Google This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project to make the world's books discoverable online. It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover. Marks, notations and other maiginalia present in the original volume will appear in this file - a reminder of this book's long journey from the publisher to a library and finally to you. Usage guidelines Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing tliis resource, we liave taken steps to prevent abuse by commercial parties, including placing technical restrictions on automated querying. We also ask that you: + Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for personal, non-commercial purposes. + Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the use of public domain materials for these purposes and may be able to help. + Maintain attributionTht GoogXt "watermark" you see on each file is essential for in forming people about this project and helping them find additional materials through Google Book Search. Please do not remove it. + Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner anywhere in the world. Copyright infringement liabili^ can be quite severe. About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I k VOCATIONAL MATHEMATICS FOR GIRLS BY WILLIAM H. DOOLEY AUTHOR OP ** VOCATIONAL MATHEMATICS*' "TBXTILB8," ETC. t J , o » D. C. HEATH & CO., PUBLISHERS BOSTON NEW YORK CHICAGO \ ^ ^^^ '^ Copyright, 1917, By D. C. Hkath & Co. IB7 u O WW bwccc w PREFACE The author has had, during the last ten years, considerable experience in organizing and conducting intermediate and sec- ondary technical schools for boys and girls. During this time he has noticed the inability of the average teacher in mathe- matics to give pupils pmctical applications of the subject. Many teachers are not familiar with the commercial and rule of thumb methods of solving mathematical problems of every- day life. Too often a girl graduates from the course in mathe- matics without being able to "commercialize" or apply her mathematical knowledge in such a way as to meet the needs of trade, commerce, and home life. It is to overcome this difficulty that the author has prepared this book on vocational mathematics for girls. He does not believe in omitting the regular secondary school course in mathematics, but offers vocational mathematics as an introduc- tion to the regular course. The problems have been used by the author during the past few years with girls of high school age. . The method of teach- ing has consisted in arousing an interest in mathematics by showing its value in daily life. Important facts^ based upon actual experience and observation, are recalled to the pupil's mind before she attempts to solve the problems. A discussion of each division of the subject usually precedes the problems. This information is provided for the regular teacher in mathematics who may not be familiar with the subject or the terms used. The book contains samples of • • • 111 4!283o IV PREFACE problems from all occupations that women are likely to enter, from the textile mill to the home. The author has received valuable suggestions from his for- mer teachers and from the following: Miss Lilian Baylies Green, Editor Ladies'^ Home Journal, Philadelphia, Pa. ; Miss Bessie Kingman, Brockton High School, Brockton, Mass. ; Mrs. Ellen B. McGowan, Teachers College, New York City ; Miss Susan Watson, Instructor at Peter Bent Brigham Hospital, Boston; Mr. Frank F. Murdock, Principal Normal School, North Adams, Mass. ; Mr. Frank Rollins, Principal Bushwick High School, Brooklyn ; Mr. George M. Lattimer, Mechanics Institute, Rochester, N. Y. ; Mr. J. J. Eaton, Director of In- dustrial Arts, Yonkers, N. Y. ; Dr. Mabel Belt, Baltimore, Md. ; Mr. Curtis J. Lewis, Philadelphia, Pa. ; Mrs. F. H. Consalus, Washington Irving High School, New York City ; Miss Griselda Ellis, Girls' Industrial School, Newark, N. J. ; Mr. J. C. Dono- hue. Technical High School, Syracuse, N. Y. ; Mr. W. E. Weafer, Hutchinson-Central High School, Buffalo, N. Y. ; The Bur- roughs Adding Machine Company ; The Women's Educational and Industrial Union ; the Department of Agriculture, Wash- ington, D. C. ; and Reports of Conference of New York State Vocational Teachers. This preface would not be complete without reference to the author's wife, Mrs. Ellen V. Dooley, who has offered many valuable suggestions and corrected both the manuscript and the proof. The author will be pleased to receive any suggestions or corrections from any teacher. CONTENTS PART I— REVIEW OF ARITHMETIC OHAPTKR PAGK I. Essentials of Arithmetic .... . . 1 Fundamental Processes; Fractions; Decimals; Com- pound Numbers ; Percentage ; Ratio and Proportion ; Involution ; Evolution. 11. Mensuration . . . . * . , . . . .64 Circles ; Triangles ; Quadrilaterals ; Polygons ; Ellipses ; Pyramid ; Cone ; Sphere ; Similar Figures. in. Interpretation of Results 80 Reading of Blue Print ; Plans of a Home ; Drawing to Scale ; Estimating Distances and Weight ; Methods of Solving Examples. PART II — PROBLEMS IN HOMEMAKING IV. The Distribution of Income 89 Incomes of American Families ; Division of Income ; Expense Account Book. V. Food ^100 Different Kinds of Food ; Kitchen Weights and Meas- ures ; Cost of Meals ; Recipes ; Economical Marketing. VI. Problems on the Construction of a House . . 128 Advantages of Different Types of Houses; Building Materials; Taxes; United States Revenue. VII. Cost of Furnishing a House 146 Different Kinds of Furniture ; Hall ; Floor Coverings ; Linen ; Living Room ; Bedroom ; Dining Room ; Value of Coal ; How to Read Gas Meters ; How to Read Elec- tric Meters ; Heating. V vi CONTENTS CHAPTXB PAOB VIII. Thrift and Investment 178 Different Institutions of Savings ; Bonds ; Stocks ; Ex- change; Insurance. PART III — DRESSMAKING AND MILLINERY IX. Problems in Dressmaking 198 Fractions of a Yard; Tucks; Hem; Rufles; Cost of Finished Garments ; Millinery Problems. X. Clothing 217 Parts of Cloth ; Materials of Yam ; Kinds ; Weight. PART IV— THE OFFICE AND THE STORE XL Arithmetic for Office Assistants .... 233 Rapid Calculations ; Invoices ; Profit and Loss ; Time Sheets and Pay Rolls. XII. Arithmetic for Salesgirls and Cashiers . . 260 Saleslips ; Extensions ; Making Change. XIIL Civil Service 268 PART V — ARITHMETIC FOR NURSES XIV. Arithmetic for Nurses 276 Apothecary ^s Weights and Measures ; Household Meas- ures; Approximate Equivalents of Metric and English Weights and Measures ; Doses ; Strength of Solutions ; Prescription Reading. PART VI — PROBLEMS ON THE FARM XV. Problems on the Farm 304 Appendix 317 Metric System ; Graphs ; Formulas ; Useful Mechanical Information. Index 365 VOCATIONAL MATHEMATICS FOR GIRLS PART I — REVIEW OF ARITHMETIC CHAPTER I Notation and Numeration A unit is one thing ; as, one book, one pencil, one inch. A number is made up of units and teUs how many units are taken. An integer is a whole number. A single figure expresses a certain number of units and is said to be in the units coiumn. For example, 6 or 8 is a single figure in the units column ; 63 is a number of two figures and has the figure 3 in the units column and the figure 6 in the tens column, for the second figure represents a certain number of tens. Each column has its own name, as shown below. •o 63 . o S ^ i I . § - alls ? • a s 1 p 2 m a 1 •c wo o •o c • a S 2 s a 1 3 8, 69 5, 40 7, 125 Reading Numbers. — For convenience in reading and writing numbers they are separated into groups of three figures each by commas, beginning at the right : 138,695,407,125. The first group is 125 units. The second group is 407 thousands. The third group is 695 millions. The fourth group is 138 billions. 1 ••: ••• • • • • •, • • • • • 2 A:icgC5AjRt6Ni/!tIw/M^^ FOR GIRLS The preceding number is read one hundred thirty-eight billion, six hundred ninety-five million, four hundred seven thousand, one hundred twenty-five ; or 138 billion, 695 million, 407 thousand, 125. Roman Numerals A knowledge of Roman numerals is very important. Dates in buildings and amounts on prescriptions are usually expressed in Roman numerals. They are also used for numbering chapters and dials. The following capital letters, seven in all, are used to express Roman numerals : I II V X L C D M One Two Five Ten Fifty 100 600 1000 All other numbers are expressed by combining the letters according to the following principles ; 1. When a letter is repeated, the value is repeated. Thus, II represents 2 ; XXX, 30. 2. When a letter of less value is placed after one of greater value, the lesser is added to the greater. Thus, VII, 7 — two added to five. 3. When a letter of less value is placed before one of greater value, the lesser is taken from the greater. Thus, IX, 9 — ten less one. Read the following Roman numeral? according to the above rules: 1. Ill 9. XIX 17. LXVI 2. XXX 10. LXXVII 18. MDC 3. ccc 11. DCCCVII 19. LXXII 4. MMM 12. XL 20. CCLI 5. VII 13. XC 21. DCLXVI 6. LXXX 14. IX 22. DCXIV 7. XXII 15. XD 23. MD^LVI 8. XVIII 16. XM 24. Mi>CCXXIX REVIEW OF ARITHMETIC Express the following numbers in Roman numerals : 1. 14 4. 81 7. 281 10. 314 13. 1837 2. 42 5. 73 a 509 11. 573 14. 1789 3. 69 6. 67 9. 812 12. 874 15. 80,003 Standard Mathematics Sheet. — To avoid errors in solving problems the work should be done in such a way as to show each step and to make it easy to check the answer when found. Paper of standard size, 8| in. by 11 in., should be used. Rule each sheet as in the following diagram, set down each example with its proper number in the margin, and clearly show the different steps required for the solution. To show that the answer obtained is correct, the proof should follow the example. Standard Mathematics Sheet 8i in. • 1. t Mary Smith — 100 Vocational Mathematics 10-2-12 No. 10 1,203 2^ 2,672 2j;f 31,118 23 480 1^ 39 19,883 '66,396 Ans. 2. Proof: 9 3. . The pupil should write or print her name and class, the date when the problem is finished, and the number of the problem on the Standard 4 VOCATIONAL MATHEMATICS FOR GIRLS Mathematics Sheet. If the question contains several divisions or prob- lems, they should be tabulated — (a), (6), etc. — at the left of the prob- lems inside the margin line. A line should be drawn between problems to separate them. Addition Addition is the process of finding the sum of two or more numbers. The result obtained by this process is called the sum or amount The sign of addition is an upright cross, +, called plus. The sign is placed between the two numbers to be added. Thus, 9 inches + 7 inches (read nine inches plus seven inches). The sign of equality is two short horizontal parallel lines, =, and means equals or is equal to. Thus, the statement that 8 feet + 6 feet = 14 feet, means that six feet added to eight feet (or 8 feet plus 6 feet) equals fourteen feet. To find the sum or amount of two or more numbers. Example. — An agent for a flour mill sold the following num- ber of barrels of flour during the day : 1203, 2672, 31,118, 480, 39, and 19,883 bbl. How many barrels did he sell during the day? [The abbreviation for barrels is bbl.] The sum of the units column is 3 + 9 + + 8 + 2 + 8 = 26 units, or 20 and 6 more ; 20 is tens, so leave the 5 under the units column and add the 2 tens in the tens column. The sum of the tens column 182 + 8 + 3 + 8 + 1+7 + = 29 tens. 29 tens equal 2 hun- dreds and 9 tens. Place the 9 tens under Sum 55,395 bbl. the tens column and add the 2 hundreds to the hundreds column. 2+8 + 4 + 1+6 + 2 = 23 hundreds ; 23 hundreds are equal to 2 thousands and 3 hundreds. Place the 3 hundreds under the hundreds column and add the 2 thousands to the next column. 2 + 9 + 1 + 2+1 = 15 thousands, or 1 ten-thousand and 6 thousands. Add the 1 ten-thousand to the ten-thousands column 1,203 2^ 2,672 2? 31,118 2^ 480 ^ 39 19,883 REVIEW OF ARITHMETIC 5 and the sum is 1 + 1 + 3 = 5. Write the 5 in the ten-thousands column. Hence, the sum is 55,396 bbl. Test. — Repeat the process, beginning at the top of the right-hand column. 4 Exactness is very important in arithmetic. There is only one correct answer. Therefore it is necessary to be accurate in performing the numerical calculations. A check of some kind should be made on the work. The simplest check is to estimate the answer before solving the problem. If there is a great discrepancy between the estimated answer and the answer in the solutioiij the work is probably wrong. It is also necessary to be exact in reading the problem. EXAMPLES 1. Write the following numbers as figures and add them : Seventy-five thousand three hundred eight ; seven million two hundred five thousand eight hundred forty-nine. 2. In a certain year the total output of copper from the mines was worth $ 58,638,277.86. Express this amount in words. 3. Solve the following : 386 + 5289 + 53666 + 3001 + 291 + 38 = ? 4. The cost of the Panama Canal was estimated in 1912 to be $ 375,000,000. Express this amount in words. 5. A farmer's wife received the following number of eggs in four successive weeks ; 692, 712, 684, and 705 eggs. How many eggs were received during the four weeks ? 6. A woman buys a two-family house for $6511.00. She makes the following repairs : mason-work, $ 112.00 ; plumb- ing, $ 146.00 ; carpenter work, $ 208.00 ; painting and decora- ting, $ 319.00. How much does the house cost her ? 7. Add the following numbers, left to right : a. 108, 219, 374, 876, 763, 489, 531, 681, 104 ; h. 3846, 5811, 6014, 8911, 7900, 3842, 5879. 6 VOCATIONAL MATHEMATICS FOR GIRLS 8. According to the census of 1910 the population of the United States, exclusive of the outlying possessions, consisted of 47,332,277 males and 44,639,989 females. What was the total population? 9. Wire for electric lights was run around four sides of three rooms. If the first room was 13 ft. long and 9 ft. wide ; the second 18 ft. long and 18 ft. wide ; and the third 12 ft. long and 7 ft. wide, what was the total length of wire re- quired? Remember that electric lights require two wires. 10. Find the sum : 46 lb. + 136 lb. + 72 lb. + 39 lb. + 427 lb. + 64 lb. + 139 lb. Subtraction Subtraction is the process of finding the difference between two numbers, or of finding what number must be added to a given number to equal a given sum. The minuend is the num- ber from which we subtract; the subtrahend is the number subtracted ; and the difference or remainder is the result of the subtraction. The sign of subtraction is a short horizontal line, — , called minus, and is placed before the number to be subtracted. Thus, 12 — 8 = 4 is read twelve minus (or less) eight equals four. To find the difference of two numbers. Example. — A house was purchased for $ 8074.00 twenty- five years ago. It was recently sold at auction for $ 4869.00. What was the loss ? Write the smaller number under the Minuend $8074.00 greater, with units of the same order in Subtrahend $4869.00 ^® same vertical line. 9 cannot be taken Remainder $3205.00 ^""^^ ^» ^ ^^^^^^ ^ *®° ^ ^°^*«- '^^ ^ ten that was changed from the 7 tens makes 10 units, which added to the 4 units makes 14 units. Take 9 from the 14 units and 6 units remain. Write the 6 under the unit col- umn. Since 1 ten was changed from 7 tens, there are 6 tens left, and 6 from 6 leaves 0. Write under the tens column. Next, 8 hundred can- REVIEW OF ARITHMETIC 7 not be taken from hundred, so 1 thousand (ten hundred) is changed from the thousands column. 8 hundred from 10 hundred leave 2 hun- dred. Write the 2 under the hundreds column. Since 1 thousand has been taken from the 8 thousand, there are left 7 thousand to subtract the 4 thousand from, which leaves 3 thousand. Write 3 under the thousand column. The whole remainder is $3206.00. Proof. — If the sum of the subtrahend and the remainder equals the minuend, the answer is correct. EXAMPLES 1. Subtract 1001 from 79,999. 2. A box contained one gross (144) of wood screws. If 48 screws were used on a job, how many screws were left in the box? 3. What number must be added to 3001 to produce a sum of 98,322 ? 4. Barrels are usually marked with the gross weight and tare (weight of empty barrel). If a barrel of sugar is marked 329 lb. gross weight and 19 lb. tare, find the net weight of sugar. 5. A box contains a gross (144) of pencils. If 109 are removed, how many remain? 6. A farmer received 1247 quarts of milk in October and 1189 quarts in November. What was the difference ? 7. A housewife purchases a $ 800.00 baby grand piano for $ 719.00. How much does she save ? 8. No. 1 cotton yarn contains 840 yards to the pound, while No. 1 worsted yarn contains 560 yards to the pound. What is the difference in length? 9. A young lady saved $453.00 during five years. She spent ^189.00 on a sea trip. How much remained? 10. 69,221-3008=? 11. The population of New York City in 1900 was 3,437,202 and in 1910 was 4,766,883. What was the increase from 1900 to 1910 ? 8 VOCATIONAL MATHEMATICS FOR GIRLS 12. If there are 374,819 wage-earning women in a certain city having a total population of 3,366,416 persons, how many of the residents are not wage-earning women ? 13. In the year 1820 only 8385 immigrants arrived in the United States. In 1842, 104,565 immigrants arrived. How many more arrived in 1842 than in 1820 ? 14. The first great shoemaker settled in Lynn, Mass., in 1636. How many years is it since he arrived in Lynn ? » Multiplication Multiplication is the process of finding the product of two numbers. ThuSf 8x3 may be read 8 multiplied by 3, or 8 times 3, and means 8 added to itself 3 times, or 8 + 8 + 8 = 24 and 8 x 3 = 24. The numbers multiplied together are called factors. The mvltiplicand is the number multiplied; the multiplier is the number multiplied by ; and the result is called the product. The sign of multiplication is an oblique cross, x, which means multiplied by or tim£8. Thus, 7x4 may bq read 7 multiplied by 4, or 7 times 4. To find the product of two numbers. Example. — A certain set of books weighs 24 lb. What is the weight of 17 sets ? Write the multiplier under the multipli- Multiplicand 24 lb. cand, units under units, tens under tens. Multiplier 17 etc. 7 times 4 units equal 28 units, which J^ are 2 tens and 8 units. Place the 8 under OA the units column. The 2 tens are to be T^ 7 ^ 17\o ii_ added to the tens product. 7 times 2 tens Product 408 lb. - . , . .v o * i« * i are 14 tens + the 2 tens are 16 tens, or 1 hundred and 6 tens. Place the 6 tens in the tens column and the 1 hun- dred in the hundreds column. 168 is a partial product. To multiply by the 1, proceed as before, but as 1 is a ten, write the first number, which is 4 of this partial product, under the tens column, and the next number under the hundreds column, and so on. Add the partial products, and their sum is tte whole product, or 408 lb. REVIEW OF ARITHMETIC 9 « EXAMPLES 1. A milliner ordered 58 spools of wire, each spool contain- ing 100 yards. How many yards did she order ill all ? 2. Each shoe box contains 12 pairs of shoes. How many pairs in 423 boxes ? 3. Multiply 839 by 291. 4. A mechanic sent in the following order for bolts : 12 bolts, 6 lb. each ; 9 bolts, 7 lb. each ; 11 bolts, 3 lb. each ; 6 bolts, 2 lb. each; and 20 bolts, 3 lb. each. What was the total weight of the order ? 5. Find the product of 1683 and «09. To multiply by 10, 100, 1000, ete., annex as many ciphers to the multiplicand as there are ciphers in the mvUiplier. Example. — 864 x 100 = 86,400. EXAMPLES Multiply and read the answers to the following : 1. 869 X 10 a 100 X 500 2. 1011 X 100 9. 1000 X 900 3. 10,389 X 1000 10. 10,000 x 500 4. 11,298x30,000 11. 10,000x6000 5. 58,999 X 400 12. 1,000,000 x 6000 6. 681,719x10 13. 1,891,717x400 7. 801,369 X 100 14. 10,000,059 x 78,911 Division Diyision is the process of finding how many times one num- ber is contained in another. The dividend is the number to be divided; the divisor is the number by which the dividend is divided; the quotient is the result of the division. When a number is not contained an equal number of times in another number, what is left over is called a remainder. 12 VOCATIONAL MATHEMATICS FOR GIRLS REVIEW EXAMPLES 1. A farmer's daughter raised on the farm 5 loads of pota- toes containing 38 bu., 29 bu., 43 bu., 39 bu., and 29 bu. respectively. She sold 12 bu. to each of three families, and 34 bu. to each of four families. How many bushels were left ? 2. Five pieces of cloth are placed end to end. If each piece contains 38 yards, what is the total length ? 3. I bought a chair for $ 3, a mat for $ 1, a table for $4, and gave in payment a $20 bill. What change did I receive ? 4. A teacupful contains 4 fluid ounces. How many teacup- fuls in 64 fluid ounces ? 5. No. 30 cotton yarn contains 25,200 yards to a pound. How many pounds of yarn in 630,000 yards ? 6. The consumption of water in a city during the month of December was 116,891,213 gallons and during January 115,819,729 gallons. How much was the decrease in con- sumption ? 7. An order to a machine shop called for 598 sewing machines each weighing 75 pounds. What was the total weight ? 8. If a strip of carpet weighs 4 lb. per foot of length, find the weight of one measuring 16' 9" in length. 9. Multiply 641 and 225. 10. Divide 24,566 by 319. 11. An order was given for ties for a railroad 847 miles long. If each' mile required 3017 ties, how many ties would be needed ? 12. How many gallons of milk are used every day by two hospitals, if one uses 25 gallons per day and the other 6 gallons less ? REVIEW OF ARITHMETIC 13 Factors The factors of a number are the integers which when multi- plied together produce that number. Thus, 21 is the product of 3 and 7 ; hence, 3 and 7 are the factors of 21. Separating a number into its factors is called factoring, A number that has no factors but itself and 1 is a prime number. The prime numbers up to 25 are 2, 3, 5, 7, 11, 13, 17, 19 and 23. A prime number used as a factor is 2l priyne factor. Thus, 3 and 5 are prime factors of 15. Every prime number except 2 and 5 ends with 1, 3, 7, or 9. To find the prime factors of a number. Example. — Find the prime factors of 84. 2 )84 The prime number 2 divides 84 evenly, leaving the quotient 2)42 ^^» which 2 divides evenly. The next quotient is 21 which 3 3)21 divides, giving a quotient 7. 7 divided by 7 gives the last quotient 1 which is indivisible. The several divisors are the IL prime factors. So 2, 2, 3, and 7 are the prime factors 1 of 84. Pboof. — The product of the prime factors gives the number. EXAMPLES Find the prime factors : 1. 63 4. 636 7. 1155 2. 60 5. 1572 8. 7007 3. 250 6. 2800 9. 13104 Cancellatioii To reject a factor from a number divides the number by that factor ; to reject the same factors from both dividend and divisor does not affect the quotient. This process is called cancellation. This method can be used to advantage in many everyday cal- culations. Example. — Divide 12 x 18 x 30 by 6 x 9 x 4. 14 VOCATIONAL MATHEMATICS FOR GIRLS 2 2 15 By *'^^ method it is not Dividend 12 X 18 X M ,_ „ ,. , necessary to multiply be- Divisor % X 9 X 4 = ^^ Q«o«w««. fore dividing. Indicate [s ^ ^ the division by writing ^ ^ r the divisor under the divi- 1 dend with a line between. Since 6 is a factor of 6 and 12f and 9 of 9 and 18, respectively, they may be cancelled from both divisor and dividend. Since 2 in the dividend is a factor of 4 in the divisor it may be cancelled from both, leaving 2 in the divisor. Then the 2 being a factor of 30 in the dividend, is cancelled from both, leaving 15. The product of the uncancelled factors is 30. Therefore, the quotient is 30. Proof. — If the product of the divisor and the quotient equal the dividend, the answer is correct. EXAMPLES Indicate and find quotients by cancellation : 1. Divide 36 x 27 x 49 x 38 x 50 by 70 x 18 x 15. 2. What is the quotient of 36 x 48 X 16 divided by 27 X 24 X8? 3. How many pounds of tea at 50 cents a pound must be given in exchange for 15 pounds of butter at 40 cents a pound? 4. There are 16 ounces in a pound ; 30 pounds of steel will produce how many horseshoes, if each weighs 6 ounces ? 5. Divide the product of 10, 75, 9, and 96 by the product of 5, 12, 15, and 9. 6. I sold 16 dozen eggs at 30 cents a dozen and took my pay in butter at 40 cents a pound ; how many pounds did I receive ? 7. A dealer bought 16 cords of wood at $ 4 a cord and sold them for $ 96 ; find the gain per cord. REVIEW OF ARITHMETIC 15 Greatest Common Divisor The greatest common divisor of two or more numbers is the greatest number that will exactly divide each of the numbers. To find the greatest common divisor of two or more numbers. Example. — Find the greatest common divisor of 90 and 160. 90 = 2x3x5x3 2)90 150 First Method 150 =2x3x5x5 • 5 )45 75 The prime factors com- Ans. 30 = 2 X 3 X 5 3 )9 15 ™<^» ^^ both 90 and 150 Q g are 2, 3, and 5. Since 2 X 3 X 5 = 30 Ans. *^® greatest common di- visor of two or more num- 90)150(1 l>ers is the product of QQ their common factors, 30 ^vQ^/^ is. the greatest common w;^^.-^ divisor of 90 and 160. Second Method 60 GHreatest Common Divisor 30)60(2 ^Q To find the greatest — common divisor when the numbers cannot be readily factored, divide the larger by the smaller, then the last divisor by the last remainder until there is no remainder. The last divisor wiU be the greatest common divisor. If the greatest com- mon divisor is to be found of more than two numbers, find the greatest common divisor of two of them, then of this divisor and the third num- ber, and so on. The last divisor will be the greatest common divisor of all of them. EXAMPLES Find the greatest common divisor : 1. 270,810. 3. 504,560. 5. 72,153,315,2187. 2. 264,312. 4. 288,432,1152. Least Common Multiple The product of two or more numbers is called a multiple of each of them; 4, 6, 8, 12 are multiples of 2. The common 16 VOCATIONAL MATHEMATICS FOR GIRLS multiple of two or more numbers is a number that is divisible by each of the numbers without a remainder ; 60 is a common multiple of 4, 5, 6. The least common multiple of two or more numbers is the smallest common multiple of the number; 30 is the least common multiple of 3, 5, 6. To find the least common multiple of two or more numbers. Example. — Find the least common multiple of 21, 28, ^d 30. • . , „^, , .. First Method 21 = 3 X 7 Take all the factors of the first number, all of 28 = 2 X 2 X T the second not already represented in the first, etc. 30 = 2 X 3 X 5 Thus, 3 X 7 X 2 X 2 X 5 = 420 i. a JJf. Second Method 2 )21 28 30 3)21 14 15 7 )7 14 5 12 5 2 X 3 X 7 X 1 X 2 X 5 = 420 i. a JIf. Divide any two or more numbers by a prime factor contained in them, like 2 in 28 and 80. Write 21 which is not divided by the 2 for the next quotient together with the 14 and 16. 3 is a prime factor of 21 and 15 which gives a quotient of 7 and 6 with 14 written in the quotient undi- vided. 7 is a prime factor of 7 and 14 which gives a remainder of 1, 2 ; and 6 midivided is written down as before. The product 420 of all these divisors and the last quotients is the least common multiple of 21, 28, and 30. EXAMPLES Find the least common multiple : 1. 18, 27, 30. 2. 15, 60, 140, 210. 3. 24, 42, 54, 360. 4. 25, 20, 35, 40. 5. 24, 48, 96, 192. 6. What is the shortest length of rope that can be cut into pieces 32', 36', and 44' long ? REVIEW. OF ARITHMETIC 17 Fractions A fraction is one or more equal parts of a unit. If an apple be divided into two equal parts, each part is one-half of the apple, and is expressed by placing the number 1 above the number 2 with a short line between: ^. A fraction always indicates division. In ^, 1 is the dividend and 2 the divisor; 1 is called the numerator and 2 is called the denominator, A common fraction is one which is expressed by a numerator written above a line and a denominator below. The nu- merator and denominator are called the terms of the fraction, A proper fraction is a fraction whose value is less than 1 j its numerator is less than its denominator, as |, |, f,, -J^. An improper fraction is a fraction whose value is 1 or more than 1 ; its numerator is equal to or greater than its denominator, as f, \^, A number made up of an integer and a fraction is a mixed number. Read with the word and between the whole number and the fraction : 4y»^, 3|^, etc. The value of a fraction is the quotient of the numerator divided by the denominator. • EXERCISE Bead the following : 1. 1 3. 12^ s. H 7- 9^ 2. \i 4. 8J 6. 6J 8- 12^ ». i Reduction of Fractions Reduction of fractions is the process of changing their form without changing their value. To reduce a fraction to higher terms. Multiplying the denominator and the numerator of the given fraction by the same number does not change the volume of the fraction. 18 VOCATIONAL MATHEMATICS FOR GIRLS Example. — Reduce | to thirty-seconds. The denominator must be multiplied by 4 to 5 X - = — - Ana, obtain 32 ; so the numerator must be multiplied o 4 32 by the same number in order that the value of the fraction may not be changed. EXAMPLES Change the following : 1. |to27ths. 6. ^to75th8. 2. l^toGOths. 7. iJtol44ths. 3. |to40ths. a f^tol68ths. 4. Jto56ths. 9. ||to522ds. 5. ^^to50ths. 10. ^to9375ths. A fraction is said to be in its lowest terms when the numera- tor and the denominator are prime to each other. To reduce a fraction to its lowest terms. Dividing the numerator and the denominator of a fraction by the same number does not change the value of the fraction. The process of dividing the numerator and denominator of a fraction by a number common to both may be continued until the terms are prime to each other. Example. — Reduce || to fourths. The denominator must be divided by 4 to give 12 __ 3 J the new denominator 4 ; then the numerator must be 16 4 ' divided by the same number so as not to change the value of the fraction. If the terms of a fraction are large numbers, find their greatest common divisor and divide both terms by that. Example. — Reduce |^|| to fourths. (1) 2166)2888(1 (2) 2166^3 ^ 2166 2888 4 O. a D. 722)2166(3 2166 REVIEW OF ARITHMETIC 19 EXAMPLES Reduce to lowest terms : 2. If* 4. i* 6. ^y a ifi 10. ,igy^ To reduce an integer to an improper fraction. Example. — Reduce 25 to fifths. oe .• K 1 o K >i Ij^ 1 there are 4. In 26 there must be 25 times 4 = ^4^ -4^« ok *• « i «/ » ^ 26 times |, or J-j^. To reduce a mixed number to an improper fraction. Example. — Reduce 16^ to an improper fraction. 1 sevenths Since in 1 there are ^, in 16 there must 112 be 16 times J, or i^. 4 sevenths H^ + *=^^. 116 sevenths, = J^^. EXAMPLES Reduce to improper fractions : 1. 3| 3. 17^ 5. 13| 7. 359^ 2. 16^ 4. 12^ 6. 27t^ a 482^1 9. 25^ 10. Reduce 250 to 16ths. 11. Change 156 to a fraction whose denominator shall be 12. 12. In $730 how many fourths of a dollar ? 13. Change 12f to 16ths. 14. Change 24| to 18ths. To reduce an improper fraction to an integer or mixed number divide the numerator by the denominator. Example; — Reduce ^^ to an integer or mixed number. 24 16)385 oo Since ^ equal 1, ^^ will equal as many times 1 as 16 is contained in 386, or 24^^ 65 24^1^ Ana, ^^^^ 64 1 20 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES Beduce to integers or mixed numbers : 1- H 4- ^H^ 7. VV^ 10. m& 2. 2^ 5. ^j^ a -Vj^ 11. Aiy^ 3. 1^1 6. ^ 9. -HJHH^ 12. e When fractions have the same denominator their denomi- nator is called a common denominator. " Thus, JJ, T%, if'jj have a common denominator. The smallest common denominator of two or more fractions is their least common denominator. Thus, Ht i^^t 1^ become |, }, | when changed to their least common denominator. The common denominator of two or more fractions is a common multiple of their denominators. The least common denominator of two or more fractions is the least common multiple of their denominatorp. Example. — Reduce | and ^ to fractions having a common denominator. 8^6^18 The common denominator must be a T T _ JJ common multiple of the denominators 4 ^ A t ~ 74 and 6, and since 24 is the product of the ? ™ 2 1 *^^ 8^2* denominators, it is a common multiple of them. Therefore. 24 is a common denominator of } and J. To reduce fractions to fractions having the least common denominator. Example. — Reduce f, |, and -^ to fractions having the least common denominator. '>^ ^ f) 1^ '^^ I'^^JsX, common de- ^'^ * -^ ^ nominator must be the * ir ?? ._ least common multiple of 1 1 ^ the denominators 3, 6, 12, 2 X 3 X 2 = 12 L.CM. which is 12. 1 = ^; | = |^;tV = T^- •^^- Divide the least common multiple 12 by the denom- inator of each fraction, and multiply both terms by the quotient If the REVIEW OF ARITHMETIC 21 denominators should be prime to each other, their product would be their least common denominator. EXAMPLES Reduce to fractions having a common denominator : 2. 1,1 6. iyi,i 3. hi 7. hhhi *• h !*¥> i ®- h A> f > i Reduce to fractions baying least common denominator: 1- hh^ 5. ft, A, 4 2. i i, A 6. f , f , J, I 3- A> A? i 7. Which fraction is larger, Addition of Fractions Only fractions with a common denominator can be added. If the fractions have not the same denominator, reduce them to a common denominator, add their numerators, and place their sum over the common denominator. The result should be reduced to its lowest terms. If the result is an improper fraction, it should be reduced to an integer or mixed number. Example. — Add |, |^, and ^^. 1. 2)4 6 16 ^ ,,. ^ common multi- ^)^ ^ ^ pie of the de- 13 4 48 2/. C, M. nominators is this by the de- nominator of each fraction and multiplying both terms by the quotient give }}, J5, JJ. The fractions are now like fractions, and are added by adding their numerators and placing the sum over the common denomi- nator. Hence, the sum is y^, or 2/j. 22 VOCATIONAL MATHEMATICS FOR GIRLS Example. — Add 5f , 7^, and 6/^. ^5 ~ ^'So First find the sum of the fractions, '^A = '^M which is fS, or 1|§. Add this to the 6^2^ = 6J^ sum of the integers, 18. 18 + 1}J = I8|f=19|f Arts, 1»M- EXAMPLES 1. Find the " over-all " dimension of a drawing if the separate parts measure f/\ f ", y, and f", respectively. 2. Find the sum of ^, f , ^, ^, and f^. 3. Find the sum of 3f , 4|, and 2^^. 4. A seam -^^ of an inch wide is made on both sides of a piece of cloth 27 inches wide. What is the width after the seams are made ? 5. I bought cotton cloth valued at $ 6 J, silk at $ 13f , hand- kerchiefs for $2^, and hose for $2 J. What was the whole cost ? 6. A ribbon was cut into two pieces, one 8|" and the other 5 j^/' long. If ^g-" was allowed for waste in cutting, what was the length of the ribbon ? 7. Three pieces of cloth contain 38J, 12^, and 53^ yards re- spectively. What is their total length in yards ? 8. Add : lOi, 7f 11, it. 9. Add : 136^, 184|, 416^, 125 J. Subtraction of Fractions Only fractions with a common denominator can be sub- tracted. If the fractions have not the same denominator, reduce them to a common denominator and write the differ- ence of their numerators over the common denominator. The result should be reduced to its lowest terms. REVIEW OF ARITHMETIC 23 Example. — Subtract f from f . The least common denominator of f f-| = f-| = f Ana, and f is 6. f = f, and f = f Their difference is |. Example. — From 11 J subtract 5f . -i-ii__-i/\g When the fractions are changed to yi 5 — A 5 their least common denominator, they are ^ "■ —4 llj — 4|. f cannot be subtracted from J, ^i ^^ ^2* -^'^' hence 1 is taken from 11 units, changed to sixths, and added to the ), which makes f . 10} — 4j = 6} = 6^. EXAMPLES 1. From eleven yards of cloth, If yards were cut for a jacket and S^ yards for a coat. How many yards were left ? 2. From a firkin of butter containing 27|^ lb. there were sold 3 J lb. and 11^ lb. How many pounds remained ? 3. The sum of two fractions is f . One of the fractions is ^. Find the other. 4. Laura had $7^ and gave away $2^ and $3^. How much remained ? 5. The sum of 2 numbers is 37^ and one of the numbers is 28f . Find the other number. 6. By selling goods for $ ^3^, I lost $ 27|. What was the cost? 7. A man sells 9J yards from a piece of cloth containing 34 yds. How many yards remain ? 8. Mr. Brown sold goods for $ 56^, gaining $ 12. What did they cost ? 9. A dealer had 208 tons of coal and sold 92f tons. How much remained ? 10. If I buy a ton of coal for $ 6\ and sell for $ 7^, how much do I gain ? 24 VOCATIONAL MATHEMATICS FOR GIRLS 14. There were 48|^ gallons in the tank. First 4^ gallons were used, then 5 J gallons, and last 2| gallons. How many gallons were left in the tank ? 15. What is the difference between -^ and ^ j ? 16. What is the difference between 32| and 3^ ? 17. A piece of dress goods contains 60 yd. If four cuts of 12 J^, 9 1, 18f, and 10^ yd. respectively are made, what remains ? Multiplication of Fractions To multiply fractionsy multiply the numerators together for the new numerator and multiply the denominators together for the new denominator. Cancel when possible. The word of between two fractions is equivalent to the sign of multiplication. To multiply a mixed number by an integer, multiply the whole number and the fraction separately by the integer tJien add the products. To multiply two mixed numbers, change each to an improper fraction and multiply. Example. — Multiply | by f . ^ multiplied by } is the same as } o/ f . 3 and 5 are prime to each other so that answer is f . This method of solution is the same as multiplying the numerators together for a new numerator and the denominators for a new denominator. Cancellation shortens the process. Example. — Find the product of 124f and 5. 124i p. If the fraction and integer are mul- — 3T ft 8 — If — Q 8 tiplied separately by 6, the result is 6 3f 6 X f - V - ^f ^.j^gg J _ Y _ 3j^ and 6 times 124 = 620 623| Ans. 620. 620 + 3} =623}. REVIEW OF ARITHMETIC 25 EXAMPLES 1. William earns 83^ cents a day. How much will he earn in five weeks ? 2. One bag of flour costs 75 cents. How much will three barrels cost ? A barrel h6lds 8 bags. 3. Erom a barrel of flour containing 196 lb., 24^ lb. were taken. At another time ^ of the remainder was taken. How many pounds were left ? 4. Multiply f of f by I of |. 5. Multiply 26f by 9f 6. Find the cost of 19| yd. of cloth at 16^ cents a yard. 7. At $ 12^ each, how many tables can be bought for $280? a I paid $ 6f for a barrel of flour and sold it for $ ^ more. How much did I sell it for ? 9. What is the cost of 18 yards of cloth at 15| cents a yard ? 10. If coal cost $7^ a ton, how much will 8^ tons cost ? 11. Multiply : 32f by 8|. Division of Fractions To divide one fraction by another, invert the divisor and proceed as in multiplication of fractions. Change integers and mixed numbers to improper fractions. Example. — Divide f X | by f x |. ixf-h(fxf)= 2 4 S S ^ S '^^ divisor { x f is inverted and the ^X-X^X^ = -. Ana. result obtained by the process of cancel- ^ P ^ P ^ lation. 26 VOCATIONAL MATHEMATICS FOR GIRLS Example. — Divide 3156f by 5. 631^ Ans. 5)3156| y^^^ ^^^ integer of a mixed 30 number is large, it may be 15 divided as follows : 6 in 3166}, 15 If = J ^1 times, with a remainder of g 1 J. This remainder divided by r 5 gives ^^, which is placed at the right of the quotieut. Example. — Divide 3682 by 5^. When the dividend is a large number and 5i-) 3682 the divisor a mixed number, it is useful to re- 2 2 member that multiplying both dividend and di- TT Vqaj visor by the same number does not change the i-^ quotient. In this example we can multiply """tt Ans. }yoth. dividend and divisor by 2 and then divide as with whole numbers. The quotient is 669^^. A fraction having a fraction for one or both of its terms is called a complex fraction. To reduce a complex fraction to a simple fraction. 42 Example.— Reduce ^ to a simple fraction. 7| = ^ = '^^V- = ¥xA=if Ans. Change 4J and 7f to improper fractions, ^ and y, respectively. Per- form the division indicated with the aid of cancellation and the result will be If EXAMPLES 1. Divide fl by f 7. 296-^10^=? 2. Divide ^^ by f. a 28,769 -^7|=? 3. Divide -If by i. 7j__^ 4. Divide^ by i. ' if~' 5. Divide f by |. iof^ _ 6. 384f ^5 = ? ' I Xf REVIEW OF ARITHMETIC 27 REVIEW PROBLEMS IN FRACTIONS 1. Two and one half yards of cloth cost $ 2.75. What is the price per yard ? 2. An 8^t. can of milk is bought from a farmer for .60 cents. What is the cost per quart ? 3. I paid 56 cents for f of a yard of lace. What was the price per yard ? 4. A farmer's daughter sold a weekly supply of eggs for $5.70. If she received 28^ cents a dozen, how many dozen did she sell ? 5. If a narrow piece of goods, 6J yd. long, is cut into pieces 6f inches long, how many pieces can be cut? How much remains ? Allow \ in. for waste. 6. What is the cost of 18|^ pounds of crackers at 17^ cents a pound ? 7. A gallon (U. S. Standard capacity) contains 231 cubic inches. a. Give number of cubic inches in ^ gallon. 6. Give number of cubic inches in 1 quart. c. Give number of cubic inches in 1 pint. d. Give number of cubic inches in ^ pint. 8. A woman earns $ 2^ a day. If she spends $ If, how much do6s she save ? How many weeks (six full working days) will it take to save $ 90 ? 9. I paid 56 cents for | of a yard of lace. What was the price per yard ? 10. A furniture dealer sold a table for $ 14^, a couch for $ 45|, a desk for $ llf, and some chairs for $ 27^. Eind the amount of his sales. 11. A woman had $ 200. She lost \ of it, gave away ^ the remainder, and spent $ 20|. How much had she left ? 12. I gave $ 16^ for 33 yards of cloth. How much did one yard cost ? 28 VOCATIONAL MATHEMATICS FOR GIRLS Drill in the Use of Fractions Addition 1. i + i = ? 19. i + TV = ? 37. ^ + i = ? 2. i + i = ? 20. H-i = ? 38. ^ + i = ? 3. i + i=? 21. i + J=? 39. ;fe + i = ? 4. i + TV = ? 22. H-tJj = ? 40. H + f-? 5. i + TV = ? 23. i + TV = ? «• l + i = ? 6. i + f = ? 24. i + T^ = ? 42. i + A = ? 7- i + TV = ? 25. tV + A = ? «• T^ + |=? a i + i = ? 26. T\ + i=? 44. H-tV = ? 9. i + i = ? 27. T»ff + i = ? 45. tV + H = ? 10- 1 + T»ir = ? 28. TV + T»ff = ? 46. U + -^=? 11. i + i = ? 29. T>ff + | = ? 47. A+A = ? 12. i + f = ? 30. T»ir + i = ? 48. i + Tfj = ? 13. i + i = ? 31. i + i = ? 49. i + i = ? 14. | + i = ? 32. H-i = ? 50. i+i = ? 15. | + i = ? 33. | + i = ? 51. i + i=? 16. | + T'ir = ? 34. | + tV=? 52. i + TJff = ? ". | + A = ? 35. f + A = ? 53. | + tV=? la 1 + ,^ = ? 36. i + TV = ? 5*- l+l\ = ? Subtraction 1. i-i = ? ft i-i=? 15. |-i = ? 2. i-i = ? 9. i-i = ? 16. f-i»ff=? 3. i-i = ? 10. i-T»ir=? 17. |-^=? 4. i-TV = ? "• i-l!V = ? 18- 1-5^ = ? 5- i-liV = ? 12. i-A = ? 19. 1-| = ? 6- i-|i^ = ? 13. |-i = ? 20. i-A = ? 7- i-?V = ? 1*- |-i = ? 21. i-i = ? 9 9 22. i -lV = ? 23. i -1>V = ? 24. i -T^ = ? 25. i -A = ? 26. A -tV = ? 27. t^%-tV = ? REVIEW OP ARITHMETIC 29 33. f -i =? 44. i -A = ? «• I -A = 5 9 35. I -^ = ? 46. H-l^ = ? 36. i -T^ = ? 47. ;,V-Ti^ = ^ ? 37. ii-ii^ = ? 48. ^-^ = ? ... .. 38. \ -T^ = ? 49. I -I =? 28.tV-t'f = ? 