments is greater than the interest then due, subtract their sum from the amount : but if the sum (fi the two payments is less than the interest then due, extend the time until the sum of the payments made shall exceed the interest due at the time of the last payment. 196 PARTIAL PAYMENTS. §250 The principle of tlie rule is that the payment of a part of the debt shall not increase the debt. Ex. 57. $725.50. Richmot^d, Va., Jan. 1, 1858. One day after date, I promise to pay J.- Jones, or order, seven hundred and twenty-five dollars and fifty cents, for value received. ^ (^../^ (g^ On this note were the following endorsements : Mar. 16, 1858, $100.00 May 16, 1859, 25.50 . July 1, 1861, 300.00 How much was due Oct. 8, 1862 ? SOLUTION. Original Principal, $725.50 Interest to Mar. 16, 1858,— 2ra. 15da., 9.068 Amount then due, ^734.568 Amount then paid, 100. Second Principal, $634,568 Interest from Mar. 16, 1858, to May 16, 1859, $44,419 Amount then paid (less than interest) 25.50 Interest from Mar. 16, 1858, to July 1, 1861, — 3y. 3m. 15d. 125.327 Amount then due, $759,895 Sum of the two payments, 325.50 ^ Third Principal, $434,395 Interest from July 1, 1861, to Oct. 8, 1862,— ly.'3m. 7d. 33.086 Amount due Oct. 8, 1862, $467,481 197 §250 PROPORTION PERCENTAGE. ')8. $3256.37. Lincolnton, N. C, Mar. 12, 1853. On demand I promise to pay to the order of J. Rein- liardtj three thousand two hundred and fifty-six dollars and thirty-seven cents, for value received. On this note were the following endorsements : July 12, 1855, received $654.33 Sept. 20, 1857, " $246.50 Jan. 5, 1859, " $945.87 What was the "balance due Sept. 7, 1860 ? Ans. $2755.41. 59. $108.43. Columbia, S. C, Dec. 9, 1857. With interest from date, for value received, I promise to pay J. Townsend or order oiic hundred and eight dollars and forty-three cents. ^^ C^^^/. ("^^ Endorsements. Mar. 3, 1858, received $50.04 ; Dec. 10, 1858,113.19; May 1, 1860, S?50.11. How much was due Oct. 9, 1862 ? Ans. 1^55.844 60. A note was given at Savannah, Geo., Apr, 16, 1856, for $450. On it the following endorsements were made : — Jan. 1, 1857, received ^20 ; Apr. 1, 1857, $14; July 16, 1857, %Z\ ; Dec. 25, 1857, ^10 ;^ July d:, 1858, $18. What balance was due June 1, 1859 ^ Note. — When r.o rate of interest is mentioned in a note, tiie legal rate at the place where it is given is to be used. In Louirfiana the legal rate is 5 per cent. : in Arkansas, Kentucky, Maryland, Mii-souri, North Carolina, Tennessee, and Virginia, it is 6 per cent.: in South Carolina it is 7 per cent. ; and in Alabama, Florida, Geor- gia, Mississippi, and Texas, it is 8 per cent. 198 COMPOUND INTEREST. |25l COMPOUND INTEREST. § 251. Compound Interest is the interest on both princi- pal and interest when the interest is not paid as it falls due. In ordinary business transactions it is not allowed by law; but in a few classes of debts it is required that the inter- est shall be compounded annually. In such cases, the in- terest for one year is added to the principal; this amount becomes the principal for the second year ; its amount for the third year, and so on to the last year or part of a year. The original principal subtracted from the final amount gives the compound interest. Ex. Gl. What is the 'compound interest of $525.75 for 3yr, Gmo. at 6 per cent., interest due annually ? SOLUTION. Original Principal, $525 75 Interest for the first year, . 31.545 Amount, — Second Principal, $557,295 Interest on $557,295 for the second y^car, 33.437 Amount, — Third Principal, $590.73^ • Interest on $590,732 for the third year, 35.443> Amount,— Fourth Principal, .$6267176 Interest on $626,175 for the remaining 6mo., 18.785 Total Amount at Compound Interest, $644,960 Original Principal, 525.75 Compound Interest, $119.21 Ex. 62. What is the amount at compound interest of $500 at 6 per cent, for 4yr. 3ino., interest due annually ? 63. What is the amount of $1000 for 7 years at 7 percent^, compounded annually ? 199 §252 PROPORTION — PERCENTAGE. 64. What is the amonnt of $1000 for 6 years at 6 per cent., compounded semi-annually? Ans. ^1425.76. ^. What is the interest of $1000 for 4 years at 6 per cent., compounded quarterly ? Ans. 1268.98. DISCOUNT. § 252. Discount is a deduction made for the payment of money before it is due. § 253. The present worth of a future debt is that sum which, at ordinary interest, will amount to the debt at the time it becomes due. The present worth bears the same relation to the debt, that the principal h^d^rs, to the amount. The problem to be solved, then, is, having given the amount, the time, and the rate, to find the principal and the interest. § 254. Rule. — Find the amount of $1 for the given time at the given rate. Then, as the amount of $1 is to $1, so is the amount of the debt to its present worth. To find the discount, subtract the present ivorth from t\e amount of the debt. Or say, as the amount of $1 is to its interest^ so is the amonnt of the debt to the discount. Ex. 66. What is the present worth, and what is the dis- count, of a note due 6 months hence for $550 at 6 per cent.? SOLUTION. Amount of $1 for 6 months at 6 per cent., $1.03. $1.03 : $1 :: $550 : $533.98, pres,ent worth. $550 — $533.98=$16.02=the discount. Or, $1.03 : $.03 :: $550 : $16.02, the discount. 200 DISCOUNT. §255 Ex. 67. What is the present worth of a note for $245, due 1 year hence when the rate of interest is 6 per cent.? 68. What discount should be allowed on a note for $525, if paid 3mo. before it is due, interest being at 7 ner cent,? C9. What is the present worth of a debt of $375.50, due in Tnio. 15da., if interest is at 8 per cent.? 70. What is the discount of a note for $725, due in lOmo. lOda., interest being 7 per cent.? 71. In Mobile, Ala., one man gave another his note for $247. 50j due twelve months after date. What was the present worth of the note ? 72. What discount would be allowed at New Orleans on a debt of $650, due 9 months hence ? 73. What is the present worth, at Little Rock, Ark., of a note for $769.35, due 5mo. 18da. hence? 74. What is the proper discount on a debt of $75.75, due 7mo. hence at Memphis, Tenn.? 75. What is the present worth of ^1250, due 12 months hence at Galveston, Texas ? 76. What is the discount of $250, due 8mo. hence at Lex- ington, Ky.? 77. What is the present worth of $55.55, due 7mo. benee at St. Louis, Mo.? BANK DISCOUNT. § 255. The present worth or proceeds of a note payable in bank is the remainder obtained by subtracting from its face its interest for the time it has to run, including three addi- tional days — called days of grace. Thus, if I deposit with the Cashier of the Bank of Cape 201 §25(5 PROPORTION — PERCENTAGE. Fear my note for $1000 due in 60 chxys, lie will pay me on it only $1000— the interest of $1000 for 63 days, that is, $989.50. §256. The bank discount of a note not yet due is the in- terest of the face of the note for three days more than the time it has to run. Ex. 78. What is the present worth in bank of a note for $500 due in 30 days, at 6 per cent.? SOLUTION. Face of the note, ' $500. nterest of $500 for Soda., — hank discount, 2.75' Present Worth or proceeds, Ex. 79. What is the proceeds of a note due in bank 60 da. hence for $250 at 6 per cent.? 80. What is the bank discount on a note for $750 due in bank in 90 days, at 6 per cent.? 81. What discount would a Kank require on a note lor $550.75, due 90 days hence at 8 per c(5nt.? 82. What is the present worth of a note due in bank 90da. hence for $333.33 at 6 per cent.? 83. What is the face of a note due 60da. hence, if its pres- ent worth in bank is $500, interest being at 6 per cent.? §257. The present worth of $1 : $1 :: present worth of the note : face of the note. In this case, $.9895 : $1 :: $500 : the answer. Ex. 84. What sum, payable in 90 days, will produce $750, if discounted at a bank at 6 per cent.? 85. What sum, payable in 60 days, will produce $S000, if discounted at bank at 7 per cent.? 202 DISCOUNT. §257 86. Far what amount must a note be drawn, payable in 30 days, so that, if discounted in bank at 5 per cent., the proceeds will be $250 ? 87. What must be the face of a note payable in bank in 90 days, so that, if discounted at 6 per cent., its present worth may be $75.75 1 Showing the number of days from any day of one month to the same day of any other month next following. From any (iay of To the same day of the uoxt j Jan. 385 Feb. 31 Mar. 59 Apr. May 120 June 151 July 181 Aug. 212 Sept. 243 Oct. 273 Nov. 304 Dec. 334 Jan. Feb. 334 36". 28 59 89 120 150 181 212 242 273 303 Mar. 306 337 365 31 61 92 122 153 184 214 245 275 Apr. 275 306 334 365 30 61 91 122 153 183 214 244 1 May 245 276 304 335 365 31 61 92 123 153 184 214 1 June ^14 245 273 304 334 365 30 61 92 122 153 183: July 184 215 243 274 304 335 365 31 (12 92 123 i53i Vug. 153 184 212 243 273 304 365 31 61 92 122il : Sept. 122 153 181 2111 242 303 334 365 30 61 91;' Oct. 92 123 151 182 212 243 273 304 335 365 31 61|i Nov. Gl 92 120 151 181 212 242 273 304 334 365 30i Dec. M\ 62 90 121 151 182 212 243 274 304 335 365 To find the interval of time between Sept. 3, 1862, and May 19, 1863. Find Sept. in the left hand column and May in the upper line : then at the right of Sept. and un- der May, is 242, the number of days from Sept. 3 to May 3. To this add 16, the number of days from May 3 to May 19. The sum 258 is the number of days required. Again, from Jan. 25, to Sept. 9, is (243-16) 227 da^s, 203 §258 PROMISCUOUS PROBLEMS. PROMISCUOUS PROBLEMS. 1. In what time will $100 amount to $200 at 6 per cen simple interest ? §258. As the interest of the given principal for 1 year the given interest :: 1 year : the number of years. In this case, as $6 : $100 :: lyr. : 16yr. Smo., the a' swer. 2. In what time will $200 gain $50 interest at 6 per cent, per annum ? 3. In what time will $500 gain $49 interest at 7 per cent.? 4. In what time will $1000 gain $10 simple interest at 5 per cent, per annum ? 5. At what rate will $100 gain $15 interest in 2yr. 6mo.? § 259. As the interest of the given principal at 1 per cent. : the given interest :: 1 : the rate per cent. In this case, as $2.50 : $15 :: 1 : 6j the answer. 6. At what rate will $250 gain $250 interest in lOyr,? 