39. ||-^ = ? 50. |-i=? 29. tV -T^ = ? «. .^_^ = ? 51. i -i =? *>.tV-1^ = ? *l--ife-liV = ? 52. |-tV = ? 31. i -I =? 42. if-^ = ? 53. ^ -^ = ? 32. f -i =? 43.^-11 = ? 54. I -^ = ? 1. i X i = ? 19. i X ^ = ? 37. ^ X i = ? 2.|Xi=? 20. ixi=? 38. ^Xi=? 3. iX\ =? 21. ixi=? 39. ^Xi=? 4. ^X^\ = ? 22. ^ XT»y = ? 40. ^XtV=? 5- iX^ = ? 23. ^ X^ = ? 41. ^Xjlj = ? 6. iX-^ = ? 24. i X^V = ? 42. isS-X5>j = ? 7. |Xi=? 25. TlirX^=? 43. i^Xi=f aixi=? 26. TVxi=? 44. ^ljXi=^ 9. ixi =? 27. tVxI =? 45. ^Xi =? W. 1X^3^=? 2a ^XtV = ? 46- T!^XtV=? 11. iXisV = ? 29. tJ5X^ = ? 47. ^X-5^=? 12. ix^ = ? 30. ^^x-^ = ? 4a ,Vx^V=? 13. |X^=? 31. |xi=? 49. ^Xi=? 14. |xi=? 32. fxi=? 50. |Xi=? 15. I X j = ? 33. I X i = ? 51. I X i = ? 16. |XtV = ? 34. I XtV = ? 52. i Xt'j=? 17. |X^7 = ? 35. f X^ = ? 53. i X^j=? ia|x^ = ? 36. I X^ = ? 54. |. X^V=? 9 9 ? 30 VOCATIONAL MATHEMATICS FOR GIRLS Division 1. i-^i -? 19. i ^i =? 37. l>^-^i =? 2. i^i =? 20. i -^i =? 38. A-i =? 3. 1-^i =? 21. i -^i =? 39. t^^i =? 4. i-^TV=? 22. i ->-tV = ? 40. 1!V-«-tV = ? 5. i^U^ = ? 23. i -,v = ? 41. sV • ^ — ^ 6. i-irV=? 24. i ^^T=? 42. ij^^=^ 7. i^i =? 25. tV-^1 =? 43. 1^-i =? 8. i-^i =? 26. TV-^i =? 44. A-^i =? 9. i-^i =? 27. TV-^i =? 45. T^-S-i =? 10. i-^iV = ? 28. tV-^tV = ? 46. Ti^-^T^ = ? 11. i^lJ^=? 29. 1 _t_ 1 — 9 47. ^^-h = '^ 12. i-^l^=? 30. lV-^l!»f = ? 48. ^-1^ = ? 13. I^i =? 31. 1 -i =? 49. i -^i =? 14. l-^i =? 32. 1 -i =? 50. i -^i =? 15. l-i =? 33. 1 +i =? 51. i -i =? 16. |-^iV = ? 34. f -^tV = ? 52. i *tV = ? 17. 1-^7=? 35. f -^li»J = ? 53. i +tV = ? 18. 1^1^=? 36. i +iiV=? 54. i -A = ? Decimal Fractions A power is the product of equal factors, as 10 x 10 = 100. 10 X 10 X 10 = 1000. 100 is the second power of 10. 1000 is the third power of 10. A decimal fraction or decimal is a fraction whose denominator is 10 or a power of 10. A common fraction may have any number for its denominator, but a decimal fraction must always have for its denominator 10, or a power of 10. A decimal is written at the right of a period (.), called the decimal point. A figure at the right of a decimal point is called a decimal figure. REVIEW OF ARITHMETIC 31 A mixed decimal is an integer and a decimal ; as^ 16.04. To read a decimal, read the decimal as an integer, and give it the denomination of the right-hand figure. To write a deci- mal, write the numerator, prefixing ciphers when necessary to express the denominator, and place the point at the left. There must be as many decimal places in the decimal as there are ciphers in the denominator. EXAMPLES 9 Read the following numbers : 1. .7 2. .07 3. .007 4. .700 5. .125 6. .0625 7. .4375 a .03125 9. .21875 10. .90625 11. .203125 12. .234375; 13. .0000054 14. 35.18006 15. .0005 16. 100.000104 17. 9.1632002 la 30.3303303 19. 9.999999 20. .10016 21. .000155 22. .26 23. .1 24. .80062 Express decimally : 1. Four tenths. 2. Three hundred twenty-five thousandths. 3. Seventeen thousand two hundred eleven hundred-thou- sandths. 4. Seventeen hundredths. 6. Five hundredths. 5b Fifteen thousandths. 7. Six ten-thousandths. a Eighteen and two hundred sixteen hundred-thousandths. 9. One hundred twelve hundred-thousandths. 10. 10 millionths. 11. 824 ten-thousandths. 12. Twenty-nine hundredths. 13. 324 and one hundred twenty-six millionths. 14. 7846 himdred-inillionths. 32 VOCATIONAL MATHEMATICS FOR GIRLS ^®* loooooooj TOT? 10000> A> To 008 OIF* 17. One and one tenth. 18. One and one hundred-thousandth. 19. One thousand four and twenty-nine liundredths. Reduction of Decimals Ciphers anneoced to a decimal do not change the value of the decimal; these ciphers are called decimal ciphers. For each cipher prefixed to a decimal, the value is diminished ten- fold. The denominator of a decimal — when expressed — is always 1 with as many ciphers as there are decimal places in the decimal. To reduce a decimal to a common fraction. Write the numerator of the decimal omitting the point for the numerator of the fra/ction. For the denominator wnte 1 with as many ciphers annealed, as there are decimal places in the dedrrval. Then reduce to lowest terms. Example. — Reduce .25 and .125 to common fractions. 1 Write 26 for the numerator and 95 = ^^ _« $!^ __ 1 J - 1 for the denominator with two O's, 1OO'~^00"~4 * which makes ^Jy^; ^ reduced to 4 lowest terms is J. 1 1 9^ — ^^^ — j^^^ _ 1 >!« « '126 is reduced to a common frac- ~ 1000 "" ^000 ~" 8 ' tion in the same way. 8 Example. — Reduce .37^ to a common fraction. 37^ has for its denominator 1 3Ii=,i = ^X^ = ? Ans ^^*^^'^^"^^^^^^io- 100 100 2 Ji/i^ 8 * This is a complex fraction 4 which reduced to lowest terms is}. REVIEW OF ARITHMETIC 33 EXAMPLES Reduce to common fractions : 1. .09375 a 2.26 11. .16f 16. .87^ 2. .15625 7. 16.144 12. .33^ 17. .66| 3. .016625 a 26.0000100 13. .06^ la .36J 4. .609375 9. 1084.0025 14. .140626 19. .83 ^ 5. .678125 10. .121- 15. .984376 20. .62| To reduce a common fraction to a decimal. Annex decimal ciphers to the numerator and divide by the de- nominator. Point off from the right of the quotient as many pla/ces as there are ciphers annexed. If there are not figures enough in the quotient, prefix ciphers. The division will not always be exact, i.e, ^ = .142|^ or .142"'". ExAUPIiE. — Reduce f to a decimal. .76 4)3.00 28 20 1 = .76 EXAMPLES • Beduce to decimals : 1- iu 6. i u. tV la H 21. sU 2- Th 7. H 12. ^ 17. 16i 22. 25.12J ^- shs 8. « 1^- IS^STF la 66| 23. 33i *• i 9- ^ 14. 12^ 19. M 24. A 5. I 10. 7i^ 15. ^ 20. I 25. tIy Addition of Decimals To add decimals, write them so that their decimal points are in a column. Add as in integers, and place the point in the sum directly wider the points above it. 34 VOCATIONAL MATHEMATICS FOR GIRLS Example. — Find the sum of 3,87,2.0983, 5.00831, .029, .831. 3.87 2 0983 Place these numbers, one under the other, with K 005^^1 decimal points in a column, and add as in addition ' of integers. The sum of these numbers should .UJy jjj^yg j.jjg decimal point in the same column as the '831 numbers that were added. 11.83661 Ans. EXAMPLES Find the sum : 1. 5.83, 7.016, 15.0081, and 18.3184. 2. 12.031, 0.0894, 12.0084, and 13.984. 3. .0765, .002478, .004967, .0007862, .17896. 4. 24.36, 1.358, .004, and 1632.1. 5. .175, 1.75, 17.5, 175., 1750. 6. 1., .1, .01, .001, 100, 10., 10.1, 100.001. 7. Add 5 tenths; 8063 millionths ; 25 hundred-thousandths ; 48 thousandths; 17 millionths; 95 ten-millionths ; 5, and 5 hundred-thousandths ; 17 ten-thousandths. 8. Add 24f , 17^, .0058, 7^, 93^- 9. 32.58, 28963.1, 287.531, 76398.9341. 10. 145., 14.5, 1.45, .145, .0145. Subtraction of Decimals To subtract decimals, tvrite the smaller number under the larger with the decimal point of the subtrahend directly under the decimal point of the minuend. Subtract as in integers, and place the point directly under the points above. Example. — Subtract 2.17857 from 4.3257. Write the lesser number under the greater, 4.32570 Minuend ^^^^ the decimal points under each other. 2.17857 Subtrahend Add a to the minuend, 4.3267, to give it the 2.14713 Remainder same, denominator as the subtrahend. Then subtract as in subtraction of integers. Write the remainder with decimal point under the other two points. REVIEW OF ARITHMETIC 35 EXAMPLES Subtract : 1. 69.0364-30.8691 = ? 3. .0626 - .03125 = ? 2. 48.7209-12.0039 = ? 4. .00011 - .000011 = ? 5. 10-.1 + .0001 = ? 6. From one thousand take five thousandths. 7. Take 17 hundred-thousandths from 1.2. 8. From 17.37^ take 14.16^. ' 9. Prove that ^ and .600 are equal. 10. Find the difference between -^^ and yf§^. Multiplication of Decimals To multiply decimals proceed as in integers, and give to tJie product as many decimal figures as there are in both multiplier and multiplicand. When there are not figures enough in the product, prefix ciphers. Example. — Find the product of 6.8 and .63. 6.8 Multiplicand .63 Multiplier ^'^ ^® ^^® multiplicand and .63 the multiplier. ofu Their product is 4.284 with three decimal figures, the number of decimal figures in the multiplier and multiplicand. 408 4.284 Product Example. — Find the product of .06 and .3. .06 Multiplicand The product of .06 and .3 is .016 with a cipher .3 Multiplier prefixed to make the three decimal figures re- .016 Product quired in the product. EXAMPLES Find the products : 1. 46.26 X. 126 3. .016 X. 06 2. 8.0626 X. 1875 4. 26.863 x 4^ 36 VOCATIONAL MATHEMATICS FOR GIRLS 5. 11.11x100 a .325x121 6. .5625x6.28125 9. .001542 x .0052 7. .326 X 2.78 10. 1.001 x 1.01 To multiply by 10, 100, 1000, etc,,, remove the point one place to the right for each cipher m the multiplier. This can be performed without writing the multiplier. Example.— Multiply 1.625 by 100. 1.626 X 100 = 162.6 To multiply by 200, remove the point to the right and multiply by 2, Example. — Multiply 86.44 by 200. 86.44. 2 17,288 EXAMPLES Find the product of : 1. 1 thousand by one thousandth. 2. 1 million by one millionth. 3. 700 thousands by 7 hundred-thousandths. 4. 3.894 X 3000 5. 1.892 x 2000. Division of Decimals To divide decimals proceed as in integers, and give to the quo- tient as many decimal figures as the number in the dividend ex- ceeds tliose in the divisor. Example. — Divide 12.685 by .5. The number of decimal figures in Divisor . 5)12.685 Dividend the quotient, 12.685, exceeds the num- 25.37 Quotient her of decimal figures in the divisor, .6, by two. So there must be two deci- mal figures in the quotient. REVIEW OF ARITHMETIC 37 Example. — Divide 399.552 by 192. , When the divisor is an integer, ^•^^^ Q^iOtient the point in the quotient should be Divisor 192)399.552 Dividend placed directly over the point in 384 the dividend, and the division per- 1555 formed as in integers. This may -I rog be proved by multiplying divisor — :r^ by quotient, which would give the ^^^ dividend. 192 Example. — Divide 28.78884 by 1.25. When the divisor contains 23.031 •*• Quotient decimal figures, move the point Divisor 1.25.)28. 78.884 Dividend in both divisor and dividend as 250 many places to the right as 3yg there are decimal places in the OTK divisor, which is equivalent to — — ^ multiplying both divisor and l^ dividend by the same number ^*^ and does not change the quo- 134 tient. Then place the point in 125 the quotient as if the divisor "~9 Remainder ^®^® ^'^ integer. In this ex- ample, the multiplier of both dividend and divisor is 100. EXAMPLES Find the quotients : 1. .0625 -f- .125 5. 1000 -^ .001 8. 1.225 -^ 4.9 2. 315.432 -^ .132 6. 2.496 -^. 136 9. 3.1416 -f- 27 3. .75^.0125 7. 28000^16.8 10. 8.33 -^ 5 4. 125-^12| To divide by 10, 100, 1000, e^c, remove the point one place to the left for each cipher in the divisor. To divide by 200, remove the point two places to the left, and divide by 2. \ 38 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES Find the quotients : 1. 38.64 -^ 10 6. 865.45^5000 2. 398.42 -f- 1000 7. 38.28^400 3. 1684.32 -^ 1000 8. 2.5^500 4. 1.155 -^ 100 9. .5^10 5. 386.54-- 2000 10. .001 -^ 1000 REVIEW EXAMPLES 1. Add 28.03, .1674, .08309, 7.00091, .1895. 2. Subtract 1.00894 from 13.0194. 3. Multiply 83.74 X 3.1416. 4. Divide 3.1416 by 8.5. 5. Perform the following calculations : .7854 X 35 x 7.5. 6. Perform the following calculations : 65.3 X 3.1416 X .7854 600 X 3.5 X 8.3 7. Change the following fractions to decimals : («) lAr^ W A» (c) ^ , (d) y|^, (e) ^j, (/) ^^, (g) ^. a Change the following decimals to common fractions : (a) .33^, (b) .25, (c) .125, (d) .375, (e) .437^, (/) .875. Parts of 100 or 1000 1. What part of 100 is 12^ ? 25 ? 33| ? 2. What part of 1000 is 125 ? 250 ? 333| ? 3. How much is ^ of 100 ? Of 1000? 4. How much is i of 100 ? Of 1000? 5. What is I of 100 ? Of 1000? Example. — How much is 25 times 24 ? 100 times 24 = 2400. 25 times 24 = } as much as 100 times 24 = 600. Ans. REVIEW OF ARITHMETIC 39 Short Method of Multiplication To multiply by 25, multiply by 100 and divide by 4 33J, multiply by 100 and divide by 3 16^, multiply by 100 and divide by 6 12|, multiply by 100 and divide by 8 9, multiply by 10 and subtract the multiplicand ; 11, if more than two figures, multiply by 10 and add the multiplicand to the product ; 11, if two figures, place the figure that is their sum between them. 63 X 11 = 693 74 X 11 = 814 Note that when the sum of the two figures exceeds nine, the one in the tens place is carried to the figure at the left. EXAMPLES Multiply by the short process : 1. 81 by 11 = ? 10. 68byl6f=:? 2. 75by33i = ? 11. 112 by 11 = ? 3. 128 by 12^ = ? 12. 37 by 11 = ? 4. 87 by 11 = ? 13. 4183 by 11 = ? 5. 19 by 9 = ? 14. 364by33i = ? 6. 846 by 11 = ? 15. 8712 by 12^ = ? 7. 88 by 11 = ? 16. 984byl6f = ? a 19 by 11 = ? 17. 36 by 25 = ? 9. 846byl6| = ? 18. 30by333J = ? Aliquot Parts of $ 1.00 The aliquot parts of a number are the numbers that are exactly contained in it. The aliquot parts of 100 are 5, 20, 12^, 16|, 33i, etc. The monetary unit of the United States is the dollar, con- taining one hundred cents, which are written decimally. 40 VOCATIONAL MATHEMATICS FOR GIRLS 6^ cents = $ ^V ^^ cents = $ ^ = quarter dollar 8| cents = $ ^^ ^H ^®^*s = $ | 12^ cents = $ ^ 50 cents = $ ^ = half dollar 16| cents = $ ^ 10 mills = 1 cent, ct. = $ .01 or $ 0.01 5 cents = 1 " nickel " = $ .05 10 cents = 1 dime, c?. = $ .10 10 dimes = 1 dollar, $ = $ 1.00 10 dollars = 1 eagle, E. = $ 10.00 Example. — Wliat will 69 pairs of stockings cost at 16| cents a pair ? 69 pairs will cost 69 x 16§ cts., or 69 x $ J = ^- = $!!{ = $ 11.60. Example. — At 25^ a peck, how many pecks of potatoes can be bought for $ 8.00 ? 8-4-J = 8xf = 32 pecks. Ans, Review of Decimals 1. For work on a job qne woman receives $ 13.75, a second woman $ 12.45, a third woman $ 14.21, and a fourth woman $ 21.85. What is the total amount paid for the work ? 2. A pipe has an inside diameter of 3.067 inches and an outside diameter of 3.428 inches. What is the thickness of the metal of the pipe ? 3. At 4^ cts. a pound, what will be the cost of 108 boxes of salt each weighing 29 lb. ? 4. A dressmaker receives $ 121.50 for doing a piece of work. She gives $ 12.25 to one of her helpers and $ 10.50 to another. She also pays $ 75.75 for material. How much does she make on the job ? 5. An automobile runs at the rate of 9^ miles an hour. How long will it take it to go from Lowell to Boston, a dis- tance of 26.51 miles ? REVIEW OF ARITHMETIC 41 6. A man uses a gallon of gasoline in traveling 16 miles. If a gallon costs 23 cents, what is the cost of fuel per mile ? 7. Which is cheaper, and how much, to have a 13^ cents an hour woman take 13J hours on a piece of work, or hire a 17^ cents an hour woman who can do it in 9^ hours ? 8. On Monday 1725.25 lb. of coal are used, on Tuesday 2134.43 lb., on Wednesday 1651.21 lb., on Thursday 1821.42 lb., on Friday 1958.82 lb., and on Saturday 658.32 lb. How many pounds of coal are used during the week ? 9. If, in the example above, there were 10,433.91 lb. of coal on hand at the beginning of the week, how much was left at the end of the week ? 10. The distance traveled in an automobile is measured by an instrument called a speedometer. A man travels in a week the following distances: 87.5 mi., 49.75 mi., 112.60 mi., 89.7 mi., 119.3 mi., and 93.75 mi. What is the total distance traveled ? U. An English piece of currency corresponding to our five- dollar bill is called a pound sterling and is worth $4,866^. How much more is a five-dollar bill than a pound ? 12. An alloy is made of copper and zinc. If .66 is copper and .34 is zinc, how many pounds of zinc and how many pounds of copper will there be in a casting of the alloy weighing 98 lb. ? 13. A train leaves New York at 2.10 p.m. and arrives in Philadelphia at 4.15 p.m. The distance is 90 miles. What is the average rate per hour of the train ? 14. The weight of a foot of ^" steel bar is 1.08 lb. Find the weight of a 21-foot bar. 15. A steam pump pumps 3.38 gallons of water to each stroke and the pump makes 51.1 strokes per minute. How many gallons of water will it pump in an hour ? 16. At 12^ cents per hour, what will be the pay for 23^ days if the days are 10 hours each ? 42 VOCATIONAL MATHEMATICS FOR GIRLS Compound Numbers A number composed of different kinds of concrete units that are related is a compound number : as, 3 bu. 2 pk. 1 qt. A denomination is a name given to a unit of measure or of weight. A number having one or more denominations is also called a denominate number. Reduction is the process of changing a number from one denomination to another without changing its value. Changing to a lower denomination is called reduction descend- ing : as, 2 bu. 3 pk. = 88 qt. Changing to a higher denomi- nation is called reduction ascending ; as, 88 qt. = 2 bu. 3 pk. Linear Measure is used in measuring lines or distance Table 12 inches (in.) = 1 foot, ft. 3 feet = 1 yard, yd. 5 J yards, or 16 J feet = 1 rod, rd. 820 rods, or 5280 feet = 1 mile, mi. 1 mi. = 320 rd. = 1760 yd. = 5280 ft. = 63,360 in. Square Measure is used in measuring surfaces. Table 144 square inches = 1 square foot, sq. ft. 9 square feet = 1 square yard, sq. yd. 30J square yards 1 ^ j ^^^ ^^ ^ ^ 272} square feet J 160 square rods = 1 acre, A. 640 acres = 1 square mile, sq. mi. 1 sq. mi. = 640 A. = 102,400 sq. rd. = 3,097,600 sq. yd. Cubic Measure is used in measuring volumes or solids. Table 1728 cubic inches = 1 cubic foot, cu. ft. 27 cubic feet = 1 cubic yard, cu. yd. 16 cubic feet = 1 cord foot, cd. ft. 8 cord feet, or 128 cu. ft. = 1 cord, cd. 1 cu. yd. = 27 cu. ft. = 46,656 cu. in. REVIEW OF ARITHMETIC 43 Liquid Measure is used in measuring liquids. Table 4 gills (gi.)=l pint, pt. 2 pints = 1 quart, qt. 4 quarts = 1 gallon, gal. 1 gal. = 4 qt. = 8 pt. = 82 gi. A gallon contains 231 cubic inches. The standard barrel is 31} gal., and the hogshead 63 gal. Dry Measure is used in measuring roots^ gradn^ vegetables^ etc. Table 2 pints = 1 quart, qt 8 quarts = 1 peck, pk. 4 pecks = 1 bushel, bu. 1 bu. = 4 pk. = 82 qt. = 64 pints. The bushel contains 2150.42 cubic inches; 1 dry quart contains 67.2 cu. in. A cubic foot is ff of a bushel. Ayoirdupois Weight is used in weighing all common articles ; aS; coal, groceries, hay, etc. Table 16 ounces (oz.) = 1 pound, lb. 100 pounds = 1 hundredweight, cwt. ; or cental, ctl. 20 cwt., or 2000 lb. = 1 ton, T. 1 T. = 20 cwt. = 2000 lb. = 32,000 oz. The long ton of 2240 pounds is used at the United States Custom House and in weighing coal at the mines. Measure of Time. Table 60 seconds (sec.) = 1 minute, min. 60 minutes = 1 hour, hr. 24 hours = 1 day, da. 7 days = 1 week, wk, 366 days = 1 year, yr. 366 days = 1 leap year, 100 years = 1 century. 44 VOCATIONAL MATHEMATICS FOR GIRLS Counting. Table 12 thiDgs = 1 dozen, doz. 12 dozen = 1 gross, gr. 12 gross = 1 great gross, G. gr. Paper Measure. Table 24 sheets = 1 quire 2 reams = 1 bundle 20 quires = 1 ream 6 bundles = 1 bale Reduction Descending Example. — Reduce 17 yd. 2 ft. 9 in. to inches. 1 yd. = 3 ft. 17 yd. = 17 X 3 = 51 ft. 51 + 2 = 53 ft. 1 ft. = 12 in. 63 ft. = 53 X 12 = 6.36 in. 636 + 9 = 646 in. Ans. EXAMPLES Reduce to lower denominations: 1. 46 rd. 4 yd. 2 ft. to feet. 2. 4 A. 15 sq. rd. 4 sq. ft. to square inches. 3. 16 cu. yd. 25 cu. ft. 900 cu. in. to cubic inches. 4. 15 gal. 3 qt. 1 pt. to pints. 5. 27 da. 18 hr. 49 min. to seconds. Reduction Ascending Example. — Reduce 1306 gills to higher denominations. 4 )1306 gi. Since in 1 pt. there are 4 gi., in 1306 gi. 2 )326 pt. + 2 gi. there are as many pints as 4 gi. are contained 4 )163 qt. times in 1306 gi., or 326 pt. and 2 gi. remainder. 40 gal. + 3 qt. In the same way the quarts and gallons are 40 gal. 3 qt. 2 gi. Ans. found. So there are in 1306 gi., 40 gal. 3 qt. 2gi. REVIEW OF ARITHMETIC 45 EXAMPLES Reduce to higher denominations : 1. Reduce 225,932 in. to miles, etc. 2. Change 1384 dry pints to higher denominations. 3. In 139,843 sq. in. how many square miles, rods, etc. ? 4. How many cords of wood in 3692 cu. ft. ? 5. How many bales in 24,000 sheets of paper ? A denominate fraction is a fraction of a unit of weight or measure. To reduce denominate fractions to integers of lower denominations. Change the fraction to the next lower denomination. Treat the fractional part of the product in the same way, and so pro- ceed to the required denomination. Example. — Reduce ^ of a mile to rods, yards, feet, etc. f of 320 rd. = ^Af^ rd. = 228^ rd. ^of yyd. = ttyd. =8}yd. ^ of 3 ft. = Of ft. f of 12 in. = y in. = 5^ in. j of a mile = 228 rd. 3 yd. ft. 6f in. The same process applies to denominate decimals. To reduce denominate decimals to denominate numbers. Example. — Reduce .87 bu. to pecks, quarts, etc. Change the decimal fraction to the next lower denomination. Treat the decimal part of the product in the same way, and so proceed to the re- q"q7 q4. quired denomination. 3 pk. 3 qt. 1.68 pt. Ans. .87 bu. 4 .84 qt. 2 3.48 pk. 1.68 pt. .48 8 pk. 46 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES Reduce to integers of lower denominations : 1. f of an acre. 3. ^ of a ton. 2. .3125 of a gallon. 4. .51625 of a mile. 5. Change f of a year to months and days. 6. .2364 of a ton. 7. What is the value of | of 1^ of a mile ? a Reduce f^ bu. to integers of lower denominations. 9. .375 of a month. 10. ^j acre are equal to how many square rods, etc. ? Addition of Compound Numbers Example. — Find the sum of 7 hr. 30 min. 45 sec, 12 hr» 25 min. 30 sec, 20 hr. 15 min. 33 sec, 10 hr. 27 min. 46 sec The sum of the seconds = 164 sec. = 2 min. 34 sec. Write the 34 sec. under the sec. column and add the 2 min. to the min. column. Add the other columns 50 39 34 in the same way. 60 hr. 39 min. 34 sec. Ana, Subtraction of Compound Numbers Example. — From 39 gal. 2 qt. 2 pt. 1 gi. take 16 gal. 2 qt. 3 pt. 3 gi. J ^ ^ . As 3 gi. cannot be taken from 1 gi., 4 gi. 9 9 1 or 1 pt. are borrowed from the pt. column -,(, o q q *"^<1 added to the 1 gi. Subtract 3 gi. from rr — X — T — — the 6 gi. and the remainder is 2 gi. Continue on , o .. o • ^ in the same way until all are subtracted. 22 gal. 3 qt. 2 gi. Ans. _, ^, . ^ . _ i o * /^ * o ^ ° Then the remamder is 22 gal. 3 qt. pt. 2 gi. hr. min. 7 30 46 12 26 30 20 16 33 10 27 46 REVIEW OF ARITHMETIC 47 Multiplication of Compound Numbers Example. — Multiply 4 yd. 2 ft. 8 in. by 8. 8 times 8 in. = 64 in. = 6 ft. 4 in. Place the 4 in. under the in. column, and add the 6 ft. to the product of 2 ft. by 8, which equals 21 ft. =7 yd. Add 7 yd. to the product of 4 yd. by 8 = 39 yd. yd. ft. in. 4 2 8 8 39 4 39 yd. 4 in. Ans. Division of Compound Numbers Example. — Find -^ of 42 rd. 4 yd. 2 ft. 8 in. rd. yd. ft. In. 36)42 4 36 7 2 8(1 rd. 3i 86 SSi + 4 36)4iii yd. (1 yd 36 36)24i(0 ft. 12 294 + 8 . 36)302(8Jf in. 280 3 22 22Jft. 12 1 rd. 1 yd. 8^ in. ^ of 42 rd. = 1 rd. ; re- mainder, 7 rd. = 38J yd. ; add 4 yd. = 42J yd. ^ of 42J yd. = 1 yd. ; remainder, 7i yd., = 22J ft. = 24J ft. ^ of 24J ft. = ft. 24 J ft. =294 in. ; add 8 in. =302 in. ■^ of 302 in. = m in. Difference between Dates Example. — Find the time from Jan. 25, 1842, to July 4, 1896. 1896 1842 7 1 4 26 64 yr. 6 mo. 9 da. Ans, It is customary to consider 30 days to a month. July 4, 1896, is the 1896th yr., 7th mo., 4th da., and Jan. 26, 1842, is the 1842d yr., Ist. mo., 25th da. Subtract, taking 30 da. for a month. 48 VOCATIONAL MATHEMATICS FOR GIRLS Example. — What is the exact number of days between Dec. 16, 1895, and March 12, 1896 ? Dec. 15 Do not count the first day mentioned. There Jan. 31 are 15 days in December, after the 16th. Jan- Feb. 29 uary has 81 days, February 29 (leap year), Mar. 12 and 12 days in March ; maldng 87 days. 87 days. Ana. EXAMPLES 1. How much time elapsed from the landing of the Pil- grims, Dec. 11, 1620, to the Declaration of Independence, July 4, 1776? 2. Washington was born Feb. 22, 1732, and died Dec. 14, 1799. How long did he live? 3. Mr. Smith gave a note dated Feb. 25, 1896, and paid it July 12, 1896. Find the exact number of days between its date and the time of payment. 4. A carpenter earning $ 2.50 per day commenced Wednes- day morning, April 1, 1896, and continued working every week day until June 6. How much did he earn ? 5. Find the exact number of days between Jan. 10, 1896, and May 5, 1896. 6. John goes to bed at 9.15 p.m. and gets up at 7.10 a.m. How many minutes does he spend in bed ? To multiply or divide a compound number by a fraction. To multiply by a fractiorij multiply by the numerator , and divide the product by the denominator. To divide by a fraction^ multiply by the denominator, and divide the product by the numerator. When the multiplier or divisor is a mixed number, reduce to an improper fraction^ and proceed as above. REVIEW OF ARITHMETIC 49 EXAMPLES 1. How much is f of 16 hr. 17 min. 14 sec. ? 2. A field contains 10 A. 12 sq. rd. of land, which is f of the whole farm. Find the size of the farm. 3. If a train runs 60 mi. 35 rd. 16 ft. in one hour, how far will it run in 12^ hr. at the same rate of speed ? 4. Divide 14 bu. 3 pk. 6 qt. 1 pt. by f 5. Divide 5 yr. 1 mo. 1 wk. 1 da. 1 hr. 1 min. 1 sec. by 3f . REVIEW EXAMPLES 1. A time card on a piece of work states that 2 hours and 15 minutes were spent on a skirt, 1 hour and 12 minutes on a waist, 2 hours and 45 minutes on a petticoat, and 1 hour and 30 minutes on a jacket. What was the number of hours spent on all the work ? 2. How many parts of a sewing machine, each weighing 14 oz., can be obtained from 860 lb. of metal if nothing is allowed for waste ? 3. How many feet long must a dry goods store be to hold a counter 8' 6", a bench 14' 4", a desk 4' 2", and a counter 7' 5"f placed side by side, if 3' 3" are allowed between the pieces of furniture and between the walls and the counters ? 4. How many gross in a lot of 968 buttons ? 5. Find the sum of 7 hr. 30 min. 45 sec, 12 hr. 25 min. 30 sec, 20 hr. 15 min. 33 sec, 10 hr. 27 min. 46 sec. 6. If a train is run for 8 hr. at the average rate of 50 mi. 30 rd. 10 ft. per hour, how great is the distance covered ? 7. A telephone pole is 31 ft. long. If 4 ft. 7 in. are under ground, how high (in inches) is the top of the pole above the street ? 8. If 100 bars of iron, each 2|' long, weigh 70 lb., what is the total weight of 2300 bars ? 50 VOCATIONAL MATHEMATICS FOR GIRLS 9. If a cubic foot of water weighs 62^ lb., how many ounces does it weigh ? 10. A farmer's wife made 9 pounds 7 ounces of butter and sold it at 41 cents a pound. How much did she receive ? U. A peck is what part of a bushel ? 12. A quart is what part of a bushel ? of a peck ? 13. I have 84 lb. 14 oz. of salt which I wish to put into packages of 2 lb. 6 oz. each. How many packages will there be ? 14. If one bottle holds 1 pt. 3 gi., how many dozen bottles will be required to hold 65 gal. 2 qt. 1 pt. ? 15. How many pieces 5^" long can be cut from a rod 16' 8" long, if 5" are allowed for waste ? 16. What is the entire length of a .railway consisting of five different lines measuring respectively 160 mi. 185 rd. 2 yd., 97 mi. 63 rd. 4 yd., 126 mi. 272 rd. 3 yd., 67 mi. 199 rd. 5 yd., and 48 mi. 266 rd. 5 yd. ? Percentage Percentage is a process of solving questions of relation by means of hundredths or per cent (%). Every question in percentage involves three elements : the rate per cent, the base, and the percentage. The rate per cent is the number of hundredths taken. The base is the number of which the hundredths are taken. The percentage is the result obtained by taking a certain per cent of a number. Since the percentage is the result obtained by taking a cer- tain per cent of a number, it follows that the percentage is the product of the base and the rate. The rate and base are always factors, the percentage is the product. Example. — How much is 8 % of $ 200 ? 8 % of ^200 = 200 X .08 = $ 16. (1) REVIEW OF ARITHMETIC 51 In (1) we have the three elements: 8% is the rate, $200 is the base, and $ 16 is the percentage. Since $ 200 x .08 = J| 16, the percentage ; tl6 -^ .08 = $ 200, the base ; and $ 16 -$- $ 200 = .08, the rate. If any two of these elements are given, the other may be found : Base X Rate = Percentage Perceyitage -^ Ba^e = Ba^e Percentage -j- Ba^se = Rate Per cent is commonly used in the decimal form, but many operations may be much shortened by using the common frac- tion form. 1 % = .01 = T^ i % = .00^ or .006 10%= .10 = tV 33i%=.33i = i 100 % = 1.00 = 1 8i % = .08^ = .0825 12^ % = .121. or .125 = 1 I % = .00^ = .00125 There are certain per cents that are used so frequently that we should memorize their equivalent fractions. H%=^ 33i%=i 66f%=| 10% =,1, mfo=i 76% =f 12^% =i 40%= I 80% =f 16i%=i 50%=^ 83i%=| 20%=^ 60% =1 87i%=i 25% =i 62^% = I EXAMPLES 1. Find 75 % of $ 368. 2. Find 15 % of $ 412. 3. 840 is 33^ % of what number ? 4. 616 is 16 % of what number ? 5. What per cent of 12 is 8 ? 52 VOCATIONAL MATHEMATICS FOR GIRLS 6. What per cent of a foot is 8 inches ? 11 inches ? 4 inches ? 7. A technical high school contains 896 pupils ; 476 of the pupils are girls. What per cent of the school is girls ? 8. Out of a gross of bottles of mucilage 9 were broken. What was the per cent broken ? Trade Discount Merchants and jobbers have a price list. From this list they give special discounts according to the credit of the cus- tomer and the amount of supplies purchased, etc. If they give more than one discount, it is understood that the first means the discount from the list price, while the second denotes the discount from the remainder. EXAMPLES 1. What is the price of 200 spools of cotton at $ 36.68 per M. at40% off? 2. Supplies from a dry goods store amounted to $ 68.75. If 12^ % were allowed for discount, what was the amount paid ? 3. A dealer received a bill amounting to $212.75. Suc- cessive discounts of 15%, 10%, and 5% were allowed. What was the amount to be paid ? 4. 2 % is usually discounted on bills paid within 30 days. If the following are to be paid within 30 days, what will be the amounts due ? a. $ 30.19 c. $399.16 e. $1369.99 b. 2816.49 d. 489.01 /. 918.69 5. Millinery supplies amounted to $ 127.79 with a discount of 40 % and 15 %. What was the net price ? 6. What single discount is equivalent to a discount of 45 % and 10 % ? 7. What single discount is equivalent to 20 %, and 10 % ? REVIEW OF ARITHMETIC 53 Simple Interest Money that is paid for the use of money is called interest. The money for the use of which interest is paid is called the principal, and the sum of the principal and interest is called the amount. Interest at 6 % means 6 % of the principal for 1 year ; 12 months of 30 days each are usually regarded as a year in com- puting interest. There are several methods of computing interest. Example. — What is the interest on $ 100 for 3 years at 6 % ? $100 .06 $ 6.00 interest for one year. Or, jj^ x J^ x | = $ 18. Am. 3 % 18.00 interest for 3 years. Ans. $ 100 + $ 18 = $ 118, amount. Principal x Rate X Time = Interest Example. — What is the interest on $ 297.62 for 6 yr. 3 mo. at6%? a 297 .62 3 M Or, -?- X ?^^?^X 21 = li§750:55 = 1^93.76. $17.8672 ' 100 1 i 200 6i 2 4.4643 89.2860 Note. — Final results should not include $93.7503 $93.76. Ans, mills. Mills are disregarded if less than 6, and called another cent if 5 or more. EXAMPLES 1. What is the interest on $ 586.24 for 3 months at 6 % ? 2. What is the interest on $ 816.01 for 9 months at 5 % ? 3. What is the interest on $ 314.72 for 1 year at 4 % ? 4. What is the interest on $ 876.79 for 2 yr. 3 mo. at ^ % ? 5. What is the interest on $ 2119.70 for 6 yr. -2 mo. 13 da. at 5i%? 54 VOCATIONAL MATHEMATICS FOR GIRLS The Six Per Cent Method By the 6 ^c method it is convenient to find first the interest of % 1, then multiply it by the principal. Example. — r What is the interest on % 60.24 at 6 % for 2 yr. 8 mo. 18 da. ? Interest on $ 1 for 2 yr. =2 x $ .06 = $.12 Interest on f 1 for 8 mo. = 8 x $ .OOJ = .04 Interest on $ 1 for 18 da. = 18 x $ .OOOJ = .003 Interest on $ 1 for 2 yr. 8 mo. 18 da. % .163 Interest on $ 60.24 is 60.24 times % .163 = % 8.19. Ans, Second Method. — Interest on any sum for 60 days at 6 % is yJ-^ of that sum and mxiy he expressed by momng the decimal point two places to the left. The interest for 6 days may be expressed by moving the decimal three places to the left. Example. — What is the interest on $ 394.50 for 96 days at 6%? $3.9460, interest on $394.60 for 60 days at 6 ^. 1.9726, interest on $394.60 for 30 days at 6 ^. .3946, interest on $ 394.60 for 6 days at 6 9^). $6.3120, interest on $394.60 for 96 days at 6 <^. Ana, $ 6.31. Example. — What is the interest on $ 529.70 for 78 days at 8% ? $6,297, interest on $629.70 for 60 days at 6 %. 1.689, interest on $629.70 for 18 days (6 days x 3). $6,886, interest on $629.70 for 78 days at 6 ^. .886 + $2,296 = $9,181. Am. $9.18. EXAMPLES Find the interest and amount of the following : 1. S 2350 for 1 yr. 3 mo. 6 da. at 6 %. 2. $ 125.75 for 2 yr. 5 mo. 17 da. at 7 %. 3. $ 950.63 for 3 yr. 7 mo. 21 da. at 5 %. 4. $ 625.57 for 2 yr. 8 mo. 28 da. at 8 %. REVIEW OF ARITHMETIC 55 Exact Interest When the time includes days, interest computed by the 6% method is not strictly exact, by reason of using only 30 days for a month, which makes the year only 360 days. The day is therefore reckoned as ^hf ^^ ^ 7©*^, whereas it is -^ of a year. To compute exact interest, find the exact time in days, and con- sider 1 day^s interest as ^-J-j of 1 year's interest. Example. — Find the exact interest of $ 368 for 74 days at 7%. ^368 X .07 = $26.06, 1 year's interest. 74 days' interest is ^ of 1 year's interest. ^ of $ 25.06 = $ 5.08. Ans, EXAMPLES Find the exact interest of : 1. $324 for 15 da. at 5 %. 2. $ 253 for 98 da. at 4 %. 3. $624 for 117 da. at 7 %. 4. $ 620 from Aug. 15 to Nov. 12 at 6 %. 5. $ 153.26 for 256 da. at 5| %. 6. $ 540.25 from June 12 to Sept. 14 at 8 %. Rules for Computing Interest The following will be found to be excellent rules for finding the inter- est on any principal for any number of days. Divide the principal by 100 and proceed as follows: 2 % — Multiply by number of days to run, and divide by 180. 2^ % — Multiply by number of days, and divide by 144. 3 % — Multiply by number of days, and divide by 120. S^ % — Multiply by number of days, and divide by 102.86. 56 VOCATIONAL MATHEMATICS FOR GIRLS 4 % — Multiply by number of days, and divide by 90. 5 % — Multiply by number of days, and divide by 72. 6 % — Multiply by number of days, and divide by 60. 1 fjo — Multiply by number of days, and divide by 61.43. 8 ^0 — Multiply by number of days, and divide by 45. Savings Bank Compound Interest Table Showing the amount of $ 1, from 1 year to 16 years, with compound interest added semiannually, at different rates. Pbb Cent 8 4 5 6 7 8 9 iyear 101 102 102 103 103 1 04 104 1 year 103 104 105 106 107 108 109 \\ years 104 106 107 109 110 112 114 2 years 106 108 1 10 112 114 116 119 2} years 107 1 10 113 115 1 18 121 124 3 years 109 1 12 115 1 19 122 126 130 3J years 110 114 1 18 122 127 131 136 4 years 1 12 1 17 121 126 131 136 142 4J years 114 119 124 130 136 1 42 148 5 years 1 16 121 128 134 141 148 155 6J years 1 17 124 131 138 145 153 162 6 years 1 19 126 134 142 151 160 169 6J years 121 129 137 146 1 56 166 177 7 years 123 131 141 151 161 173 185 7i years 124 134 144 156 167 180 193 8 years 126 137 148 160 173 187 2 02 8i years 128 139 162 166 179 194 2 11 9 years 130 142 1 55 170 185 2 02 220 9J years 132 146 159 176 192 2 10 2 30 10 years 134 148 163 180 198 2 19 2 41 11 years 138 164 1 72 191 2 13 2 36 2 63 12 years 142 160 180 203 2 28 2 56 2 87 13 years 147 167 190 2 16 2 44 2 77 3 14 14 years 161 173 199 2 28 2 62 2 99 3 42 15 years 166 180 2 09 2 42 2 80 3 24 3 74 REVIEW OF ARITHMETIC 57 EXAMPLES Solve the following problems by using the tables on page 56 : 1. What is the compound interest of $1 at the end of 8^ years at 6 % ? 2. What is the compound interest of $ 1 at the end of 11 years at 6 % ? 3. How long will it take $400 to double itself at 5 %, compound interest ? 4. How long will it take $580 to double itself at 6^%, compound interest ? 5. How long will it take $615 to double itself at 8 %, simple interest? 6. How long will it take $784 to double itself at 7%, simple interest ? 7. Find the interest of $684 for 94 days at 3 %. a Find the interest of $ 1217 for 37 days at 4 %. 9. Find the interest of $ 681.14 for 74 days at 4J %. 10. Find the interest of $414.50 for 65 days at 5 %. 11. Find the interest of $384.79 for 115 days at 6 %. Ratio and Proportion Ratio is the relation between two numbers. It is found by dividing one by the other. The ratio of 4 to 8 is 4 -^ 8 = |^. The terms of the ratio are the two numbers compared. The first term of a ratio is the antecedent, and the second the con- sequent. The sign of the ratio is (:). (It is the division sign with the line omitted.) Ratio may also be expressed fraction- ally, as J^ or 16 : 4 ; or ^^ or 3 : 17. A ratio formed by dividing the consequent by the antece- dent is an inverse ratio : 12 : 6 is the inverse ratio of 6 : 12. The two terms of the ratio taken together form a couplet 58 VOCATIONAL MATHEMATICS FOR GIRLS Two or more couplets taken together form a compound ratio. Thus, 2:6 6 : 11 A compound ratio may be changed to a simple ratio by- taking the product of the antecedents for a new antecedent, and the product of the consequents for a new consequent ; as, 6x2:11x6, or 12; 55. Antecedent -j- Consequent = Ratio Antecedent -r- Ratio = Consequent Ratio X Consequent = Aiitecedent To multiply or divide both terms of a ratio by the same number does not change the ratio. Thus 12 : 6 = 2 3x12:3x6 = 2 EXAMPLES Find the ratio of 1. 20 : 300 Fractions with a common de- 2. 3 bu. : 3 pk. nominator have the same 3 21 • 16 ratio as their numerators. *• 12:i 7. ^:|^,||:^,||:|J 6. 16:(?) = J Proportion An equality of ratios is a proportion. A proportion is usually expressed thus : 4 : 2 : : 12 : 6, and is read A: is to 2 as 12 is to 6. A proportion has four terms, of which the first and third are antecedents and the second and fourth are consequents. The first and fourth terms are called extremes^ and the second and third terms are called means. The product of the extremes equals the product of the means. REVIEW OF ARITHMETIC 59 To find an extreme^ divide the product of the uneans by the given extreme. To find a mean, divide theprodvjct of the extremes by the given mean, EXAMPLES Supply the missing term : 1. 1 : 836 : : 25: ( ) 4. 10 yd. : 50 yd. : : $20 : ($ ) 2. 6:24::( ):40 5. $f :$3|::( ):5 3. ( ):15::60:6 Simple Proportion An equality of two simple ratios is a simple proportion. Example. — If 12 bushels of charcoal cost $ 4, what will 60 bushels cost ? 19 . ftft . • «4 . r« ^ There is the same relation between the cost . . . ^p f\j9 ) of 12 bu. and the cost of 60 bu. as there is be- '^^* = $20. Ans, tween the 12 bu. and the 60 bu. |4is the third term. The answer is the fourth term. It must form a ratio of 12 and 60 that shall equal the ratio of $4 to the answer. Since the third term is less than the required answer, the first must be less than the second, and 12 : 60 is the first ratio. The product of the means divided by the given extreme gives the other extreme, or % 20. EXAMPLES Solve by proportion : 1. If 150 yd. of edging cost $ 6, how much will 1200 yd. cost ? 2. If 250 pounds of lead pipe cost $ 15, how much will 1200 pounds cost ? 3. If 5 men can dig a ditch in 3 days, how long will it take 2 men? 4. If 4 men can shingle a shed in 2 days, how long will it take 3 men ? 5. The ratio of Simon's pay to Matthew's is |. Simon earns $ 18 per week. What does Matthew earn ? 60 VOCATIONAL MATHEMATICS FOR GIRLS 6. What will 11 1 yards of cambric cost if 50 yards cost $ 6.76 ? 7. If it takes 7^ yards of cloth, 1 yard wide, to make a suit, how many yards of cloth, 44 inches wide, will it take to make the same suit ? 8. If 21 yards of silk cost $ 52.50, what will 35 yards cost ? 9. A farm valued at $5700 is taxed for $38.19. What should be the tax on property valued at $ 28,500 ? 10. If there are 7680 minims in a pint of water, how many pints are there in 16,843 minims ? 11. There are approximately 15 grains in a gram. How many grams in 641 grains ? 12. In a velocity diagram a line '31 in. long represents 45 ft. What would be the length of a line representing 30 ft. velocity ? 13. When a post 11.5 ft. high casts a shadow on level ground 20.6 ft. long, a telephone pole nearby casts a shadow 59.2 ft. long. How high is the pole ? 14. If 10 grams of silver nitrate dissolved in 100 cubic cen- timeters of water will form a 10 % solution, how much silver nitrate should be used in 1560 cubic centimeters of water ? 15. A ditch is dug in 14 days of 8 hours each. How many days of 10 hours each would it have taken ? 16. If in a drawing a tree 38 ft. high is represented by 1^^", what on the same scale will represent the height of a house 47 ft. high ? 17. What will be the cost of 21 motors if 15 motors cost $ 887 ? 18. If goods are bought at a discount of 25 % and are sold at the list price, what per cent is gained ? (Assume $ 1 as the list price.) REVIEW OF ARITHMETIC 61 18. If a sewing machine sews 26 inches per minute on heavy goods, how many yards will it sew in an hour ? 19. If a girl spends 28 cents a week for confectionery, how much does she spend for it in three months ? 20. If a pole 8 ft. high casts a shadow 4^ ft. long, how high is a tree which casts a shadow 48 ft. long ? Involution The product of equal factors is a power. The process of finding powers is involution. The product of two equal factors is the second power, or square, of the equal factor. The product of three equal factors is the third power, or cube, of the factor. 42 = 4x4 is 4to the second power, or the square of 4. 2^ = 2 X 2 X 2 is 2 to the third power, or the cube of 2. 3* = 3x3x3x3is3to the fourth power, or the fourth power of 4. EXAMPLES Find the powers : 1. 6« 3. 1* 5. (2^)2 7. 9» 2. 1.1^ 4. 262 6. 2* 8. .152 Evolution One of the eqiuil factors of a power is a root. One of two equal factors of a number is the square root. One of three equal factors of a number is the cube root of it. The square root of 16 = 4. The cube root of 27 = 3. The radical sign (^) placed before a number indicates that its root is to be found. The radical sign alone before a number indicates the square root. Thus, V9 = 3 is read, the square root of 9 = 3. 62 VOCATIONAL MATHEMATICS FOR GIRLS A small figure placed in the opening of the radical sign is called the index of the root, and shows what root is to be taken. Thus, \/8 = 2 is read, the cube root of 8 is 2. Square Root The square of a number composed of tens and units is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. ten^ -h 2 X tens x units + units? Example. — What is the square root of 1225 ? 