7. At what rate will $427.25 gain $143.60 in Syr. 4mo. lOda.? Ans. 10 per cent. 8. At what rate will $746 gain $83.92 in 2yr. 3mo.? 9. "What principal will gain $174.56 in lyr. 7mo. at 7 per cent.? § 260. As the interest of $1 for the given time at the given rate : the given interest :: $1 : the principal. In this case, $.11081 : $174,56 :: $1 : $1575, the an- swer. 10. What principal will gain $42 in Syr. 6mo. at 6 per cent.? Ans. $200. 11. What principal will gain $210 in Syr. at 6 per cent.t 204 PROMISCUOUS PROBLEMS. §260 • 12. What principal will gain $400 in 4 years at 8 per cent.? 13. What is the fourth root of 810000 ? Ans. 30. 14. What is the value of 2.8.5.7.11 ? 15. What are the prime factors of 1800 ? 16. A commission merchant sold goods worth $9072; what was his commission at 2^- per cent.? Ans, $220.80. 17. A capitalist sent his broker $15400 to lay out in stocks, after retaining i per cent, of the amount purchased.' How much stock did he purchase? Ans. $15801.00. 18. A gentleman laid out 83025 in stocks which were 10 per cent, below par. What was the nominal value of the stock purchased ? 19. If I buy coffee at 30ct. per lb., and sell it for 3Gct. per lb., what per cent, do I gain? Ans. 20 per cent. 20. A merchant bought 125 bushels of wheat at ^1.60 per bu., and sold it at a profit of 20 per cent. ; what did hegetfi.rit? Ans. $210. 21. If I pay $12000 for a house and lot and sell them at an advance of 25 per cent., what do I gain by the transaction '^. 22. A mei'chant gave $3.51 for an article which he is wil- ling to sell at a profit of 88^ per cent., how must he mark it? 23. By selling a tract of land for $4704 I gain 12 per cent, on ic ; how much did it cost me ? Ans. $4:i00. 24. If ocwt. ot sugar cost $23.40, what will lOcwt. 8qr. cost? 25. A merehantj failing, pays only 60ct. on the dollar of his indebtedness ; how much will a man receive to whom he owes $1800 ? Ans. $1080. 26. What cost 4G2yd. of cloth at $1.06^ per yd.? Ans. $490.87i. 27. What cost 83bu. 3pk. 2qt. of clover seed at $8 per bu.? 205 §261 AVERAGE. 28. What per cent, of $50 is $G? Ans. 12 pei- cent. 29. What is 115 per cent, of $287.50 ? Ans. $330,625. 30. At 5 per cent, commission, wkat would I receive for selling $240 worth of property ? 31. A commission merchant sells property amounting to $550. Retaining his commission of 5 per cent., he lays out the balance after deducting a commission of 2^- per cent, on the amount purchased. How much did he lay out ? 32. What amount can I retain for commission at 8 per cent, on the amount invested, if I have received $2647.08 ? AVERAGE. § 261. The average of two numbers is one half of their sum. Thus, the average of 7 and 13 is (7 + 13)-^2=10. The average of three numbers is one third of their sum. The average o^ four numbers is one fourth of their sum. And so on. § 262. The average of two dates is a date lying half way between them. Thus, in any year June 23 is the average between June 1 and July 15. Ex. 1. Find the average of 2, 4.5, 5.75, 7, and 9.25. 2. 45 5.75 § 263. Model. — Find the sum of the five 7. given numbers. (§ 159.) Divide this sum by 9.25 5. (§164.) The quotient 5.7 is their average. ' 5)28.50 5.7 This needs no explanation. 206 ALLIGATION MEDIAL. ^:^65 Ex. 2. What if? the average of 2, 3, 5, and 6? :}. What 4. What 5. What 6. What 7. What S. What 9. What 10. What 11. What 12. What 13. What 14. What 15. What s the average of 2, 5, 7, and 10 1 s the average of 25 and 32 ? s the average of 34 and 19 ? s the average of 25, 32, and 41 ? s the average of 17, 29, and Go ? s the average of 25, 170, and 195 ? s the average of 2, 102, 111, and 115? s the average of 0, 5, 7.5, 25, and 40 ? s the average of 1, 7, 15, 25.25, and 37.5 ? s the average of 3, 7.5, 5.75, 11.75, and .625? s the average of 20, 47, 35, 91.5, 79.5, and 10.01? s the average of 13, 15, 17, 29.5, 37.5, and 63.75? s the average of 0, 1, 7, 9, 25, 37, and 39 ? ALLIGATION MEDIAL. § 204. This name is given to the process of finding the moan value of a mixture, when the values of the substances composing it are known. Ex.16. If 41b. of sugar worth lOct. per lb. are mixed with 101b. worth 12ct. per lb., what is a pound of the mix- ture worth? • § 265. Model. — Multiply lOct. by4. (§183.) Multi- ply 12ct. by 10. (§ 183.)— Add the products together. (§179.) Divide the sum by 14. (§ 185.) The quotient, llf ct., is the average price per lb. 4xl0ct.= 40ct. 10x12" =1 20" 14 i60ct. 14 "20 14 6 14 llfct. 207 §266 AViCRAaE. ExPL\?fATio?^. — The whole mlstare weighs 141b., whick evidently cost 160ct. : and 1 fourteenth of this amount is the average price per lb. Rule. — Divide the whole cost by the number of articles ; the quotient will be the average cost per unit. This rule applies to several things not embraced in the definition. Ex. 17. During 24 hours the thermometer stood for 2hi. at 55°, for 3hr. at 60°, for 4hr. at 65^ for 5hr. at 70°, for 6hr, at 75°, and for 4hr. at 80^. What was the mean tena- perature of the day ? 18. A goldsmith mixes lOoz. of goid 16 cirats fine with 6oa. 17 ci]*ats fine and 8az. 19 carats fine ; what is^the fina- nes^ of the mixture ? 19. A grocer mixed 4gal. of wine worth $1 i gallon, 5gal.' worth $1.25 a gallon. ani'lOgal. worth $1.50 a gallon ; wh (t s^as the mixture worth per gal.? 20. If 30gal. of molasses at lOct., 40gii. at oOct., 70gal, at 60ct., and SOgal. at SOjt., be mixed to^^erjier, wnat ia a gallon of the mixture worth '/ Ans. 32Y\ct. 21. A farmer has 10 sheep worth $4 eaoh, 12 worth $5 eacli, and 8 worth $10 each ; what is their averag« value ? ALLIGATION ALTERNATE. §286. This consists in finding the proportional quantities of several simple substances which shall make a compound of a given mean value. It is, therefore,'the converse of the preceding. • 208 ALT.IOATION ALTFRNATE. §26T Ex. 22. In wbnt, proportions inir«t sugars worth lOct., llct., 13ct., and ISot., be mixed, that the compuund uiaj be worth 14ot.? riO— 1 § 2C7. Model.— Connect 10 1 with 15, 11 wi'h 15, and 13 1 .with 15. 10 from 14 h-avf8 4; ^^1 ^3.^ , L15ir_ .4 + 3 + 1 = 8 set 4 opposite 15: 11 from 14 leaves 8; ^et 3 opposie 15 : 13 from 14 leaves 1 ; set 1 opposite 15: — 14 tr<*iij 15 Uaves 1; pet 1 opposite 10, 11, an«l 13. ll« nee there muft he lib. ai lOct., lib. at Hot., and lib. at 13ot., to 81b. at 15cf. Explanation. — After arranging the several prices as in the model, atid placitig the mean price on the left, we con- nect each price below the mean with ont> above it, and t^ach price above the mean with one below it. Then taking the dififorence between each price and the mean, we set this dif- ference opposite the price with which this price is connect- ed ; observing during the operation to consider all the prices as abstract numbers. The reason for all this is evident when we consider that tach pound at lOct. falls 4ct. below the mean, while each pound at 15ct. is only let. above it. To average these two values, therefore, we mu.«t have 41b, of the sugar at Inct, to every one at l^cf. For a siniilar reason, it requires 31b. at 15ct. to counterbalance lib. afc llct. And as the mean price is equidistant between 13ct. and 15et., these two qualities must be taken in equal quan- tities. So that to bring the three i ferior qualities up to the required average, it is neces^ary to take 4-|-3-f 1, /. <?., 8lb. of the superior quality to lib. of each of the inferior qualities. Rule. I. — JJrrange the several prices in a vertical coi- umriy and place the mean price on the left, N 209 §267 AVERAGE. Connect each price below the mean with one above it, and each price above the mean with one below it. Find the difference between each price and the mean, and set it opposite the price luith which it is connected. If only one difference stands opposite any price, it denotes the j)^o- portion of that value ; but if several differences stand oppo- site any price, their sum denotes the proportion of that value. II. If it is required to have a specified quantity OF ANT VALUE. — Find the proportions as above. Then say, As the proportion found for this value : the quantity reqiiir- ed for it : : the proportion for any other value : the quanti- ty required for it. III. If the whole quantity oi' the mixture is speci- fied. — Find the proportions as ab JVC. Then take the sum of the proportional numbers, and say, As the sum of the proportional numbers : the required quantity of the mixture :: the proportion for any value '. the quantity required for that value. Proof. — By Alligation Medial. Ex. 23. In what proportions may gold of 10, 13, 14, and 22 carats fine, be mixed so that the compound may be 17 carats fine ? 24. A grocer having brandy worth $1 a gallon, wishes to mix it with water so that he can sell the mixture at 80ct. a gallon. In what proportions must he mix them ? 25. In what proportions may liquors worth respectively $1, $1.20, $1.40, and $1.50 be mixed, that the mixture may be worth $1.25 ? 26. A farmer wishes to mix 14bu. of wheat worth $1 per bu. with such a quantity worth $1.24 as will make the mix- ture worth $1.03 ; how much must he take ? 210 EQUATION OF PAYMENTS. §269 27. How much tea at 80ct., 70ct,j and 60c t., respectively, should be mixed with 901b. at 90ct., so that the mix- ture may be worth 75ct. per lb.? 28. A merchant having 1001b. of sugar worth lOct. per lb., mixed it with other sugar worth respectively 5, 8, and 9ct., and sold the mixture at S^ct. How much of each quality was there in the mixture ? 29. How much sugar at lOct., and how much at 15ct. per lb., must be taken to make 601b. worjth $7.20 ? Ans. 361b. at lOct., and 241b. at 15ct. W, A grocer mixes 1441b. of sugars worth respectively 12, 10, 0, and 4ct. per lb., and sells the mixture at 8ot. per lb. ,• how much of each quality does he take ? 31. A man paid S165 to 55 persons — men, women, and boys; to each man he paid $5, to each woman ^1, to each boy 50ct. ; how many were there of each ? Ans. 30 men, 5 women, 20 boys. EQUATION OF PAYMENTS. § 268. This consists in finding the average date at which several amounts due at different times may all be paid, so that no interest shall be either gained or lost. Ex. 32. A owes B S25 due in 4mo., $50 due in 6mo., and $75 due in 8mo. ; what is the mean time of payment? 25x4mo. = 100mo. §269. Model.— Multiply 4mo. 50x6 " =300 '^ by 25. (§183.) Multiply 6mo. 75x8 " =600 '- by 50. Multiply 8mo. by 75.— 150 15,0)100,0mo. ^^^ ^^^ products together. Add — Wf- — the multipliers together. Divide Ogmo. iQQQ^^Q^ Ijy i^Q rp^g quotient 6|mo. is the mean time of payment. 