12'25(30 + 6 = 86 Separating re»w2, 302 =900 into periods of 2 X tens = 2 x 30 = 60 326 two figures 2 X tens + units = 2 x 30 + 6 = 66 325 each, by a check mark ('), beginning at units, we have 12'26. Since there are two periods in the power, there must be two figures in the root^ tens and units. The greatest square of even tens contained in 1226 is 000, and its square root is 30 (3 tens). Subtracting the square of the tens, 000, the remainder consists of 2 x (tens x units) + units. 326, therefore, is composed of two factors, units being one of them, and 2 x tens — units being the other. But the greater part of this factor is 2 X tens (2 x 30 = 60). By trial we divide 326 by 60 to find the other factor (units), which is 6, if correct. Completing the factor, we have 2 X tens + units = 66, which, multiplied by the other factor, 6, gives 826. Therefore the square root is 30 + 6 = 36. The area of every square surface is the product of two equal factors, length, and width. Finding the square root of a number, therefore, is equivalent to finding the length of one side of a square surface, its area being given. 1. Length x Width =Area 2. Area -i- Length = Width 3. Area -i- Width = Length REVIEW OF ARITHMETIC 63 Short Method Example. — Find the square root of 1306.0996. 13^06.09^96 (36.14 Beginning at the decimal point, separate the 9 number into periods of two figures each, point- 66) 406 ing whole numbers to the left and decimals to 896 the right. Find the greatest square in the left- 721)1009 hand period, and write its root at the right. 721 Subtract the square from the left-hand period, 7224)28896 and bring down the next period for a dividend. 28896 Divide the dividend, with its right-hand figure omitted, by twice the root already found, and annex the quotient to the root, and to the divisor. Multiply this complete divisor by the last root figure, and bring down the next period for a dividend, as before. Proceed in this manner till all the periods are exhausted. When occurs in the root, annex to the trial divisor, bring down the next period, and divide as before. If there Is a remainder after all the periods are exhausted, annex deci- mal periods. If, after multiplying by any root figure, the product is larger than the dividend, the root figure is too large and must be diminished. Also the last figure in the complete divisor must be diminished. For every decimal period in the power, there must be a decimal figure in the root. If the last decimal period does not contain two figures, supply the deficiency by annexing a cipher. EXAMPLES Find the square root of : 1. 8836 5. yjl^l 9. V3.532 ^ 6.28 2. 370881 6. 72.5 10. V625+1296 3. 29.0521 7. .009^9^ 11. J_x:^ 4. Am56 8. 1684.298431 12. V9 3969 5625 13. What.is the length of one side of a square field that has an area equal to a field 75 rd. long and 45 rd. wide ? CHAPTER II MENSURATION The Circle A circle is a plane figure bounded by a curved line, called the circumference, every point of which is equidistant from the center. The diameter is a straight line drawn, from one point of the circumference to another and passing through- the center. The ratio of the circumference to the diameter of any circle is always a constant number, 3.1416+, approxi- mately S^, which is represented by the Greek letter ir {pi). C = Circumference D = Diameter The radius is a straight line drawn from the center to the circumference. Any portion of the circumference is an arc. By drawing a number of radii a circle may be cut into a series of figures, each one of which is called a sector. The area of each sector is equal to one half the product of the arc and radius. Therefore the area of the circle is equal to one half of the product of the circumference and radius. 1 See Appendix for explanation and directions concerning the use of formulas. 64 MENSURATION 65 2 In this formula A equals area, ir = 3.1416, and -B* = tlie radius squared. In this formula D equals the diameter and G the circum- ference, 4 4 Example. — What is the area of a circle whose radius is 3ft.? xZ)2 A-rB?^ ^ = 4 9 -4=irx9 ^=^ = ir9 = 28.27 sq.ft. Ans, Example. — What is the area of a circle whose circumfer- ence is 10 ft. ? 8.1416 2 2 ix-i5_ xlx 10 = -^5_ = 7.1 sq.ft. ^ns. 2 3.1416 2 3.1416 ^ Area of a Ring. — On examining a flat iron ring it is clear that the area of one side of the ring may be found by subtracting the area of the inside circle from the area of the outside circle. Let D = outside diameter d = inside diameter A = area of outside circle a = area of inside circle (1) ^ = :^=.7854Z>» 66 VOCATIONAL MATHEMATICS FOR GIRLS (2) . (3) A Let a = -a B B 4 =.7854 (P 4 4 area of circular ring =s A — a -cP) Example. — If the outside diameter of a flat ring is 9" and the inside diameter 7", what is the area of one side of the ring? B = .7854 (2>2 - cP) B = .7864 (81 - 49) = .7854 x 32 = 25.1328 sq. in. Am. Angles We make two common uses of angles : (1) to measure a cir- cular movement, and (2) to measure a difference in direction. A circle contains 360°, and the angles at the center of the circle contain as many degrees as their corresponding arcs on the circumference. Angle FOE has as many degrees as arc PE. A right angle is measured by a quarter of the circumference of the circle, which is90^ The angle AOG is a right angle. The angle ACy made with half the cir- cumference of the circle, is a straight angle, and the two right angles, AOO and GOC, which it contains, are supplementary to each other. When the sura of two angles is equal to 90®, they are said to be complementary angles, and one is the com- plement of the other. When the sum of two angles equals 180**, they are supplementary angles, and one is said to be the supple- ment of the other. MENSURATION 67 The number of degrees in an angle may be measured by a protractor. The distance around a semicircular protractor is Pbotractob— Semicircular, having 180°. divided into 180 parts, each division measuring a degree. It is used by placing the center of the protractor on the vertex and the base of the protractor on one side of the angle to be measured. Where the other side of the angle cuts the circular piece of the protractor, the size of the angle may be read in degrees. EXAMPLES 1. What is the area of a circular piece of velvet 8" in diameter ? 2. What is the distance around the edge of a hat 6" in diameter ? 3. Name the complements of angles of 30^ 45^ 65^ 70°, 85°. 4. Name the supplements of angles of 55°, 140°, 69°, 98° 44', 81° 19^. 5. What is the diameter of a wheel that is 12' 6" in circum- ference ? 68 VOCATIONAL MATHEMATICS FOR GIRLS 6. What is the area of one side of a flat iron ring 14" inside diameter and 18" outside diameter ? 7. The wheel of a child's carriage is 30" in diameter. What is the length of the rubber tire that fits it ? 8. How much ribbon is needed to bind the edge of a circu- lar cloth that exactly covers the top of a center table 28" in diameter ? 9. A straw hat measures 30" around the rim. What is the diameter of the hat ? 10. If a circular dining room table measures 12' 6" in cir- cumference, what is the greatest distance across the table ? Triangles A triangle is a plane figure bounded by three straight lines. Triangles are classified according to the relative lengths of their sides and the size of their angles. A triangle having equal sides is called equilateral. One having two sides equal is isosceles. A triangle having no sides equal is called scalene. If the angles of a triangle are equal, the triangle is equi- angular. If one of the angles of a triangle is a right angle, the tri- angle is a right triangle. In a right triangle the side opposite the right angle is called the hypotenuse and is the longest side. The other two sides of the right triangle are the legs, and are at right angles to each other. Equilateral Isosceles Scalene BlQHT MENSURATION 69 Kinds of Triangles Right Triangles In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides or legs. If the length of the hy- potenuse and one leg of a right triangle is known, the other side may be found by squaring the hypotenuse and squaring the leg, and extracting the square root of their dif- ference. Example. — If the hypotenuse of a right angle triangle is 30" and the base is 18'', what is the altitude ? A 802 = 30 X 30 182 = 18 X 18 900-324 VEw 900 324 576 24". Ans, JS" Areas of Triangles The area of a triangle may be found when the length of the thi'ee sides is given by adding the three sides together, divid- ing by 2, and subtracting from this sum each side separately. Multiply the four results together and find the square root of their product. 70 VOCATIONAL MATHEMATICS FOR GIRLS Example. — What is the area of a triangle whose sides measure 15, 16, and 17 inches, respectively ? V24x 9 x8'x7 = \/l2096 V12096 = 109.98 sq. in. Ana, 16 16 17 )48 24- 15 = 9 24- 16 = 8 24- 17 = 7 Area of a Triangle = ^ Ba^e X Altitzide Example. — What is the area of a triangle whose base is 17" and altitude 10"? 1 ^ ^ = - X 17 X ^jJ = 85 sq. in. Ans. EXAMPLES 1. A ladder 17 ft. long standing on level ground reached to a window 12 ft. from the ground. If it is assumed that the wall is perpendicular, how far is the foot of the ladder from the base of the wall ? 2. Find the area of a triangular piece of cloth having the base 81" and the height measured from the opposite angle 56". 3. Find the length of the hypotenuse of a right triangle with equal legs and having an area of 280 sq. in. 4. Find the length of a side of a right triangle with equal legs and an area of 72 sq. in. 5. Find the hypotenuse of a right triangle with a base of 8" and the altitude of 7". 6. What is the area of a triangle whose sides measure 12, 19, and 21 inches ? 7. What is the altitude of an isosceles triangle having sides 8 ft. long and a base 6 ft. long ? MENSURATION 71 Quadrilaterals Four-sided plane figures are called quadrilaterals. Among them are the trapezoid, trapezium, rectangle, rhombus, and rhom- boid. Squaab Bbctanqle Rhomboid 7/7 Rhombus Tbapbzium Tbapbzoid Paballblogbam Kinds of Quadrilaterals A rectangle is a quadrilateral which has its opposite sides parallel and its angles right angles. Its area equals the prod- uct of its base and altitude. A= ha A trapezoid is a quadrilateral having only two sides parallel. Its area is equal to the product of the altitude by one half the sum of the bases. A = (b + c)x ^a In this formula c = length of longest side b = length of shortest side a = altitude ni\ A trapezium is a four-sided figure with no two sides parallel. The area of a trapezium is found by dividing the trapezium into triangles by means of a diagonal. Then the area may be found if the diagonal and perpendicular heights of the triangles are known. 72 VOCATIONAL MATHEMATICS FOR GIRLS Example. — In the trapezium ABOD if the diagonal is 43' and the perpendiculars 11' and 17', respectively, what is the area of the trapezium ? 43 X V- =^p = 236| sq. ft., area of ABC 43 X 4^ = 4^=365^ sq. ft., area of ADG 602 sq. ft., total area Arts, To find the areas of irregular figures, draw the longest diagonal and upon this diagonal drop perpendiculars from the ver- tices of the figure. These perpendiculars will form trapezoids and right triangles whose areas may be determined by the pre- ceding rules. The sum of the areas of the separate figures will give the area of the whole irregular figure. Polygons A plane figure bounded by straight lines is a polygon. A polygon which has equal sides and equal angles is a regular polygon. The apothem of a regular polygon is the line drawn from the center of the polygon perpen- dicular to one of the sides. A five-sided polygon is a pentagon. A six-sided polygon is a hexagon. An eight-sided polygon is an octagon. The shortest distance between the opposite sides of a regu- lar hexagon is the perpendicular distance between them, and is equal to the diameter of the inscribed circle. The diameter of the circumscribed circle is the long diame- ter of a regular hexagon. The perimeter of a polygon is the sum of all its sides. Pentagon Hexagon MENSURATION 73 The area of a regular polygon equals one half the product of the apothem and the perimeter. Formula -4 = i dP In this formula P = perimeter a = apothem Ellipse Only the approximate circumference of an ellipse can be ob- tained. The circumference of an ellipse equals one half the product of the sum of two diameters and ir. If di = major diameter dj = minor diameter (7= circumference then C = ^±^,r The area of an ellipse is equal to one fourth the product of the major and minor diameters by w. If A= area di = major diameter dz = minor diameter then A = w^ 4 EXAMPLES 1. Find the area of a trapezium if the diagonal is 93' and the perpendiculars are 19' and 33'. 2. What is the area of a trapezoid whose parallel sides are 18 ft and 12 ft., and the altitude 8 ft. ? 3. What is the distance around an ellipse whose major diameter is 14" and minor diameter 8" ? 74 VOCATIONAL MATHEMATICS FOR GIRLS 4. In the map of a country a district is found to have two of its boundaries approximately parallel and equal to 276 and 216 miles. If the breadth is 100 miles, what is its area ? 5. If the greater and lesser diameters of an elliptical man- hole door are 2' 9" and 2' &', what is its area ? 6. Find the area of a trapezium if the diagonal is 78'' and the perpendiculars 18" and 27". 7. The greater diameter of an elliptical funnel is 4 ft. 6 in., and the lesser diameter is 4 ft. What is its area ? 8. Find the perimeter of a hexagon having each side 15" long. 9. What is the area of a pentagon whose apothem is 4y and whose side is 5" ? Volumes The volume of a rectangular-shaped bar is found by multi- plying the area of the base by the length. If the area is in square inches, the length must be in inches. The volume of a cube is equal to the cube of an edge. The contents or volume of a cylindrical solid is equal to the product of the area of the base by the height. If S = contents or capacity of cylinder R = radius of base H= height of cylinder V = 3.1416+ or ^ (approx.) 8 = irB'H Example. — Find the contents of a cylindrical tank whose inside diameter is 14" and height 6'. S = tR^H H= 6' = 72" ^ = yx7x7x72 = 11,088 cu. in. MENSURATION 75 The Pyramid The volume of a pyramid equals one third of the product of the area of the base and the altitude. V=\ba The volume of a frustum of a pyramid equals the product of one third the alti- tude and the sum of the two bases and the square root of the product of the bases. The surface of a regular pyramid is equal to the product of the perimeter of the bases and one half the slant height. S^Pxish The Cone A cone is a solid generated by a right triangle revolving on one of its legs as an axis. The altitude of the cone is the perpendicular distance from the base to the apex. The volume of a cone equals the product of the area of the base and one third of the altitude. or F= .2618 L^H Example. — What is the volume of a cone 1^" in diameter and 4" high ? Area of base = .7864 x } 7.0686 = 1.7671 sq. in. F=.2618D2fi- = .2618 X J X 4 = 2.3662 cu. in. Ans. The lateral surface of a cone equals one half the product of the perimeter of the base by the slant height. 76 VOCATIONAL MATHEMATICS FOR GIRLS Example. — What is the surface of a cone having a slant height of 36 in., and a diameter of 14 in. ? C = irZ> = 14 X V = 44" ^^^ = 792 sq. in. Ans, Frustum of a Cone The frustum of a cone is the part of a cone included between the base and a plane or upper base which is parallel to the lower base. The volume of a frustum of a cone equals the product of one third of the altitude and the sum of the two bases and the square root of their product. When altitude upper base lower base H R F= \ H(B-^B' + VBB') The lateral surface of a fi^stum of a cone equals one half the product of the slant height and the sum of the perimeters of the bases. The Sphere The volume of a sphere is equal to 3 where R is the radius. The surface of a sphere is equal to The Barrel To find the cubical contents of a barrel, (1) multiply the square of the largest diameter by 2, (2) add to this product MENSURATION 77 the square of the head diameter, and (3) multiply this sum by the length of the barrel and that product by .2618. Example. — Find the cubical contents of a barrel whose largest diameter is 21" and head diameter 18", and whose length is 33". 212 = 441 X 2 = 882 V= [(Z>2 x 2) + ^2] x Z X .2618 182 = 324 _^ 39798 1206 .2618 33 10419.11 cu. in. 3618 3618 10419.11 39798 231 = 45.10 gal. Ans, Similar Figures Similar figures are figures that have exactly the same shape. The areas of similar figures have the same ratio as the squares of their corresponding dimensions. Example. — If two boilers are 15' and 20' in length, what is the ratio of their surfaces ? JJ = }, ratio of lengths — = — , ratio of surfaces 42 16 One boiler is ^^ as large as the other. Ans. The volumes of similar figures are to each other as the cubes of their corresponding dimensions. Example. — If Jiwo iron balls have 8" and 12" diameters, respectively, what is the ratio of their volumes ? j^ = I, ratio of diameters = ^, ratio of their volumes. Ans, One ball weighs ^ as much as the other. 78 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES 1. Find the volume of a rectangular box with the following inside dimensions : 8" by 10" and 4' long. 2. The radius of the small end of a bucket is 4 in. Water stands in the bucket to a depth of 9 in., and the radius of the surface of the water is 6 in. (1) Find the volume of the water in cubic inches. (2) Find the volume of the water in gallons if a cubic foot contains 7.48 gal. 3. What is the volume of a steel cone 2^" in diameter and 6" high ? 4. Find the contents of a barrel whose largest diameter is 22", head diameter 18", and height 35". 5. What is the volume of a sphere 8" in diameter ? 6. What is the volume of a pyramid with a square base, 4" on a side and 11" high ? 7. What is the surface of a wooden cone with a 6" diameter and 14" slant height ? 8. Find the surface of a pyramid with a perimeter of 18" and a slant height of 11". 9. Find the volume of a cask whose height is 3J-' and the greatest radius 16", and the least radius 12", respectively. 10. How many gallons of water will a round tank hold which is 4 ft. in diameter at the top, 5 ft. in diameter at the bottom, and 8 ft. deep ? (231 cu. in. = 1 gal.) 11. What is the volume of a cylindrical ring having an outside diameter of 6y, an inside diameter of 5^^", and a height of 3f " ? What is its outside area ? 12. If 9 tons of wild hay occupy a cube 7' x 7' x 7', how many cubic feet in one ton of hay ? 13. A sphere has a circumference of 8.2467". (a) What is its area ? (b) What is its volume ? MENSURATION 79 14. If it is desired to make a conical can with a base 3.5" in diameter to contain ^ pint, what must the height be ? 15. What is the area of one side of a flat ring if the inside diameter is 2|-" and the outside diameter 4 J" ? 16. There are two balls of the same material with diameters 4" and 1", respectively. If the smaller one weighs 3 lb., how much does the larger one weigh ? 17. If the inside diameter of a ring is 5 in., what must the outside diameter be if the area of the ring is 6.9 sq. in. ? 18. How much less paint will it take to paint a wooden ball 4" in diameter than one 10" in diameter ? 19. What is the weight of a brass ball 3^" in diameter if brass weighs .303 lb. per cubic inch ? 20. A cube is 19" on its edge, (a) Find its total area. (b) Its volume. 21. If a barrel of water contains about 4 cu. ft., what is the approximate weight of the barrel of water? (1 cu. ft. of water weighs 62.5 lb.) 22. A conical funnel has an inside diameter of 19.25" at the base and is 43" high inside, (a) Find its total area, (b) Find its cubical contents. 23. A pointed heap of corn is in the shape of a cone. How many bushels in a heap 10' high, with a base 20' in diameter ? A bushel contains 2150.42 cu. in. 24. Find the capacity of a rectangular bin 6 ft. wide, 5 ft. 6 in. deep, and 8 ft. 3 in. long. 25. Find the capacity of a berry box with sloping sides 5.1" by 5.1" on top, 4.3" by 4.3" at the bottom, and 2.9" in depth. 26. Find the capacity of a cylindrical measure 13" in diameter and 6" deep. 27. How many tons of nut coal are in a bin 5 ft. wide and 8 ft. long if filled evenly to a depth of 4 ft. ? Average nut coal weighs 52 lb. to a cubic foot. CHAPTER III INTERPRETATION OF R£STTLTS Reading a Blue Print. — Everyone aliould know how to read a blue print, which iB the name given to working plana and drawings with white lines upon a blue background. The blue print ia the language which the architect uses to the builder, the machinist to the pattern maker, the engineer to the foreman EiTBBiOE View of Cohplbtkd Hoosb of construction, and the designer to the workman. Through following the directions of the blue print the carpenter, metal worker, and mechanic are able to produce the object wanted by the employer and his designer or draftsman. Blue Print of a House. — An architect, in drawing the plans of a house, usually represents the following views : the ex- terior views to show the appearance of the house when it ia finished ; views of each floor, including the basement, to show INTERPRETATION OF RESULTS r/agtiont Cap West Elbtation East ELKViTiOK the location of rooms, windows, doors, and stairs. Detailed plans of sections are drawn for the contractors to show the method of construction. 82 VOCATIONAL MATHEMATICS FOR GIRLS //jg^ton* Ctft North Elevation ^/"/agilone. Cap South Elevation Pupils should be able to form a mental picture of the appear- ance of a building constructed from any blue print plan set before them. They should have practice in reading the plans of the house and in computing the size of the rooms directly from the blue print. INTERPRETATION OP RESULTS 1. "What is the height of the rooms on the tirst floor? 2. What is the height of the rooms on the second floor? 3. What ia the height of the cellar, first, and second floors ? GsouND FLOOa Plan 84 VOCATIONAL MATHEMATICS FOR GIRLS 1. What is the frontage of the house ? 2. What is the depth of the house ? 3. What is the length and width of the front porch ? 4. What is the length and width of the living room ? the dining room ? the kitchen ? rf^fy-Tfry ■ShinffU ea Shia^ki JBed £oom C 4oyr ® BedJioomB Second Floor Plan 1. What is the size of each of the bedrooms? with aid of ground floor plan.) 2. What are the dimensions of the bathroom ? 3. How large is the storage room ? (Compute Two views are usually necessary in every working drawing, one the plan or top view obtained by looking down upon the object, and the other the elevation or front view. When an INTERPRETATION OF RESULTS 85 object is very complicated, a third view, called an end or profile view is shown. All the information, such as dimensions, etc., necessary to construct whatever is represented by the blue print must be supplied on the draw- ing. If the blue print represents a machine, it is necessary to show all the parts of the machine put together in their proper places. This is called an assembly drawing. Then there must be a drawing for each part of the building or the machine, giving information as to the size, shape, and number of the pieces. Then if there are interior sections, these must be represented in section drawings. Drawing to Scale. — As it is impossible to draw most objects full size on paper, it is necessary to make the drawings pro- portionately smaller. This is done by making all the dimen- sions of the drawing a certain fraction of the true dimensions of the object. A drawing made in this way is said to be drawn to scale, \\ \\ \\\ \\\\ \\\\\\.\ \\y \\ y \\ \\\ \y \ \ \\\ \\.\ \\ y \\ \ II 2 Vg I z \\\M\\l\\\ \ \ \ \ \ Triangular Scale The dimensions on the drawing designate the actual size of the object — not of the drawing. If a drawing were made of an iron door 25 inches long, it would be inconvenient to repre- sent the actual size of the door, and the drawing might be made half or quarter the size of the door, but on the drawing the length would read 25 inches. In making a drawing " to scale," it becomes very tedious to be obliged to calculate all the small dimensions. In order to obviate this work a triangular scale is used. It is a rule with the different scales marked on it. By practice the student will be able to use the scale with as much ease as the ordinary rule. QUESTIONS AND EXAMPLES 1. Tell what is the scale and the length of the drawing of each of the following : 86 VOCATIONAL MATHEMATICS FOR GIRLS a. An object 14" long drawn half size. b. An object 26" long di*awn quarter size. c. An object 34" long drawn one third size. d. An object 41" long drawn one twelfth size. 2. If a drawing made to the scale of |" = 1 ft. is reduced ^ in size, what will the new scale be ? 3. A drawing is made \ size. If the scale is doubled, how many inches to the foot will the new scale be ? 4. On the ^^" scale, how many feet are there in 18 inches ? 5. On. the Y^ scale, how many feet are there in 26 inches ? 6. On the J" scale, how many feet are there in 27 inches ? 7. If the drawing of a door is made \ size and the length of the drawing is 8^", what will it measure if made to scale 3" = lft.? 8. What will be the dimensions of the drawing of a banquet hall 582' by 195' if it is made to a scale of yV" = 1 f t. ? Estimating Distances. — Everyone meets occasions in daily life when it is of utmost importance that distance or weight should be correctly estimated. Few people have a clear conception of even our common standards of measurements. This is due to the fact that the average person has never given the proper attention to them. Improvement will be noticed after a small amount of drill. To illustrate : if the distances of one inch, one foot, one yard, six feet, and ten feet are measured off in a classroom so that an actual view of standard distances is obtained, and then pupils are asked to estimate other and unknown distances, they will estimate with a greater degree of accuracy. Pupils should be able to estimate within J inch any distance up to a yard. The power of estimating longer distances, such as the distance between buildings, across streets, or between streets, may be developed by laying off on a straight road one hundred feet, three hundred feet, and five hun- dred feet sections, with the proper distance marked on each. The same plan applies to heights of buildings, etc. Standards of alti- tude may thus be established. Pupils should measure in their homes pieces of furniture and wall openings so that they may develop an eye for estimating distances. INTERPRETATION OF RESULTS 87 1. Estimate the length and width of the schoolroom. Verify this estimate by actual measurement and express the accuracy of your estimate in per cent. 2. Estimate the height and width of the school door. Verify this estimate by actual measurement and express the accuracy of the estimate in per cent. 3. Estimate the width and length of the window panes; the width and length of the window sill. Estimating Weights. — What is true concerning the advan- tage of being able to estimate distances applies equally well to weights. In this, guesswork may be largely eliminated. A little mental figur- ing on the part of the pupil will usually produce clear results. Weight depends not only on volume but also on the density of the material. Regular blocks of wood are excellent to begin with, and later small spheres and rectangular blocks of different metals afford good material. 1. Select blocks of wood, coal, iron, lead, tin, or copper, and estimate their respective weights. 2. Estimate the weight of a chair. 3. Estimate the weight of different persons. Methods of Solving Examples. — Every commercial, household, or mechanical problem or operation has two distinct sides : the collecting of data, and the solving of the problem. The first part, the collecting of data, demands a knowledge of the materials and conditions under which the problem is given, and calls for the exercise of judgment as to the neces- sary accuracy of the work. There are three ways by which a problem may be solved : 1. Exact method. 2. Rule of thumb method, by the use of a formula or a rule committed to memory. 3. By means of tables. The exact method of solving a problem in arithmetic is the one usually taught in school and is the method obtained by 88 VOCATIONAL MATHEMATICS FOR GIRLS analysis. Everyone should be able to solve a problem by the exact method. The Rule of Thumb Method. — Many of the problems that arise in home, office, and industrial life have been met before, and very careful judgment has been exercised in solving them. As the result of this experience and the tendency to abbreviate and devise shorter methods that give 'sufficiently accurate re- sults, we find many rule of thumb methods used by the house- wife, the storekeeper, the nurse, etc. The exact method would require considerable time and the use of pencil and paper, whereas in cases that are not too complicated the estimates, based on experience or rule, give a quick and accurate result. In solving problems involving the addition and subtraction of fractions, use the yardstick or tape to carry on the compu- tation. To illustrate : if we desire to add ^ and ^ of a yard, place the thumb over ^ of a yard divisions, then slide (move) the thumb along the divisions corresponding to ^ of a yard, and then read the number of divisions passed over by the thumb. In this case the result is 21 inches. The Use of Tables. — In the commercial world the tendency is to do everything in the quickest and the most economical way. To illustrate : hand labor is more costly than machine work, so, whenever possible, machine work is substituted for hand labor. The same tendency applies to calculations in the dressmaking shop or the office. The exact methods of doing examples are not the quickest, nor are they more easily under- stood and performed by the ordinary girl than the shorter methods. Since a great many of the problems in calculation that arise in the daily experiences of the office assistant, the housewife, the dressmaker, the nurse, etc. are about ordinary things and repeat themselves often, it is not necessary to work them anew each time, if, when they are once solved, results are kept on file in the form of tables. See pages 220, 222, and 254 for tables used in this book. PART II — PROBLEMS IN HOMEMAKING CHAPTER IV THE DISTRIBUTION OF INCOME The economic standing of every person in the community depends upon three things : (1) (earning capacity, (2) spend- ing ability, and (3) the saving habit. The first regulates the amount of income; the second determines the purchasing power after the amount is earned; the third paves the way to independence. The welfare of every person, whether single or married, depends upon the systematic and careful regulation of each of these three items. No matter how large or small his wages or salary, if he does not spend his money wisely and carefully, or save each week or month a certain per cent of his earnings, a young man or woman is not likely to make a success of life. A young woman usually has more to do with the spending of money than a young man. The wife is really the spender and the husband the earner in the ordinary home. Therefore, it becomes necessary for every young woman to know how to get one hundred cents out of a dollar. In order to do this, she must know how to distribute the income over such items as rent, food, clothing, incidental expenses due to sickness, pleas- ure, or self-improvement. The proportion spent for each item should be carefully regulated. Incomes of American Families The average family income of both foreign and native born heads is about $725 a year; that of families with native bom heads alone is about $ 800. Not more than one-fourth have incomes exceeding $ 1000. The daily wages of adult men range from $ 1.60 to $ 5.00. This amounts on the average from $460 to $1500 a year. The family, the head of which earns only a few hundred dollars a year, must either be contented with comparatively low standards of liv- 89 90 VOCATIONAL MATHEMATICS FOR GIRI^ ing or obtain additional income, either titrongh tlie labor of children or from boarders or lodgere. The foreign-bom workers resort to the labor of children and mothers more than do the native Americana. The second course la quite often adopted ao that the average incoma of workingmen's families is considerably greater than the average earnings of the heads of the families. ExPBHDrrUBES EXAMPLES 1. The average workingman's family spends at leaat two- fifths of ita income for food. What per cent is spent for food ? 2. If the income of a workingman's family is $ 800, and the amount spent for food is $ 350, what per cent ia spent for food ? 3. One-fifth of the expenditure of workingmen's families ia for rent. What per cent ? 4. A &.niily with an income of S 800 spends $ 12.50 a month for rent. What per cent of the income is spent for rent? Is this too much? 5. A family's income is $760. The father contributes $601. What per cent of the income is contributed by the father? 6. A family of six has an income of $ 840. The father contributes $592, mother $112, and one child the balance. What per cent is contributed by the mother and child? THE DISTRIBUTION OF INCOME 91 7. A man and wife have an income of $ 971. The husband earns $ 514, the wife keeps boarders and lodgers, and provides the rest of the income. What per cent of the income is con- tributed by the boarders and lodgers ? Cost of Subsistence Shelter, warmth, and food demand from two-thirds to three- fourths of the income of most workingmen's families. This leaves for everything else — clothing, furniture, sickness, death, insurance, religion, education, amusements, savings — only one- third or one-fourth of the income. Between $ 200 and $ 250 a year may be considered the usual outlay of workingmen's fami- lies for all these purposes combined. It is in these respects that the greatest difference appears between the families of the comparatively poor and the families of the well-to-do. * The well-to-do spend not only more in absolute amount, but also a larger proportion of their incomes on these, in general, less absolutely necessary things. Clothes. — On the average, approximately one-eighth of the income in workingmen's families goes for clothes. To those who keep abreast of the fashions and who dress with some elegance, it may seem quite preposterous that a family of five should spend only $ 100 or less a year for clothing, but multi- tudes of working-class families are clad with warmth and with decency on such an expenditure. EXAMPLES 1. If two-thirds of the average workingman's income is spent for shelter, warmth, and food, what per cent is used ? 2. A family, receiving an income of $ 847, spends $ 579 for shelter, warmth, and food. What per cent is used ? 3. If one-eighth of the income of the average workingmen's family is spent for clothes, what is the per cent ? 4. A family receives an income of $ 768, and $ 94 is spent for clothes. What per cent is spent for clothes ? 92 VOCATIONAL MATHEMATICS FOR GIRLS The High Cost of Living The average cost of living represents the amount that must be expended during a given period by the average family- depending on an average income. The maximum or minimum cost, however, is another phase of the problem. It no longer involves the amount of dollars and cents necessary to buy and pay for life's necessaries, but involves questions of home management and housekeeping skill, which cannot be stand- ardized. About 1907 food and other necessities of life began to increase in cost — and this has continued to the present day. EXAMPLES 1. In 1906 a ton of stove coal cost $ 5.75, and in 1915 $ 8.75. What was the per cent of increase in the cost of coal ? 2. In 1907 a suit of clothes cost $ 15. The same suit in 1912 cost $ 19.75. What was the per cent of increase ? 3. In 1908 a barrel of flour cost $ 6.10. The same barrel of flour cost in 1914 $ 8.25. What was the per cent of increase ? Division of Income A girl should always consider her income for the entire year and divide it with some idea of time and relative proportion. If she earns a good salary for only ten months of. the year, she must save enough during those months to tide her over the other two. For instance, if a teacher earns $ 60 a month for 10 months of the year, her actual monthly income is $50. The milliner, the trained nurse, the actress, and sometimes even the girl working in the mill have the same problem to confront. No girl has a right to spend nearly all she earns on clothing, neither should she spend too much for amusement. We find from investigations that have been made that girls earning $ 8 or $ 10 a week usually spend about half their income on board and laundry. Girls earning a larger income may pay more for board, but not quite so great a fractional part. In these THE DISTRIBUTION OF INCOME 93 days, when the cost of living is so* high, a girl should consider carefully a position that includes her board and laundry, for in such a position she will be better off financially at the end of the year than her higher salaried sister, who has to pay for the cost of her own living. The housegirl can save about twice as much as the average stenographer. We find that the average girl needs to spend about one-fifth of her income for clothing. A poor manager will often spend as much as one-third and not be very well dressed at that, because she buys cheap materials, that have to be frequently replaced, and follows every passing fad and style. Choose medium styles and good materials and you will look more richly dressed. Keep the shoes shined, straight at the heel, and the strings fresh. Keep gloves mended, and as clean as possible. If you spend more on clothing than the allotted one- fifth, you will have to go without something else. It may be spending money, or it may be gift or charity money, and quite often it is the bank account that suffers. Every person should save some part of his income. One never knows when sickness, lack of employment, or ill health may come. Saving money is a habit and one that should be acquired the very first year that a person earns his own living. EXAMPLES 1. A girl earns $ 12 a week for 42 weeks, and in this time spends $ 144 for clothing. Is she living within the per cent of her income that should be spent for clothing ? 2. A salesgirl earns $ 8 a week. She spends $ 98 a year for clothes. Is she living within her income ? 3. A girl earns $ 5 a week and pays half of it to her home. She has two car fares and a 14-cent lunch each day. How much should she spend on clothing each year ? How much has she for spending money each week ? Should she save any money ? 4. Which girl is the better off financially, one earning $ 6 a week as a housemaid or one earning $ 7 a week in a store ? 94 VOCATIONAL MATHEMATICS FOR GIRLS Buying; Christmas Gifts Let the gift be something useful. Do not be tempted by the display of fancy Christmas articles, for it is on these that the merchant makes his profit for extra decorations and light. Think of the person for whom you are buying. She may not have the same tastes as you have, so give something that she will like rather than something she ought to like. For in- stance, a certain girl may be very fond of light hair-ribbons when you know that dark ones would be much more sensible, but at Christmas give the light ones. The stores always show an extra supply of fancy neckwear. A collar cannot be worn more than three days without becom- ing soiled, so even 25 cents is too much to pay for something that cannot be cleansed. Over-trimmed Dutch collars and jabots easily rip apart. Choose the plain ordinary ones that you would be glad to wear any day. You see whole counters of handkerchiefs displayed with embroidery, lace, and ruffles. A linen handkerchief, even of very coarse texture, is more suitable. Be careful also about bright colors, for everything about the store is so gay that ordinary things appear dull, but when you get them out against the white snow, they will be bright enough. EXAMPLES 1. Shortly before Christmas I purchased ^ doz. handker- chiefs for $ 1.50. One month later I purchased the same kind of handkerchiefs at 16f cents each or 6 for $ 1.00. What per cent did I save on the second purchase ? 2. I also bought a chiffon scarf for which I paid $2.25. Early in the fall I saw similar scarfs selling for $ 1.50. How much did I lose by not making my purchase at that time? What per cent did I lose ? 3. I bought at Christmas two pairs of silk stockings at $ 1.50 per pair. If I had purchased the stockings in October THE DISTRIBUTION OF INCOME 95 they would have cost me $ 1.12^ per pair. How much would I have saved ? What per cent would I have saved ? An Expense Account Book Every person and every family should keep an expense account showing each year's record of receipts and expendi- tures. A sample form is shown in page 96. Rule sheets in a similar manner for the solution of the problems that follow. At the end of the year a summary should be made of receipts and dis- bursements in some such form as the following : Yearly Summary HeceipU DiahuraemenU Receipts Cash on hand January 1 Salary, etc. Other Income Disbursements Savings and Insurance Rent Food Clothing Laundry Car fares Stamps and Stationery Health Recreation Education Gifts, Church, Charity Incidentals Balance on Hand December 31 Totals Rule similar sheets for the solution of the following problems. 96 VOCATIONAL MATHEMATICS FOR GIRLS Details of Disbursements 00 © 00 =^3 B O 00 OQ 1 g CO > a a -SQ-^ j4 g fe si a 3 ^1 so 1 .c: ^1 H n CD ^ bo 5^ P V En P4 . • s i o 1 00 1 3 THE DISTRIBUTION OF INCOME 97 EXAMPLES 1. A man and wife have an income of $ 1000 a year. The disbursements for the month of October are as follows : Kentf $15 ^ Tooth paste, $.18 Telephone, 1.46 Provisions, 6.86 Repair on coat, 6.80 Life insurance, 7.40 Gas, .76 Coal, 7.60 Dinner, 1.60 Outlook (1 year), 8.00 Stationery, 2.61 Assistance, .60. Fares, .86 Shampoo, .60 Groceries, 10.36 Fares, .60 Fruit, .30 Rubbers, .90 Theater, .60 Soap, .10 Papers, .06 Church, .26 Church, .26 Milk, .66 Milk, .71 Ice, .40 Ice, .46 Papers, .11 Electricity, 1.00 Vase for D., .75 Laundry, .60 Peroxide, .26 Enter the above disbursements in the expense account book. Find the total amount. What per cent was spent for food? house? clothing? housekeeping ? luxuries ? savings ? 2. The items of expense for the month of January, 1915, are : rent and water, $ 15 ; operating expense : light and heat, $ 11 ; food : meats, groceries, $ 30 ; labor or services, $ 16.65 ; cloth- ing, $ 15 ; physician, $ 1 ; carfares, % 2.85 ; books, $ 1.00 ; amusements, % 4 ; cigars, $ 1.00 ; gifts, $ 1.00 ; sundry ex- penses, $ 1.50. Treat as in Ex. 1. 