211 §269 AVERAGJ5. Explanation. — The irifere^t of 625 dollars for 4 months IS e(jn-il to the interest of 1 dollar for 100 months: the in- fere.-st of 350 for 6iiio. = the interest of SI forSOOuio.: the in- ter es of $75 tor Sin!t.=: the interest of ^ I for GJOoOo Hence the interest, of the several am >airth for their respectiv;e times Ln equal to the interest of $1 for l0(»0ii!o., and this i» equal to the interest of $150 for Cfmo. Oence it is fair that tb« whole amount .should be paid in 6|nio. IvUi.K. — Mnlfiplf^ e<i(}i teiDi of cn'dlt hij the number oj Wtita ill ih> correypititdiuij juiijmeht, iinil divide the .sum oj thf pmdnrts hy the aiim of (kr niu(i'plitr» : the quotient wilf %e ihe medv tim" of pai/mcnr, ^ WjS^^ '"I-^. a Qian owes an other $500 due in 8 mo., 6400 .dn»^ ii» 6mo., and SGOO due in 9ino. ; what is the averagj* term of er<t;dit for the three debt?*? 84. liou^ht goods a ■< follows : ^h)0 on a credit of 6rao.; ,$200 on o|tiio.; and $560 on 6mo. ; what average credit f^hould lie allowed me on the «hole? B5. B'U|ihi $iOOO worth of goods lo be paid for as follows; $200 on the day of purchase, $400 in 5mo., and $400 in I5m(>. What average credit &hould be allowed in* ot) the whole ? 86. Ill what time should the f(*llowing amounts be paid ali at. once : $1000 due in 5mo., $1200 in Gmo., and $1200 in 8mo.? 87. I owe $100 to be paid Jan. 15 $200 due Feb. 15, and $300 due Mar. 9 ; on what day may the whole debt be paid at once ? jV'o/*? — Felec' }• owe day from wh'ch the periods of credit nifty be supi 08^•d to • omnieiicts In this i.saiKe. Jan 15 in the mosi conv.uJtiit Kind ihe interval ehi|s:iiK t>. tweeti tUis date wnd each 0*" th** o»h» r«. }0(d then proceed according to the rule. Consider <a*ch manth 80 days. 212 FQUATrOV np PATMFNTJ3. §270 ZH. A Tnai) Awe^ liis jieiii'hho'* $12f)0 duo in 8tn<>. : but at the <vi<l of 3m'). be pfi^ys $250, and \\\ 'Iwak more be pa)' $'5''; whut oxteiisioti of credit sboiild be allowed on the remainder ? 250x5nio. = l 50aM). 270. M(.nKL. — Multiply 5 150x3 '* = 450 " mo. b> 2:)!). (^ 1>^3 ) xMulti'pIj 850)17007n7r. ^'>»''. *'> ^^'^' Add tbe pind- ucrs t<tfri;rhor. l)ivide ITOOnio, '^"**'- by 850. The quotient 2mo. i« the extension of cre<^it:. EXPL>NAT(()N.— The debtor, having pqid $2^0 (S — 3) 5mo before it was due, is entitled to a credit of l25!)ino. on $1 : and, having paid SI 50, 8'no. before it was due, ii therefore entitb'd to a credit of 45'hiio. on 8'. For both prepayments be is entirliMl to a credit eipiivMlent to $1 for ITOOmo. Twe remainder unpaid is 81250 — ($25!) + 3l r>0) =:$'*'50 : nnd a credit of ITOOuio. on $1 is equal t<j a credit of 2mo ofi S85'>. Ex. 811. I owe $1000 due in 12mo. Tf T pny $100 at the end of Bnin., ar.d $100 at the end of 4nio.. bow long be- yond the r2ino. should my creditor wait for tb« jiayment of the 'balance ? 40. I owe $2 100 due in 6.no. If I pny 8500 down, ?3"0 i^ .2mo., and S200 in omo., in bow many mOnth-i from thfc contraction of the debt should I pay the bjilnnce? 41. A niercbaiit owes SI 200, of which §200 is to be paid in 4 mon'hs, 8400 in 10 mo-nths, and the remainder in Id months: if he pays the whole at ouce^ at what time rDu.-'t he ni'tke the payment? 42. A merchant owes $ISOh to be paid in 1.2 months, 8240O to be [>aid in 6 mou'hs, and 82700 to. be p.iid iu 0^- mouths: what is the averaj^e tuue of; payment ? 2ia PROMISCUOUS PROBLEMS. PROMISCUOUS PROBLEMS. 1. Reduce ,£19, 8|s. to pence. 2. Reduce 9oz. 16|-dwt. to grains. 3. Reduce - of ~ of IG-J- to its simplest form. ■]r 12 ^ ^ . Ql. Qi. 4i 4. Reduce 4-^, -zrrr-, tt-^ j and I of -:^ to their least com- '^ 20f 26 -" 9 mon denominator. 5. Add 900.01, 450.037, and 696.9 together. 6. Add 2y\, 6|-, and 12if together. 7. Add -^ of -*- of 20, -f- of -f- of 24^, and ^,- of 2-i- together. 6a 2i 8. From -^ take -^. 9. From -f- of f of 3i take f of f. 10. From $49f take i4.75 + $5^4-$9.30. 8-1- 17 11. Multiply -4 by -7-. 5 17 12. What is the product of -fi by --- ? 50 J 2Q_:2. 5 13. Multiply ^g-^^ by -^V of if of 5f 14. Divide f of 3-| by -f- of 6i. 15. Divide 1301|- by 161.3. 16. Divide $1843i by 368f, 17. What is the insurance on $3125 at 6i per cent.? 18. A commission merchant sold 19 firkins at 45ct. per lb., and retained 5 per cent., commission ; how much did he return to the owner ? ^ .19. What is the par value of two certificates of stock; one ^ 214 PROMISCUOUS PROBLEMS. for S350 at 2} per cent, discount, the other for $527-.- 60 at 5 per cent, advance ? 20. What is the amount of ^1054, 10s. 9d. for 2yr. 9mo. at 4 per cent, per annum ? 21. If the interest of a certain amount of money at 6 per cent, is $241.80, what is the interest of the same sum , for the same time at 7^- per cent.? 22. At what rate per cent, per annum will jei8295 10s. amount to £1898, 2s. l^d. in 9 months] 23. What is the greatest common measure of 560, 880, 1028, and 1296? 24. What is the least common multiple of 36, 18, 33, 11, and 6 ? 25. What is the greatest common measure of 56, 154, and 182? 26. What is the least common multiple of 2, 4, 10, 7, 14, 15, and 21? 27. Resolve 528 into its prime factors. 