3. A family of two receives an income of $ 1200 a year. They spend per week $ 6.93 for food, $ 3.51 for rent, % 3.49 for operating expenses, $ 5.81 for contingency. What is the amount for each itiem per month (4 weeks) ? per year (52 weeks) ? What is the per cent of each item ? 98 VOCATIONAL MATHEMATICS FOR GIRLS 4. A young married woman has an income of $ 1200 a year — $ 100 a month. The following represents the way she spends her income a month : Savings bank, $5.00 Rent, 25.00 Insurance, 5.00 Groceries, 4.70 Meat and fish, 11.15 Milk, 2.79 Clothing, 12.00 Heat and light, 5.00 Laundry and supplies, 2.00 Help, 4.00 Repairs and replenishing, .50 Ice (set aside in the winter months for the summer), f 0.25 Necessary carfares (the house is located in the country), 2.70 Recreation, 3.00 Avocation, 3.00 Literature, .50 Church and charity, .80 Remembrances, .75 Fire insurance, .09 Enter the above in the expense account book. How much was left toward the next month's expense ? Can you improve on this? What is the per cent for food? Clothing ? 5. A girl 14 years of age has cost her parents an amount equal to the following items : Fo6d, .$597.16 Clothing, 339.66 Furniture, 88.65 Carfare to school, 1 11.00 Doctor, 70.00 Dentist, 10.00 What is the per cent for each item ? 6. A family of seven — three grown people and four chil- dren — lived in a southern city on $ 600 — $ 50 per month. The expenses each month were as follows ; House rent, ^ 12.00 Groceries, 12.00 Washing, 5.00 Bread, $2.50 Beef, 2.60 Vegetables, 2.00 What is the balance for clothing and fuel ? What is the per cent of income spent for food ? Clothing ? Rent ? Suggest points of saving. THE DISTRIBUTION OF INCOME 99 7. A girl in New York City lives on $ 10 a week. The ex- penses are as follows : Board and washing, $ 300.00 Luncheons and carfare, 100.00 Clothing and vacation, $96.00 Church and charity, 10.00 How much can she save ? What is the per cent for cloth- ing ? Incidentals ? Can you suggest any improvements in the distribution of her income ? 8. A family of four lives on $ 750 a year. The expenses are as follows : Rent, 1 180.00 Fuel, 62.00 Meat, oysters, cheese, 96.00 Groceries, including vegetables, butter, eggs, milk, kerosene, soap, etc., $241.00 Clothing, about 146.00 How much is left for the savings bank ? What per cent for rent ? Food ? Clothing ? Operating expenses ? CHAPTER V FOOD Since half of the income of the average family is spent for food, it is important that this expenditure should be made intelligently. Experts of the United States Department of Agriculture estimate that the food waste in a great many of the American homes is as high as 20 % . The causes of waste may be classified as follows : Needlessly expensive material, providing little nutrition ; failure to select according to season ; great amounts thrown away; poor preparation; badly con- structed ovens. If this waste were checked, it would afford an increase in the purchasing power of the income which would appreciably lift the family to a higher plane of efficiency. The efficiency of every person depends upon the energy and constant repair of the body. A woman should know the cost of food and real- ize what food value she is receiving for her money. It may seem strange, but it is true, that "a Eoman feast, a Lenten fast, a Delmonico dinner, and the lunch of the wayside beggar contain the same few elements of nutrition." The art of cooking can transform the common but nutritious foods into the most appetizing dishes. Further than this, the freshness and attractiveness of the food, the way in which it is served, the sur- roundings — all affect the appetite and the power to digest. Cost, which must always be considered in the limited income In relation to the nutritive value of food, is influenced by an equally important factor — waste. The problem of feeding our bodies is primarily a question of supply and demand. Of course, the element of pleasure in eating is a properly 100 POOD normal one, for enjoyment aids digestion. We must, however, eat to live rather than live to eat. Every motion of the body and every thought in the brain destroy cer- tain tissues. This material must be replaced from the food we eat. The two objects of eating are tissue repair in the adult and tissue repair plus growth in the child. As soon as we realize that these two pur- poses should determine the character of the food we eat, then we shall know the importance of intelligent instead of haphazard choice of food. To repair the*body means to supply the elements needed to renew the tissues that are worn or destroyed. We can sepiurate the elements of the human body broadly into water, protein, carbohydrates, fats, and ash. Water composes 60 per cent of a normal man's body. In other words, a 200-pound man is composed of only 80 pounds of solids, of which 18 per cent is composed of protein, 16 per cent of fat, 1 per cent of carbo- hydrates, and 6 per cent of ash. Water aids digestion and is necessary in the economy of life. Protein is the basis of bone, muscle and other tissues, and essential to the body structure. Fat is an important source of energy in the form of heat and muscular power. Carbohydrates are transformed into fat and are important constituents, though in small proportions in the human body. Ash is composed of potash, soda, and phosphates of lime, and is necessary in forming bone. The diet best fitted to supply all the needs of the healthy human organism must contain a correct proportion of these elements ; it is called the balanced ration. Nutritive Ingredients (or Nutrients) of Food What has thus far been said about the ingredients of food and the ways in which they are used in the body may be briefly summarized in the following manner : f Water Food as purchased contains Edible portion . . . e.g. flesh of meat, yolk and white of eggs, wheat, flour, etc. Refuse. 6.^. bones, entrails, shells, bran, etc. Nutrients ' Protein Fats Carbohydrates Mineral mat- ters. •I • • • ••• • • • r • • • • • • : •1(12 :• tVWCJA^iONAt MATHEMATICS FOR GIRLS .•VTi^ I •• • ••' Are stored in the body as fat Uses of Nutrients in the Body Protein Forms tissue .... e g. white (albumen) of eggs, curd (casein) of milk, lean meat, gluten of wheat, etc. £ a vS • • . ' . . • e.g. fat of meat, butter, olive oil, oils of com and wheat, etc. Carbohydrates .... e.g. sugar, starch, etc. Mineral matters (ash) . e,g. phosphates of lime, potash, soda, etc. Are transformed into fat . Repairs tissues All serve as fuel to yield energy in the forms of heat and muscular power. Share in forming bone, assist in digestion, etc. The views thus presented lead to the following definitions : (1) Food is that which, taken into the body, builds tissue or yields energy. (2) The most healthful food is that which best fills the needs of the man. (8) The cheapest food is that which furnishes the largest amount of nutriment at the least cost. (4 ) The best food is that which is at the same time most healthful and cheapest. EXAMPLES Carbohydrates are present in large proportions in all the cereals, bread, and potatoes, and are almost 100 % pure in sugar. There is a widespread notion that starch, which is a fat-producing element, is mostly contained in potatoes, and many people who wish to reduce flesh omit potatoes and substitute rice or larger quantities of bread or cereals. The fact is that the white potato contains only 15 % carbohydrates and the sweet potato 27 %, while rice has 77 % and the breads range from whole wheat bread at 49 % to soda crackers at 73 % and the cereals from oats at 69 % to rye, 78 %. 1. How many ounces of carbohydrates in f lb. of white potatoes ? ,2. Find the number of pounds of carbohydrates in 4 lb. of rice. 3. Find the number of ounces of carbohydrates in a loaf of whole wheat bread (| lb.) FOOD 103 4. How many ounces of carbohydrates in a 2 lb. package of Quaker Oats ? 5. How many ounces of carbohydrates in 4 oz. of soda crackers ? EXAMPLES The proportion of ash in foods is small, and as the body requires 6 per cent, we must be sure to supply it in the food. Salt cod has the largest proportion, 26 per cent, and we find it in good quantities in butter, dried beef, smoked herring, and bacon. Of the proteins, lean meat is the one most easily digested and assim-> ilated. Curiously enough, dried beef has the largest proportion of pro- tein of any flesh meat, 30 per cent, while next in range are leg of lamb, beef steak, roast beef, and fowl, with about 18 per cent. Let us note the protein value of fish. Smoked herring, despised by many, contains 36 per cent of protein, salt cod and canned salmon 21 per cent, and perch, halibut, mackerel, and fresh cod average 18 per cent, equal in food value per pound to the best beef and fowl. The peanut has 27 per cent of its weight protein, and peanut butter 29 per cent. Fat is found in large proportion in nuts, especially in walnuts, which contain 63 per cent, and cocoanuts 67 per cent. Bacon contains 67 per cent of fat, and butter 86 per cent. 1. If a man weighs 189 pounds, how many pounds of water in his weight? How many pounds of solids? How many pounds of fat ? protein ? carbohydrates ? 2. How many ounces of protein in a pound and a half dried beef? 3. Give the number of ounces of protein in a pound fowl. 4. Give the number of ounces of protein in 1^^ lb. of herring. 5. How many ounces of fat in 1^ lb. of shelled walnuts ? Kitchen Weights and Measures Correct measurements are absolutely necessary to insure successful results in cooking. Sift flour, meal, powdered sugar and soda before measuring. Many articles settle hard or in lumps and should be stirred and pulverized before measuring. 104 VOCATIONAL MATHEMATICS FOR GIRLS Measure all materials level full, leveling with knife. Do not pack powdered articles. Butter, lard, etc., should be packed in measure and leveled. A half spoonful should be taken lengthwise and not crosswise. A quarter spoonful should be taken by dividing a half crosswise. A third spoonful is obtained by dividing twice crosswise. Equivalents To Make One Pound 8 teaspoons equal 1 tablespoon. 4 cups flour. 4 tablespoons equal ^ cup. 2f cups com meal. 2 cups equal 1 pint. 2} cups oatmeal. 2 pints equal 1 quart. 6 cups rolled oats. 4 quai'ts equal 1 gallon. 4| cups rye meal. 4 cups flour equal 1 lb. 2 cups rice. 2 cups sugar equal 1 lb. 2 cups granulated sugar. 16 tablespoons dry ingredients 2} cups brown sugar. equal 1 cup. 2f cups powdered sugar. 12 tablespoons liquid equal 1 cup. 3^ cups confectioner's sugar. 2 cups butter. 2 cups chopped meat. . 4| cups ground coffee. t. is the abbreviation for teaspoonful; and tb., f or tablespoonful. EXAMPLES 1. How many teaspoons in 4 tablespoons ? 2. How many tablespoons in f cup ? 3. How many cups are equivalent to 5 pints ? 4. Give the number of tablespoons in a pint. 5. Give the number of teaspoons in 3 quarts and 1 pint. 6. A cup of flour weighs how many ounces ? What part of a pound ? 7. How many cups will 56 tablespoons of baking soda fill ? 8. A pint of liquid contains how many tablespoons ? 9. A cup of sugar weighs how many ounces ? 10. Two cups of corn meal is what part of a pound ? FOOD 105 11. Give the weight in ounces and fractions of a pound of the following quantities : — (a) 1 cup of butter. (/) 1 .cup of powdered sugar. (b) 1 cup of rice, (g) 1 cup of brown sugar. (c) 3 cups chopped meat. (h) 1 cup of rye meal. (d) 1 cup of coffee. (i) 1 cup of oatmeal. (e) 1 cup of conf . sugar. 12. What is the cost of each of the following : (a) 1 cup of oatmeal at 5 cts. a pound ? (b) 2 cups of sugar at 5 lbs. for 33 cts.? (c) 2^ cups of rice at 5 cts. a pound ? (d) ^ cup of milk at 8 cts. a quart ? (e) f cup of butter at 35 cts. a pound ? (/) 2 eggs at 38 cts. a dozen ? (g) I peck of potatoes at 72 cts. a bushel ? (h) 3^ level teaspoons of sugar at 8 cts. a pound ? (i) ^ cup of vinegar at 9 cts. a quart. Cost of Food In order to calculate the cost of food it is necessary to know price per pound, price per cupful, and even price per teaspoon- ful. The price should be calculated to three decimal places and the results tabulated as follows : Cost per pound or quart. Number of cupfuls in pound or quart. Cost per cupful. Cost per tablespoonful. Cost per teaspoonful. When it is once calculated the data may be used from day to day in calculating the cost of food. EXAMPLES 1. What is the cost per teaspoonful of cocoa at 38 cents a pound ? (4 cups in a pound.) 106 VOCATIONAL MATHEMATICS FOR GIRLS 2. What is the cost of a third of a cup of powdered sugar at 10 cents a pound ? (2f cups in a pound.) 3. What is the cost of a tablespoonful of cream at 23 cents a pint ? 4. What is the cost of 2 teaspoonfuls of sugar at 6^ cents a pound? (2 cups in a pound.) 5. What is the cost of 6 nuts at 20 cents a pound ? (29 walnuts in a pound.) 6. What is the cost of 6 tablespoons of coffee at 35 cents a pound ? (4^ cups of coffee in a pound.) 7. What is the cost of 3 slices of toast at 5 cents a loaf ? (10 slices in a loaf.) 8. What is the cost of a pat of butter at 38 cents a pound ? (16 pats of butter in a pound.) 9. What is the cost of an ordinary serving of cornflakes at 10 cents a pound ? (15 servings in a pound.) 10. What is the cost of a serving of cream (| of cup) at 24 cents a pint ? 11. What is the cost of an ordinary serving of macaroni at 12 cents a pound ? (11 servings to the pound.) 12. What is the cost of a serving of cheese at 20 cents a pound ? (9 servings in a pound.) 13. What is the cost of a serving of cabbage salad if cab- bage is 3 cents a pound and two servings in a pound? (A tablespoonful of salad dressing of equal parts of oil and vinegar. Oil is 25 cents a half pint. Vinegar is 10 cents a quart.) 14. What is the cost of a serving of stewed apricots at 18 cents a pound ? (A half pound of sugar at 7 cents is added to the apricots. Nine servings in a pound.) FOOD 107 15. What is . the cost of an ordinary serving of mashed potatoes at 25 cents a half peck ? (A teaspoonf ul of milk at 8 cents a quart to each serving. A half pat of butter at 37 cents a pound, sixteen pats in a pound. Twenty-one servings in a half peck.) 16. What is the cost of a serving of grape jelly {^ oz.) at 13 cents a pound ? Girls should know how to make out a tabulation of stand- ard food materials, the current price for such material at the local stores, and the cost of quantities commonly used in cook- ing, as one cup or one tablespoonful. They should, in addi- tion, know how to take common recipes that are used in cook- ing classes •and reckon the cost. This will aid in reckoning the cost of meals iand arranging them economically. In a like manner, the cost of meals for one day and for one week may be calculated to see how near they are living within their income. EXAMPLES 1. A supper consisting of the following is served for 14 people : codfish in tomato sauce, cereal muffins, cookies, and tea. What is the cost per person if the codfish costs 30 cents, the muffins 24 cents, the cookies 34 cents, the tea 10 cents, and fuel 5 cents ? 2. What is the cost per person for the following meal when 14 are served ? The meal consists of milk toast, stewed prunes with lemon, chocolate layer cake, and tea. The milk toast costs 25 cents, the prunes 25 cents, the cake 50 cents, the tea 10 cents, and fuel 10 cents. 3. In a pound of rolled oats, costing 8 cents, there are 4 cups. What is the cost of a serving (\ cup) of rolled oats ? 4. In a package of cream of wheat costing 14 cents there are 4^ cups. One eighth of a cup is necessary for a serving. What is the cost of a serving ? 108 VOCATIONAL MATHEMATICS FOR GIRLS 5. Compute the cost of a cup of white sauce from the fol- lowing recipe : 1 cup milk 4i t; flour 4^ t. butter } t. salt milk, Oc a quart flour, 6c a pound butter, 39c a pound salt Ic a cup 6. A dinner consisting of mashed potatoes, peas, rib roast, rolls, jelly, fruit salad, krummel torte, coffee, cream and sugar is served for six. What is the cost per person? Estimate portions from amounts given. Dishes Mashed potatoes Peas Kib roast Gravy Rolls Jelly Apple and grape salad Saratoga flakes Salad dressing Krummel Torte Coffee Sugar Sugar total Butter total AMouirre J pk. potatoes at 46c per pk. 1} cups milk at 2c per cup 8 tablespoons butter at 40c per lb. 6 tablespoons or ^ can at 13c per can f of 4-lb. roast at 28c per lb. 3 cups flour at .IJc per cup J cup milk at 2c per cup 3 tablespoons sugar, 2 tablespoons butter, } yeast cake at 2c i glass at 10c per glass 3 apples at 8c per doz. } lb. grapes at 10c per lb. \ package at 15c per package J cup vinegar at 8c per qt. 1 egg at 30c per doz. 3 tablespoons sugar at 7c per lb. 1 tablespoon butter at 40c per lb. I package dates at 10c per package } cup nuts at 70c per lb., 3 cups per lb. IJ eggs at 30c per doz. i pt. whipping cream for torte as well as for coffee at 15o }pt. 6 tablespoons at 34c per lb. 3 teaspoons 7 tablespoons = ^ cup at 7c per lb. 6 tablespoons = } cup at 40c per lb. FOOD 109 7. The following breakfast and luncheon were served for six. What was the cost of each meal per person? Dishes Amount psk Pebson Orange 1 medium-sized, 30c a doz. Toast 2 thin slices, ^c a slice Butter 1 ordinary pat, Jc Egg 1 medium-sized, 3c Com flakes 1 ordinary serving, .2c Cream f cup, 16c J pt. Sugar 3 J level teaspoons, 7c lb. Coffee 2^ tb. at 34c per lb Macaroni and cheese Cabbage salad Cooked dressing Stewed apricots Doughnut Milk a The following cost per person ? Dishes Bib roast 1 Brown gravy j Butter Mashed potatoes Peas Jelly Buns Apple and grape salad Cooked dressing Saratoga flakes Krummel Torte Dates Nuts Whipping cream Sugar Cream Luncheon Ordinary serving, 4c } cu. in., .02c Large serving, .02c 1 tablespoon, .OOJc 1 serving, 2c 1 large, Ic 1 cup, 3c dinner was served for six. Amount per Person What is the Fairly large serving, 9c Ordinary pat, Jc Ordinary serving, Jc Very small, Jc Very small, Ic 2 buns, Ic apiece } apple, }c i oz. of grapes, Ic 1 tablespoon, ^c 3 portions, lOc 0, pkg. (12 portions) 6 dates, 10c a lb. (30 dates in a lb.) 6 nuts, 18c a lb. (22 nuts in a lb.) 1 tablespoon, 16c J pt. 2 teaspoons, 7c a lb. 1 tablespoon, 25c a pt. 110 VOCATIONAL MATHEMATICS FOR GIRLS 9. The recipe for potato soup for a family of man, wife, and two cliildren is : 3 large potatoes 2 tb. flour 1 qt. milk 1^ t. salt 2 slices onion dash pepper 3 tb. butter 1 1. chopped parsley a. What is the recipe for five men, two women, and three children (consider a child's diet one-half a man's diet, and a woman's equal to a man's)? b. What is the recipe for one person (adult) ? 10. The recipe for a vegetable soup for a family of husband, wife, and two children is : Beeff 1 lb. 1^ qt. water Bones, 1 lb. 5 tb. butter ^ cup carrot 1 tb. finely chopped parsley ^ cup turnip salt 1^ cups potato pepper \ onion a. What is the recipe for a family of three men, two women, and a child ? 6. What is the recipe for a child ? 11. The recipe for sour-milk biscuits for a family of hus- band, wife, and two children is : 2i c. flour 1 tb. fat (lard or butter) 1 1. salt 1 c. sour milk, or ^ c. sour milk ^ t. soda ^ c water a. What is the recipe for a man, wife, two boarders, and five children ? 6. What is the recipe for one adult ? Food Values The heating value of food is measured by the amount of heat given off when burned. The food taken into the human system is oxidized slowly in order to give us the ability to do work ; FOOD 111 the number of heat units that food is capable of giving a body represents the quantity of energy the food will provide. There are two units for measuring heat : the English and metric unit. The English unit is called a Calorie, and it represents the quantity of heat necessary to raise a pound of water four degrees on the Fahrenheit scale. The metric unit is also called a calorie and is the amount of heat neces- sary to raise a gram of water one degree on the Centigrade scale. The English unit is called a large Calorie and is represented by the large letter C while the metric unit is called a small calorie and is represented by the small letter c. All scientific experiments are conducted in the metric system, while our measurements in daily life are in the English system. It is only nec- essary to know the English unit, which is used in this book. The United States Department of Agriculture The Department Bulletin No. 28 gives the fuel value of foods. It may be well to know how the fuel value is determined. To illustrate : What is the fuel value of flour ? Careful experiments by the Department of Agriculture show that flour is composed of 10.6 ^o protein, 1.1 <j^ fat, and 76.6 ^ carbohydrates. Other experiments have shown that : 1 gram of protein yields 4100 Calories (C) 1 gram of fat yields 9300 Calories (C) 1 gram of carbohydrates yields 4100 Calories (C) or 1 ounce of protein yields 113 Calories (C) 1 ounce of fat yields 256 Calories (C) 1 ounce of carbohydrates yields 113 Calories (C) Then each ounce of flour contains 0.106 ounce protein 0.011 ounce fat 0.763 ounce carbohydrates Since each ounce of protein yields 1 13 C, 0. 106 oz. will yield 113 x 0. 106, or 11.98 C. 0.011 oz. of fat will yield 256 x 0.011, or 2.81 C. 0.763 oz. of carbohydrates will yield 0.763 x 113, or 86.22 C. EXAMPLES 1. Rice contains 6% protein, 79.5% carbohydrates, and 0.7 % fat. What is the fuel value of 3 oz. rice ? 112 VOCATIONAL MATHEMATICS FOR GIRLS 2. Milk contains 4 % protein, 5 % carbohydrates, and 4 % fat. What is the fuel value of 8 oz. milk ? 3. Beans contain 23.1 % protein, 57 % carbohydrates, and 2 % fat. What is the fuel value of 5 oz. beans ? 4. Chicken contains 24.4 % protein and 2 % fat. What is the fuel value of 7 oz. chicken ? 5. Pork (shoulder) contains 16 % protein and 32.8 % fat. What is the fuel value of 2 lb. pork ? 6. Butter contain^ 0.6 % protein, 0.5 % carbohydrates, and 85 % fat. What is the fuel value of | lb. butter ? 7. Eggs contain 14.9 % protein and 10.5 % fat. What is the fuel value of 7 oz. eggs ? 8. Cornmeal contains 9.2% protein, 70.6% carbohydrates, and 3.8 % fat. What is the fuel value of 3 lb. cornmeal ? 9. English walnuts contain 16.6% protein, 63.4% carbo- hydrates, and 16.1% fat. W^hat is the fuel value of \ lb. nuts? It is clear that a balanced ration need not be an expensive one. The amount of heat required by the body varies from 2000 to 3600 calories approximately, dependent upon age, occupation, and sex. A family of a working man, wife, and three children under sixteen years of age requires 12,000 total calories, 1200 to 1800 of protein, or from 10 to 26 9^ of the total amount required. The quantity of food taken at each meal may vary, provided the total quantities each day reach the standard required. Some authorities suggest about four-twelfths for breakfast, three-twelfths for luncheon, and five-twelfths for dinner. There are two defects in American diet. First, we fail to have a bal- anced ration and, second, we think that the richer the food the more nourishing it is, and that its goodness is in proportion to the hours spent in its preparation. The protein is the most valuable and expensive part of the food supply and it is wise to have a list of proteins so that one can substitute the lesser for the more expensive. Protein, of which we need 18 %, is found more generally in fish than in meat, and the inexpensive peanut is an appe- tizing substitute; fat, of which we need 16 9^, can be had from th^ FOOD 113 fat of all meats, and carbohydrates are better obtained from potatoes than from rich cakes, confectionery, and jellies. We are indebted' to modem inventions for a wide list of cooked and partially cooked foods which have economized the time of the busy housewife and which have enriched our breakfast table beyond that of othe.r nations. The breakfast cereals and the grains from which they are made, white bread, potatoes, sugar, butter, and other fats may be classed as carbohydrates, while meat, fish, eggs, milk, cheese, peas, beans, and cabbage are some of the repre- sentatives of the protein group. These carbohydrates and nitrogenous substances are not wholly such, but are more or less a mixture of other things. Making Up Menus In making up menus it is necessary to have them balance evenly. One should not have too much fat one day, too much starch the next, etc. The menus for each day should hold part of each kind of food, one meat (fish or eggs), one fat, one starch, one tonic vegetable, and one laxative vegetable or fruit. The summer menus must be compiled most carefully, for too much fat or too much meat tends to heat the body at an excessive rate and should therefore be avoided. Of the different food materials which are palatable, nutritious, and otherwise suited for nourishment, we should select those that furnish the largest amounts of available nutrients at the lowest cost. To do this it is necessary to take into account not only the price per pound, quart, or bushel of the different materials, but also the kinds and amounts of the actual nutrients they contain and their fitness to meet the demands of the body for nourishment. The cheapest food is that which supplies the most nutriment for the least money. The most economical food is that which is cheapest and at the same time best adapted to the needs of the user. There are various ways of comparing food materials with respect to the relative cheapness or expeusiveness of their nutritive ingredients. The best way of estimating the relative pecuniary economy of different food mate- rials is found in a comparison of the quantities of nutrients and energy which can be obtained for a given sum, say 10 cents, at current prices. This is illustrated in the table which follows : 114 VOCATIONAL MATHEMATICS FOR GIRLS Comparative Cost op Digestible Nutrients and Energy in DippERENT Food Materials at Average Prices [It is estimated tliat a man at light to moderate muscular work requires about 0.23 pounds of protein and 3060 Calories of energy per day.] 1 • r1 « Amoukt fob 10 Cbnts Kind of Food Matbbial 5 S ^•3 %2 00 OS OST OF TBI a 1 t5 Cents ^ Cents H^ Lbs. Lbs, Lbs. H Dollars Lis. Calories Beef, sirloin .... 26 1.60 26 0.40 0.06 0.06 -^ 410 Do 20 1.28 20 .50 .08 .08 —^ 615 Do 15 .96 15 .67 .10 .11 — 685 Beef, round .... 16 .87 18 .63 .11 .08 — . 560 Do 14 .76 16 .71 .13 .09 .^ 630 Do 12 .66 13 .83 .15 .10 .^ 740 Beef, shoulder clod . . 12 .75 17 .83 .13 .08 — 696 Do .57 13 1.11 .18 .10 — . 796 Beef, stew meat . . . 5 .35 7 2 .29 .23 — . 1,530 Beef, dried, chipped . . 25 .98 32 .40 .10 .03 315 Mutton chops, loin . . 16 1.22 11 .63 .08 .17 -^ 890 Mutton, leg .... 20 1.37 22 .50 .07 .07 _^ 445 Do 16 1.10 18 .63 .09 .09 — 560 Roast pork, loin . . . 12 .92 10 .83 .11 .19 — 1,036 Pork, smoked ham . . 22 1.60 13 .45 .06 .14 — 735 Do 18 1.30 11 .56 .08 .18 — i 916 Pork, fat salt .... 12 6.67 3 .83 .02 .68 — • 2,960 Codfish, dressed, fresh . 10 .93 46 1 .11 _^ — . 220 Halibut, fresh .... 18 1.22 38 .56 .08 .02 — i 265 Cod, salt 7 .46 22 1.43 .22 .01 — • 466 Mackerel, salt, dressed . 10 .74 9 1 .13 .20 —^ 1,136 Salmon, canned . . . 12 .57 13 .83 .18 .10 — . 760 Oysters, solids, 50 cents per quart 25 4.30 111 .40 .02 — .01 90 Oysters, solids, 36 cents per quart 18 3.10 80 .66 .03 .01 .02 126 1 The cost of 1 pound of protein means the cost of enough of the given ma- terial to furnish 1 pound of protein, without regard to the amounts of the other nutrients present. Likewise the cost of energy means the cost of enough ma- terial to furnish 1000 Calories, without reference to the kinds and proportions of nutrients in which the energy is supplied. These estimates of the cost of protein and energy are thus incorrect in that neither gives credit for the value of the other. FOOD 115 COMPARATIVB CoST OP DiOBSTIBLB NUTRIBNTS AND EnBROT IN DiFFBRBNT FooD MATERIALS AT Ayeragb Pricbs — (Continued) Kind of Food Matxbial Lobster, canned . . . Butter Do Do Eggs, 36 cents per doz. . Eggs, 24 cents per doz. . Eggs, 12 cents per doz. . Cheese Milk, 7 cents per quart . Milk, 6 cents per quart . Wheat flour .... Do Com meal, granular . . Wheat breakfast food . Oat breakfast food . . Oatmeal Rice Wheat bread .... Do Do Rye bread Beans, white, dried . . Cabbage Celery Com, canned .... Potatoes, 90 cents per bu. Potatoes, 60 cents per bu. Potatoes, 45 cents per bu. Turnips Apples Bananas Oranges Strawberries .... Sugar 1 o ^'^ Ajiount fob 10 Cit £ ^3 v« 5 04 5s O ig tl a 1 H C hS p s i Cents O* Cents 1 Lbs. 2 Lbs. Lbs. Dollars Lbs. 18 1.02 46 .56 .10 .01 —. 20 20.00 6 .50 .01 .40 — 25 25.00 7 .40 — . .32 — ~ 30 80.00 9 .33 ~— .27 — 24 2.09 39 .42 .05 .04 16 1.39 26 .63 .07 .06 8 .70 13 1.25 .14 .11 — 16 .64 8 .63 .16 .20 .02 ^ 1.09 11 2.85 .09 .11 .14 3 .94 10 3.33 .11 .13 .17 3 .31 2 3.33 .32 .03 2.46 2} .26 2 4 .39 .04 2.94 ^ .32 2 4 .31 .07 2.96 7i .73 4 1.33 .13 .02 .98 7i .53 4 1.33 .19 .09 .86 4 .29 2 2.50 .34 .16 1.66 8 1.18 5 1.25 .08 — .97 6 .77 5 1.67 .13 .02 .87 5 .64 4 2 .16 .02 1.04 4 .51 3 2.50 .20 .03 1.30 5 .65 4 2 .15 .01 1.04 6 .29 3 2 .35 .03 1.16 2i 2.08 22 4 .05 .01 .18 5 6.65 77 2 .02 — .06 10 4.21 23 1 .02 .01 .18 H 1.00 5 6.67 a .01 .93 1 .67 3 10 .15 .01 1.40 i .50 3 13.33 .20 .01 1.87 1 1.33 8 10 .08 .01, .54 H 5.00 8 6.67 .02 .02 .65 7 10.00 27 1.43 .01 .01 .18 6 12.00 40 1.67 .01 .13 7 8.75 47 1.43 .01 .01 .09 6 — 3 1.67 ^— — 1.67 I CcUoriss 225 1,705 1,365 1,125 260 385 770 1,186 885 1,030 5,440 6,540 6,540 2,235 2,395 4,500 2,025 2,000 2,400 8,000 2,340 3,040 460 130 430 1,970 2,950 3,936 1,200 1,270 370 250 216 2,920 116 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES 1. What is the most economical part of beef for a soup ? 2. What is the most economical part of mutton for boiling ? 3. What is the most economical part of pork for a roast ? 4. Is fresh or salt codfish more economical ? 5. What is the fuel value of 3 oz. oatmeal ? 6. What is the fuel value of 3 oz. rice ? 7. What is the fuel value of 4 oz. strawberries ? a What is the fuel value of 6 oz. milk ? EXAMPLES Since several hundred Calories are required each day for a person's diet, it is most convenient in computing meals to think of our foods in 100-Calorie portions. Therefore it is desirable to know how to compute this portion and tabulate it for future reference. 1. 42 qt. of milk give 36,841 Calories. What is the weight of a 100-C portion ? 2. 3^ lb. of flour give 1610.5 Calories. What is the weight of a 100-C portion ? 3. ^ lb. of dates give 393.75 Calories. What is the weight of a 100-C portion ? 4. If J of a cup of flaked breakfast food gives approximately 100 C, what is the food value of 1 lb.? 5. If ^ of a cup of skimmed milk gives approximately 100 C, what is the food value of 1 qt.? 6. A teaspoonful of fat gives 100 C. What is the food value of 1 lb. lard ? 7. If ^ of a medium-sized egg gives a food value of 100 C, what is the food value of an egg ? 8. 4 thin slices of bacon (1 oz.) give a food value of 100 C. What is the food value of 9 lb. of bacon ? FOOD 117 9. If I oz. of sweet chocolate has a food value of 100 C, what is the food value of ^ lb.? 10. Ten large pears have the value of 100 C, which is the same as for 2 doz. raisins. What is the food value of a single raisin? 11. Eind the individual cost of feeding the following families per week and per day. Find the number of Calories per indi- vidual per day. (Arrange results in a column as suggested.) Family No. in Family Total Cost Total Calories A 6 B 7 C • 8 D 8 E 7 F 6 G 7 H 4 I 4 J 6 K 8 L 6 M 7 N 14 O 6 Economical Use of Heat It is important to reduce waste by using as much as possible of the bone, fat, and trimmings, not usually served with the meat. If nothing better can be done with them, the bones and trimmings are profitably used in the soup kettle, and the fat can be saved for cooking to be used in place of more expensive butter and lard. The bits of meat not served with the main dish, or remaining after the first serving, may be seasoned and recooked in many palatable ways. Or they can be combined with vegetables, pie crust, or other materials, thus extending the meat flavor over a large quantity of less expensive food. 13.60 86224 15.06 90928.64 11.21 101966.75 6.68 33744.14 15.01 130657.04 12.89 93456.34 17.77 11063.91 11.86 90891.3 10.28 50490 16.47 69385.9 10.37 112197.3 16.08 930262 30.89 86006.8 32.91 141517 12.31 85582.8 118 VOCATIONAL MATHEMATICS FOR GIRLS Different kinds and cuts vary considerably in price. Sometimes the cheaper cuts contain a larger proportion of refuse than the more expen- sive, and the apparent cost is really more than the actual cost of the more edible portion. Aside from this advantage, that of the more ex- pensive cuts lies in the tenderness and flavor, rather than in the nutritive value. Tenderness depends on the character of the muscle fibers and connective tissues of which the meat is composed. Flavor depends partly on the fat present in the tissues, but mainly on nitrogenous bodies known as extractives, which are usually more abundant or of more agreeable flavor in the more tender parts of the animal. The heat of cooking dissolves the connective tissues of tough meat and in a measure makes it more tender, but heat above the boiling point or even a little lower tends to change the texture of muscle fibers. Hence tough meats must be carefully cooked at low heat long applied in order to soften the connective tissue. For this purpose the fireless cooker may be used to great advantage. Steers and Beef Steers are bought from the farmer by the hundredweight (cwt.). They are inspected and then weighed. After they are killed and dressed, they are washed several times and sent to the cooler. The carcass must be left in the cooler several days before it can be cut. It is then divided into eight standard cuts and each piece weighed separately. Sixty per cent of the meat used in this country is produced in the Federally inspected slaughtering and packing houses, of which there are nearly 900, located in 240 cities. EXAMPLES 1. A steer weighing 1093 lb. was purchased for $ 7.42 per cwt. What was paid for him ? 2. The live weight of a steer is 1099 lb. ; the dressed weight 641 lb. What is the difference ? What is the percentage of beef in the animal ? 3. A steer with a dressed weight of 677 lb. was cut into the following parts : two ribs weighing 61 lb. each, 2 loins 103 lb., 2 rounds 154 lb., and suet 21 lb. What was the percentage of each part to the total amount ? FOOD 119 4. A steer with a dressed weight of 644 lb. was sold at $ 10.51 per ewt. What was paid ? 5. If the value of ribs is 18^, loins 18^^, rounds 9|^, what is the value of cuts in problem 3 ? 6. A housewife buys 8f lb. of meat every Monday, 9^ lb. on Wednesday^ and 10} lb. on Saturday. What is the total amount of meat purchased in a week ? 7. The live weight of a low-grade steer was 947 lb. and dressed weight 475 lb. What is the per cent of dressed to live weight? .What did the steer sell for at 6^ cts. live weight ? What was the selling price per cwt. ? a A high-grade steer weighed live weight 1314 lb. and dressed weight 897 lb. What is the per cent of "dressed to live weight? What did the steer sell for at 9 cts. a pound live weight? What was the selling price per cwt.? Note the difference in the price between low- and high-grade steers due to the fact that the latter have a greater proportion of the higher priced cuts. 9. A steer was killed weighing 632 lb. and sold for $ 10.38 cwt. a. What was the selling price ? b. What was the average price per pound ? c. What was the percentage of each cut to total value ? d. What was the total value of each cut ? Cuts Weight Pbioi p»b Pound (Wholesale) 2 Ribs 681b. 1.17 2 Loins 100 .18J 2 Rounds 160 .09} 2 Chucks 160 .08 2 Fla.nkR 30 .06} 2 Shanks 26 .06 2 Briskets 32 .08} Navel End 46 .06 Neck Piece 8 .01} 2 Kidneys 2 .06 Suet 20 .08 632 lb. 120 VOCATIONAL MATHEMATICS FOR GIRLS Cuts of Beef The cuta of beef differ with the locality and the packing house. The general method of cutting up a aide of beef is illustrated in the following figure. Stahdikd Bbef Cuts — Chicago Style 1 — Jtound Rump Roast RouDd Steak Corned Beef Hambarger Sleak Dried Beef Shank — Soup Bone 2 — Loin Slrlotn Steak Porterhonse Sleak Club Steak Beef Tenderloin S — Flank Flank Steak Hamburger Steak Corned Beet S — XanelEnd Short Riba Corned Beef Soap Meat G — BrUket Comed Bee( Soup Meat Pot Roast t — Fore Shank Soup Bone i — Chuck Shoulder Steak Shoulder Roast Pot Boaat Slews Standard Fork Cctb — Chicaoo Stvlb -Boston Bull Pickled Potk Pork Sboalder Fork Steak 6 — Belly Spare Ribs Brisket Bacon Salt Pork Pock Eoasl Pork Chops , Fork Tenderloin 7 — Fat Back Paprika Bacon Dry Salt Fat Backs Barrel Fork EXAMPLES Hogs are usaallj killed when Dine or ten months old. The velght iB 76% to 80% of live weight. The method of cutting up aside of pork differs considerably from that employed with other meats, A large por- tion of the carcfias of a dressed pig consists of almost clear fat. This fur- nishes the cuts wliicb are used for salt pork and bacon. 1. A hog weighed at the end of 9 months 249 lb. When he was killed and dressed, he weighed 203 lb. What was the per cent of dressed to live weight? 2. A hog weighing 261 lb. was sold for 8\ cents live weight. When he was dressed, he weighed 204 lb. What should he sell for per cwt. (dressed) in order to cover the price of purchase ? 122 VOCATIONAL MATHEMATICS FOR Q-IRLS 3. Sugar-cured hams and bacons are made by rubbing salt into the pieces and placing a brine solution of the following proportions over them in a barrel, before smoking them; 8 lb. salt, 21 lb. brown sugar, 2 oz. saltpeter in four gallons of water for every 100 lb. of meat. What percentage of the solu- tion is salt? Sugar? (Consider a pint of water equal to a pound.) 4. Sausages are made by mixing pork trimmings from the ham with fat and spices, and placing in casings. If 3 lb. of ham are added to 1 lb. fat pork, what is the percentage of lean pork? St&ndabd Mutton Cuts — Chicaoo Sttlb l~Leg Leg □( Mutton Mutton Chops 2— Loin Loin Roast Mutton Cbopfl 3— Hotel Back Rib Chops '0 — Chuck Shoulder Boast Stow Shoulder Chops EXAMPLES 1. A butcher buys 169 sheep at S5.76 a head. He sells them so as to receive on the average $6.12^ for each. What does he gain ? FOOD 123 2. A market man bought 19 dressed sheep for $81.75. What was the average price ? 3. A sheep weighed 138 lb. live weight and 72 lb. dressed. What was the per cent of dressed to live weight ? 4. A dressed sheep when cut weighed as follows ; Leg 23.11b. each Neck 3.41b. Breast 8.2 1b. Loin 18.4 lb. each Shoulder 6.1 lb. each Shank 5.3 lb. each Ribs 15.3 lb. each What was the total dressed weight ? What was the percent- age of each cut to the dressed weight ? Length of Time Required to Cook Mutton Boiling Mutton, per pound 16 minutes Baking Mutton, leg, rare, per pound ... 10 minutes Mutton, leg, well done, per pound . 15 minutes Mutton, loin, rare, per pound . . 8 minutes Mutton, shoulder, stuffed, per pound 16 minutes Mutton, saddle, rare, per pound . . 9 minutes Lamb, well done, per pound ... 15 minutes Broiling Mutton chops, French 8 minutes Mutton chops, English 10 minutes EXAMPLES Give the fraction of an hour required (a) To boil mutton (2 lb.). (b) To bake leg of mutton (3 lb.). (c) To bake loin of mutton (4 lb.). (d) To broil mutton chops (French). (e) To broil mutton chops (English). {f) To bake shoulder of mutton (5 lb.). 124 VOCATIONAL MATHEMATICS FOR GIRLS Fish is a very economical kind of food. It can be obtained fresh at a reasonable figure in seacoast towns. 1. During the year 1913, 170,000,000 lb. of fish were brought into Boston, and sold for $ 7,000,000. What was the average price per pound ? 2. If 528,000,000 lb. of fish were caught in the waters of New England during the year 1913, it would represent one- quarter of the catch of the entire country. What is the catch of the entire country ? 3. A pound of smoked ham at 24 cents contains 16 % protein, while a pound of haddock at 7 cents contains 18% protein. How much more protein in a pound of haddock than in a pound of ham ? (In ounces.) 4. For the same value, how much more protein can you pur- chase in the haddock than in the ham ? 5. A pound of pork chops at 25 cents contains 17 % protein ; a pound of herring at 8 cents contains 19 %. How much more protein is there in the pound of herring than in the pork chops ? 6. For the same value, how much more protein can be pur- chased in the pound of herring than in pork chops ? 7. A pound of sirloin steak at 30 cents gives the same amount of protein as the pork chops in example 6. For the same value how much more protein can be obtained from haddock than from the steak ? What per cent of protein per pound in had- dock ? Use data in Example 3. 8. If fish can be purchased at any time at not over 12 cents per pound, and meats at not less than 20 cents per pound, what is the per cent of saving by buying fish ? 9. If 5.3 % of the total expenses for foodstuffs is for fish, and 22 % of the family earnings goes for food, what is the amount spent for each ? Family income $ 894. POOD 125 Economical Marketing The most economical way to purchase food is to buy in bulk. Fancy packages with elaborate labels must be paid for by the consumer. All realize the convenience of package goods, the saving in cost of preparation and cooking and the ease with which they are kept clean and wholesome, but the additional expense is enormous, in some instances as high as 300 ^ . EXAMPLES 1. If the retail price of dried beef is 50 cents a pound, how much more per pound do I pay for dried beef, when I purchase a package weighing S^ oz. for 18 cents? What per cent more do I pay ? 2. Wheat costs the farmer or producer 1^ cents per pound. I purchase a package of wheat preparation weighing 5 oz. for 10 cents. How much more do I pay for wheat per pound than it costs to produce it ? What per cent more do I pay ? 3. Good apples cost $ 2.75 per barrel. If I purchase a peck for 50 cents, at what rate am I paying for apples per barrel ? (A standard apple barrel contains 2^ bushels.) How much would I save a peck, if a few families in the neighborhood joined me in purchasing a barrel ? 4. Codfish retails at 17 cents a pound. A group of families sent one of their members to the wharf and she purchased for 60 cents a codfish weighing 6 lb. How much was saved per pound ? What per cent ? 5. Print butter is molded by placing a quantity of tub butter in a mold. If the tub butter costs 34 cts. a pound and the print butter 42 cts. a pound, how much cheaper (per cent) is the tub butter than print butter? Does it afford the same nourishment? 6. A pint can of evaporated milk costs 10 cents and con- tains the food element of 2^ quarts of fresh milk at 8 cents a quart. What is the saving per quart of milk ? 