28. What prime factors are common to 360, 420, and 840 ? 29. I sold 125A. 2R. 20P. of land for ^2050 ; how much did I gain or lose, if I gave $15.50 per A. for the land? 30. I bought a lot of English paper for .£698, 10s. 6d.,and sold it at a proJGit of 75 per cent. ; how much did I re- ceive for it in Federal currency ? 31. What is the amount of £300, 10s. for 2yr. 3mo. at in- terest compounded semi-annually, at 8 per cent, per annum ? 32. What is the square root of 509796? 33. What is the cube root of 16003008 ? 34. Find the greatest common measure of 1538, 2307, and 3845. 215 PROMISCUOUS PROBLEMS. 35. I sold i of my land to A, ^ of ifc to B, and retained 20i.'A. for myself; how much had I at first? ii6. A, B, and C trade in p'^rtnersliip. A invests SIOOO 'for 12 mon'hs; B, fj^l 500 for 10 months ; and C, ?2000 for months. How bhali their profit of SlUOO be di- vided ? 37. How many barrels of potatoes at S2.50 per bbl. should be exchanged for a Intgshead of sngnr weighing IST-^^lb. groBP, worth 115.00 a hundred pounds net, tare being 8 per cent.? 8?. How many firkins of butter, at 25ct. per lb., can bo bought fur 9iiiO. interest of §800 at 7 per cent, per annum? 89. Having been engaged in merchandise with a capital of $19500, I realized a profit of 33^^ per cen^t., which I immediately inve>ted in land at $16.50 per A.; how many acres did I buy 1 40. If [ owe three notes, one for ^630 due 3mo. hence, an other for ^800 due 6mo. h'nce, and the o-her for Si 000 due 15rao. hence, in what timjc might I fairly pay the three notes together? 41. If i2lb. of tea @81.->0, 151b.@$1.44, and 18ib.@S1.80, be mixed together, what irs liae value of 111b. of the ntixture ? 42. What i.s the 4th power of 7^? 43. What is the cube of 3.5? 44. What is the c.ibo root of ]9.Gi»3? 45 What is the 8(iuare root of 7l.)3^g ? 46*. Required to fill a hogshead with two kinds of wine worth $1.20 and $1.05 fier gal. respectively, so that the mixture wil' be worth ?I.15 per gal.; how mariy gal- lons of each kind will be rcijuired ? 216 pnOMISrrOUS P'CtBLFMS. 47. The total Mock in n UnilnKaJ is SIOOOoOO tb'^ net m- cctme for a veur is $501'00 ; wlia dividend will I re- ceive f r SlOOOn woith of .-tuck ? 18 I exebaficed a houf^e jmd lot worth £500, 15^ fnr land * « • ? worth SIU.50 per A.; how tuuch l;iud did I r^'C'•ue • 49. A pedlar exclxintred a pit-ce of calico, rated at 2 let. per yd., ff^r a fii kin of butter worih '2'2}ji5t. per lb; how many yards of calico were tb re? 50. I imported V):")?. of iron wor-b S82 per T, ; what was th« duty o!i it at 3*3^ |>er c<Mit.? 51. ^Vlnjt is il»e net weight ot* 275 bags of coffee, weighing each 78tb. irross, tnre being 4 per cent.? .Vi. What cost 7UA. nn. 25P. of land at $.\5.75 per A.? 33. What co.-t 51'. IGcwf. 3.|r. o iron at $1:.125 per cwt.? i4. If, by se'ling a tract of land for St)450, I lo>=e 4 per cent, of what it cost nie, for what would I have had to sell it, to jjuin C^\ f>^»r cent.? 55. Bought 2')T. IGcwr. of iron a^ =£1 1, 1 Gs. per T. ; and oM the whole for 32 JOO , what did [ gaiu or lose p«r T? ^iiy^ What IS the present worth of $-00n, due in 2yr. Huio. 1.') Ia.» ijitere:-t beiiig at fi per cetit. p^r annum r* 57. What is the discount on £iXO^ due in Syr. Omo., inter- est hfiiig at 8 per cent, per annuu) ? 58. I wish to borrow 81 150 iti bank : interest being at 6 percent, per tinnum, what ujust be the fa<^e of t!) ; proper note at 90 days? 59. In what tiute will £4o2, 15s., at C percent, per annum, amount to £502, 1 1 -. '^d.? 6D. If the insurance of :B25ajO is S400, wb-it is the rate per C'-'Iit.? ($1. What per cent, of 00 is 1,25? Q2. Wluit per cent, of 75 is 125 ? 217 PROMISCUOUS PROBLSMS, 63. What is S-^\r per cent, of S11755 ? 64. If 5|-A. of land cost $144.50, what will 17A. 3E. 19.- 375P. cost ? 65. If 1.37gal. of sorghum molasses cost $1.4375, what will I3.7gal. cost? 66. Divide 17mi. 5fur. 25rd. by 1.5. 67. Multiply 3deg. 17min. 45sec. by 2.03. 68. Dividend is £1, 18s. 9.5d., divisor is 4.9, what is the quotient? 69. Dividend is Ibu. 3pk. 4.5qt., e^uotient is 75bu. 2pk, 4qt., what is the divisor ? 70. From 1.475T. take 17cwt. Iqr. 19.29lb. 71. Add together 4.75gal., 3.07qt., 7.45pt., and 6.19g). 72. What cost 17bbl. flour at $10 per 1001b., 3bu. salt at $1.25 per bu., and 677.51b. pork at $0,065 per lb.? 73. If 75 persons eat 800bu. corn in 1 year, how long will 600bu. last 90 persons ? 74. If 150 copies of a book of 200 pages require 6rm. 4qr, of paper, how many reams will 15000 copies of a book of 224 pages require ? 75. If Irra. of paper weigh 301b. and cost 30ct. per lb., what will the paper cost for an edition of 1000 copies of a book which requires 5rm. lOqr. for 96 copies? 76. If 83iT. of coal cost $405.50, what will 17T. 3cwt. Iqr. bring at 16|- per cent, advance ? 77. The second, third, and fourth terms of a proportion are -|, 1^, and 2.5, respectively ; what is the first term ? 78. If the first, third, and fourth terms of a proportion are $64.96, 7cwt. Iqr., and 4cwt. 2qr., respectively, what is the second term ? 79. Multiplicand is 94; product is .66; what is the multi- plier ? 218 PROMISCUOUS PROBLEMS. 80. I bought 6251b. of cheese for $62.50, and sold it at 12ict. per lb. ; how much per cent, did I gain? 81. I own I of a ship worth 820000, and have insured it at 2.375 per cent. ; what insurance do I pay ? 82. What is the amount of $2169.845 for lyr. lOmo. 17da. at 7 per cent, per annum ? 83. What are the prime factors of 7825? 84. What are the common prime factors of 875 and 1750 .'' 85. How many hours will there be in the year 11)00 ? 86. The Mecklenburg Declaration of Independence was made May 20, 1775; North Carolina unanimously seceded from the United States May 20, 1861 ; how many days elapsed between these two great events 1 87. What cost 30001b. of corn at S3.00 per bbl.] 88. What cost 5.25bbl. of flour at $ .04 per lb.? 89. An officer, in pursuit of a criminal, goes lOmi. per hr.; the criminal, who has 36mi. the start, goes 7mi. per hr.; how far must the officer go, to catch the criminal ? 90. Bought 40gal. wine at $2.50 per gal. ; lost 5gal. by leakage : how must I sell the remainder per gal. so as to gain 25 per cent, on the whole ? 91. A vessel laden with 3000bu. wheat, found it necessary to throw 25 per cent, of her cargo overboard ; what was her loss at $1.25 per bushel 1 92. What is the value in Avoirdupois weight of 161b. 5oz. lOdwt. 12gr. Troy? 93. How many sheets in 7 reams of paper? 94. If 7 silver spoons weigh lib. 2oz. 3dwt., what will each spoon weigh ? 95. If 2A. produce 45bu. 3pk. 6c][t. Ipt, of corn, how much will 32A. produce ? 219 PR.OM.'SCUOUS TRrELEMS. 96. A.li together i of i of an Acrp, 75^^?. -fll., and |A.r 97. What part f.f u futhom is Sifr.V 98. What is the anionnt of $30U0 for Gmo. 24Ja. at 7^- per ceiif. per arirnim ? 99. A ari.l H [)iirclj;iPe(l a house for $3000, of which A paid 31800, how shall they divide a rent of $350 ? iOO. Wliat it^ the scjuare root of 57(5? 101. What is the 4rh root of GoGl ? I'v2. WhMt is the cube root of yf-^iouo ^ 103. How much stock at 7 perceut. advance niaj be bought fur Sf)350? 104. Ij.'uglit lOrm. of paper at ?3.50 per rm., and sold it at $ .25 per quire, h -vv luuch did I gain or lose on it all? 105. Bought 30ubhl. of flour for $2250, sold I of it at $0 per hbl., jiiid the remainder at $8 per bbh, how much did T receive for the whole? 106. Reduce 26^ to a decimal form. 107. Miihiply four rhousjindths by five hundredth!*. 108. Multiply fdur hundred ati({ fifty by two hundredths. 109. Diviile -even tenths by one hundredth. liO. \\''hai is the difference between thirty-five hundredths^ and thirty-five thouHa'jdrh.s ? 111. What is the 2nd term of h proportion whcse T.-t, 3rd^ atid 4(h terms are 7, 1.3, and 19, respectively ? 112. If one jicre of lar»d costs jC2, 15s. 4d., what will be th» cost (.f 173 A. 2R. 14P. at the same rate ? 113. A geijtlemah's estate is worth dC42 !5, Is. a year : what may he spend per day and yet save i^lOOO per annum? 114. A father left his son a foitune, -^ of which he ran through in 8 months, -^ of the remainder lasted him 12 months longer, when he had barely £820 left: what hum did his father leave htm ? 220 PROMlSCUOye PROBLEMS. 115. Ther? are 1000 men be>ie«^ed in a to'-vn with j)rovi" einiis for 5 weeks, allowing each uian 10 ounces a. day. If rh-y art> reinf.rrced by 500 more and no relief can be afforded til! the end of 8 wi-ek.s, Low many ouiKces n\\ii\t be given daily to each man '{ 116. A fjitlier pave -{^ of his eMate to one Hon, and ^'^ of the remainder to another, leaving the rest to his widow. The difference of the childjen'w h-gaeies was £514, 6s. Si\. : what was the widow's portion ? 117. If Hcwt. 2(jr. of sugar cost S12l\92, what will be tlie price of Ocwt.? 118. If the freight of 80 tierces of Kiigar, each weighing Sicwt.. 150. miles, cost S8[, what must be paid fur the freight of 30hhd of sugar, each weighing 12cwl., 50 miles ? 119. If one pound of tea be equal in value to 50 ornnges, nnd 70 oranges be worth 8* lemons, what is the value of a pound of tea when a lemon is worth 2 cents? 120. If 60 bushels of oats wilbserve 2( horses f.r 40 dajR, how long will^iO bushels swerve 48 horses at the same rate ? 131. W!,at will be the cost of 2hhd. 5ga]. 3qt. 2gi. of mo- Jasse^, at 12} cents p^r (jnart? 122. Wliat is the interest of 53153.82 for years, at 4} per cent, per annum ? 123. What is the interest of 831573.25 for 10 months at 6 per cent, per annum ? 124. What will be the amount of $9537.15 for 11 years, 2 months, and 18 days at 7 per cent, per annum ? 125. What will be the amount of 83758.56 for 3 years at 7 per cent., th© interest being compounded semi-annu- ally ? 221 PROMISCUOUS PROBLEMS. 126. If I buy 895 gallons of molasses and lose it per cent, by leakage, how much have I left ? 127. Bought a piece of cloth containing 150 yards for $650: what must it be sold for per yard, in order to gain $300? 128. What is the bank discount on a note of $556.27 pay- able in 60 days, discounted at 6 per cent, per annum ? 129. The sum of two numbers is 5330, their difference is 1999 : what are the two numbers ? 130. How many scholars are there in a class, to which if 11 be added the number will be augmented one-sixteenth? 