126 VOCATIONAL MATHEMATICS FOR GIRLS Every housewife should possess the following articles in the kitchen so as to be able to verify everything she buys : 1 good 20-lb. scale 1 dry quart measure 1 peck measure 1 liquid quart measure 1 half-peck measure 60-inch steel tape 1 quarter peck measure - 8-oz. graduate The above should be tested and "sealed" by the Super- intendent of Weights and Measures. Check the goods bought and see if weight and volume agree with what was ordered. EXAMPLES 1. If a gallon contains 231 cu. in., how many cubic inches are there in a quart ? 2. If a bushel contains 2150.42 cu. in., how many cubic inches are there in a dry quart ? 3. If a half-bushel basket or box, heaping measure, must contain five-eighths bushel, stricken ^ measure, how many cubic inches does the basket contain ? 4. A box 12 by 14 by 16 inches when stricken full will hold a heaping bushel. How many cubic inches in the box ? 5. A dealer often sells dry commodities by liquid measure. If a quart of string beans were sold by liquid measure for 16 cts., how much would the customer lose ? What is the differ- ence in per cent between liquid and dry quart measure ? 6. A grocer sold a peck of apples to a housewife. As he was about to place the apples in the basket, the woman called his attention to the fact that the measure was not " heaping." He placed four more apples in the measure. When she reached home she counted 24 apples. What would have been the per cent loss if she had not called his attention to the measure ? ^ Stricken measure is measure that is not heaped, hut even full. FOOD 127 7. A " five-pound " pail of lard was found to weigh 4 lb. 11 oz. What per cent was lost to the customer ? 8. A package (supposed to be a pound) sold for 12 cents and was found to weigh 14^ ounces. How much did the consumer lose ? 9. A quart of ice cream was bought for 40 cents. The box was found to be 12^ % short. How much did the consumer lose? 10. A girl bought a quart of berries for ten cents. J The box was found to contain 54.5 cu. in. How much was lost ? 11. A pound of print butter cost 39 cents and was found to weigh 14^ ounces. How much did the consumer lose ? CHAPTER VI PROBLEMS ON THE CONSTRUCTION OF A HOUSE Most people live either in a flat or a house. Each has its advantages and its disadvantages. The work of a flat is all on one floor ; there are no stairs, halls, cellars, furnaces, and side- walks to care for, and when the building is heated by steam, there is only the kitchen fire or a gas range to look after. These are the advantages and they reduce the work of the home to very simple proportions. Then, too, it is possible to find comfortable flats at a moderate price in a neighborhood where it would be impossible to build a small house. How- ever, in these flats some of the rooms are not well lighted and ventilated, and one is dependent upon the janitor for many services which are not always pleasantly performed, though fees are constantly expected. The long flights of stairs are a great drawback, because people will not go out as much as they should, on account of the exhausting climb on their return. The small house, in country or city, brings more healthful mental and physical surroundings than the flat. Perfect venti- lation, light, sunshine, and freedom from all petty restrictions give a more vigorous tone to body and mind. If the house is in the suburbs and there is some land with it, where a few vegetables and flowers can be cultivated, it has an added charm and blessing in the form of healthful outdoor work : furnace, cellar, and grounds for the husband's . share ; house, from garret to cellar, for the wife's share. In a flat a man can escape nearly all duties about the house, but in the little house he must bear his share. If one lives in the suburbs, the time and money spent in going to and from the city is quite an item, but the cheaper rent usually more than balances the traveling expense. A person should not pay over 26% of income for rent. In case a person receives an income of $ 1600 or over, and has a savings bank deposit of about $ 1600, it is usually better to 128 CONSTRUCTION OF A HOUSE ' 129 purchase a house than to rent. Money may be borrowed from either the cooperative bank or the savings bank. The total rent of a house a year should be at least 10 % of the value of the house and land : 6 % represents interest on the investment, and 4 % covers taxes and depreciation. In a flat the middle floor should cost approximately 10 % more than the first floor, and the top 10 % less than the first floor. EXAMPLES 1. A single house and land cost $ 2800. Wliat should be considered the rent per year ? 2. A two-family house cost $5600. (a) What should be the rent per month ? (6) What should be the rent of each flat ? 3. A three-family house costs $ 6500. What should be the rent of each floor ? 4. A family desires to build a cottage-style, gambrel roof house containing seven rooms, bath, reception hall, cemented cellar, and small storage attic. It is finished inside with North Carolina pine and has hard-pine floors, fir doors, open plumb- ing, two coats of plaster, furnace heat, and electric light. The first floor has three rooms and a reception hall. The second floor has three chambers, bath, and sewing room over the hall. The architect finds that the cost of materials in the summer and late fall varies as follows : Amount sa.vbd Itkm Summer BT BUILDING IN THE Fall Mason work $200 a 16.00 Brick and cement 90 7.20 Lumber 600 60.00 Finish 126 12.60 Plumbing 226 22.60 Heat (furnace) 100 10.00 Paint and paper 200 20.00 Plastering 200 16.00 Electric wiring 40 3.20 Electric light fixtures 40 4.00 Labor (carpenters) 460 Profit to contractor 213 27.62 130 VOCATIONAL MATHEMATICS FOR GIRLS (a) What is the total cost in each case ? (b) What is the difference in per cent ? What is the per cent difference in each item ? Economy of Space it Many persons who build houses, barns, and other buildings do not understand the fundamental fact that there is more space in a square building than in a long one, and that the further they depart from the square form the more their build- ing will cost in proportion to its size. For instance, a building 20' by 20' has 400 square feet of floor space and requires 80 feet of outside wall, while one 10' by 40' will, with the same floor space, require 100 feet of wall. Accordingly more material and work will be required for the longer one. In many cases, of course, there are objections to having a building square. The longer building, for instance, gives more wall space and more light, and these may be desired items. The roof and floor items are about the same in either case. Preparation of Wood for Building Purposes In winter the forest trees are cut and in the spring the logs are floated down the rivers to sawmills, where they are sawed into boards of different thicknesses. To square the log, four slabs are first sawed off. After these slabs are off, the remainder is sawed into boards. As soon as the boards or planks are sawed from the logs, they are piled on prepared foundations in the open air to season. Each layer is sepa- rated from the one above by a crosspiece, called a strip, in order to allow free circulation of air about each board to dry it quickly and evenly. If lumber were piled up without the strips, one board upon another, the ends of the pile would dry and the center would rot. This seasoning or drying out of the sap usually requires several months. Wood that is to be subject to a warm atmosphere has to be artificially dried. This artificially dried or kiln-dried lumber has to be dried to a point in excess of that of the atmosphere in which it is to be placed after being removed from the kiln. This process of drying must be done grad- ually and evenly or the boards may warp and then be unmarketable. CONSTRUCTION OF A HOUSE 131 Definitions Board Measure. — A board one inch or less in thickness is said to have as many board feet as there are square feet in its surface. If it is more than one inch thick, the number of board feet is found by multiplying the number of square feet in its surface by its thickness measured in inches and fractions of an inch. The number of board feet = length {in feet) x width {in feet) x thick- ness {in inches). Board measure is used for plank meafiure. A plank 2" thick, 10" wide, and 15' long, contains twice as many square feet (board measure) as a board 1'' thick of the same width and length. Boards are sold at a certain price per hundred (C) or per thousand (M) board feet. The term luinber is applied to pieces not more than four inches thick ; timber to pieces more than four inches thick ; but a large amount taken together often goes by the general name of lumber, A piece of lumber less than an inch and a half thick is called a board and a piece from one inch and a half to four inches thick is called a plank. Rough Stock is lumber the surface of which has not been dressed or planed. The standard lengths of pieces of lumber are 10, 12, 14, 16, 18 feet, etc. EXAMPLES 1. How many board feet in a board 1 in. thick, 15 in. wide, and 15 ft. long ? 2. How many board feet of 2-inch planking will it take to make a walk 3 feet wide and 4 feet long ? 3. A plank 19' long, 3" thick, 10" wide at one end and 12" wide at the other, contains how many board feet ? 4. Find the cost of 7 2-inch planks 12 ft. long, 16 in. wide at one end, and 12 in. at the other, at $ 0.08 a board foot. 5. At $ 12 per M, what will be the cost of 2-inch plank for a 3 ft. 6 in. sidewalk on the street sides of a rectangular corner lot 56 ft. by 106 ft. 6 in. ? 132 VOCATIONAL MATHEMATICS FOR GXHLS Frame and Koof After the excavation ia finished and the foundation laid, ttie ci tion of the l)UJldiDg itself Is begun. On the top of the foundation a lar) timber called a sill is placed. The timbers running at right angles to tt front sill are called side sills. The sills are joined at the comers by half-lap joint and held together by spikes. a. Outside studding de. Second-floor joisia t. Sheathing b. Rafters d^. First-floor joists j. Partition studs c. Plates g. Qirder or cross sill k. Partition heads d. Ceiling joists h. Sllla I. Piers The walls of the building are framed by placing eorner posls 4" by 6" on the four corners. Between these comer pneta there are placed smaller timbers called studding, 2" by 4", 13" apart. !Later the laths, 4' long, are nailed to this stndding. The upright timbers are often mortised into the sills at the bottom. When these uprights are all in position, a timber, called a plate, is placed on the top of tliem and they are spiked together. On the top of the plate is placed the roof. The principal timbers of the roof are tlie rafters. Different roofs have a different pitch or slope — that Is, form different angles with the plate. To obtain the desired pitch the carpenter uses the steel square. CONSTRUCTION OF A HOUSE 133 A roof with one half pitch means that the height of the ridge of the roof above the level of the plate is equal to one half the width of the building. This illustrates a roof with one-half pitch. EXAMPLES Give the height of the ridge of the roof above the level of the plate of the following building : Pitch Width op Bciloino 1. One-half 32' 2. One-fourth 40' 3. One-third 36' 4. One-sixth 48' Building Materials Besides wood many materials enter into the construction of buildings; among these are mortar, cement, stone, bricks, marble, slate, etc. Mortar is a paste formed by mixing lime with water and sand in the correct proportions. (Common mortar is generally made of 1 part of lime to 5 parts of sand.) It is used to hold bricks, etc., together, and when stones or bricks are covered with this paste and placed together, the moisture in the mortar evaporates and the mixture ** sets *' by the absorption of the carbon dioxide from the air. Mortar is strengthened by adding cow's hair when it is used to plaster a house ; in such mortar there is sometimes half as much lime as sand. Plaster is a mixture of a cheap grade of gypsum (calcium sulphate), sand, and hair. When the plaster is mixed with water, the water com- bines with the gypsum and the minute crystals in forming interlace and cause the plaster to " set." When masons plaster a house, they estimate the amount of work to be done by the square yard. Nearly all masons use the following rule : Calculate the total area of walls and ceil- 134 VOCATIONAL MATHEMATICS FOR GIRLS ings and deduct from this total area one-half the area of open- ings such as doors and windows. A bushel of mortar will cover about 3 sq. yd. with two coats. Example. — How many square yards of plastering are nec- essary to plaster walls and ceiling of a room 28' by 32' and 12' high? Areas of the front and back walls are 28 x 12 x 2 = 672 sq. ft. Areas of the side walls are 32 x 12 x 2 = 768 sq. ft. Area of the ceiling is 28 x 32 = 896 sq. ft. 2336 sq. ft. 2386 sq. ft. = ^^ sq. yd. = 259^ sq. yd. 260 sq. yd. Ans. EXAMPLES 1. What will it cost to plaster a wall 10 ft. by 13 ft. at $ 0.30 per square yard ? ' 2. What will it cost to plaster a room 28' 6" by 32' 4" and 9' 6" high, at 29 cents a square yard, if one-half its area is allowed for openings and there are two doors 8' by 3^' and three windows 6' by 3' 3" ? 3. What will it cost to plaster an attic 22' 4" by 16' 8" and 9' 4" high, at a cost of 32 cents a square yard ? Bricks used in Building Brickwork is estimated by the thousand, and for various thicknesses of wall the number required is as follows : 8J-inch wall, or 1 brick in thickness, 14 bricks per superficial foot. 12f-inch wall, or 1 J bricks in thickness, 21 bricks per superficial foot. 17-inch wall, or 2 bricks in thickness, 28 bricks per superficial foot. 21^inch wall, or 2^ bricks in thickness, 35 bricks per superficial foot. .EXAMPLES From the above table solve the following examples : 1. How much brickwork is in a 17" wall (that is, 2 bricks in thickness) 180' long by 6' high ? CONSTRUCTION OF A HOUSE 135 2. How many bricks in an 8 ^ wall, 164' 6" long by 6' 4" ? 3. How many bricks in a 17" wall, 48' 3" long by 4' 8" ? 4. How many bricks in a 21^" wall, 36' 4" long by 3' 6" ? 5. How many bricks in a 12f" wall, 38' 3" long by 4' 2"? 6. At $ 19 per thousand find the cost of bricks for a build- ing 48' long, 31' wide, 23' high, with walls 12f " thick. There are 5 windows (7' X 3') and 4 doors (4' x 8^'). To estimate the number of bricks in a wall it is customary to find the number of cubic feet and then multiply by 22, which is the number of bricks in a cubic foot with mortar. * 7. How many bricks are necessary to build a partition wall 36' long, 22' wide, and 18" thick ? 8. How many bricks will be required for a wall 28' 6" long, 16' 8" wide, and 6' 5" high ? 9. How many cubic yards of masonry will be necessary to build a wall 18' 4" long and 12' 2" wide ^nd 4" thick? 10. At $ 19 per thousand, how much will the bricks cost to build an 8^", or one-brick wall, 28' 4" long and 8' 3" high ? 11. At $20.60 per thousand, how much will the bricks cost to build a 12f " wall, 52' 6" long and 14' 8" high ? 12. A house is 45' x 34' x 18', the walls are 1 foot thick, the windows and doors occupy 368 cu. ft. ; how many bricks will be required to build the house ? 13. What will it cost to lay 250,000 bricks, if the cost per thousand is $ 8.90 for the bricks ; $ 3 for mortar ; laying, $ 8 ; and staging, $ 1.25 ? Stonework Stonework, like brickwork, is measured by the cubic foot or by the perch (16 J' x 1^ x 1') or cord. Practical men usu- ally consider 24 cubic feet to the perch and 120 cubic feet to the cord. The cord and perch are not much used. 136 VOCATIONAL MATHEMATICS FOR GIRLS The usual way is to measure the distance around the cellar on the out- side for the length. This includes the corners twice, but owing to the extra work in making corners this is considered proper. * No allowance is made for openings unless they are very large, when one-half is deducted. The four walls may be considered as one wall with the same height. Example. — If the outside dimensions of a wall are 44' by 31', 10' 6" high and 8" thick, find the number of cubic feet. 44 25 31 «« 2 •"^^ ;^jai X ~ X i = 1050 cu. ft. Ans. _2 . ^^ ^ ;;2 150 ft. length. ^ Cement Some buildings have their columns and beams made of concrete. Wooden forms are first set up and the concrete is poured into them. The concrete consists of Portland cement, sand, and broken stone, usually in the proportion of 1 part cement to 2 parts sand and 4 parts stone. The average weight of this mixture is 150 pounds per cubic foot. After the con- crete has " set," the wooden boxes or forms are removed. Within a few years twisted steel rods have been placed in the forms and the concrete poured around them. This is called reenforced con- crete and makes a stronger and safer combination than the whole concrete. It is used in walls, sewers, and arches. It takes a long time for the con- crete to reach its highest compressive and tensile strength. Cement is also used for walls and floors where a waterproof surface is desired. When the cement "sets," it forms a layer like stone, through which water cannot pass. If the cement is inferior or not properly made, it will not be waterproof and water will pass through it and in time destroy it. EXAMPLES 1. If one bag (cubic foot) of cement and one bag of sand will cover 2f sq. yd. one inch thick, how many bags of cement and of sand will be required to cover 30 sq. yd. 2^" thick ? CONSTRUCTION OF A HOUSE 137 2. How many bags of cement and of sand will be required to lay a foundation 1" thick on a sidewalk 20' by 8' ? 3. How many bags of cement and of sand will it take to cover a walk, 34' by 8' 6", |" thick ? 4. If one bag of cement and two of sand will cover 6 J sq. yd. I" thick, how much of each will it take to cover 128 sq. ft. ? 5. How much of a mixture of one part cement, two parts sand, and four parts cracked stone will be needed to cover a floor 28' by 32' and 8" deep ? How much of each will be used ? Shingles Shingles for roofs are figured as being 16" by 4" and are sold by the thousand. The widths vary from 2" upward. They are put in bundles of 250 each. When shingles are laid on the roof of a building, they overlap so that only part of each is exposed to the weather. EXAMPLES 1. How much will it cost for shingles to shingle a roof 60 ft. by 40 ft., if 1000 shingles are allowed for 125 sq. ft. and the shingles cost $ 1.18 per bundle ? 2. Find the cost of shingling a roof 38 ft. by 74 ft., 4" to the weather, if the shingles cost $ 1.47 a bundle, and a pound and a half of cut nails at 6 cents a pound are used with each bundle. 3. How many shingles would be needed for a roof having four sides, two in the shape of a trapezoid with bases 30 ft. and 10 ft., and altitude 15 ft., and two (front and back) in the shape of a triangle with base 20 ft. and altitude 15 ft.? (1000 shingles will cover 120 sq. ft.) Slate Roofing In order to make the exterior of a house fireproof the roof should be tile or slate. Slates make a good-looking and durable 138 VOCATIONAL MATHEMATICS FOR GIRLS roof. They are put on, like shingles, with nails. Estimates for slate roofing are made on 100 sq. ft. of the roof.^ The following are typical data for building a slate roof : A square of No. 10 x 20 Monson slate costs about 9 8.35. Two pounds of galvanized nails cost $0.16 per pound. Labor, $ 3 per square. Tar paper, at 2} cents per pound, 1 J lb. per square yard. EXAMPLES Using the above data, give the cost of making slate roofs for the following : 1. What is the cost of laying a square of slate ? 2. What is the cost of laying slate on a roof 112' by 44' ? 3. What is the cost of laying slate on a roof 166' by 64'? 4. What is the cost of laying slate on a roof 118' by 52' ? 5. What is the cost of laying slate on a roof 284' by 78' ? Clapboards Clapboards are used to cover the outside walls of frame buildings. Most clapboards are 4' long and 6" wide. They are sold in bundles of twenty-five. Three bundles will cover 100 square feet if they are laid 4" to the weather. To find the number of clapboards required to cover a given area, find the area in square feet and divide by 1^, Allowance may be made for openings by deducting from area, EXAMPLES 1. How many clapboards will be required to cover an area of 40 ft. by 30 ft.? 2. How many clapboards will be necessary to cover an area of 38' by 42' if 66 sq. ft. are allowed for doors and windows ? 3. How many clapboards will a barn 60 ft. by 50 ft. require if 10 % is allowed for openings and the distance from founda- tion to the plate is 17 ft. and the gable 10 ft. high ? ^ Called a square. ^^ CONSTRUCTION OF A HOUSE 139 Flooring Most floors in houses are made of oak, maple, birch, or pine. This flooring is grooved so that the boards fit closely together without cracks between them. The accompanying figure shows the ends of c c g l^ pieces of matched flooring. Matched boards are also used for ceilings and walls. In estimating for matched flooring enough stock must be added to make up for what is cut away from the width in matching. This amount varies from J" to |" on each board ac- cording to its size. Some is also wasted in squaring ends, cutting up, and fitting to exact lengths. A common floor is made of unmatched boards and is usually used as an under floor. Not more than \^' per board is allowed for waste. Example. — A room 12 ft. square is to have a floor laid of unmatched boards 1|-" wide ; one-third is to be added for waste. What is the number of square feet in the floor ? What is the number of board feet required for laying the floor? 12 X 12 = 144 sq. ft. = area. 144 x J = 48 144. Ans, 144 192 board measure for unmatched floor. 192. Ans. EXAMPLES 1. How much 1^ in. matched flooring 3" wide will be re- quired to lay a floor 16 ft. by 18 ft. ? One-fourth more is al- lowed for matching and 3 % for squaring ends. 2. How much hard pine matched flooring |^" thick and 1^" wide will be required for a floor 13' 6" x 14' 10" ? Allow \ for matching and add 4 % for waste. 3. An office floor is 10' 6" wide at one end and 9' 6" wide at the other (trapezoid) and 11' 7" long. What will the material cost for an unmatched maple floor |^" thick and IJ" wide at $ 60 per M, if 4 sq. ft. are allowed for waste ? 140 VOCATIONAL MATHEMATICS FOR GIRLS 4. How many square feet of sheathing are required for the outside, including the top, of a freight car 34' long, 8' wide, and 7 J^' high, if 37^% covers all allowances ? 5. In a room 60' long and 20' wide flooring is to be laid ; how many feet (board measure) will be required if the stock is I" X 3" and \ allowance for waste is made ? Stairs The perpendicular distance between two floors of a building is called the rise of a flight of stairs. The width of all the steps is called the run. The perpendicular dis- tance between steps is called the width of risers. Nosing is the slight pro- jection on the front of each step. The board on each step is the tread. To find the number of stairs necessary to reach from one floor to another : Measure the rise first. S^^'^« Divide this by 8 inches, which is the most comfortable riser for stairs. The run should be 8J^ inches or more to allow for a tread of 9f inches with a nosing of 1 J inches. Example. — How many steps will be required, and what will be the riser, if the distance between floors is 118 inches ? 118 H- 8 = 14i or 16 steps. 118 -H 15 = 7{i inches each riser. Ans, EXAMPLES . 1. How many steps will be required, and what will be the riser, (a) if the distance between floors is 8' ? (6) If the dis- tance is 9 feet ? CONSTRUCTION OF A HOUSE 141 2. How many steps will be required, and what will be the riser, (a) if the distance between floors is 12' ? (b) If the dis- tance is 8' 8"? Lathing Laths are thin pieces of wood, 4 ft. long and 1|- in. wide, upon which the plastering of a house is laid. They are usu- ally put up in bundles of one hundred. They are nailed | in. apart and fifty will cover about 30 sq. ft. EXAMPLES 1. At 30 cents per square yard what will it cost to lath and plaster a wall 12 ft. by 15 ft. ? 2. At 45 cents per square yard what will it cost to lath and plaster a wall 18 ft. by 16 ft. ? 3. What will it cost to lath and plaster a room (including walls and ceiling) 16 ft. square by 12 ft. high, allowing 34 sq. ft. for windows and doors, at 40 cents per square yard ? 4. What will it cost to lath and plaster the following rooms at 41|^ cents per square yard ? a. 16' X 14' X 11' high with a door 8' x 2^ and 2 windows 2^ X 5'. b. 18' X 15' X 11' high with a door 10' X 3' and 4 windows 2^ X 5'. c. 20' X 18' X 12' high with a door 11' X 3' and 4 windows 2|'x 4'. d. 28' X 32' X 16' high with a door 10' X 3' and 4 windows 3' x5'. e. 28' X 30' X 15' high with a door 10' X 3' and 3 windows 3' x5'. Painting Paint, which is composed of dry coloring matter or pigment mixed with oil, drier, etc., is applied to the surface of wood by means of a brush to preserve the wood. The paint must be composed of materials which will render it impervious to water, or rain would wash it from the exterior of houses. It should thoroughly conceal the surface to which it is applied. The unit of painting is one square yard. In painting wooden houses two coats are usually applied. 142 VOCATIONAL MATHEMATICS FOR GIRLS It is often estimated that one pound of paint will cover 4 sq. yd. for the first coat and 6 sq. yd. for the second coat. Some allowance is made for openings ; usually about one-half the area of openings is deducted, for considerable paint is used in painting around them. Table 1 gallon of paint will cover on concrete . . . 300 to 376 superficial feet 1 gallon of paint will cover on stone or brick work 190 to 226 superficial feet 1 gallon of paint will cover on wood .... 376 to 625 superficial feet 1 gallon of paint will cover on well-painted sur- face or iron 600 superficial feet 1 gallon of tar will cover on first coat ... 90 superficial feet 1 gallon of tar will cover on second coat . . 160 superficial feet EXAMPLES 1. How many gallons of paint will it take to paint a fence 6' high and 5(y long, if one gallon of paint is required for every 350 sq. ft.? 2. What will be the cost of varnishing a floor 22' long and 16' wide, if it tak^s a pint of varnish for every four square yards of flooring and the varnish costs $2.65 per gallon ? 3. What will it cost to paint a ceiling 36' by 29' at 21 cents per square yard ? 4. What will be the cost of painting a house which is 52' long, 31' wide, 21' high, if it takes one gallon of paint to cover 300 sq. ft. and the paint costs $ 1.65 per gallon ? (House has a flat roof.) Papering Wall paper is 18" wide and may be bought in single rolls 8 yards long or double rolls 16 yards long. When you get a price on paper, be sure that you know whether it is by the single or double roll. It is usually more economical to buy a double roll. There is considerable waste in cutting and match- ing paper, hence it is difficult to estimate the exact amount. CONSTRUCTION OF A HOUSE 143 A fraction of a roll is not sold, — there are various rules pro- vided. The border, called frieze, is usually sold by the yard. Find the perimeter of the room in feet, and divide this by the width of the paper (which is 18" or 1^'). The quotient obtained equals the number of strips of paper required. Then divide the length of the roll by the height of the room in order to obtain the number of strips in the roll. The number of rolls required is found by dividing the strips in the room by the strips in the roll. Another rule is : Find the perimeter of the room in yards, multiply that by 2, and you have the number of strips. Find the length of each strip. How many whole strips can you cut from a double roll ? How many rolls will it take ? To allow for doors and windows deduct 1 yard from the perimeter for each window and each door. EXAMPLES 1. A paper hanger is asked to paper a square room 18' by 18' with a door and three windows. The door is 3' by 7' and tht windows 2' by 4'. How many double rolls of paper will he use ? (Consider all rooms 9' high.) 2. How much paper will be required to paper a room 18' by 14'? 3. How much paper will be required to paper a room 18' 6" by 16' 4" with 2 doors and 2 windows ? 4. How much will it cost to paper a room 19' 6" by 16' 4" with 2 doors and 2 windows. The paper costs 49^ a roll to place it on the wall. Taxes Find out where the money comes from to support the schools, police, library, etc. in your city or town. How is it obtained ? What is real estate ? What is personal property ? What is a poll tax ? A tax is the sum of money assessed on persons and property to defray the expenses of the community. 144 VOCATIONAL MATHEMATICS FOR GIRLS * The tax rate is usually expressed as so many dollars per thousand of valuation, generally between $10 and $20. In some places it is expressed as a certain number of mills on $ 1 or cents on $ 100. The tax rate, or the amount on each thousand dollars of property, is determined by dividing the whole tax by the num- ber of thousand dollars of taxable property in the community. To illustrate : In a certain community the whole tax is $1,942,409.73. The taxable property is $ 97,945,162.00. $ 1,942,409.73 ^..qoo 97,945 =^^^-^- EXAMPLES 1. If the tax rate is $ 21.85, what are the taxes paid by a family of women owning property worth $ 16,000 ? 2. What is the tax on $ 34,697 in your town or city ? 3. A man owns real estate worth $ 84,313, and has personal property worth $ 16,584. What is his tax bill, if the tax rate is $ 1.75 per hundred and a poll tax is $ 2 ? 4. A dwelling house is valued at $8500 and the tax rate is $ 17.52 per thousand. What is the tax ? 5. What is the tax on a house valued at $ 3500, if the tax rate is $ 23.45 ? 6. The taxable property of a city is $ 97,945,162.00 ; and the expenses (taxes) necessary to run the city are $ 1,900,136.14. Obtain the tax rate. United States Revenue The town or city derives revenue from taxes levied on real and personal property. The county and state derive part of their revenue from a tax imposed upon the towns and cities. The United States government derives a great part of its rev- CONSTRUCTION OF A HOUSE 145 enue from a tax placed on tobacco and liquor sold within its boundaries and from a tax, called customs duties, imposed upon articles imported from other countries. Some articles are admitted into the country free; these are said to be on the free list. The others are subject to one or both of the follow- ing duties : a duty placed on the weight or quantity of an article without regard to value (called specific duty), or a duty based upon the value of the article (expressed in per cent and called ad valorem duty). When goods are received into this country, they are examined by an officer (called a customs officer). The goods are accompanied by a written statement of the quantity and value (called manifest or invoice). Sometimes the goods are liquid, and in this case the weight of the bar- rel (called tare) must be subtracted from the total weight to obtain the net weight on which duty is imposed. In case bottles are broken and liquids have escaped, due allowance must be made before imposing duty. This is called leakage or breakage, EXAMPLES 1. What is the duty on bronze worth % 8760 Sit^5%? 2. What is the duty on goods valued at $ 3115 at 35 % ? 3. What is the duty on 3843 sq. ft. of plate glass, duty $ 0.09 per square foot ? 4. What is the duty on jewelry valued at $ 8376 at 40 % ? 5. What is the duty on cotton handkerchiefs valued at $ 834 at 45 % ? 6. What is the duty on woolen knit goods valued at $ 1643, 41 cts. per pound plus 50 % ? 7. What is the duty on rugs (Brussels), 120 yards, 27" wide, invoiced at $ 1.80 a yard, at 29 cts. per square yard and 45 % ad valorem ? CHAPTER VII COST OF FURNISHING A HOUSE When about to furnish a house, one of the first things to consider is the amount of money to be devoted to the purpose. This amount should depend on the income. A person with a salary of $ 1000 a year should have saved at least $ 250 toward the equipment of his home before starting house- keeping. This is sufficient to purchase the essentials of a simply furnished apartment or small house. After one has lived in the house for a short time, it will be easy to study the possibilities and necessities of each room, and as time, opportunity, and money permit, one can add such other things as are needed. In this way the purchase of undesirable and inharmonious articles may be avoided. There are many different styles and grades of furniture. The cost depends upon the kind of wood used, and the care with which it is put together and finished. The most inexpensive furniture is not the cheapest in the end. It is made of inferior wood and with so little care that it is neither durable nor attractive. The medium grades are gen- erally made of birch, oak, or willow, are durable, and may be found in styles that are permanently satisfactory. The best grades are made of mahogany and other expensive woods, and those whose income consists only of wages or a salary cannot usually afford to buy more than a few pieces of this kind. Furniture that is well made, of good material, and free from striking peculiarities of design and of decoration is chosen by all people of good taste and good judgment. Furnishing the Hall The only furniture necessary for the vestibule is a rack for umbrellas. The walls should be painted with oil paint in some warm color, and the floor should be tiled or covered with inlaid linoleum in tile or mosaic 146 COST OF FURNISHING A HOUSE 147 design. If the vestibule serves also as the only hall, it should contain a rug, a small table or chair, and a mirror. ^ A panel of filet lace is suitable to use across the glass in the front door. Through the front door one gets one's first impression of the occupants of the house. The furnishings of the hall should therefore be carefully chosen. It is a passageway rather than a room, and requires very little furniture. The walls may be done in a landscape paper, if one wishes to make the room appear larger, or in plain colonial yellow, if a bright effect is desired. If the size of the hall will permit, it is best to furnish it as a reception room; it may be made an attractive meeting place for the family and friends ; but if it is one of the narrow passages so often found in city houses, one must be content with the regulation hall stand, or a mirror and a narrow table, and possibly one chair. Price List of Hall Furniture • g H M Boo b 39 ij £ t^ S M CO M « ^ J, H §5 2o CD 0^3 Design IN Oa Inamel Design inBba 1^ B3 ^ ^K Q ^ S ew a K u !S «^ o 2 -^ H io * H -4 LONIAL PBODUCB Bieou » 9 S 2 25 LONIAL PRODUCE LHOGANT H < is 1 •J o H oa o wT 53 1 it ^ Umbrella rack . .$ 1.25 ^.00 ^7.25 $8.50 $10.00 $5.00 $7.50 Table .... 3.75 6.75 8.25 9.76 20.00 10.00 87.50 Mirror . . . 3.00 3.00 3.40 3.75 30.00 7.50 Straight chair . 2.75 4.50 6.50 6.60 25.00 6.60 8.00 Chest .... 13.50 13.50 16.60 19.50 60.00 40.00 Sofa .... 50.00 85.00 16.00 Tall clock . . 60.00 60.00 160.00 75.00 Settle .... 18.00 18.00 22.50 27.00 32.00 82.00 21.00 Telephone stand 6.75 6.75 8.25 9.75 10.50 5.50 15.00 Clothes rack 3.50 3.50 4.15 4.90 5.00 7.00 8.25 EXAMPLES 1. What is the complete cost of furnishing a hall with willow furniture ? 148 VOCATIONAL MATHEMATICS FOR GIRLS 2. Compare the cost of furnishing a hall with mahogany or birch. 3. If a family receives an income of $1400 a year and lives in a single cottage house, what kind of furniture should be selected? What should the cost not exceed for the hall furniture ? 4. A hall was furnished with the following articles. What did it cost ? What kind of furniture was probably purchased ? Seat, $ 11.85 Rug, $ 0.86 Mirror, $ 2.15 China umbrella stand, $ 2.10 Table, $ 2.20 Table cover (one yard of felt), $ 1. 15 Two chairs, ^ 7.40 Pole, $ 2.20 With hardwood or stained floors the furnishing and care of a house are much simplified. If one must have carpets, the colors should be neutral. The best quality of Canton or Japanese matting is satisfactory ; it is a yard wide and costs fifty cents a yard. Next to matting, the most sani- tary and economical carpet is good body Brussels. It wears well, and the dust does not get under it. A cheap, loosely woven matting or woolen carpet is always unsatisfactory. Floor Coverings In selecting floor coverings there are several important considerations. The design and quality should be governed by the treatment the rug will necessarily have. Hall A hall rug or carpet will receive hard wear ; therefore, the quality should be good. A small all-over symmetrical design in two tones of one color or in several harmonizing colors will show dust and wear less than a plain surface would do. Rag rug, machine made, 8 by 6 feet $2.18 Hand- woven rag rug, 3 by 6 feet 7.50 Scotch wool rug, 3 by 6 feet 4.00 Hand-woven wool rug, 3 by 6 feet 6.00 East India drugget, 3 by 6 feet 8.00 Saxony, 3 by 6 feet 9.00 Brussels rug, 8 by 6 feet 9.00 Oriental rug, 8 by 6 feet 85.00 COST OF FURNISHING A HOUSE 149 ' Living Boom In a living room the floor covering will be worn all over equally. Since there is always a variety of colors and forms in a living room, it is well to keep the floor covering as plain as possible. A rug with a plain center and a darker border of the some color is excellent in this room, particularly if the walls or hangings are figured. If they are plain, the rug or carpet may have a small, indefinite figure. If several domestic rugs are used in the same room, they should be exactly alike in design and color. If small Oriental rugs are used, they will, of course, differ in design, but they should be as nearly as possible in the same tone. Good Living-room Bugs Crex or grass rug, 9 by 12 feet $8.60 Rag rugs, 9 by 12 feet $ 10.00 to 46.00 Scotch wool rug, 9 by 12 feet $ 14.60 to 26.00 Brussels, 9 by 12 feet 32.76 Hand-woYen wool rug, 9 by 12 feet 36.00 East India drugget, 9 by 12 feet 43.00 Saxony, 9 by 12 feet 60.00 Oriental, 9 by 12 feet 200.00 up Dining Boom A dining-room rug gets very hard wear in spots. It should, therefore, be selected in as good quality as one can afford. It is not well to have a perfectly plain rug in a dining room, as a plain surface shows crumbs and spots too readily. There is no objection to having a dining-room floor quite bare, if the floor is well finished. Inlaid linoleum also makes an excellent floor covering for a dining room that receives very hard usage. The best coverings for this room are : Crex ingrain rug, 9 by 12 feet ....... $8.60 Rag rug, 9 by 12 feet $ 10.00 to 46.00 Brussels, 9 by 12 feet 32.76 East India drugget, 9 by 12 feet 36.00 Saxony, 9 by 12 feet 60.00 Oriental, 9 by 12 feet 200.00 up Bedroom and Sewing Room On account of the lint which accumulates in bedrooms, it is a good plan to keep the space under the beds bare, so that it may be dusted every day. Small rugs laid where most needed are more hygienic in sleeping 150 VOCATIONAL MATHEMATICS FOR GIRLS rooms than are large rugs and carpets. Plain Chinese matting makes a clean floor covering when the boards are not in good condition. Although it is in good taste to use a carpet or one large rug in a bedroom, the preference lies among the foilowing : Small rag rugs, 8 by 6 feet $ 1.76 Oval braided rag rugs, 3 by 6 feet 2.50 East India drugget, 3 by 6 feet 8.00 Saxony, 3 by 6 feet 8.00 Oriental, 3 by 6 feet 36.00 EXAMPLES 1. A family has an income of $ 1400. They buy a Brussels rug 3' X 6' for $ 9. Are they extravagant ? 2. How much cheaper is a crex rug, 9 by 12 feet, than a Brussels the same size ? What per cent cheaper ? 3. A dining-room rug is purchased for $ 49.75. What kind of a rug is it ? Is it suitable for a family with an income of $2500? 4. An oval braided rag rug 3' x 6' costs $ 2.50 and will last twice as long as a small rag rug that costs $ 1.75 for the bed- room. Which is more economical to purchase ? How much more economical is it? The Living Room In houses or apartments of but five or six rooms there is usually but one living room. This room should represent the tastes which the members of the family have in common. The first requisite of such a room is that it should be restful. It is, therefore, advisable to use a wall covering that is plain in effect. Tan is good in a room that is inclined to be dark ; gray-green or gray itself in a very bright living room. One large rug in two tones of one color, preferably the same color as the walls, is better than a figured rug for this room. Chairs are an important part of the furnishing of a living room. It is well to have comfortable armchairs, upholstered COST OF FURNISHING A HOUSE 151 in plain material, or willow chairs with cushions of chintz, if this material is used aa curtains. A roomy table with a good reading lamp ia essential, while open bookshelves, a writing desk or table, a sofa, a sewing table, and a piano are all appro- priate furnishings for this room. A HARUONlOUSLr FUBNIBHKD LlYING ROOU The curtains may be of figured materials, such as chintz or cretonne. Plain scrim or net curtains may be used over cur- tains of plain-colored material or of chintz simply to give the necessary warmth and color to the sides of the room. Valances are used to reduce the apparent height of a window and to give a low cozy look to the room. Plants are always appropriate to use in sunny windows, and pictures of common interest, framed in polished wood or dull gilt frames, help to make the living room attractive. Use very little bric-a-brac. Nothing which does not actually contribute to the beauty of the room should be allowed to find a place there. 162 VOCATIONAL MATHEMATICS FOR GIRLS Price List of Living-room Furniture Table . . Chair . . Sofa . . . Armchair . Desk chair Desk . . Bookcase . Sewing table Tea table . Footstool . Wood box or rack Magazine stand . Piano .... Music cabinet « ^ J o3 E? M >« fi^ 0£St £05 S ^ J 5? as •J 1 Se; ►J 2 » 5 ■< « LONIA PEODl BiRC ill -^ g «< cSScS ' n tf 0«0 a^a $4.50 $15.00 $17.00 $50.00 17.00 22.50 25.00 45.00 65.00 20.00 38.00 2.75 6.75 7.75 15.00 0.75 19.60 21.75 90.00 9.00 9.00 11.25 100.00 5.00 5.00 6.00 17.00 1.50 1.50 2.00 35.00 2.25 3.75 3.00 6.00 5.00 • 5.00 6.00 6.75 8.25 10.00 200.00 250.00 450.00 6.75 6.75 8.26 28.00 . I to Hi $35.00 $59.00 30.60 50.00 68.00 100.00 82.00 65.00 4.75 15.00 28.00 90.00 25.00 100.00 18.50 12.00 4.50 6.00 6.00 6.00 8.50 8.50 450.00 10.00 S S $12.00 12.75 23.50 9.76 8.25 37.50 13.50 13.50 7.25 6.25 3.50 12.76 ^'^^^ICoal, 17.00 Wood or coal $16.00 Franklin grate or andirons, 25.50 wood or coal $35.00 EXAMPLES 1. How much more will it cost to furnish a living room with library furniture than with willow furniture ? 2. How much more will it cost to furnish a living room with hand-made oak furniture than with colonial designs in oak ? 3. A living room was furnished with the following furni- ture. Ascertain from the price list what kind of furniture it is. Large round table and small Curtains and shades for three table, $7.95 Six chairs and couch, $ 51.15 Bookcase or shelves, $ 9.85 What is the cost ? windows, $6.15 Rug and draperies, $ 34. 15 Incidentals, $ 24.65 COST OP FURNISHING A HOUSE 153 The Bedroom When one stops to think that about one-third of one's life k spent in aleep, it is easy to understand that the first requisite in the fumiBhing of the bedroom is that it be fresh and clean. A COHFURTABLB BeDBOOM UnlesB the room must be used as a study or sitting room in the daytime, the amount of furniture should be reduced aa much as possible. The necessary pieces are a bed, a dressing case which should be generous in drawers and mirror, a wash- stand, a toilet set, towel-rack, one easychair and one plain one, a small table, a rug, and window shades. If space and money permit, a couch is desirable. Naturally, a writing desk, book- ahelveSj and pictures all add to the attractiveness of such a room. If one cannot have bare floors, the next best thing is good matting. A woolen carpet is not desirable for a sleeping room. All draperies should be of materials that will hold neither dust nor odor. 154 VOCATIONAL MATHEMATICS FOR GIRLS The bed is the most important article in the room. The springs and mattress should be firm enough to support all parts of the body when it is in a horizontal position. The walls should be light in color and the woodwork white if possible. The furniture also may be white, although dull- finished mahogany in colonial designs, with small rugs on the floor, makes a charming bedroom. One set of draw cur- tains, of figured chintz if the walls are plain, and of plain-colored material if the walls have a small figure, is enough for each window. The furnishings of a young girl's bedroom should be carried out in her favorite color, and to the usual bedroom furniture should be added a desk, lamp, worktable, and bookshelves. The bedroom for a growing boy should be his own sitting room and study as well ; a place where he can entertain his friends, do his studying, and develop his hobbies. The walls, hangings, couch cover, etc., should be very plain, as a boy usually has a collection of trophies which need' the plainest sort of a background in order to prevent the room from looking cluttered. Instead of the usual bed he should have an iron- framed couch, which in the daytime may be made up with a plain dark cover with cushions, to be used as a couch. A chif- fonier, an armchair, bookshelves, writing table, and one or two small rugs will complete the furnishings of the boy's bedroom. EXAMPLES 1. Sheets should be of ample length and breadth. The finished sheets should be nearly three yards long. How many inches long ? 2. The supply of bedroom linen, blankets, and counterpanes for a small house is as follows : 12 sheets @ $ 0.85 4 pairs blankets @ $8.00 12 pillow cases @ $ .40 2 counterpanes @ $ 2.50 24 towels @$ 0.50 What is the total cost ? COST OP FURNISHING A HOUSE 155 Price List of Bedroom Furniture Ml 2 -"l M Colonial Designs Kepboduced in Oak OB BiBCU 1 Colonial Designs Kepboduced in Oak — Gloss Enamel Colonial Designs Kepboduced in Oak — Kubbed Enamel 1 Colonial Designs Kepboduced in Keal Mahogany Hand-made Fubni- TUBE IN Oak H a s s .J Bed .... 19.75 .$16.50 118.75 $21.00 $55.00 $30.00 $56.00 Mattress . . 3.35 16.00 16.00 16.00 36.00 36.00 36.00 to to to to 16.00 25.00 25.19 25.00 Box spring . . 20.00 20.00 20.00 20.00 20.00 20.00 20.00 Crib (iron) . . 12.75 12.75 12.75 12.75 12.75 12.75 Crib mattress . 3.75 9.00 9.00 9.00 9.00 9.00 Pillows (pair) . 1.25 2.10 2.10 2.10 6.00 6.25 5.25 Bureau . . . 0.75 22.50 25.00 27.60 75.00 50.00 67.50 Washstand 1.50 2.00 2.75 3.50 6.00 (enamel iron) 10.00 Dressing table 9.00 12.57 14.25 15.75 55.00 26.00 48.00 Chiffonier (no 9.00 12.00 14.25 16.'50 100.00 39.00 60.00 mirror) . . (high- boy) Chair . . . 2.75 4.50 5.25 6.00 10.00 6.50 8.00 Rocking chair . 2.75 6.75 7.75 8.75 9.00 6.50 8.25 Waist box . . Home- made 2.50 3.50 4.50 20.00 16.00 4.50 i^esK . • . . 4.50 9.75 10.75 11.76 60.00 20.00 28.50 Armchair . . 6.75 7.75 8.75 24.00 8.00 7.60 Couch , . . 6.00 (iron frame) 13.26 (box) 60.00 50.00 25.00 Bookshelves . Home- made 9.00 10.50 12.00 (built in) 21.50 13.50 Cheval glass . 11.25 15.50 16.50 18.00 50.00 25.00 qx^ / Gas, $5.00 Wood . . .$15.50 Franklin grate or andirons, DTOves <^ ^^^^^ ^jQQ ^^^ Qj. ^Q^j 25.50 wood or coal . $35.00 156 VOCATIONAL MATHEMATICS FOR GIRLS 3. If a person spends one-third of a life in a bedroom, how many hours a day are spent in the bedroom ? 4. A bedroom is furnished with the following furniture : Enameled bedstead with springs, Dimity for draping bed , washstand 97.50 and two windows, twenty-one A dressing case, $ 15.00 yards, $ 3.15 A plain wooden table to be Enameled cloth for washstand, $.65 used as washstand, $ 1.00 Two pillows, |4.00 A small table $ 2.00 Toilet set, $ 3.00 • Chair, $ 2.00 Shades for two windows, $ 1.00 Mattress, $ 5.00 Towel rack, $ .75 Bug, $3.00 What is the total cost ? 5. What will it cost to furnish a bedroom with simple cot- tage furniture as provided above ? 6. What will it cost to furnish a bedroom with the good grade of oak furniture in gloss enamel ? What is the least income a family should have in order to buy this furniture ? 7. What will it cost to furnish a bedroom with real mahog- any furniture ? What is*the least income one should have in order to buy this furniture ? The Dining Room The dining room does not require a great deal of furniture, but what there is should be of the most substantial kind. Mahogany and oak are the woods to be preferred. The table should be broad, stand well, with the legs so placed that they will not interfere with the comfort of any one seated at the table. The chairs should be well made, with broad, deep seats and high, straight backs. Unless one can afford the right kind of a sideboard it is better to purchase a sideboard table in simple design. A piece of Japanese matting in the center of the dining room floor is quite satisfactory when the floor is stained. COST OF FURNISHING A HOUSE 157 The room in which the family assemblea several times each day to enjoy its meals together should be the most cheerful room in the house. An Attbactivb Dinihq Roou Because there ia bo much attractive bhie-and-nbite china in use, lUAUj peraons want dining rooms with blue walls. Tbie is usually a mis- take, as tilue used in large quantitiea absorbs the light and makes a room gloomy, particularly on dark days and at night. By using colonial yel- low on. the walla, with hangings, rug, and decorative china in blue and white, one has an almost ideal arrangement. There are many charming landscape and foliage papers on the market which, used without pictures against them, but with bulbs or plants blooming on the windowsills and with hangings of plain, semi transparent, colored material make most delightful rooms. Plat« rails or racks reduce the apparent height of an oyer-high ceiling. It is better to use a rimple flat molding than to crowd a plate rail full of inharmonious objects. Ugly glass domes on lamps are tieing replaced by silk ones with deep silk fringe or, better still, the center light is abandoned iu faror of side wall natures in all of the rooms. Candles, prettily shaded, are used on the table at night, with a jar of flowers or fruit ai 158 VOCATIONAL JVIATHEMATICS FOR GIRLS EXAMPLES 1. What will the following cottage dining-room furniture cost ? (Include the items given in the price list below.) 2. What will the following oak dining-room furniture cost ? 3. What will the following real mahogany dining-room fur- niture cost ? Price List of Dining-room Furniture Table . . Chair . . Armchair . Serving table Buffet . . China closet Serving table wheels . Screen . . High chair on HUM £ « S M £ M 00 ,j MOO ^ < ^ <« (B ■< >3 £ -< ^ RNITU BiR r Col 20 20u 2o« O H to 2 as fi A S qH M "MP M ►J P M 3 u a - o 2 :^toS ^ Q « J O C i§5 5® 2 *^ w Q SCO OJS (A ox o Colon Repro — Glo Colon Kbfro — Rub Colon Repro Mauoc Hand- TURE 1 $9.00 $30.00 $10.50 112.00 $85.00 $21.00 2.76 4.50 5.50 6.50 10.00 6.50 2.75 6.76 7.75 8.75 15.00 10.00 8.26 9.00 10.50 12.76 35.00 18.00 18.00 27.50 21.00 24.00 126.00 34.00 15.00 30.00 34.50 39.00 60.00 46.00 16.75 16.75 30.50 34.00 27.00 27.00 3.76 5.00 4.50 5.25 25.00 20.00 2.50 2.50 4.15 5.50 10.00 9.00 S3 t3 S pi M $16.00 8.25 28.00 82.50 24.00 8.00 Sfnvfts/^^' $5.00 Wood. . .$15.50 Franklin grate or andirons, \ Coal, 17.00 Wood or coal 25.00 wood or coal . . $36.00 4. What will it cost to furnish a home on a moderate scale with china of the following amounts and kinds : J dozen soup plates (to be used for cereals also) . . $2.36 J dozen dinner plates 2.26 1 dozen lunch plates (used also for breakfast and for salads) 3.85 } dozen dessert plates . 1.60 COST OF FURNISHING A HOUSE 159 ) dozen bread-and-butter plates $0.70 I dozen coffee cups and saucers 3.30 i dozen tea cups and saucers 2.80 ^ dozen after-dinner coffee cups and saucers . . . 2.35 1 teapot . 1.90 1 coffee pot 2.00 1 covered hot-milk jug or chocolate pot 2.60 1 large cream pitcher 70 1 small platter or chop platter 2.50 3 odd plates for cheese, butter, etc 95 Covered dish 2.80 I dozen egg cups 1.60 5. What will it cost to furnish a home on' a moderate scale with glass, colonial period, of the following amounts and kinds : } dozen tumblers f^OM i dozen sherbet glasses 35 I dozen dessert plates . .* 1.25 J dozen finger bowls 75 Sugar bowl and cream pitcher 50 Dish for lemons 60 Dish for nuts 25 Pitcher 60 Candlesticks 65 Vinegar and oil cruets 50 Berry dish 25 } dozen iced-tea glasses 76 } dozen individual salt cellars .60 6. What will it cost to furnish a home on a moderate scale with silver, pilgrim pattern, of the following amounts and kinds : 1 dozen teaspoons $14.00 } dozen dessert spoons (used for soup also) . . . 9.60 4 tablespoons 9.60 1 dozen dessert forks (used also for breakfast, lunch, salad, pie, fmit, etc.) 19.00 I dozen dessert knives 11.00 160 VOCATIONAL MATHEMATICS FOR GIRLS 1 dozen table knives with steel blades and ivoroid handles $2.00 Carving set to match steel knives 4.00 J dozen table forks 12.00 2 fancy spoons for jellies, bonbons, etc. ($ 1.60 each) 3.00 2 fancy forks for olives, lemons, etc. (|1.60 each) . 3.00 ^ dozen after-dinner coffee spoons 5.00 i dozen bouillon spoons 8.00 } dozen butter spreaders 1.60 1 gravy ladle 4.76 Saltspoon .20 Sugar tongs 2.26 7. What will it cost to furnish a home on a moderate scale with silver-plated ware of the following amounts and kinds : Covered vegetable dish (cover may be used as a dish by removing handle) $10.00 Platter 11.60 Pitcher 12.00 Coffeepot 12.60 Toast rack 4.60 Small tray 6.60 Sandwich plate 6.00 SUverbowl 9.00 Egg steamer 8.00 Bread or fruit tray 6.60 Tea strainer 1.00 Candlesticks, each 3.76 Household Linen The quality of linen in every household should be the best that one can possibly afford. The breakfast runners and napkins are to be made by hand, of unbleached linen such as one buys for dish towels. With insets of imitation filet lace these are very attractive, durable, and easy to launder. 1. What is the cost of supplying the following amount of table and bed linen for a couple with an average income of $ 1400, who are about to begin housekeeping ? COST OF FURNISHING A HOUSE 161 Table Linen 2 dozen 22-inch napkins, at $3.00 a dozen. 2 dozen 12-inch luncheon napkins, at $4.50 a dozen. (Luncheon napkins at $1.00 a dozen if made by hand of coarse linen.) 2 two-yard square tablecloths, at $1.25 a yard. Two-yard square asbestos or cotton flannel pad for table, at $ 1.00. J dozen square tea cloths, $12.00. i dozen table runners for breakfast, at $2.40. 1 dozen white fringed napkins, at $1.20. 4 tray covers, at 65 cts. 1 dozen finger-bowl doilies, at $3.00. 1 dozen plate doilies, at $3.00. Bed Linen 4 sheets (extra long) for each bed, at $ 1.10. 4 pillow cases for each pillow, at 20 cts. 1 mattress protector for each bed, with one extra one in the house, at $1.50. 2 spreads for each bed, at $ 2.50. 1 down or lamb's-wool comforter for each bed, at $ 6. 1 pair of blankets for each bed, with 2 extra pairs in the house, at $ 8. } dozen plain huckaback towels for each person, at 25 cts. 3 bath towels for each person, at 30 cts. } dozen washcloths for each person, at 11 cts. 1 bath mat in the bathroom, 2 in reserve, at $ 1.50. The Sewing Room Even in a small house there is sometimes an extra room which may be fitted up as a sewing room in such a way as to be very convenient and practical, and at the same time so attractive as to serve occasionally as an extra bedroom. This room should be kept as light as possible and should be so furnished that it may easily be kept clean. EXAMPLE 1. What will it cost to furnish a sewing room with the fol- lowing articles ? Sewing machine with flat top to be Used as a dressing table . . $ 20.00 Chair 1.25 Box couch 13.25 Chiffonier 9.00 162 VOCATIONAL MATHEMATICS FOR GIRLS Mirror againat a door 811.25 Low rocking-chair without arms l.&O Cutting table, box underneatli ; tilt top to be used 6.76 Clothes tree 3.88 Tbe Kitchen The room in which the average housekeeper spenda the greater part of her time ia usually the least attraetive room in the house, whereas it should be made — and we learn by visiting foreign kitchena that it may be made — a picturesque setting for one of the finest arts — the art of cookery. A Convenient EiTcaEN The woodwork should be light in color, the walls should be painted with oil paint, or covered with washable material, this also in a light color. A limited number of well-made, carefutt; selected utensils will be found more useful than a large supply purchased without due con- sideration as to their real valne and the need of them. Of course, the style of living and the size of the family must to aome extent control the number, size, and kind of utensils that are required in each kitehen. As in all the other fiirnishingB, the beginner will do well to purchase only the essential articles until time demonstrates the need of otbeis. COST OF FURNISHING A HOUSE 163 EXAMPLES 1. What will it cost to furnish your kitchen ? ^ Stoves — Gas $2.60, $10.00, ^30.00 Blue-flame kerosene 10.25 Coal, wood, gas 86.00 Coal and wood 49.75 Small electric 33.00 Table . . . $2.10; $9.00 (drop leaf) ; $11.25 (white enamel on steel) Chair $1.87, $6.75 Ice chest $7.00, $15.00, $40.00 (white enamel) Kitchen cabinet $28.00, $29.00 (white enamel on steel) Linoleum . . . 60c. square yard, printed ; $ 1.60 square yard, inlaid 2. What will the following $0.85 .35 Small-sized ironing board Small glass washboard Clothesline and pins . 2 irons, holder and stand 2-gallon kerosene can Small bread board Hack for dish towels 6 large canisters . Wooden salt box . 1 iron skillet . . 1 double boiler . . Dish drainer . . 2 dish mops . . . Wire bottle washer Small rolling pin . Chopping machine Large saucepan 3 graduated copper, enam- eled or nickel handled dishes 50 2 covered earthenware or enameled casseroles . . 1.50 2 pie plates enameled . . .20 Alarm clock 1.00 .59 .70 .45 .15 .10 .60 .10 .30 1.00 .25 .10 .10 .10 1.10 .30 small kitchen furnishings cost ? Small covered garbage pail . .35 Scrubbing brush 20 Broom and brushes ... .60 1 quart ice-cream freezer . 1.75 Roller for towel 10 Bread box 50 4 small canisters 40 2 sheet-iron pans to use as roasting pans 20 Dishpan (fiber) ^ . . . .50 Plate scraper 15 Soap shaker 10 Vegetable brush 05 Muffin tins 25 Granite soup kettle ... .45 3 graduated small saucepans .30 Glass butter jar 35 6 popover or custard cups . .30 Soap dish 25 Knives, forks, egg beater, lemon squeezer, etc. . . 5.50 Sink strainer, brush, and shovel 50 Galvanized-iron scrub pail . .30 ^ Consider income of family and size of kitchen. 164 VOCATIONAL MATHEMATICS FOR GIRLS SXAMPLES IN LAYING OUT FURNITURE Considerable practice should be given in laying out furniture according to scale. 1. A bedroom 12' x 10' 6" faces the south, and has 2 win- dows, 3' 6" wide, 1 window, 3' 6", two feet from corner of west sidej and a door 3' wide two feet from east wall. This room is to contain the following furniture : 1 bed, 6' 6'' X 4' 1 dresser, 3' x 1'6" , p^' , s j'-*~ ,»• ^ D 1 >• C ' £ \<^ bJJ t -/a- J9 Solution 1 dining table, 5' in diameter 1 buffet, 4' X 2' 1 table, 2' 6" x 3' 2 chairs, 1' 6" x 2' Draw a plan showing the most artistic arrangement of furniture. Scale y = 1'. 2. A dining room 15' X 18' faces the east, and has two windows 3' 6" wide on the east side, 2 windows 3' 6" on the north side, folding doors 6 wide in the center, on the south side. Draw a plan and place the following furniture in it in the most artistic manner : 6 chairs, 2' x 1' 6" Scale, 1" = 1' 3. A living room 15' x 18' faces the north and has 2 win- dows 3' 6" wide on the north side, 2 windows 3' 6" on the west side, and folding doors on the south side. Draw a plan and place the following furniture in the most artistic manner : 1 settee 1 table 1 desk and chair 2 easy-chairs 2 rockers Scale \" = V 4. A kitchen 12' x 10' 6" faces the south and has 2 windows 3' 6" wide on the south side, 1 on the west side, two feet COST OF FURNISHING A HOUSE 165 from the north comer, a door 3' wide, two feet from the north- east comer that leads into the dining room. Draw a plan and place the furniture in proper places : 1 kitchen range 1 table 1 sink 2 chairs • 2 set tubs Scale f' = l' ^ REVIEW EXAMPLES 1. A living room was fitted out with the furniture in the list below. What kind of furniture is it ? What is the cost ? Large round table and small Curtains and shades for three table, $8.00 windows, $ 6.30 Six chairs and couch, $ 60.00 Rug and draperies, ^ 84.00 Bookcase or shelves, $ 10.00 Incidentals, $25.00 2. A hall was furnished with the following articles. What was the total cost ? What kind of furniture was used ? Seat, $ 12.00 Rug, $ 10.00 Mirror, $ 2.00 Umbrella stand, $ 2.00 Table, 1 2.00 Table cover, $ 1.00 Two chairs, ij^ 7.50 Pole, ^ 3.00 3. A family of seven — three grown people and four chil- dren — lived in a southern city on $ 600 a year. The monthly expense was as follows : House rent, $ 12.00 Bread, $3.50 Groceries, k 12.00 Beef, $3.60 Washing, $5.00 Vegetables, $3.00 What is the balance from the monthly income of $ 60 for clothing and fuel ? 4. What is the cost of the following kitchen furniture ? 1 kitchen chair, $ 1.25 1 broom, 50 cents 1 table, $ 1.50 Kitchen utensils, $8.50 166 VOCATIONAL MATHEMATICS FOR GIRLS 5. What is the cost of the following living-room furniture ? How much income should a family receive to buy this furniture ? Overstuffed chair, $ 12.60 2 willow chairs, $ 6 each 1 willow stool, $4.26 1 rag rug, $ 9.60 1 newspaper basket, $2.26 12 yards of cretonne, 36 cents a yard 1 green pottery lamp bowl, f 3.00 1 wire shade frame, 60 cents 7 yards of linen, at 60 cents a yd. 10 yards of cotton fringe, at 6 cents a yd. 6 yards of net, at 26 cents a yd. Table, 48 by 30 inches, $ 7.00 6. What is the cost of the following bedroom furniture? How much income should a family have to buy this furniture ? 1 bed spring, $ 3.60 1 single cotton mattress, $4.26 1 chiffonier, $6.50 1 dressing Uble, $2.26 1 mirror, $2.76 1 armchair, $4.00 1 rag rug, $3.26 2 pillows, 76 cents each 1 bed pillow, $ 1.00 10 yards of white Swiss, at 26 cento a yd. 8 yards of pink linen, at 60 cento a yd. 1 comfortable, $ 4.26 Sheeto and blanketo for one bed, $6.00 3 yards of cretonne, at 36 cento a yd. 7. What is the cost of the following bedroom furniture ? How much income should a family have to warrant buying this furniture ? 2 white iron beds, at $ 4.26 each 2 single springs, at $ 2.60 each 2 cotton mattresses, at $4.26 each. 2 bed pillows, at $ 1.00 each 1 dressing table, $ 6.60 1 white desk, $6.76 1 chiffonier, $6.60 1 dressing-table mirror, $ 3.26 1 chiffonier mirror, $ 1.60 1 rag rug, $3.26 I wastepaper basket, .60 II yards of cretonne, at 36 cento a yd. 6 yards of yellow sateen, at 26 cento a yd. 2 comfortables, at $ 4.26 each 10 yards of cream sateen, at 26 cento a yd. 16 yards of cotton fringe, at 6 cento a yd. 1 willow chair, $ 6.00 1 cushion, 76 cents 4 yards of net, at 26 cento a yd. Sheeto and blanketo for two beds, $12.00 1 dressing table chair, $4.60 HEAT AND LIGHT 167 8. What is the cost of the following dining-room furniture ? What income should one receive to buy this furniture ? 6 dining-room chairs, $ 4.50 1 dining table, $6.75 1 serving table, $ 6.25 1 rag rug, $ 0.50 1 set of dishes, $ 9.75 10 yards of cretonne, at 35 cents a yd. One wire shade frame, 50 cents Table linen, $8.00 Silverware, $7.50 1 willow tray, $3.25 HEAT AND LIGHT Value of Coal to Produce Heat Several different kinds of coal are used for fuel. Some grades of the same coal give off more heat in burning than others. The heating value of a coal may be determined in three ways : (1) by chemical analysis to determine the amount of carbon ; (2) by burning a definite amount in a calorimeter (a vessel immersed in water) and noting the rise in tempera- ture of the water ; (3) by actual trial in a stove or under a steam boiler. The first two methods give a theoretical value ; the third gives the real result under the actual conditions of draft, heating surface, combustion, etc. The coal generally used for household purposes in the Eastern states comes from the anthracite fields of Pennsylvania. This coal, as shipped from the mines, is divided into several different grades according to size. The standard screening sizes of one of the leading coal-mining districts are as follows : Broken, through 4 J" round Egg, through 2f" square Stove, through 2" square Nut, through If" square Pea, through }" square Buckwheat, through i" square Rice, through f " round Barley, through J" round The last three sizes given above are too small for household use and are usually purchased for generating steam in large power-plant boilers. Coke is used to some extent in localities where it can be obtained at a reasonable price in sizes suitable for domestic purposes. The grades of coke generally used for this purpose are known as nut and pea. The use of coke in the household has one principal objection. It bums up quickly and the fires, therefore, require more attention. This is due to the fact that a given volume of coke weighs less and therefore contains less heat than other fuel occupying the same space in the stove or furnace. 168 VOCATIONAL MATHEMATICS FOR GIRLS The chief qualities which determine the value of domestic coal are its percentage of ash and its behavior when burned. Coal may contain an excessive amount of impurities such as stone and slate, which may be easily observed by inspection of the supply. The quality of domestic coke depends entirely upon the grade of coal from which it has been made, and may vary as much as 100 ^ in the amount of impurities contained. Aside from the chemical characteristics of domestic coal, the most im- portant factor to consider in selecting fuel for a given purpose is the size which will best suit the range or heater. This depends on the amount of grate surface, the size of the fire-box, and the amount of draft. EXAMPLES 1. Hard coal of good quality has at least 90 % of carbon. How much carbon in 9 tons of hard coal ? 2. A common coal hod holds 30 pounds of coal. How many hods in a ton ? 3. If coal sells for $8.25 in June and for $ 9.00 in January, what per cent is gained by buying it in Jime rather than in January ? When is the most economical time to buy coal ? 4. The housewife buys kerosene by the gallon. If the price per gallon is 13 cts. and live gallons cost 6b cts., what is the per cent gained by buying in 6-gallon can lots ? 5. If kerosene sells for $ 4.60 a bg^rrel, what is the price per gallon by the barrel? What per cett is gained over single gallons at 13 cts. retail ? What is the most economical way to buy kerosene ? (A barrel contains 42 gallons.) How to Read a Gas Meter ^ 1. Each division on the right-hand circle denotes 100 feet; on the center circle 1000 feet; and on the left-hand circle 10,000 feet. Read from left- hand dial to right, always tak- ing the figures which the hands have passed, viz. : The above dials register 3, 4, 6, adding ^ Gas is measured in cubic feet. HEAT AND LIGHT 1^9 two ciphers for the hundreds, making 34,600 feet registered. To ascertain the amount of gas used in a given time, deduct the previous register from the present, viz. : Register by above dials 34,600 Register by previous statement 18,200 Given number of feet registered 16,400 16,400 feet @ 90 cts. per 1000 costs what amount ^ 2. If a gas meter at the pre- .^^^^^\^ ^^^^*^^ J^^^T^ vious reading registered 82,700 ^?^\^^^''^^^ feet, and to-day the dials read W ^ ysA^v \ Aa^V \ /3/ as follows, ^^^^3^^ ^^^^^^^ X!l3j>^ what is the cost of the gas at 95 cts. per 1000? 3. What is the cost of the gas used during the month from the reading on this meter, if ^-,hoos^^ o'******'*^ i^^^r^ the previous reading was 6100 /x^\\ //^~\\ Z/^^\\ feet ? The rate is $ 1.00 per r(\ jYf • [Tfj^ Yj 1000 cu. ft. less ten per cent, \^--jr''^\k''^^ if paid before the 12th of the ^-^2.-^ ^^--i-^^ month. Give two answers. si^p^^^ ^^^^S^d n?^^^5^^ 4. What is the cost of gas pf^^Y\rf ^r\r/>^ ^^ registered by this meter at wl yTvV ^>7 \ V yv 85 cts. per 1000 cu. ft.^ ^^is:^ x^l^^ ^k:3^ How to Read an Electric Meter (See the subject of the electricity in the Appendix) There are three terms used in connection with electricity which it is important to understand; namely, the volt, the ampere, and the watt or kilowatt. (1) The volt is the unit of Electromotive Force or electrical pressure. It is the pressure necessary to force a current of one ampere through a resistance of one ohm, (2) The imit of electric current strength is the ampere. It 170 VOCATIONAL MATHEMATICS FOR GIRLS is the amount of current flowing through a resistance of one ohm under a pressure of one volt. (3) The watt is the unit of electrical power ; it is the prod- uct of volts (of electromotive force) and current (amperes) in the circuit, when their values are respectively one volt and one ampere. That is to say, if we have an electrical device operated at 3 amperes, on a line voltage of 115 volts, the amount of current consumed is equal to 116 X 3 = 346 watts, which, if operated continuously for one hour, will register on the electric meter as 345 watt hours, or .345 kilowatt hours (a kilowatt hour being equal to 1000 watt hours). All electrically operated devices are stamped with the ampere and voltage rating. This stamping may be found on the name-plate or bottom of the device. By multiplying the voltage of the circuit upon which the device is to be operated by the amperes as found stamped on the device, we can quickly determine the wattage consumption of the latter, as ex- plained under the definition of the watt, and as shown above. The line voltage which is most extensively supplied by Electric Lighting com- panies in this country is 115 volts, and where this voltage is in operation, the devices are stamped for voltage thus : V. 110-126. This means that the device may be used on a circuit where the voltage does not drop below 110 volts or rise above 126 volts. By operating a device with the above stamping on a circuit of 106 volts the life of the device would be very much longer, but the results desired from it would be secured much more slowly. Again, if the same device were used on a circuit oper- ating at 130 volts, the life of the device would be very short, although the results desired from it would be brought about much more quickly. Be- fore attempting to operate an electrically heated or lighted device, if in doubt about the voltage of the circuit, it is best to call upon the Electric Company with which you are doing business and ask the voltage of their lines. Incandescent electric lamps, while known to the average user as lamps of a certain "candle-power,'* are all labeled with their proper wattage consumption. Mazda lamps, suitable for household use and obtainable at all lighting companies, are made in 16, 26, 40, 60, and 100 watt sizes. For commercial use, lamps of 1000 watts and known as the nitrogen-filled lamps are on the market. Nitrogen lamps are made in sizes of 200 watts and upwards. HEAT AND LIGHT 171 The rate by which current coDsumed for lighting and small heating is figured in some cities is known as the <* sliding scale rate,^* and current is charged for each month, as follows : The first 200 kw. hrs. used @ 10^ per kilowatt hour. The next 300 kw. hrs. used @ 8^ per kilowatt hour. The next 600 kw. hrs. used @ 7 f per kilowatt hour. The next 1000 kw. hrs. used @ 6 ^ per kilowatt hour. The next 3000 kw. hrs. used @ 6 ^ per kilowatt hour. All over 6000 kw. hrs. used @ 4 ^ per kilowatt hour. Less 6% discount, if bill is paid within 16 days from date of issue. Under the sliding-scale rate the more electricity that is consumed, the cheaper it becomes. But it is also readily seen that the customer who uses a large amount of electricity pays in exactly the same way as the small consumer pays for his consumption. If a person uses less than 200 kw. hrs. per month, he pays for his con- sumption at the rate of 10 ^ per kilowatt hour ; if he uses 201 kw. hrs. of electricity per month, he pays for his first 200 kw. hrs. at the first step, namely 10 ^, and for the remaining 1 kw. hr. he pays 8 ^ per kilowatt hour. If a meter reads ** 1000 kw. hrs.,'* the bill is not figured at 6^ direct, but must be figured step by step as shown in the examples below. For convenience in figuring, the amount of power used by various electrically operated devices is given in the following table. By figuring the cost of each per hour, it will be seen that these electric servants work very cheaply. Appabatdb "Watts used What FEB 18 Cost HOUB* (a) Disk stove 200 • ? (6) 61b. iron 440 ? (c) A'ir heater, small 1000 ? (d) Toaster-stove 600 ? (e) Heating pad 66 ? (/) Sewing-machine motor 60 (average) ? (gr) 26 watt (16 c p.) lamp 26 ? (h) Chafing dish 600 ? (i) Washing-machine motor 200 (average) f Example. — Suppose a customer in one month used 6120 kilowatt hours of electricity, what is the amount of his bill ^ Based on 10 cents per kilowatt hour. 172 VOCATIONAL MATHEMATICS FOR GIRLS with 5 % deducted if the bill is paid within the discount period of 15 days from date of issue ? Solution. — 6120 kw. hrs. = total amount used. First 200kw. hrs. @10)^ = 6920 $ 20.00 Next 800 kw. hrs. @ 8^ = 6620 24.00 Next 600 kw. hrs. @ 7 ^ = 6120 36.00 Next 1000 kw. hrs. @ 6^ = 4120 60.00 Next 8000 kw. hrs. @ 5^ = 160.00 We have now figured for 6000 kw. hrs., and as our rate states that all over 6000 kw. hrs., is figured at 4 ^ per kilowatt hours, we have > 1120kw. hrs. @ 4j^ = | 44.80 $383.80 = gross bill Assuming that the bill is paid within the given discount period, we deduct 6% from the gross bill, which equals $ 16.69 $317.11 = net bill EXAMPLES 1. A customer uses in one month 300 kw. hr. of electricity. What is the amount of his bill if 5 % is deducted for payment within 15 days ? 2. What is the amount of bill, with 5 % deducted, for 15 kw. hr. of electricity ? An electric meter is read in the same way that a gas meter is read. In deciding the reading of a pointer, the pointer before it (to the right) must be consulted. Unless the pointer to the right has reached or passed zero, or, in other words, completed a revolution, the other has not com- pleted the division upon which it may appear to rest. Figure 1 reads 11 kw. hrs., as the pointer to the extreme right has made one complete revolution, thus advancing the second pointer to the first digit and has itself passed the first digit on its dial. HEAT AND LIGHT 173 Fig. 1. — Reading 11 kw. hrs. KpiOOO 1.000 too 10 KIL0WATT-t40URS Fig. 2. — What is the Reading? 10,000 KILOWATT- HOUKfi Fig. 3. — Reading 424 kw. hrs. laooo 1.000 100 10 KILO WATT- HOilBS Fig. 4. — What is the Reading? laooo KILOWATT-MOUPS Fig. 6. — What is the Reading ? 174 VOCATIONAL MATHEMATICS FOR GIRLS 1. What is the cost of electricity in Eig. 1, using the rates on page 171 ? 2. What is the cost of electricity in Fig. 2, using the rates on page 171, with the discount ? 3. What is the cost of electricity in Fig. 3, using the rates on page 171, with the discount ? EXAMPLES 1. What is the cost of maintaining ten 25-watt Mazda lamps, burning 30 hours at 10 cents per kw. hr. ? 2. What will it cost to run a sewing machine by a motor (50 watts) for 15 hours at 9 cents per kw. hr. ? 3. A 6-lb. electric flatiron is marked 110 V. and 4 amperes. What will it cost to use the iron for 20 hours at 8 cents per kw. hr. ? 4. An electric washing machine is marked 110 V. and 2 amperes. What will it cost to run it 15 hours at 8^ cents per kw. hr. ? 5. An electric toaster stove is marked 115 volts and 3^ am- peres. What will it cost to run it for a month (thirty break- fasts) 15 hours at 8f cents per kw. hr. ? If a discount of 5 % is allowed for prompt payment, what is the net amount of the bill ? Methods of Heating Houses are heated by hot air, hot water, or steam. In the hot-water system of heating, hot water passes through coils of pipes from the heater in the basement to radiators in the rooms. The water is heated in the boiler, and the portion of the fluid heated expands and is pushed upward by the adjacent colder water. A vertical circulation of the water is set up and the hot water passes from the boiler to the radiators and gives off its heat to the radiators, which in turn give it off to the surrounding air in the room. The convection currents HEAT AND LIGHT 175 carry heat through the room and at the same time provide for ventilation. In the hot-air method the heat passes from the furnace through openings in the floor called registers. This method frequently fails to heat a house uniformly be- cause there is no way for the air in certain rooms to escape so as to per- mit fresh and heated air to enter. Steam heating consists in allowing steam from a boiler in the basement to circulate through coils or radiators. The steam gives off its heat to the xx * tt ° , . , . Hot Aib Heating System radiators, which in turn give it off to the surrounding air. Room-heating Calculations Hot Water Heating System In order to insure comfort and health, every housewife should be able to select an efficient room-heating appli- ance, or be able to tell whether the existing heating appara- tus is performing the required service in the most econom- ical manner. In order to do this, it is necessary to know how to determine the re- quirements for individual room heating. 176 VOCATIONAL MATHEMATICS FOR GIRLS For Steam Heating Allow 1 sq. ft. of radiator surface for each 80 cu. ft. of volume of room. 13 sq. ft. of exposed wall surface. 3 sq. ft. of exposed glass surface (single window). 6 sq. ft. of exposed glass surface (double window). For Hot-water Heating Add 60 per cent to the amount of radiator surface obtained by the above calculation. For Oas Heaters having no Flue Connection Allow 1 cu. ft. of gas per hour for each 215 cu. ft. of volume of room. 36 sq. ft. of exposed wall surface. 9 sq. ft. of exposed glass surface (single window). 18 sq. ft. of exposed glass surface (double window). The results obtained must be further increased by one or more of the following factors if the corresponding conditions are present. Northern exposure 1.8 Eastern or western exposure 1.2 Poor frame construction 2.5 Fair frame 2.0 Good frame or 12-inch brick 1.2 Room heated in day time only 1.1 Room heated only occasionally 1.3-1.4 Cold cellar below or attic above . . . . . 1.1 Example. — How much radiating surface, for steam heating, is necessary to heat a bathroom containing 485 cu. ft. ? The bathroom is on the north side of the house. y^ = 6^ sq. ft. of radiating surface 6^ X 1.3 = IJ X i* = 7itt sq. ft. eVs + 7Hi = 6^^ + 7 Hi = 13iH sq. ft. or approx. 14 sq. ft. Ana, EXAMPLES 1. How much radiating surface, for steam heating, is re- quired for a bathroom 12' x 6' x 10' on an eastern exposure ? 2. How much radiating surface, for hot-water heating, is required for the bathroom in example 1 ? COST OP FURNISHING A HOUSE 177 3. How large a gas heater should be used for heating the bathroom in example 1 ? 4. (a) How much radiating surface is required for steam heating, in a living room 18' x 16^' x 10', with three single windows 2' x 5^' ? The room is exposed to the north. (b) How much radiating surface for hot-water heating ? (c) How much gas should be provided to heat the room in example (a) ? 5. (a) How much radiating surface is required for steam heating a bedroom 19' x 17' x 11' with two single windows 2' X 5^' ? The house is of poor frame construction. (b) How much radiating surface for hot-water heating ? (c) How much gas should be provided to heat room in example (a) ? CHAPTER VIII THRIFT AND INVESTMENT It is not only necessary to increase your earning capacity, but also to develop systematically and regularly the saving habit. A dollar saved is much more than two dollars earned. For a dollar put at interest is a faithful friend, earning twenty-four hours a day, while a spent dollar is like a lost friend — gone forever. Histories of successful men show that fortune's ladder rests on a foundation of small savings ; it rises higher and higher by the added power of interest. The secret of success lies in regularly setting aside a fixed portion of one's earnings, for instance 10 % ; better still, 10 % for a definite object, such as a home or a competency. In every community one will find various agencies by which savings can be systematically encouraged and most success- fully promoted. These institutions promote habits of thrift, encourage people to become prudent and wise in the use of money and time. They help people to buy or build homes for themselves or to accumulate a fund for use in an emergency or for maintenance in old age. Banks Working people should save part of their earnings in order to have something for old age, or for a time of sickness, when they are unable to work. This money is deposited in banks — savings, National, cooper- ative, and trust companies. National Banks National banks pay no interest on small deposits, but give the depositor a check book, which is a great convenience in business. National banks require that a fixed sum should be left on deposit, $ 100 or more, and some of them charge a certain amount each month for taking care of the money. 178 THRIFT AND ESTVESTMENT 179 Trust Companies Trast companies receive money on deposit and allow a customer to draw it out by means of a check. They usually pay a small interest on deposits that maintain a balance over $ 500. CoQperative Banks When a person takes out shares in a cobperative bank, he pledges him- self to deposit a fixed amount each month. If he deposits $5, he is said to have live shares. No person is permitted to have more than twenty- five shares. The rate of interest is much higher than in other banks, and when the shares mature, which is usually at the end of about eleven years, all the money must be taken out. Many people build their home through the cooperative bank, for, like every other bank, it lends money. When a person borrows money from a cooperative bank, he has to give a mortgage on real estate as security, and must pay back a certain amount each month. Savings Banks The most common form of banking is that carried on by the Savings Bank. People place their money in a savings bank for safe keeping and for interest. The bank makes its money by lending at a higher interest than it pays its depositors. There is a fixed date in each bank when money deposited begins to draw interest. Some banks pay quarterly and some semi-annually. At different times banks pay different rates of interest ; and often in the same community there are different rates of in- terest paid by different banks. Every bank is obliged to open its books for inspection by special officers who are appointed for that work. If these men did their work carefully and often enough, there would be almost no chance of loss in putting money in a bank. Banks fail when they lend money to too many people who are unable to pay it back. EXAMPLES (Review interest on page 60) 1. I place $ 400 in a savings bank that pays 4 % on Jan. 1, 1916. Money goes on interest April 1 and at each successive quarter. How much money have I to my credit at the begin- ning of the third quarter ? 2. A man with a small business places his savings, $ 1683, 180 VOCATIONAL MATHEMATICS FOR GIRLS ■ in a trust company so lie can pay his bills by check. The bank pays 2 % for all deposits over $ 500. He draws checks for $ 430 and $ 215 within a few days. At the end of a month he will receive how much interest ? 3. Practically 10 % of the entire population of the United States, including children, have savings-bank accounts. If* the population is 92,818,726, how many people have savings bank deposits ? 4. On April 1, 1910, a woman deposited $ 513 in a savings bank which pays 4 ^ interest. Interest begins April 1 and at each succeeding quarter. Dividends are declared Jan. 1 and July 1. What is the total amount of her deposit at the present date ? The savings bank is not adapted to the needs of those ^ith large sums to place at interest. It is a place where small sums may be deposited with absolute safety, earn a modest amount, and be used by the depositor at short notice. The savings bank lends money on mortgages and re- ceives about 6 ^0, It pays its depositor either 3J % or 4 %. The differ- ence goes to pay expenses and to provide a surplus fund to protect depositors. The question may be asked, " Why cannot the ordinary depositor lend his money on mortgages and receive 5 % ? " He can, if he is willing to assume the risk. When you receive 4 % interest, you are paying 1 9^ to 1^ % in return for absolute safety and freedom from the necessity of selecting securities. Mortgages A mortgage is the pledging of property as a secuilty for a debt. Mr. Allen owns a farm and wants some money to buy cattle for it. He goes to Mr. Jones and borrows $ 1000 from him, and Mr. Jones requires him to give as surety a mortgage on his farm. That is, Mr. Allen agrees that if he does not pay back the f 1000, the farm, or such part as is necessary to cover the debt, shall belong to Mr. Jones. Under present law, if a man wishes to foreclose a mortgage, — that is, compel its payment when due, — he cannot take the property, but it must be sold at public auction. From the money received at the sale the man who holds the mortgage receives his full amount, and anything that is left belongs to the man who owned the property. THRIFT AND INVESTMENT 181 Notes A promissory note is a paper signed by the borrower promising to repay borrowed money. Notes should state value received, date, the amount borrowed (called the face), the rate, to whom payable, and the time and place of payment. Notes are due at the expiration of the specified time. The rate of interest varies in different parts of the country. The United States has to pay about 2 %. Savings banks pay S^o or 4^o, Individuals borrowing on good security pay from 4^o to 6 %. In order to make the one who loans the money secure, the borrower, called the maker of the note, often has to get a friend to indorse or sign this note. The indorser must own some sort of property and if, at the end of six months or the time specified, the maker cannot pay the note, he is notified by written order, called a protest, and may, later, be called upon to repay the note. A man is asking a great deal when he asks another man to sign a note for him. Unless you have more money than you need, it is better busi- ness policy to refuse the favor. Always be sure that you know exactly what you are signing and that you know the responsibility attached. If you are a stenographer or a clerk in an office, you will often be called upon to witness a signature and then to sign your own name to prove that you have witnessed it. Always insist upon reading enough of the document to be sure that you know just what your signature means. EXAMPLES 1. My house is worth $ 4000 and the bank holds a mortgage on it for one-half its value. They charge 5 % interest, which must be paid semi-annually. How much do I pay each time ? 