132. Sound travels about 1142 feet in a second. If then the flash of a cannon be seen at the moment it is fired, and the report heard 45 seconds after, what distance would the observer be from the gun ? • 133. What number is that which being augmented by 85, and this sum divided by 9, will give 25 for the quotient? 134. One -fifth of an army was killed in battle, i part was taken prisoners, and J^ died by sickness : if 4000 m.en were left, how many men^did the army at first consist of? 135. The greatest of two numbers is 15. and the sum of their squares is 346 : what are the two numbers ? 136. At what rate per cent, will $1720.75 amount to $2325.- 86 in 7 years ? 137. In what time will $2377.50 amount to $2852.42 at 4 per cent, per annum ? 138. What principal put at interest for 7 years, at 5 per cent, per annum, will amount^to $2327.89? 139. What is the greatest common measure of 945, 1660, and 22683 ? 140. What is the greatest common measure of 204, 1190, 1445, and 2006 ? 141. Find the least common multiple of 6, 9, 4, 14, and 16. 222 PROMISCUOUS PROBLEMS. 142. What is the least common multiple of 11, 17, 19, 23, and 7? 143. What is the least common multiple of 7, 15, 21, 28, 35,100,125? 144. Reduce ^-^^*- to a mixed number. 145. Reduce 149|^ to an improper fraction. 146. Reduce 375|^ to an improper fraction. 147. Reduce 174947^^^ f|-g to an improper fraction. 148. Reduce fig- to its lowest terms. 149. Reduce -tt-Vt *^ ^^^ lowest terms. 150. Reduce 14-1^- to its lowest terms. 151. Reduce %, -^, and -j\- to their least common denomi- nator. 152. Pteduce ^V? ^\y ^^^ v *^ their lea.<^t common denomi- nator. 153. Find the least common denominator and add the frac- tions, yV? h h and t, 154. Find the least common denominator and add /^, }f ^-, and -V- 155. Multiply 5^- by ■}. 156. Multiply f5 by | of 9. 157. If 80 yards of cloth cost $340, what will 050 yards cost ? 158. If 120 sheep yield 330 pounds of wool, how many pounds will be obtained from 1200 sheep 1 159. If 6 gallons of molasses cost SI. 95, what will 6 hogs- heads cost ? 160. If *- of a yard of cloth cost ^1|, what will 7^ yards cost ? 161. What is the cost of 28|- yards of cloth, at $4f per yard ? i 223 PUOMISrUOUS PROBLTfMS. 162. Whnt. is the interest of $1914.16 for 18 years at 3^ per c#Tit. per animni ?' ICS. What is the a»n>uni:, of $7958.70 for 9 raonths at (^ por ctMit per unniuij ? 164. A Fnerch-mt h-d^ 1-00 barrels of fl )ur ; he shipped 64 per c^uf. of it and sold the remainder: how much did lie sell? 165. Two men had eacli $240. Que of them Rpend8l4per cent., and the otiier 1<Sl per cent. : how ir)a[»y dollar* more ditJ one >pend than the n<her? 166. Wh;)t is the diife'vnce between f)^- per cent, of $800 and 6*- per cent of Sl<'50 ? 167. What is the square root of 1.519:^^9!? 168. What is the Hj.nn-e root, of :^'^:i7296l ? 1C>9. What is the cube r .ot of 18 28541 ? 170. What is the cnhe root of 2;<>5408-6008 ? 171. if a portion re(;='ivt\s ?1 iur ± of a djy's work, how innrh is that a d^y i 112. \V\u\i nmnher \<i rh»f of whicii i, -\, and |- added to- gether, will muke 6-3? Date Due tilG Tir«~ L. B. Cat. No. 1 137 \ 511.02 L255 23448 r^Ouri Own Series of School Books.l PKEPARED BY ^ Ricl^'ard Sterling, A. M.. & J. U. Campbell, A. iili. i CONSIST 'N-G OF si y li Oixr O^vn I*rimer. For the Uoe of children, pp. 34, 12mo OixrO^vxi Spelling Book. For schools, pp.112. *~^ III. <^J-r Gtvn Fii'st R,eader. Arranged to follow the Pn.mcr ^ / and CO- taining easy lessons for those beginning to read, witk sptlliug leeson? ^ ** accompanying each, pp. 72, 12mo. . <8 I"V". Oixr O^vn. Second I?.eacler. The lessons in this book S liave been carefully selected to suit the progress of thoso who liave finished the ^ ») ifirst Reader, and at the sanio time to fr.rnish both entertainment and iustruc- ^^^ ^ tion, pp. 16S, 12iao. V ■ %'V. G XV Own Tiiird Header. In this book the most difficult ^ <• words uf each Ji^bsou arc dc^'ncd, to aid tl j pupil in understanding what ho rcad°^;^ X Each lesson ip also folio (*fd'by an Exej^se to be written on the black-boa; .;' v'^ 4^ slate, for the : u: oi e "' teaching tV iJ.'ul to b^iell and punctuate, pp.200. ^ * ^' "VI. Oixr , ..3wn t'oui'tlx Header. The lessons ;i this book will % ^ bo found to possess high literary merit, witliout leing above the capacity of the ^ (^ young reader. Definitions of \i< ■^'.e difficuii'vrord>' are continued. -h 7» VII. Our Own Fiftli. K.eader. A Khetorical Reader for the '^ Y more advanced classes in Academies and High Sc-ht •'■' 4 % OTJH OT\^J>f SCHOOL OH A^IMIMAHS. % BY C. W SMYTIIE, A. MI I t •^ Our O^vn I*rinaary Orammar. Besigned for the use of begin- (^ «^ ners, and containing only the simple facts of the language, pp. 72, 12mo. _: i^ Our Own Elementary Oramr lar, Besigned for our Schools and ^^ \^ Academie?!, as a sequel to the Primary Grammai, and embracing a complete •) >^ elementary vatemcnt oi the subject. ^ •) ^ BY S. LANBER, A. M. Oixr O^vn JPrimary J^i-ithmelic. For the nee of beginners, pp. 96, 12 mo. Ov.r 0\vn jMental ^VritlimelAO. In course of preparation. Onr O^vn School AritliiTietic. pp. 221, l£jio. U I iT> Fot tlio use of Schools, with Exercises and Vocabularies. -. (* IJingliom's Oeesar, with Notes and Yocabular}. SterliaiS", Can^pbell & -A..lbrignt, FuUisJiers, GREENSBORO, N. C. % t •:? M,