2. A bank holds a mortgage of $ 2500 on a house. The in- terest is 5 % payable semi:annually. How much is paid for interest at the end of three years ? 3. A man buys property worth $ 3000. He gives a $ 2000 mortgage and pays 5^ % interest. What will be the interest on the mortgage at the end of the year ? Suppose he does not pay the interest, how long can he hold the property ? 182 VOCATIONAL MATHEMATICS FOR GIRLS Different Kinds of Promissory Notes $ Montgomery, Ala 191 after date for value received promise to pay to the order of .Dollars a4> iWedjanicg National JSank* JVb Due A Common Note $ St, Paul, Minn 19 after date for value received we jointly and severally promise to pay to the order of .Dollars a4> fHerijanicg National JSanfe. J^o Due Joint Note $ Fall RiYEB, Mass. 191 after date for value received promise to pay to the order of The Mechanics National Bank of Fall River, Mass. Dollars, at said Bank, and interest for such further time as said principal sum or any part thereof shall remain unpaid at the rate of per cent per annum, having deposited with the said Mechanics National Bank, as General Col- lateral Security, for the payment of any of liahilities to said Bank due, or to become due, direct or indirect, joint or several, individual or firm, now or hereafter contracted or incurred, at the option of said Bank, the following property, viz. : and hereby authorize said Bank or its assigns to sell and transfer said property or any part thereof without notice, at public or private sale, at the THRIFT AND INVESTMENT 183 option of said Bank or its assigns, on the non-payment of any of the liabili- ties aforesaid, and to apply the proceeds of said sale or sales, after deducting all the expenses thereof, interest, all costs and charges of enforcing this pledge and all damages, to the payment of any of the liabilities aforesaid, giving credit for any balanqe that may remain. Said Bank or its assigns shall at all times have the right to require the undersigned to deposit as general collateral security for the liabilities aforesaid, approved additional securities to an amount satisfactory to said Bank or its assigns, and hereby agree to deposit on demand (which may be made by notice in writing deposited in the post office and addressed to at last known residence or place of business) such additional collateral. Upon fail- ing to deposit such additional security, the liabilities aforesaid shall be deemed to be due and payable forthwith, anything hereinbefore or elsewhere ex- pressed to the contrary notwithstanding, and the holder or holders may immediately reimburse themselves by public or private sale of the security aforesaid ; and it is hereby agreed that said Bank or any of its officers, agents, or assigns may purchase said collateral or any part thereof at such sale. In case of any exchange of or addition to the above described collateral, the provisions hereof shall apply to said new or additional collateral. COLLATEBAL NOTE 4. On Jan. 2, 1915, Mr. Lewis gave his note for $2400, payable on Feb. 27, with interest at 6 %. On Feb. 2, he paid $ 600. How much was due Mar. 2, 1915 ? Solution. — In the case of notes running for less than a year, exact days are counted ; from Jan. 2 to Feb. 2 is 31 days. Interest Jan. 2 to Feb. 2, 31 days, $12.00 for 30 days .40 for 1 day $ 12.40 31 days Amount due Feb. 2, $ 2400 + 12.40 = $ 2412.40. $ 2412.40 - 600 = $ 1812.40. Interest Feb. 2 to March 2, 28 days, $6.0413 20 days 1.8124 6 days .6041 2 days $8.4678 or $8.46 1812.40 Amount due March 2, $ 1820.86 Ans. 184 VOCATIONAL MATHEMATICS FOR GIRLS Money lenders may discount their notes at banks and thus obtain their money before the note comes due. But the banks, in return for this serv- ice, deduct from the full amount of the note interest at a legal rate on the full amount for such time as remains between the day of discount and the day when the note comes due. To illustrate : A man has a note for $ 600 due in three months at 6 9^ interest. At the end of a month he presents the note at a bank and returns the difference between the amount at maturity, $600, and the interest on $609 for two months, the remaining time, at legal rate 69^, $6.09 or $609 — 6.09 = $602.91. 5. On June 1, 1914, Mr. Smith givea his note for $ 1200, payable on demand with interest at 6%. The following pay- ments are made on the note : Aug. 1, 1914, $ 140 ; Oct. 1, 1914, $ 100 ; Dec. 1, 1914, $ 100 ; and Feb. 1, 1914, $ 160. How much was due May 1, 1915 ? 6. A merchant buys paper amounting to $ 945. He gives his note for this amount, payable in three months at 6 % . The paper dealer desires to turn the note into cash immedi- ately. He therefore discounts it at the bank for 6%. How much does he receive ? Stocks It often happens that one man or a group of men desire to engage in a business that requires more money than they alone are able or willing to invest in it. They obtain more money by organizing a stock company, in which they themselves buy as many shares as they choose, and then they induce others to pay for enough more shares to make up the capital that is needed or authorized for the business. A stock company consists of a number of persons, organized under a general law or by special charter, and empowered to transact business as a single individual. The capital stock of a company is the amount named in its charter. A share is one of the equal parts into which the capital stock of a company is divided (generally $ 100). The par value of a share of stock is its original or face value ; the market value of a share of stock is the price for which the share will sell in the market. The market values of leading stocks vary from day to day, and are quoted in the daily papers ; e.g. *' N. Y. C, 131 " means that the stock of the New York Central R. R. Co. is selling to-day at $ 131 a share. THRIFT AND INVESTMENT 185 DiTidends are the net profits of a, stock company divided among the BtockholderH according lo llie amount of stock tbey own. Stock compantea often issue two kinda of stock, namely : preferred stoclc, wbich consists of a certain number of shares on wliicli dividends are paid at a fixed rate, and commOD stock, which consists of the re- maining shares, among wbich are apportioned whatever protiU there are remaining after payment of the required dividends on the preferred stock. Cbrtificatb of Stock Stocks are generally bought and sold by brokers, who act as agents for the owners of the stock. Broken receive as their compensation a certain per cent of the par value of the stock bongbt Qr sold. This Is called brokerage. The usual brokerage is } ^ of the par value ; e.g. if a broker sells 10 shares of stock for me, his brokerage is ) 9b of 1 1000, or 1 1.26. Example. — What is the cost of 20 shares of No. Butte 30J ? *30i + J i 1 = f 30t, cost of 1 share. (30} x20 = i000+ $12^ = $012.60, total cost. 1 1 of 1 % of 8 100 = I of $ 1, broker's charge pet share. 186 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES 1. The par value of a certain stock is $100. It is quoted on the market at $87^.. What is the difference in price per share between the market value and the par value ? 2. What is the cost of 40 shares of Copper Range at 53 ? 3. What is the cost of 53 shares of Calumet and Hecla at 680 ? 4. I have 50 shares of Anaconda. How much shall I re- ceive if I sell at 66^ ? 5. I buy 60 shares of Anaconda at 66^. It pays a quarterly dividend of $ 1.50. What interest am . I receiving on my money ? Bonds Corporations and national, state, county, and town govjBmments often need to borrow money in order to meet extraordinary expenditures. When a corporation wishes to borrow a large sum of money for several years, it usually mortgages its property to a person or bank called a trustee. The amount of the mortgage is divided into parts called bonds, and these are sold to investors. The interest on the bonds is at a fixed I'ate and is generally payable semi-annually. Shares of stock represent the property of a corporation, while bonds represent debts of the corporation ; stock- holders are owners of the property of the corporation, while bondholders are its creditors. Bonds of large corporations whose earnings are fairly stable and regu- lar, like steam railroads, street railways, and electric power and gas plants, whose property must be employed for public necessities regardless of the ability of the managers, are usually good investments. Well-secured bonds are safer than stocks, as the interest on the bonds must be paid re- gardless of the condition of the business. For the widow who is obliged to live on the income from a moderate amount of capital, it is better to invest in bonds and farm mortgages than in stock. THRIFT AND INVESTMENT 187 A Sample Bond 188 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES 1. A man put $ 200 in the Postal Savings Bank and received 2 ^0 interest. What would have been the difference in his income for a year if he had taken it to a savings bank that paid 3| % ? 2. A widow had a principal of $ 18,000. She placed it in a group of savings banks that paid 3f %. The next year she purchased farm mortgages and secured h\ ^o* What was the difference in her income for the two years ? 3. Two sons were left % 15,000 each. One placed it in first- class bonds paying 5^ %. The other placed it in savings banks and averaged 4^%. What was the difference in income per year ? Fire Insurance Household furniture, books, apparel^ etc., can be insured at a low rate. While it will not make a man less careful in protecting his home from fire, it will make him more comfortable in the thought that if fire should come, the family will not be left without the means of clothing themselves and refurnishing the house. One of the first duties then, after the home is established, is to secure insurance. Insurance companies issue a policy for 1, 3, or 5 years. There is an advantage in buying a policy for more than one year, for on the 3- or 6- year policy there is a saving of about 20 9^ in premiums. Rules of per- centage apply to problems in insurance. Example. — A house worth $8400 is insured for its full value at 28 cents per $ 100. What is the cost of premium ? Solution. $ 8400 is the value of the policy or base. 28 cents is the rate of premium or rate. The premium or interest is the amount to be found. 84 X $0.28 = $23.52, premium. EXAMPLES 1. Find the insurance upon a dwelling house valued at $ 3800 at $ 2.80 per $ 1000 if the policy is on 80 % of the value of the house. THRIFT AND INVESTMENT 189 2. Mr. Jones takes out $ 800 insurance on his automobile at 2 fo' What is the cost of the premium ? 3. The furniture in one tenement of a three-family house is valued at $ 1000. What premium is paid, if it is insured at the rate of 1 % for 5 years ? 4. If the premium on the same furniture in a two-family house in a different city is $ 7.50, what is the rate, expressed in per cent ? Life Insurance Every indostrious and thrifty person lays aside it certain amount regularly for old age or future necessities, or in case of death to provide sufficient amount for the support of the family. This is usually done by taking out life insurance from a corporation called an insurance company. This corporation is obliged to obtain a charter from the state, and is regularly inspected by a proper state oflBcer. The poliqf or contract which id made by the company with the member, fixing the amount to be paid in the event of his death, is called a life insurance policy, and the person to whom the amount is payable is termed the beneficiary. The contribution to be made by the member to the common fund, as stipulated in the policy, is termed the pr^mtttm, and is usually payable in yearly, half-yearly, or quarterly installments. There are different kinds of insurance policies : the simplest is the ordinary life policy. Before entering into a contract of this, kind, it is necessary to fix the amount of the premium, which must be large enough to enable the company to meet the necessary expense of conduct- ing the businesd and to accumulate a fund sufficient to pay the amount of the policy when the latter matures by the death of the insured. Making the Premium. — If it were known to a certainty just how long the policy holder would live, anyone could compute the amount of the necessary premium. Let us suppose, for illustration, that the face of the policy is $ 1000, and that the policyholder will live just twenty years. Let us assume that the business is conducted without expense, and that the premiums are all to be invested at interest from date of payment. We do not know to a certainty what rate of interest can be earned during the whole period, and we shall therefore assume one that we can safely depend upon, say three per cent. A yearly payment of .$86.13 invested at three per cent compound interest will amount to $ 1000 in twenty years. 190 VOCATIONAL MATHEMATICS FOR GIRLS JVo.j218649^ $5000 Sftje "^ovth Mvcv ^ntnvd %iU %nsnvvcntt In Consideration of the application for this Policy, a copy of which is attached hereto and made a part hereof, and in further consideration of the payment of ^m j^utilyreH Ctytrtg^giflbt^^^^-^^^^-^^-v^N^^^^^s Dollars 100 ' the receipt whereof is hereby acknowledged, and of the_™55?Lpayment of a like simi to the said Company, on or before the irtrgt ^^.y of _5555?3_in every year during the continuance of this Policy, promises to pay at its of&ce in Milwaukee, Wisconsin, unto Itlarg wot ~ , Beneficiar__J__, . CTliCe of 3ot;n IBoe ^hf. Insured, of JBeg fHotneg in the State of ^^^^ subject to tf|g rtg!|t of ttje ffneureH, tyerefeg resnrbeH, t0 diange ttye ISeneficiatg or iSntefictgrteg the sum nf .^tbg grtjottganH Dniiara, upon receipt and approval of proof of the death of said Insured while this Policy is in full force, the balance of the year's premium, if any, and any other indebtedness on account of this Policy being first deducted there- from; provided, however, that if no Beneficiary shall survive the said Insured, then such payment shall be made to the executors, administra- tors or assigns of the said Insured. In Witness Whereof, THE NORTH STAR MUTUAL LIFE INSURANCE COMPANY, at its office in Milwaukee, Wisconsin, has by its President and Secretary, executed this contract, this £ll^ ^day of ^Tanuary ^^^ thousand nine hundred and sixteen. S. A. Hawkins, Secretary. L. H. Perkins, President. Ordinary Life Insurance Policy THRIFT AND INVESTMENT 191 If it were certain that the policyholder would live just twenty years, and that his premiums would earn just three per cent interest, and that the business could be conducted without expense, the necessary premium would be $36.13. But there are certain other contingencies that should be provided for; such as, for example, a loss of invested funds, or a failure to earn the full amount of three per cent interest. To meet these expenses and contingencies something should be added to the premium. Let us estimate as sufficient for this purpose the sum of $7. This v^U make the gross yearly premium $43.13, the original pay- ment ($36.13) being the net premium, while the amount added thereto for expenses, etc. ($7.00), is termed the loading. The net premium is the amount which is mathematically necessary for the creation of a fund sufficient to enable the company to pay the policy in full at maturity. The loading is the amount added to the net premium to provide for expenses and contingencies. The net premium and loading combined make up the gross premium^ or the total amount to be paid each year by the insured. Mortality Tables. — Although it is impossible, as in the illustration given above, to predict in advance the length of any individual life, there is a law governing the mortality of the race by which we may determine the average lifetime of a large number of persons of a given age. We cannot predict in what year the particular individual will die, but we may determine with approximate accuracy how many out of a given number will die at any specified age. By means of this law it becomes possible to compute the premium that should be charged at any given age with almost as much exactness as in the example given, in which the length of life remaining to the individual was assumed to be just twenty years. Let us suppose, for example, that observations cover a period of time sufficient to include the history of 100,000 lives. Of these, you will find a certain number dying at the age of thirty, a larger number at the age of forty, and so on at the various ages, the extreme limit of life reached being in the neighborhood of one hundred years. The mortuary records of other groups of 100,000, living where conditions are practically the same, would give approximately the same results — the same number of deaths at each age in 100,000 bom. The variation would not be great, and the larger the number of lives under observation, the nearer the number of deaths at the several ages by the several records would ap- proach to uniformity. In this manner mortality tables have been constructed which show how many in any large number of persons bom, or starting at a certain age, will live to age thirty, how many to age forty, how many to any other 192 VOCATIONAL MATHEMATICS FOR GIRLS age, and likewise the number that will die at each age, with the average lifetime remaining to those still alive. The insurance companies from these tables construct tables of premiums, varying according to the amount and kind of insurance and the age at which the policy is taken out. Kinds of Policies. — An endowment policy is essentially for persons who must force themselves to save. It is ah expensive form of insurance, but one that affords the young man or woman an incentive for saving, and that matures at a time when the individual has, as a result of long experience, better opportunities to make profitable investments. This policy also has a larger loan value than any other, and this sometimes be- comes an advantage to the young person. However, the chief advantage of the endowment policy is its incentive to save. A limited payment policy^ such as the fvoenty-payment life, appeals most directly to those who desire to pay for life insurance only within the productive period of their life. This policy should attract the young man who is uncertain of an income after a given period, or who does not wish insurance premiums to be a burden upon him after middle life. Out of the relatively large and certain income of his early productive years he pays for his insurance. This policy also appeals to the man of middle age who has neglected to purchase life insurance but who wishes to buy it and pay for it before he becomes actually old. The Annuity An annuity is a specific sum of money to be paid yearly to some designated person. The one to whom the money is to be paid is termed the annuitant. If the payment is to be made every year until the annui- tant dies, it is termed a life annuity. For example, a life insurance company or other financial institution, in consideration of the payment to it of a specified amount, say $ 1000, will enter into a contract to pay a designated annuitant a stated sum, say $ 70, on a specified day in every year so long as the annuitant continues to live. The latter may live to draw his annuity for many years, until he has received in aggregate several times the original amount paid by him, or he may die after having collected but a single payment. In either case, the contract expires and the annuity terminates with the death of the annuitant. The amount of the yearly income or annuity which can be purchased with $ 1000 will depend, of course, upon the age of the annuitant That sum will buy a larger income for the man of seventy than for one of fifty-six, for the reason that the former has, on the average, a much shorter time yet to live. The net cost of an annuity, that is, the net THRIFT AND INVESTMENT 193 amount to be paid in one sum, and which is termed the value of the annuity, is not a matter of estimate, but, like the life insurance premium, is determined by mathematical computation, based upon the mortality table. The process is quite as simple as the computation of the single premium. Many men who insure their lives choose a form of policy under which the beneficiary, instead of receiving the full amount of the insurance at the death of the insured, is paid an annuity for a period of years or throughout life. The amount of annuity paid in such cases is exactly equal to the amount that could be bought for a sum equal to the value of the policy when it falls due. EXAMPLES 1. A young man at 26 years of age takes out a straight life policy of $ 1000, for which he pays $ 17.03 a year as long as he lives, and his estate receives $ 1000 at his death. If he dies at 46 years of age, how much has he paid in ? How much more than he has paid does his estate receive then ? 2. Another young man at the same age takes out a twenty- payment life policy and pays $ 24.85 for twenty years. At the end of the twenty years, how much has he paid in ? Does he receive anything in return at the end of the twenty years ? 3. Another form of insurance, called an endowment, is taken out by another young man at twenty-six years of age. He pays $ 41.94 a year. At the end of twenty years he receives $ 1000 from the insurance company. How much has he paid in ?. Where is the difference between these two amounts ? Exchange Bzchange is the process of making payment at a distant place without the risk and expense of sending money itself. Funds may be remitted from one place to another in the same country in six different ways : Postal money order, express money order, telegraphic money order, bank draft, check, and si^ht draft. The largest amount for which one can obtain a postal money order is $ 100. It is drawn up by the postmaster after an application has been duly made out. An express money order is similar to a postal money order, but may be 194 VOCATIONAL MATHEMATICS FOR GIRLS drawn for any number of dollars at the same rate as the post office order. This is issued at express offices. A telegraphic money order is an order drawn by a telegraph agent at any office, instructing the agent at some other office to pay the person named in the message the sum specified. The rates are high, and in addition one must pay the actual cost of sending the telegram according to distance and number of words. A bank draft is an order written by one bank directing another bank to pay a specified sum of money to a third party. This order looks much like a check. A check is an order on a bank to pay the sum named and deduct the amount from the deposit of the person who signs the check. A sight dj'aft is an order on a debtor to pay to a bank the sum named by the creditor who signs the draft. Foreign exchange is a system for transmitting money to another country. By this means the people of different countries may pay their debts. The most common methods of foreign exchange for an ordinary traveler are letters of credit or travelers' cheques. A letter of credit is a circular letter issued by a banking house to a person who desires to travel abroad. The letter directs certain banks in foreign countries to furnish the traveler such sums as he may require up to the amount named in the letter. Fees For Money Orders Domestic Bates When payable in Bahamas, Bermuda, British Guiana, British Hon- duras, Canada, Canal Zone, Cuba, Martinique, Mexico, Newfoundland, The PJiilippine Islands, The United States Postal Agency at Shanghai (China), and certain islands in the West Indies, listed in the register of money order offices. For Orders from $00.01 to $2.50 Scents From $ 2.51 to $ 5 5 cents From $ 5.01 to $ 10 8 cents From $10.01 to $20 10 cents From $ 20.01 to $ 30 12 cents From $30.01 to $40 15 cents From $40.01 to $60 18 cents From $ 50.01 to $ 60 20 cents From $60.01 to $76 25 cents From $ 75.01 to $ 100 30 cents THRIFT AND INVESTMENT 195 International Bates When payable in Asia, Austria, Belgium, Bolivia, Chile, Costa Rica, Denmark, Egypt, France, Germany, Great Britain and Ireland, Greece, Honduras, Hongkong, Hungary, Italy, Japan, Liberia, Luxemburg, Netherlands, New South Wales, New Zealand, Norway, Peru, Portugal, Queensland, Russia, Salvador, South Australia, Sweden, Switzerland, Tasmania, Union of South Africa, Uruguay, and Victoria. For Orders from $00.01 to 310 10 cents From S 10.01 to $20 20 cents From $20.01 to $30 30 cents From $30.01 to $40 40 cents From $40.01 to $50 60 cents From $60.01 to $60 60 cents From $ 60.01 to $ 70 70 cents From $70.01 to $80 80 cents From $80.01 to $90 90 cents From $90.01 to $100 • . • 1 dollar Rates for Honey Transferred by Telegraph The Western Union charges for the transfer of money by telegraph to its offices in the United Stales the following : First: For $ 25.00 or less 25 cents $ 25.01 to $ 60.00 36 cents $50.01 to $ 75.00 60 cents $ 75.01 to $ 100.00 85 cents For amounts above $ 100.00 add (to the $ 100.00 rate) 25 cents per hundred (or any part of $ 100.00) up to $ 3000.00. For amounts above $ 3000.00 add (to the $ 3000.00 rate) 20 cents per hundred (or any part of $ 100.00). Second : To the above charges are to be added the tolls for a fifteen word message from the office of deposit to the office of payment. Express rates are the same as postal rates. EXAMPLES 1. A young woman in California desires to send $ 20 to her mother in Maine. What is the most economical way to send it, and what will it cost ? 2. A young lady, traveling in this country, finds that she 196 VOCATIONAL MATHEMATICS FOR GIRLS needs money immediately. What is the quickest and most economical way for her to obtain $ 275 from her brother who lives 1000. miles distant ? 3. A merchant in Boston buys a bank draft of $ 3480 for Chicago. The bank charges | of 1 % for exchange. How much must he pay the bank ? 4. A domestic in this country sends to her mother in Ireland 5 pounds for a Christmas present. What will it cost her, if $ 4.865 = £ 1 ? A commission of ^ of 1 % is charged. Claims If a person traveling by boat, electric or steam railway is injured by an accident which is the fault of the company, it is bound to repair the finan* cial loss. The company is not responsible for the carelessness of passen- gers or for the action of the elements. When an accident occurs, the injured persons are interviewed by a claim agent, whom all large com- panies employ, and he offers to settle with you for a certain amount. If you are not satisfied with this amount, you may put in your claim and the case goes to court, where you may lose or win according to the decision of the jury. When a wreck occurs on a railroad, a claim agent and a doctor are brought to the scene as soon as possible. They take the name and address of each person in the accident and try to settle the case at once, because it is expensive to go to court and the newspaper notoriety injures the reputation of the company. If you are not seriously hurt, the claim agent tries to persuade you to sign a paper which relieves the Com- pany from any responsibility forever after. For instance, in a collision you seem to be only shaken up, not injured. The claim agent perhaps offers to pay you 1 26. You think that is an easy way to get $ 25, so you take it, but in turn you must sign a paper which states that the company has settled in full with you for any claim that you may have against it for that accident. Now it may prove later that you have an internal injury which you did not realize at the time, and that an operation costing $ 600 is necessary. Can you compel the company to pay the bill ? People who are not hurt at all in an accident and to whom the claim agent offers nothing are also asked to sign a paper relieving the company from all responsibility. Do not sign such a paper. The company cannot compel you to, you gain nothing by it, and may lose much if it proves later that you are internally injured. THRIFT AND INVESTMENT 197 EXAMPLES 1. A woman was riding in an electric car that collided with another. She was cut with flying glass and was obliged to hire a servant for four weeks at $8. Doctor's bills amounted to $24.50, medicine, etc., $8.75. She settled at the time of the accident for $50. Did she lose or gain ? 2. A man working in a mill was injured in an elevator acci- dent. The insurance company paid his wages and medical bills for 8 weeks at $13.50 per week. A year later he was out of work for three weeks for the same injury and did not receive any compensation. Would it have been better for him to have settled for $100 at the beginning ? 3. A saleslady tripped on a staircase and sprained her ankle. She was out of work for two weeks and two days at $8.75 per week. Her medical supplies cost $9.75. She settled for $45. How much did she gain ? PART m — DRESSMAKING AND MILLINERY CHAPTER IX PROBLEMS IN DRESSMAKING The yardstick is much used for measuring cloth, carpets, and fabrics. The yardstick is divided into halves, quarters, and eighths. Dressmakers should know the fractional equiva- lents of yards in inches and the fractional equivalents of dollars in cents. It is wise to buy to the nearest eighth of a yard unless the cost per yard is so small that an eighth would cost as much as a quarter. EXAMPLES 1. Give the equivalent in inches of the following : (a) 1 yd. W H yd. (c) H yd. (d) 2i yd. (e) 3f yd. 2. A piece of cloth is 12 yd. long. How many pieces are needed for 16 aprons requiring 1^ yd. each ? 3. A piece of lawn cloth is 28 yd. long. How many pieces are needed for 20 aprons requiring 1| yd. each ? 4. Give the value in cents of the following fractions of a dollar : (/) 4| yd. (*) i yd. (9) H yd. (0 i yd. w n yd. (m) tV yd. (t) If yd. (n) ^Jjyd. U) i yd. (0) A yd. («)il («)il (0 A (m) ^^ (P) 1 (/•)! (j)i («) i\ Wi (9) i W i\ (o)i WH W A iO^ (Jp)i 198 ARITHMETIC FOR DRESSMAKERS 199 5. If 16" is cut from 1| yd. of cloth, how much remains ? 6. If 1^ of a yard of lawn is cut from a piece 40 in. long, what part of a yard is left ? 7. I bought 9| yd. of silk for a dress. If If yd. remained, how much was used ? « 8. A towel is 33 inches long and and a dishcloth 13 inches. (a) Find the length of both. (Allow |^" for each hem.) (b) Find the number of yards used for both. (c) Find the number of inches used by a class of 24. (d) Find the number of yards used by a class of 24. (e) Find the cost per pupil at 6 cts. per yard. (J) Find the cost for a class of 24 at 6 cts. per yard. 9. If it took 72 yards of material for a dishcloth and towel for two classes of 24 (48 in all), find the amount used by each pupil. 10. Jf 45| yards of material were used for a class of 42, find the amount used by each pupil. 11. (a) Reduce 75 inches to yards, {b) Find the number of inches in 3^ yards, (c) From 2f yards cut 40 inches. Tucks A tuck is a fold in the cloth for the purpose of shortening garments or for trimming or dec- oration. A tuck takes up twice its own depth ; that is, a 1" tuck takes up 2" of cloth. EXAMPLES 1. Before tucking, a piece of goods was I yd. long ; after tuck- ing, it was f yd. long. How many y tucks were made ? Mbasurinq fob Tucks from Fold to Fold 200 VOCATIONAL MATHEMATICS FOR GIRLS 2. How mucli lawn is taken up in 3 groups of tucks, the first group containing 6 one-inch tucks, the second group 6 one- half-inch tucks, and the third group 12 one-eighth-inch tucks ? 3. A piece of muslin 29 inches wide was tucked and when returned to the teacher was only 14 inches wide. How many \" tucks were made in it ? 4. Before tucking, a piece of goods was f yd. long ; after tucking, it was ^ yd. long. How many ^" tucks were made ? Hem Hem Turned A hem on a piece of cloth is an edge turned over to form a border or finish. In making a hem an edge must always be turned to prevent fraying; ex- cept for very heavy or very loosely woven cloth this is usu- ally y. For an inch hem you would have to allow 1\", EXAMPLES 1. I wish to put three y tucks in a skirt/ which is to be 40" long. How long must the skirt be cut to allow for the tucks and ^" hem ? 2. My cloth for a ruffle is 10" deep. It is to have a ly hem, and five \" tucks. How long will it be when finished ? 3. If. a girl can hem 2\ inches in five minutes, how long will she take to hem 2 yards ? 4. At the rate of | of an inch per minute, how long will it take a girl to hem 2 yards ? 10 yards ? 5. At the rate of 5|^ inches per ten minutes, how long will it take to hem 3^^ yards ? ARITHMETIC FOR DRESSMAKERS 201 6. A girl can hem 3 inches in five minutes. How much in an hour ? 7. How long will she take to hem 90 inches ? 8. At 6 cents per hour, how much can she earn by hemming 190 inches ? 9. How long will it take a girl to hem 2\ yards if she can hem 5^ inches in ten minutes ? Ruffle A ruffle is a strip of cloth gathered in narrow folds on one edge and used for the trimming or decoration. Different pro- portions of material are allowed according to the use to which it is to be put. For the ordinary ruffle at the bottom of a skirt, drawers, apron, etc., allow once and a half. Once and a quarter is Rufplb enough to allow for trimming for a corset cover or for other places where only a scant ruffle is desirable. A plaiting requires three times the amount. EXAMPLES 1. How much hamburg would you buy to make a ruffle for a petticoat which measures 3 yd. around, if once and a half the width is necessary for fullness ? 2. How much lace 2^ inches wide would you buy to have plaited for sleeve finish, if the sleeve measures 8 inches around the wrist — allowing three times the amount for plaiting ? 3. A skirt measuring S\ yd. around is to have two 5-inch ruffles of organdie flouncing. Allowing twice the width of skirt for lower ruffle, and once and three quarters for the upper one, how much flouncing would you buy, and what would be the cost at $ .87^ per yard for organdie ? 202 VOCATIONAL MATHEMATICS FOR GIRLS 4. How deep must a ruffle be cut to be 6" deep when finished, if there is to be a 1^" hem on the bottom and three ^" tucks above the hem ? 5. How deep a ruffle can be made from a strip of lawn 16" deep, if a 2" hem is on the bottom and above it three J" tucks ? 6. How many yards of cloth 36" wide are needed for S^ yd. of ruffling which is to be cut 6" deep? 7. How many widths for ruffling can be Ivpv^v-::^ cut from 4 yd. of lawn 36" wide, if the t^^Vp;;'^/-;;^^^ ^^^^ .g g„ finisiie^i^ ^j^^ i^ a |" hem and five y tucks ? Note. — Allowance must be made for joining a ruffle to a skirt, usu- ally i". 8. How deep must a ruffle be cut to be 6" deep when finished, if there is to be a 1^" hem on the bottom, and five y tucks above the hem ? 9. How many yards of ruffling are needed for a petticoat 2 J yd. around the bottom ? EXAMPLES m FINDING COST OP MATERIALS 1. What is the cost of hamburg and insertion for one pair of drawers ? 32 in. around each leg. Hamburg at 16 cents a yard. Insertion at 15 cents a yard. 2. What is the cost of hamburg and insertion for one pair of drawers ? 36 in. around each leg. Hamburg at 18^ cents a yard. Insertion at 16} cents a yard. 3. What is the cost of hamburg and insertion for a petticoat ? 6 yd. around. Hamburg at 25 cents a yard. Insertion at 15 cents a yard. ARITHMETIC FOR DRESSMAKERS 203 4. What is the cost of hamburg and insertion for a petti- coat? SJ yd. around. Hamburg at 27^ cents a yard. Insertion at 16} cents a yard. 5. What is the cost of trimming for a corset cover ? 38 in. around top. 13 in. around armhole. Lace at 10 cents a yard. 6. What is the cost of trimming for a corset cover ? 41 in. around top. 13} in. around armhole. Lace at 12} cents a yard. 7. What is the cost of lace for neck and sleeves at 12}^ cents a yard ? Neck, 13 in., sleeves, 8 in. 8. What is the cost of lace for neck and sleeves at 16 cents a yard ? Neck, 14 in., sleeves, 8} in. 9. What is the cost of a petticoat requiring 2^ yd. long- cloth at 12 j^ cents a yard, and 2\ yd. hamburg at 16}^ cents a yard? 10. What is the cost of a petticoat requiring 2f yd. long- cloth at 13} cents a yard, and 2^ yd. hamburg at 15^ cents a yard ? 11. What is the cost of a nightdress requiring S^ yd. of cambric at 25 cents a yard and 3 skeins of D. M. C. em- broidery cotton which sells at 5 cents for 2 skeins, and 1}- yd. |-inch ribbon at 9 cents a yard ? 12. What is the cost of the following material for a corset cover ? 1} yd. longcloth at 15 cents a yard. 2\ yd; hamburg at 8 cents a yard. 6 buttons at 12} cents a dozen. 204 VOCATIONAL MATHEMATICS FOR GIRLS 13. What is the cost of the following material for a skirt ? 7 yd. silk at 79 cents a yard. 1} yd. lining at 35 cents a yard. 14. What is the cost of the following material for a corset cover ? 1} yd. longcloth at 16 cents a yard. 2^ yd. hamburg at 8} cents a yard. 4 buttons at 12^ cents a dozen. 15. What is the cost of the following material for a corset cover ? 1} yd. longcloth at 16| cents a yard. 2} yd. hamburg at 25| cents a yard. 2| yd. insertion at 19} cents a yard. 4 buttons at 15 cents a dozen. 16. What is the cost of the following material for a corset cover ? 1} yd. longcloth at 14} cents a yard. 1} yd. hamburg at 17} cents a yard. 17. What is the cost of the following material for a skirt ? 7} yd. silk at 83} cents a yard. 1} yd. lining at 37} cents a yard. 18. Find the cost of a corset cover that requires 1 yd. cambric at 12} cents a yard, f yd. bias binding at 2 cents a yard. } doz. buttons at 12 cents a dozen. 1} yd. lace at 10 cents a yard. } spool thread at 5 cents a spool. 19. Find the cost of an apron that requires 1 yd. lawn at 12} cents a yard. 2} yd. lace at 10 cents a yard. } spool thread at 5 cents a spool. ARITHMETIC FOR DRESSMAKERS 205 20. Find the cost of a nightgown containing 3} yd. cambric at 12} cents a yard. 2 yd. lace at 5 cents a yard. 3 yd. ribbon at 3 cents a yard. i spool thread at 5 cents a spool. 21. Find the cost of drawers containing 2 yd. cambric at 12} cents a yard. 1} yd. finishing braid at 5 cents a yard. 1 spool thread at 5 cents a spool. 2 buttons at 10 cents a dozen. 22. What is the cost of a waist made of the following ? 2f yd. shirting, 32 inches wide, at 23 cents a yard. Sewing cotton, buttons, and pattern, 25 cents. 23. What is the cost of 7^ yd. chiffon faille, 36 inches wide, at $ 1.49 a yard ? 24. How many yards of ruffling are needed for 1 dozen aprons if each apron is one yard wide and half the width of the apron is added for fullness ? 25. How many pieces of lawn-36 inches wide are needed for the ruffle for one apron ? For eight aprons ? 26. A skirt measures 2^ yards around the bottom. How much material is needed for ruffling if the material is one yard wide and ruffle is to be cut 7 inches wide ? 27. How deep would you cut a cambric ruffle that when finished will measure 12^", including the hamburg edge which measures 4", two clusters of 6 tucks ^" deep, and allowing 1' for making? 28. Find the cost of a poplin suit made of the following : Silk poplin, 40 inches wide : 5} yards, at $ 1.79 a yard. Satin facing for collar, revers, and cuffs, 21 inches wide : 1 yard, at 11.26 a yard. Coat lining, 36 inches wide : 2J yards, at $ 1.60 a yard. Buttons, braid, sewing silk, two patterns, $ .64. 206 VOCATIONAL MATHEMATICS FOR GlRLS Cloths of Different Widths There are in common use cloths of several different widths and at various prices. It is often important to know which is the most economical cloth to buy. This may be calculated by finding the cost per square yard, 36" by 36". To illustrate : which is less expensive, broadcloth 56" wide, at $2.25 per yard, or 50" wide, at $1.75 per yard ? ?^JiM X 2.26 = 9 1.44^ per square yard. 66 X 3^ ^^^ X 1.76 = $ 1.26 per square yard. EXAMPLES Find the cost per square yard and the relative economy in purchasing : (a) Prunella, 46" wide, at $ 1.60 a yard. Prunella, 44'' wide, at $ 1.36 a yard. (6) Serge, 64'' wide, at $ 1.26 a yard. Poplin, 42" wide, at $ 1.00 a yard. (c) Serge, 42" wide, at 49 cents a yard. Serge, 37" wide, at 39 cents a yard. (d) Shepherd check, 64" wide, at $ 1.76 a yard. Shepherd check, 62" wide, at $ 1.60 a yard. Shepherd check, 42" wide, at $ 1.00 a yard. (e) Taffeta, 19" wide, at 89 cents a yard. TafEeta, 36" wide, at ^ 1.26 a yard. (/) Cashmere, 42" wide, at § 1.00 a yard. Nuns veiling 44" wide, at 76 cents a yard. (g) Cheviot, 67" wide, at $ 1.60 a yard. Diagonal, 64" wide, at ^2.00 a yard. (A) Messaline, 26 " wide, at 69 cents a yard. Messaline, 36 " wide, at $1.26 a yard. ARITHMETIC FOR DRESSMAKERS 207 PROBLEMS IN TRADE DISCOUNT Illustrative Example. — A dressmaker bought $ 125 worth of material, receiving 6 % discount for cash. She sold the material for 20 % more than the original price. What was the gain? Solution. — $ 126.00 original price $126.00 .06 7.50 $ 7.50 discount $ 117.50 price paid for material. 6 1^126 original cost $ 150.00 selling price $25 20 % gain 117.50 p rice paid $ 160 selling price $ 32.50 gain. Ans, EXAMPLES 1. A dressmaker bought 25 yd. of hamburg at 50 cents per yard, receiving 6 % discount for cash. She then sold the ham- burg to her custonxers at 60 cents per yard. What was the price paid for hamburg, and what per cent did she make ? 2. A dressmaker bought $ 325 worth of goods, receiving 6 % discount for cash. She sold the goods for 25 % more than the original price. What was the gain ? 3. A milliner bought $200 worth of ribbons, velvets, and flowers, receiving 5 % discount for cash. She then sold the materials for 30 % more than the original price. What was the gain ? REVIEW EXAMPLES 1. A dressmaker bought 30 yd. of silk at $ 1.25 per yard. She received a discount of 10 %. She sold the silk for $ 1.89 per yard. How much did she gain on the 30 yards ? 2. A merchant bought 50 yd. of lawn at 12^ cents a yard, and received a discount of 6 % for cash. How much did the lawn cost ? 3. A piece of crinoline containing 45 yd. was bought for $ 18. It was made into dress models of 5 yd. each. What was the cost of the crinoline in each model ? 208 VOCATIONAL MATHEMATICS FOR GIRLS 4. A dressmaker bought $ 175 worth of silk, receiving 6 % discount for cash. She sold the silk for 25% more than the original price. What was the gain per cent ? 5. A dressmaker bought 24| yd. of silk, at $ 1.10 per yard. From it she made three dresses, and had 13f yd. left. How much did the silk for one dress cost ? 6. Find the cost of 36 yd. of Valenciennes lace at 7^ cents a yard, 12 yd. of insertion at 6J cents a yard, and 12 yd. of beading at 7 cents a yard. What is the net cost, when 2 % discount is given ? 7. How many lingerie shirtwaists, each containing 2| yd., can be made from 49 yd, of batiste? What is the cost of material for one waist, if the whole piece cost $ 9.80, less 5 % discount ? 8. A dressmaker bought 2^ yd. of crepe at 29 cents a yard, for a shirtwaist, 3 yd. of beading at 12^ cents a yard, 6 crochet buttons at 35 cents a dozen. What did the material for the waist cost ? 9. A woman bought 9^ yd. of foulard silk, at $ 1.10 a yard, for a dress. If yd. of net at $ 1.50 a yard, and f yd. of plain silk at $ 1.25 a yard. What was the cost of material ? 10. A dressmaker bought 50 yd. of taffeta silk for $ 45.00. She sold 8^ yd. to one customer for $ 1.25 a yard, 15^ yd. at $ 1.00 a yard to another customer, and the remainder at cost. What did she gain on the entire piece ? What was the gain per cent ? 11. Two and one-half yards of cloth cost $ 2.75. What was the price per yard ? 12. A dressmaker bought 50 yd. of handmade lace abroad and paid $ 75 for it. She paid 60 % duty on the lace and sold it at a gain of 33^ % . What was the selling price per yard ? ARITHMETIC FOR DRESSMAKERS 209 13. A dressmaker bought 20 yd. of foulard silk at 90 cents a yard. She received 6 % discount. She sold it for lOJ % more than the original price. How much did she gain on the sale ? What per cent did she gain ? 14. A dressmaker bought the following materials for a customer : 4^ yd, of broadcloth at % 2.75 a yard, 6^ yd. of silk at % 3 J5 a yard, 2\ yd. of trimming at $ 2.50 a yard. She received a dressmaker's discount of 6 %, and 5 % discount for cash payment. What did she pay for the materials? She charged the retail price for them. How much did she gain ? What per cent ? 15. A dressmaker bought a 7^yd. remnant of broadcloth for $22.50. She sold 6 yd. to a customer at $3.50 a yard, but the remainder could not be sold. Did she gain or lose ? What per cent ? 16. A dressmaker bought in France three 15-yd. pieces of dress silk at 25^ cents a yard. After paying 60% duty on them, she sold two pieces to one customer at 48 % gain, and the third piece to another customer at 35 % gain. What was the gain on the three pieces ? . 17. A dressmaker furnished the materials for a lingerie dress and charged $25 for it. For the materials she paid the following : 10 yd. of dimity at 45 cents a yard, \2\ yd. Cluny insertion at 25 cents a yard, findings, $2. If she charged $12 for making, how much did she gain on the material ? Make a bill for the same and receipt it. 18. The materials for a dress cost a dressmaker $14.50. She sold them for 10 % more than cost and charged $ 15 for making. She paid her helper 20% of the amount received. What was the gain per cent ? 19. If it takes 6^ yards of cloth 52 inches wide to make a dress, how many yards of cloth 22 inches wide will be needed to make the same dress ? 210 VOCATIONAL MATHEMATICS FOR GIRLS 20. A dressmaker agreed to make a dress for a customer for $25. She paid 2 assistants $1.25 a day each for 3^ days of work. The dress was returned for alterations, and the assistants were paid for one more day's work. How much did the dressmaker receive for her own work ? 21. A dressmaker bought $1.50 worth of silk, receiving 6 % discount for cash. She sold the silk for 40 % more than the original price. What was the gain per cent ? 22. A dressmaker has an order for three summer dresses, for which 31J yd. of batiste are needed. She can buy three remnants of 10^ yd. each for 25 cents a yard, or she can buy a piece of 35 yd. for 25 cents a yard and receive 4 ^ discount for cash. Which is the better plan ? 23. (a) How many inches in | yd. ? (b) How many inches in \ yd. ? (c) How many inches in | yd. ? (d) How many inches in | yd. ? (e) How many inches in | yd. ? (/) How many inches in ^ yd. ? (g) How many inches in ^ yd. ? 24. Find the cost of each of the above lengths in lace at $ .12| a yard. 25. Find the cost of 4^ yd. of lace at $1.95 per piece (ouq piece = 12 yd.). 26. A dressmaker bought 2 pieces of white lining taffeta, one piece 42 yd. and another 48|^ yd., at $ .42^ a yard. What was the total cost ? 27. A piece of crinoline containing 42^ yd. that cost $ 1.70 a yard was made into dress models of 8^ yd. each. What was the cost of the crinoline in each model? 28. What is the cost of a child's petticoat containing : 2 J yd. longcloth at 15 cents a yard, 1} yd. hamburg at 19 cents a yard, li yd. insertion at 15 cents a yard ? ARITHMETIC FOR DRESSMAKERS 211 29. What is the cost of two petticoats requiring for one : 2} yd. longcloth at 19 cents a yard, 8 yd. hamburg at 25 cents a yard, 2^ yd. insertion at 19 cents a yard ? 30. What is the cost of a petticoat requiring : 8 yd. longcloth at 12^ cents a yard, 8J yd. hamburg at 17 cents a yard ? 31. What is the total cost of the following ? Wedding gloves, J| 2.75. Slippers and stockings, $5.00. Six undervests, at 19 cents each. Six pairs of stockings, at 38} cents a pair. Two pairs of shoes, at $5.00 a pair. One pair of rubbers, 75 cents. One pair long silk gloves, $2.00. One pair of long lisle gloves, $ 1.00. Two pair^ of short silk gloves, $ 1.00. Veils and handkerchiefs, $5.00. Two hats, $ 10.00. Corsets, $3.00. Wedding veil of 3 yards of tulle, 2 yards wide, at 89 cents a yard. 32. What is the cost of the following material for a top coat? Cotton corduroy, 32 inches wide : 4 J yards at 75 cents a yard. Lining, 36 inches wide : 4 J yards at $ 1.50 a yard. Buttons, sewing silk, pattern, 27 cents. Velvet for collar facing, J yard, at $1.50 a yard. 33. What is the cost of the following dressmaking supplies ? } yard of China silk, 27 inches wide, at 49 cents a yard (for the lining). 1} yard of mousseline de soie interlining 40 inches wide, at 80 cents a yard. I yard of all-over lace 86 inches wide, at $ 1.48 for front and lower back. i yard of organdie at $1.00, 32 or more inches wide, for collar and vest. Sewing silk, hooks and eyes, pattern, at 32 cents. 212 VOCATIONAL MATHEMATICS FOR GIRLS 34. What is the cost of the following ? Cotton gabardine, 36 inches wide : 5} yards at 39 cents a yard. Sewing cotton, braid, buttons, pattern, at 35 cents. 35. Which of the following fabrics is the most economical to buy? Crepe meteor, 44" wide, at 3 3.26 a yard. Faille Fran9aise, 42" wide, at $3.00. Charmeuse, 40'' wide, at $2.25. Louisine, 38" wide, at $2.00. Armure, 20" wide, at $1.50. Satin duchesse, 21" wide, at $1.25. MILLINER7 PROBLEMS 1. What would a hat cost with the following trimmings ? IJ yd. velvet, at $2.50 a yard. \ yd. satin for facing, at $ 1.98 a yard. 2 feathers, at $ 5.50 each. Frame and work, at $ 2.50. Make out a bill. (See lesson on Invoice, Chapter XI, page 243.) 2. A leghorn hat cost $6.98. Four bunches of fadeless roses at $2.98, 2 bunches of foliage at $.98, and 1^ yd. of velvet ribbon at $ 1.49 were used for trimming. The milliner charged 76 cents for her work. How much did the hat cost ?• 3. A milliner used the following trimmings on a child's bonnet: 1 piece straw braid, at $1.49. 2 yd. maline, at 25 cents a yard. 4 bunches flowers, at 69 cents each. 4 bunches foliage, at 49 cents each. Work, at $2.00. What was the total cost of the hat ? Make out a bill and receipt it. ARITHMETIC FOR MILLINERS 213 4. An old lady's bonnet was trimmed with the following : 3 yd. silk, at $ 1.50 a yard. 1 piece of jet, $3.00. 2 small aigrettes, at $ 1.50 each. Ties, 76 cents. Work, $1.60. How much did the finished bonnet cost ? 5. What was the total cost of a hat with the following trim- mings? 2 pieces straw braid, at $ 2.50 each. 2 yd. velvet ribbon, at 98 cents a yard. 5 flowers, at 59 cents. 4 foliage, at 49 cents. Frame and work, at $ 2.60. 6. A milliner charged $2.00 for renovating an old hat. She used 2 yd. satin at $ 1.60 a yard and charged $ 2.25 for an ornament. . How much did the hat cost ? 7. The following trimmings were used on a child's hat : 3 yd. velvet, at $ 1.60 a yard. 8 yd. lace, at 15 cents a yard. 2 bunches buds, at 49 cents a bunch. Work, $2.00. How much did the hat cost ? 8. A milliner charged $ 6.00 for renovating three feathers, $2.60 for a fancy band, $4.75 for a hat, and 75 cents for work. How much did the customer pay for her hat ? 9. A lady bought a hat with the following trimmings : 2 yd. satin, at $ 1.76 a yard. 2 bunches grapes, at $ 1.59 a bunch. 2 J yd. ribbon, at 69 cents a yard. Work, 75 cents. How much did the hat cost ? 214 VOCATIONAL MATHEMATICS FOR GIRLS 10. What would a hat cost with the following trimmings ? 2 pieces straw braid, at $ 1.98 each. 3 yd. ribbon, at 89 cents a yard. Fancy feather, $6,98. Frame and work, $ 2.50. 11. Estimate the cost of a hat using the following materials : 2^ yd. plush, at $ 2.25 a yard. 2 yd. ribbon, at 26 cents a yard, f yd. buckram, at 25 cents a yard. I yd. tarlatan, at 10 cents a yard. 1 band fur, 75 cents. Foliage, 10 cents. Labor, $2.00. 12. If the true bias from selvedge to selvedge is about ^ longer than the width of the goods, how many bias strips must be cut from velvet 18" wide in order to have a three-yard bias strip ? 13. The edge of a hat measures 46 inches in circumference; the velvet is 16 inches wide. How many bias strips of velvet would it take to fit the brim? Wire Hat Frame ^^ ^^^^ amount of velvet would be needed to cover brim if each strip cut measured f of a yard along the selvedge ? 15. Give the number of 13^-in. strips that can be cut from 3^ yards of material ; also the number of inches of waste. 16. How many 22^in. scrips can be cut from 2^ yd. of material ? 17. What length bias strip can be made from 1^ yd. of silk, each strip 1 yd. 10 in. long and 1^ in. wide? 18. How many six-petal roses can be made from 1 ysird of velvet 18 inches wide, each petal cut 3 inches square ? ARITHMETIC FOR MILLINERS 215 19. Estimate the total cost of roses, if velvet is $ 1.50 a yard, centers 18 cents a dozen, sprays 12 cents a dozen, stem- ming 6 cents a yard, using ^ of a yard for each flower. 20. Find the cost of one flower; the cost of ^ of a dozen flowers, using the figures given above. 21. What amount of velvet will be needed to fit a plain-top facing and crown of hat, width of brim 6 inches, diameter of headsize 7|- inches, diameter of crown 16^ inches, allowing 8^ inches on brim for turning over edges ? 22. If the circumference of the brim measures 56 inches, what amount of silk will it take for a shirred facing made of silk 22 inohes wide, allowing twice around the hat for fullness, and also allowing 1 inch on depth of silk for casings ? 23. At the wholesale rate of eight frames for one dollar, what is the cost of five dozen frames ? of twelve dozen ? 24. A milliner had 2^ dozen buckram frames at $3.60 a dozen. She sold ^ of them at 75 cents each, but the others were not sold. Did she gain or lose and what per cent ? 25. Flowers that were bought at $ 5.50 a dozen bunches were sold at 75 cents a bunch. What was the gain on 1^ dozen bunches ? 26. A milliner bought ten rolls of ribbon, ten yards to the roll, for $ 8.50. Ten per cent of the ribbon was not salable. The remainder was sold at 19 cents a yard. How much was the gain ? what per cent ? 27. A piece of velvet containing twelve yards was bought for $28.20 and sold for $2.75 per yard. How much was gained on the piece? 28. A thirty-six yard piece of maline cost $ 7.02 and was sold at 29 cents a yard. One yard was lost in cutting. How much was gained on the piece ? 216 VOCATIONAL MATHEMATICS FOR GIRLS 29. Find cost of a velvet hat requiring IJ yd. of velvet, at $ 1.50 a yard. } yd. of fur band, at $4.00 a yard. 1 feather ornament, at $3.00. Hat frame, 50 cents. Edge wire, 10 cents. Taffeta lining, 25 cents. Making, $ 2.50. 30. A milliner charged $ 8.37 for a hat. She paid 37 cents for the frame, $ 2.80 for the trimming, and $ 1.60 for labor. What was the per cent profit ? 31. A child's hat of organdie has two ruffles edged with Valenciennes lace. The lower ruffle is 3" wide ; the upper ruffle, 2J". 2f yd. lace edging cost 12^ cents a yard, 2 yd. of 3" ribbon cost 25 cents a yard, 1^ yd. of organdie cost 25 cents a yard, the hat frame cost 36 cents, and the lining cost 10 cents. Find the total cost. 32. How much velvet at $2.00 a yard would you buy to put a snap binding on a hat that measures 43" around the edge ? Should the velvet be bias or straight ? CHAPTER X CLOTHING Since about one oighth of the income in the average working- man's family is spent for clothing, this is a very important subject. The housewife purchases the linen for the house and her own wearing apparel. It is not uncommon for her to have considerable to say about the clothing of the men, particularly about the underclothing. Therefore she should know some- thing about what constitutes a good piece of cloth, and be able to make an intelligent selection of the best and most economical fabric for a particular purpose. The cheapest is not always the best, although it is in some cases. All kinds of cloth are made by the interlacing (weaving) of the sets of thread (called yarn). The thread running length- wise is the strongest and is called the warp. The other thread is called thQ filling. Such fabrics as knitted materials and lace are made by the interlacing of a single thread. Threads (yarn) are made by lengthening and twisting (called spinning) short fibers. Since the fibers vary in such qualities as firmness, length, curl, and softness, the resulting cloth varies in the same way. This is the reason why we have high-grade, medium- grade, and low-grade fabrics. The principal fabrics are wool, silk, mohair, cotton, and flax (linen). The consumer is often tempted to buy the cheaper fabrics and wonders why there is such a difference in price. This difference is due in part to the cost of raw material and in part to the care in manufacturing. For example, raw silk costs from $1.35 to $5.00 a pound, according to the nature and 217 218 VOCATIONAL MATHEMATICS FOR GIRLS quality of the silk. The cost of preparing the raw silk aver- ages about 65 cents a pound, according to the nature of the twist, which is regulated by the kind of cloth into which it is to enter. The cost of dyeing varies from 55 cents to $ 1.50 a pound. Weavers are paid from 2 to 60 cents a yard for weav- ing, the price varying according to the desirability of the cloth. When we compare the relative values of similar goods produced by different manufacturers, there are a few general principles by which good construction can easily be determined. The density of a fabric is determined by the number of warp yarn and filling yarn to the inch. This is usually determined by means of a magnifying glass with a \" opening. To illus- trate : If there are 36 threads in the filling and 42 threads in the warp to \", what is the density of the cloth to the inch ? Solution. — 36 X 4 = 144 threads in the filling. 42 X 4 = 168 threads in the warp. EXAMPLES 1. A 25-cent summer undervest (knitted fabric) will outwear two of the flimsy 15-cent variety in addition to retaining better shape. What is the gain, in wear, over the 15-cent variety ? 2. A 50-cent undervest will outwear three of the 25-cent variety. What is gained by purchasing the 50-cent style ? 3. A cotton dress for young girls, costing 75 cents ready made, will last one season. A similar dress of better material costs 94 cents, but will last two seasons. Why is the latter the better dress to buy ? What is gained ? 4. A linen tablecloth (not full bleached) costing $1.04 a yard, will last twice as long as a bleached linen at % 1.25 a yard. Which is the better investment ? 5. A sheer stocking at 50 cents will wear just half as long as a thicker stocking at 35 cents. What is gained in wear ? What kind of stockings should be selected for wear ? CLOTHING 219 SHOES Our grandfathers and grandmothers wore handmade shoes, and wore them until they had passed their period of usef uhiess. At that time the consumption of leather did not equal its pro- duction. But, since the appearance of machine-made shoes, different styles are placed on the market at different seasons to correspond to the change in the style of clothing, and are often discarded before they are worn out. Thus far we have not been able to utilize cast-off leather as the shoddy mill uses cast-off wool and silk. The result is that the demand for leather is above the production ; therefore, as in the case of textiles, substitutes must be used. In shoe materials there is at present an astonishing diversity and variety of leather and its substitutes. Every known leather from kid to cowhide is used, and such textile fabrics as satins, velvets, and serges have rapidly grown in favor, especially in the making of women's and children's shoes. Of course, we must bear in mind that for wearing qualities there is nothing equal to leather. In buying a pair of shoes we should try to combine both wearing qualities and simple style as far as possible. EXAMPLE 1. A pair of shoes at $ 1.75 was purchased for a boy. The shoes required 80 cents worth of mending in two months. If a $3.00 pair were purchased, they would last three times as long with 95 cents worth of mending. How much is gained by purchasing a $ 3.00 pair of shoes ? YARNS Worsted Tarns. — All kinds of yarns used in the manufacture of cloth are divided into sizes which are based on the relation between weight and length. To illustrate : Worsted yams are made from combed wools, and the size, technically called the 220 VOCATIONAL MATHEMATICS FOR GIRLS counts, is Itased upon the number of lengths (called hanks') of d60 yards required to weigh one pound. ROVnjQ OB YABM SCALB8 These gcalea will weigh one pound by tenths ot graios ot ooe Beyenty-thon- Bandch part of one ponnd aToirdapol8, rendering them well adapted for ubb in coDnectioD with yarn reels, for the onmberlng of yarn from the weight of hank, giving the weight in lentbe of grains to compare with tables. ThnB, il one hank weighs one pound, the yam will be number ona counts, while if 20 hanks are required for one pound, the yam is the 20's, etc. The greater the number of hanks necessary to weigh one pound, the higher the counts and the finer the yam. The hank, or 500 yards, is the unit of measuretnent for all worsted yams. Lenqth for Wobstbd Tabhs .0. k"l^. No. PK^'l^. No. PIB™. No. fti™. 1 660 6 2800 B 6040 13 7280 2 1120 6 3360 10 6600 14 7840 3 1680 7 3820 11 6100 15 8400 4 2240 8 4480 12 6720 16 8060 Woolen Yams, — In worsted yarna the fibers lie parallel to ea«h other, while in woolen yams the fibers are entangled. CLOTHING 221 This difference is due entirely to the different methods used in working up the raw stock. In woolen yarna there is a great diversity of systems of grading, Taiy- ing according to the districts in which the grading is done. Among the many eysteniB are the Engiish sktin, which differs in various parts of Eng- land ; the Scotch sjiyn^le; the American rvii; the Philadelphia cut; EUid others. In these lessons the run system will be used unless otherwise stated. This Is the system used in New England. The run is based upon 100 yards per ounce, or 1600 yards to the pound. Thus, if 100 yards of woolen yarn weigh one ounce, or if 1600 yards weigh one pound, it is technically termed a No. 1 run ; and if 1300 yards weigh one ounce, or 4800 yards weigh one pound, the size will be No. 3 run. The finer the yam, or the greater the number of yards necessary 1« weigh one pound, the higher tbe run. Yarn Rbbi. Foi reeling and measuring lengths of cotton, woolen, and worsted yams. Lbnqth fob Woolbh Yabns (Run Ststbh) No. Yabi.« No. TlBIW PRH Ln. ... Yard. rH Ld. No. .K. i 200 1 1600 2 3200 8 4800 i 400 H 2000 ^ 3«00 31 6200 i 800 1* 2400 n 4000 H 5600 i 1200 n 2800 2J 4400 222 VOCATIONAL MATHEMATICS FOR GIRLS Raw Silk Tarns. — For raw silk yarns the table of weights is; 16 drams = 1 ounce 16 ounces = 1 pound 266 drams = 1 pound The unit of measure for raw silk is 266,000 yards per pound. Thus, if 1000 yards — one skein — of raw silk weigh one dram, or if 266,000 yards weigh one pound, it is known as 1-dram silk, and if 1000 yards weigh two drams, the yarn is called 2-dram silk ; hence the following table is made: 1-dram silk = 1000 yards per dram, or 256,000 yards per lb. 2-dram silk = 1000 yards per 2 drams, or 128,000 yards per lb. 4-dram silk = 1000 yards per 4 drams, or 64,000 yards per lb. Dbahb riB 1000 Takds YABD8 PER Pound Yards per Ounce 1 266,000 16,000 U 204,800 12,800 li ? ? ^ 146,286 9143 2 128,000 8000 2J 113,777 7111 2i 102,400 6400 2J 93,091 6818 3 ? ? 3J 78,769 4923 3i 73,143 4671 Linen Tarns. — The sizes of linen yarns are based on the lea or cuts per pound and the pounds per spindle. A cut is 300 yards and a spindle 14,000 yards. A continuous thread of several cuts is a hank, as a 10-cut hank, which is 10 X 300 = 3000 yards per hank. The number of cuts per pound, or the leas, is the number of the yarn, as 30's, indicating 30 x 300 = 9000 yards per pound. Eight-pound yarn means that a spindle weighs 8 pounds or that the yarn is 6-lea (14,400 -r- 8) -f- 300 = 6. CLOTHING 223 Cotton Tarns. — The sizes of cotton yarns are based upon the system of 840 yards to 1 hank. That is, 840 yards of cotton yarn weighing 1 pound is called No. 1 counts. Spun Silk. — Spun silk yams are graded on the same basis as that used for cotton (840 yards per pound), and the same rules and calculations that apply to cotton apply also to spun silk yams. Two or More Ply Yams. — Yams are frequently produced in two or more ply ; that is, two or more individual threads are twisted together, making a double twist yam. In this case the size is given as follows : 2/30*8 means 2 threads of SO^s counts twisted together, and 8/30*s would mean 3 threads, each a 80's counts, twisted together. (The figure before the line denotes the number of threads twisted to- gether, and the figure following the line the size of each thread.) Thus when two threads are twisted together, the resultant yam is heavier, and a smaller number of yards are required to weigh one pound. For example : 30's 'worsted yarn equals 16,800 yd. per lb., but a two- ply thread of SO's, expressed 2/30' s, requires only 8400 yards to the pound, or is equal to a 15*s ; and a three-ply thread of 30*s would be equal to a lO's. When a yam is a two-ply, or more than a two-ply, and made up of several threads of equal counts, divide the number of the single yarn in the required counts by the number of the ply, and the result will be the equivalent in a single thread. To Find the Weight in Orains of a Given Number of Yards of Worsted Yarn of a Known Count Example. — Find the weight in grains of 125 yards of 20's worsted yams. No. I's worsted yam = 560 yards to a lb. No. 20'8 worsted yam = 11,200 yards to a lb. 1 Jb. worsted yam = 7000 grains. 224 VOCATIONAL MATHEMATICS FOR GIRLS 125 11,200 If 11,200 yards of 20's worsted yarn weigh 7000 grains, then of 7000 = -i?5_ X 7000 = — = 78.126 grains. 11,200 8 * Note. — Another method : Multiply the given number of yards by 7000, and divide the result by the number of yards per pound of the given count. 126 X 7000 = 876,000. 1 pound 20's= 11,200. 875,000 -f- 11,200 = 78.126 grains. Ans. To Find the Weight in Grains of a Given Number of Yards of Cotton Yarn of a Known Count Example. — Find the weight in grains of 80 yards of 20's cotton yarn. No. I's cotton = 840 yards to a lb. No. 20's cotton = 16,800 yards to a lb. 1 lb. = 7000 grains. 7000 1 yd. 20's cotton = -^— — grains. 16,800 80 yd. 20's cotton = ^525. x 80 = 152 = 33.38 grains. Ans. ^ 16,800 21 ^ It is customary to solve examples that occur in daily practice by rule. The rule for the preceding example is as follows : Multiply the given number of yards by 7000 and divide the result by the number of yards per pound of the given count. 80 X 7000 = 560,000. 560,000 -5- (20 X 840) = 33.33 grains. Aiis. Note. — 7000 is always a multiplier and 840 a divisor. To find the weight in ounces of a given number of yards of cotton yarn of a known count, multiply the given number of yards by 16, and divide the result by the yards per pound of the known count. To find the weight in pounds of a given number of yards of cotton yarn of a known count, divide the given number of yards by the yards per pound of the known count. CLOTHING 225 To find the weight in ounces of a given number of yards of woolen yam (run system), divide the given number of yards by the number of runs, and multiply the quotient by 100. Note. — Calculations on the run basis are much simplified, owing to the fact that the standard number (1600) is exactly 100 times the number of ounces contained in 1 pound.- Example. — Find the weight in ounces of 6400 yards of 5-run woolen yarn. 6400-^ (6 X 100)= 12.8 oz. Ana, To find the weight in pounds of a given number of yards of woolen yam (run system) the above calculation may be used, and the result divided by 16 will give the weight in pounds. To find the weight in grains of a given number of yards of woolen yarn (run system), multiply the given number of yards by 7000 (the number of grains in > pound) and divide the result by the number of yards per pound in the given run, and the quotient will be the weight in grains. EXAMPLES 1. How many ounces are tiiere (a) in 6324 grains? (6) in 34^ pounds ? 2. How many pounds are there in 9332 grains ? 3. How many grains are there (a) in 168^ pounds ? (6) in 2112 ounces ? 4. Give the lengths per pound of the following worsted yarns : (a) 41's ; (6) 55's ; (c) 105's ; {d) 115's ; (e) 93's. 5. Give the lengths per pound of the following woolen yarns (run system): (a) 9^'s ; (h) 6's; (c) 19's ; {d) 17's ; (e) H's. 6. Give the lengths per pound of the following raw silk yams : (a) l^'s j (6) 3's ; (c) 3f 's ; (d) 20's ; (e) 28's. 7. Give the lengths per ounce of the following raw silk yams : (a) 4|'s ; (6) 6i 's ; (c) 8's ; (d) 9's ; {e) 14's. 226 VOCATIONAL MATHEMATICS FOR GIRLS a What are the lengths of linen yarns per pound : (a) 8's ; (6) 25's ; (c) 32's ; (d) 28's ; (e) 45'8 ? 9. What are the lengths per pound of the following cotton yams : (a) lO's ; (b) 32's ; (c) 54's ; (d) 80's ; (e) 160's ? 10. What are the lengths per pound of the following spun silk yarns : (a) 30's ; (b) 46's ; (c) 38's ; (d) 29's ; (e) 42's ? 11. Make a table of lengths per ounce of spun silk yams from I's to 20's. 12. Find the weight in grains of 144 inches of 2/20*3 worsted yam. 13. Find the weight in grains of 25 yards of 3/30's worsted yarn. 14. Find the weight in ounces of 24,000 yards of 2/40^8 cotton yam. 15. Find the weight m pounds of 2,840,000 yards of 2/60's cotton yam. 16. Find the weight in ounces of 650 yards of l^run woolen yarn. 17. Find the weight in grains of 80 yards of ^run woolen yam. . 18. Find the weight in pounds of 64,000 yards of 5-run woolen yarn. Solve the following examples, first by analysis and then by rule; 19. Find the weight in grains of 165 yards of 35's worsted. 20. Find the weight in grains of 212 yards of 40's worsted. 21. Find the weight in grains of 118 yards of 25's cotton. 22. Find the weight in grains of 920 yards of 18's cotton. 23. Find the weight in pounds of 616 yards of 16^'s woolen. 24. Find the weight in grains of 318 yards of 184's cotton. 25. Find the weight in grains of 25 yards of 30's linen. CLOTHING 227 26. Find the weight in pounds of 601 yards of 60's spun silk. 27. Find the weight in grains of 119 yards of 118's cotton. 28. Find the weight in grains of 38 yards of 64's cotton. 29. Find the weight in grains of 69 yards of 39's worsted. 30. Find the weight in grains of 74 yards of 40's worsted. 31. Find the weight in grains of 113 yards of 1^'s woolen. 32. Find the weight in grains of 147 yards of If 's woolen. 33. Find the weight in grains of 293 yards of 8's woolen. 34. Find the weight in grains of 184 yards of 16J's worsted. 35. Find th^ weight in grains of 91 yards of 44's worsted. 36. Find the weight in grains of 194 yards of 68's cotton. 37. Find the weight in pounds of 394 yards of 180's cotton. 38. Find the weight in pounds of 612 yards of 60's cotton. 39. Find the weight in grains of 118 yards of 44's linen. 40. Find the weight in pounds of 315 yg^rds of 32's linen. 41. Find the weight in grains of 84 yards of 25's worsted. 42. Find the weight in grains of 112 yards of 20's woolen. 43. Find the weight in grains of 197 yards of 16's woolen. 44. Find the weight in grains of 183 yards of 18's cotton. 45. Find the weight in grains of 134 yards of 28's worsted. 46. Find the weight in grains of 225 yards of 34's linen. 47. Find the weight in pounds of 369 yards of 16's spun silk. 48. Find the weight in pounds of 484 yards of 18's spun silk. To Find the Size or the Counts of Cotton Tarn of Known Weight and Length Example. — Find the size or counts ot 84 yards of cotton yam weighing 40 grains. 228 VOCATIONAL MATHEMATICS FOR GIRLS Since the counts are the number of hanks to the pound, 7000 X 84 = 14,700 yd. in 1 lb. 40 14,700 -4- 840 = 17.6 counts. Ans. Rule. — Divide 840 by the given number of yards ; divide 7000 by the quotient obtained ; then divide this result by the weight in grains of the given number of yards, and the quotient will be the counts. 840 ^ 84 = 10. 7000 -^ 10 = 700. 700 -^ 40 = 17.5 counts. Ans. To Find the Run of a Woolen Thread of Known Length and Weight Example. — If 50 yards of woolen yarn weigh 77.77 grains, what is the run ? 1600 -f- 60 = 32. 7000 - 32 = 218.76. 218.76 -^ 77.77 = 2.812-run yarn. Ans, Rule. — Divide 1600 (the number of yards per pound of 1- run woolen yarn) by the given number of yards ; then divide 7000 (the grains per pound) by the quotient; divide this quotient by the given weight in grains and the result will be the run. To Find the Weight in Ounces for a Given Number of Yards of Worsted Yarn of a Known Count Example. — What is the weight in ounces of 12,650 yards of 30's worsted yarn ? 12,660 X 16 = 202,400. 202,400 H- 16,800 = 12.047 oz. Ans, Rule. — Multiply the given number of yards by 16, and divide the result by the yards per pound of the given count, and the quotient will be the weight in ounces. CLOTHING 229 To Find the Weight in Pounds for a Given Number of Yards of Worsted Yam of a Known Count Example. — Find the weight in pounds of 1,500,800 yards of 40's worsted yarn. 1,600,800 -T- 22,400 = 67 lb. Ans, Rule. — Divide the given number of yards by the number of yards per pound of the known count, and the quotient will be the desired weight. EXAMPLES 1. If 108 inches of cotton yam weigh 1.5 grains, find the counts. 2. Find the size of a woolen thread 72 inches long which weighs 2.5 grains. 3. Find the weight in ounces of 12,650 yards of 2/30's worsted yam. 4. Find the weight in ounces of 12,650 yards of 40's worsted yam. 5. Find the weight in pounds of 1,500,800 yards of 40's worsted yarn. 6. Find the weight in pounds of 789,600 yards of 2/30's worsted yam. 7. What is the weight in pounds of 851,200 yards of 3/60's worsted yarn ? 8. If 33,600 yards of cotton yam weigh 5 pounds, find the counts of cotton. Buying Yarn, Cotton, Wool, and Rags Every fabric is made of yam of definite quality and quan- tity. Therefore, it is necessary for every mill man to buy yam or fiber of different kinds and grades. Many small mills buy cotton, wool, yarn, and rags from brokers who deal in these commodities. The prices rise and fall from day to day 230 VOCATIONAL MATHEMATICS FOR GIRLS according to the law of demand and supply. Price lists giving the quotations are sent out weekly and sometimes daily by agents as the prices of materials rise or fall. The following are quotations of different grades of cotton, wool, and shoddy, quoted from a market list : QUANTITT PBIOS FEB Lb. 8103 lb. white yam shoddy (best all wool) f 0.485 8164 lb. white knit stock (best all wool) 366 2896 lb. pure indigo blue 316 1110 lb. fine dark merino wool shoddy 226 410 lb. fine light merino woolen rags 115 718 lb. cloakings (cotton warp) 02 872 lb. wool bat rags 035 96 lb. 2/20's worsted (Bradford) yam 725 408 lb. 2/40's Australian yam 1.35 598 lb. 1/50's delaine yam 1.20 987 lb. 16-cut merino yarn (50 % wool and 50 % shoddy) . . .285 697 lb. carpet yarn, 60 yd. double reel, wool filling 235 Find the total cost of the above quantities and grades of textiles. EXAMPLES 1. The weight of the fleece on the average sheep is 8 lb. Wyoming has at least 4,600,000 sheep ; what is the weight of wool raised in a year in this state ? 2. A colored man picks 155 lb. of cotton a day ; how much cotton will he pick in a week (6 days) ? . 3. The average yield is 558 lb. per acre ; how much cotton will be raised on a farm of 165 acres ? 4. The standard size of a cotton bale in the United States is 54 X 27 X 27 inches ; what is the cubical contents of a bale ? 5. In purchasing cotton an allowance of 4 % is made for tare. How much cotton would be paid for in 96 bales, 500 lb. to each bale ? CLOTHING 231 6. Broadcloth was first woven in 1641. How many years has it been in use ? 7. The length of "Upland" cotton varies from three- fourths to one and one-sixteenth inches. What is the differ- ence in length from smallest to largest ? a If a sample of 110 lb. of cotton entered a* mill and GS* lb. were made into fine yarn, what is the per cent of waste ? 9. If a yard of buckram weighs 1.8 ounces, how many yards to the pound ? 10. If a calico printing machine turns out 95 fifty-yard pieces a day, how many are printed per hour in a ten-hour day ? 11. If a sample of linen weighing one pound and a half absorbs 12 % moisture, what is the weight after absorption ? 12. A piece of silk weighing 3 lb. 4 oz. is " weighted " 175% ; what is the total weight ? 13. If the textile industry in a certain year pays out $ 500,000,000 to 994,875 people, what is the wage per capita ? 14. How much dyestuff, etc., will be required to dye 5 lb. of cotton by the following receipt ? 6 ^ brown color, afterwards treated with 1.6 ^ sulphate of copper, 1,6^0 bichromate of potash, 8 ^ acetic acid. 15. How many square yards of cloth weighing 8 oz. per sq. yd. may be woven from 1050 lb. of yarn, the loss in waste be- ing 5 per cent ? 16. A piece of union cloth has a warp of 12's cotton and is wefted with 30's linen yam, there being the same number of threads per inch in both warp and weft ; what percentage of cotton and what of linen is there in the cloth ? 17. A sample of calico 3 in. by 4 in. weighs 30 grains. What is the weight in pounds of a 70-yard piece, 36 in. wide ? \ 232 VOCATIONAL MATHEMATICS FOR GIRLS 18. 4 yd. of a certain cloth contains 2 lb. of worsted at 67 cents a pound and 1\ lb. of cotton at 18 cents a pound. Each is what per cent of the total cost of material ? 19. A bale of worsted weighing 75 lb. loses 8 oz. in reeling off ; what is the per cent of loss ? 20. If Ex. 19 worsted gains 0.45 lb. to the 75 lb. bale in dye- ing, what is the per cent of gain ? 21. This 75 lb. cost $ 50.25 and it lost 4 oz. in the fulling mill, what was the value of the part lost ? 22. The total loss is what per cent of the original weight ? What is its value at 67 cents a pound ? PARt IV — THE OFFICE AND THE STORE CHAPTER XI ARITHICETIC FOR OFFICE ASSISTANTS EvEBY office assistant should be quick at figures — that is, should be able to add, subtract, multiply, and divide accurately and quickly. In order to do this one should practice, addition, subtraction, multiplication, and division until all combinations are thoroughly mastered. An office assistant should make figures neatly so that there need be no hesitation or uncertainty in reading them. Rapid Calculations Add the following columns and check the results. Compare the time required for the different examples. 1. 27 2. 37 3. 471 4. 668 5. 1,039 12 20 296 284 679 8 11 194 187 381 18 20 327 341 668 12 16 287 272 669 8 12 191 184 376 8 16 237 • 193 430 8. 9 194 166 360 7 12 169 166 336 11 16 247 232 479 12 13 194 180 374 2 3 27 26 62 12 17 263 240 493 11 14 241 212 463 12 20 366 367 722 12 14 244 222 466 8 11 93 79 172 10 16 208 233 213 421 234 VOCATIONAL MATHEMATICS FOR GIRLS 7 7. 7 8. 169 9. 162 10. 311 2 6 60 78 138 4 7 HI 88 199 6 10 173 121 294 4 6 112 84 196 4 4 88 76 164 4 6 104 83 187 4 6 96 104 200 4 7 120 97 217 8 9 144 123 267 4 5 60 101 161 4 5 73 92 166 8 10 186 176 362 4 4 64 76 139 4 6 114 113 227 4 4 89 88 177 6 7 91 80 171 8 204 170 374 4 13 176 166 341 4 4 73 77 160 4 7 119 127 246 4 6 84 103 187 8 11 177 166 342 6 8 166 136 292 8 4 94 61 166 12 18 310 293 603 8 12 191 189 380 8 13 268 198 466 2 2 17 17 34 4 8 122 137 269 8 193 186 378 1 . 4 6 10 1 9 16 24 2 16 16 32 1 11 16 26 2 34 44 78 2 27 34 61 2 26 63 79 2 36 41 76 2 17 10 27 2 38 22 60 ARITHMETIC FOR OFFICE ASSISTANTS 235 11. $162.24 12. $37,000.00 13. $31.26 14. $8,627.08 15. $630.33 266.46 300,000.00 73.70 2,907.31 408.32 277.66 410,000.00 2.00 3,262.68 399.99 12,171.44 82,000.00 4 26 8,096.90 28.00 17.72 .89 9,369.21 644.15 6.00 51,000.00 31.16 2,177.30 18,000.00 83.16 40,000.00 3.20 8,386.50 32.86 23.66 16.75 7,229.20 164.66 3.18 34,500.00 4.61 8,462.38 82.35 3,066.34 1,768.13 617.60 17,000.00 2,666.76 6,236.32 25.00 1128.13 6,147.42 639.24 36.00 16,600.00 3.20 4,443.88 2.60 30.00 3,386.72 79.90 4.00 6,600.00 3,927.78 1,143.00 289.22 1,000.00 29.12 4,797.46 266.60 2,612.00 727.00 17.82 70,500.00 1.00 2,476.31 141.33 199.87 33.27 3,706.00 2314.76 10,000.00 19.09 6,417.42 3,091.72 2.40 12,500.00 720.00 1,574.60 1,049.96 9.26 1,500.00 28.80 3,121.97 166.64 66.80 300.00 96.00 120.00 494.03 3.41 26,146.93 1,483.84 18.00 800.00 6.00 51.397.19 657.62 1.66 50.00 7.37 99.56 1,416.68 3.16 100.00 3.60 3,606.93 135.50 2.65 200.00 22,830.14 208.33 4,010.92 250.00 9.08 85,706.13 42.84 126.46 300.00 36,361.19 362.26 2.26 4.60 39,056.23 234.47 162.70 2,000.00 30,000.00 31.60 10.26 36.84 179,346.77 49.76 3.62 1,000.00 3,375.31 160.22 4.00 2.00 12,638.86 2.64 111.10 1,200.00 3.50 30,992.76 2.40 324.83 11.06 179,346.77 22.60 302.10 114,360.00 .74 3,376.31 8.92 346.04 40,000.00 7.26 12,638.86 176.91 301.10 120,000.00 6.00 30,992.76 11.30 1.20 9,476.00 3.00 16,503.48 17.00 236 VOCATIONAL MATHEMATICS FOR GIRLS 16. $437.58 17. $81.33 18. $144.40 19. $61.45 2.75 31.66 16.00 14.50 1.40 9.91 1,124.04 L80 70.06 20.00 110.69 2.00 3.54 23.25 44.83 24.17 396.89 129.99 318.40 272.90 33.00 9.01 22.35 5.13 18.24 208.01 757.00 482.09 6.75 160.98 674.37 .50 68.70 14.60 220.50 1.53 10.60 27.30 36.60 9.20 280.00 6.50 .90 3.60 83.78 .32 98.95 2.60 36.90 216.60 117.13 31.00 246.00 40.00 192.71 91.87 481.30 542.25 58.43 18.97 67.96 2.11 26.49 59.35 53.07 2.92 3.76 2.54 43.34 8.14 1,863.74 36.08 5.80 165.70 21.25 108.81 22.38 1,076.82 1.75 6.47 8,699.46 449.85 10.10 132.28 4,437.97 3.25 4.00 391.00 394.48 881.69 3.00 72.00 82.80 24.00 35.00 85.12 .75 10.00 310.49 47.90 3.00 10.40 1,078.50 31.68 26.50 .85 49.50 19.04 37.70 77.91 39.76 2.24 64.43 17.21 19.50 2.20 168.26 2,676.35 186.99 1.50 25.25 2.40 6.00 .70 53.49 8.62 36.53 2.50. 7.50 3.85 3.60 1.70 5.05 23.65 3.00 2.00 7.60 259.00 168.66 .70 2.00 701.47 67.60 92.00 11^ 3,148.00 ARITHMETIC FOR OFFICE ASSISTANTS 237 Horizontal Addition Reports, invoices, sales sheets, etc., are often written in such a way as to make it necessary to add figures horizontally. In adding figures horizontally, it is customary to add from left to right and check the answer by adding from right to left. EXAMPLES Add the following horizontally : 1. 38 + 76 + 49 = 2. 11 + 43 + 29 = • 3. 27 + 57 + 15 = 4. 34 + 16 + 23 = 5. 47 + 89 + 37 = 6. 53 + 74 + 42 = 7. 94 + 17 + 67 = a 79 + 37 + 69 = 9. 83 + 49 + 74 = 10*. 19 + 38 + 49 = . Add the following and check by adding the horizontal and vertical totals : 11. 36 + 74 + 19 + 47 = 29 + 63 + 49 + 36 = + + + = 12. 74 + 34 + 87 + 27 = 37 + 19 + 73 + 34 = + + + = 13. 178+ 74 + 109+ 83 = 39 + 111 + 381 + 127 = + + + = 14. 217 + 589 + 784 = 309 + 611 + 983 = + + = 238 VOCATIONAL MATHEMATICS FOR GIRLS 15. ^ 1118 + 3719 + 8910 3001 + 5316 + 6715 + + Add the following and check by adding horizontal and verti- cal totals. Compare the time required for the different examples. 16. 17. 18. $702,000 $14,040 $370,000 $6,475.00 $1,072,000 $20,516.00 626,000 10,600 20,000 350.00 666,000 11,300.00 1,267,500 25,360 447,260 7,826.88 1,724,750 33,401.88 333,000 6,660 340,000 5,960.00 833,000 16,022.60 380,000 7,600 351,000 6,142.50 790,000 16,070.00 1,077,000 21,640 60,000 875.00 1,127,000 22,415.00 702,000 14,040 370,000 6,475.00 1,072,000 20,516.00 625,000 10,600 20,000 350.00 665,000 11,300.00 1,264,600 25,290 447,260 7,826.87 1,721,760 33,341.87 333,000 6,660 200,000 3,600.00 693,009 13,672.60 356,000 7,100 348,000 6,090.00 768,000 14,427.60 1,072,000 21,440 50,000 876.00 1,122,000 22,315.00 318,143 28,760 9.04 491.86 189.64 77,751,393 295,187 18,363 6.22 498.23 188.74 78,426,000 300,789 23,398 7.96 479.80 187.88 76,180,746 279,736 22,290 7.97 611.43 187.24 79,864,039 302,737 28,699 9.48 523.56 187.80 82,001,180 302,338 22,149 7.33 678.00 188.83 91,025,879 341,086 27,766 8.14 664.30 192.87 89,161,101 336,776 24,080 7.17 534.23 192.13 85,603,137 311,730 20,366 6.63 521.79 192.17 83,627,195 336,360 21,299 6.36 524.17 192.76 84,266,576 281,481 18,032 6.41 600.09 194.89 81,283,747 306,370 20,866 6.83 496.12 196.06 81,122,670 380,782,161 461,880,223 620,781,017 889,692,401 1,743,186,792 452,491,808 480,722,907 637. ,837,674 481,628,491 1,962,580,780 71,709,667 28,842,684 17, 066,567 91,836,090 209,444,988 1,686 600 317 1,907 1,102 283,448,988 282,640,795 326,283,015 291,835,161 1,184,157,949 6,264 5,879 6,066 6,061 6,068 97,333,163 169,239,428 194,548,002 97,867,260 668,977,843 ARITHMETIC FOR OFFICE ASSISTANTS 239 19. 3,200,000 17,000,000 28,000,000 7,000,000 66,700,000 27,200,000 26,000,000 31,400,000 23,000,000 106,600,000 6,100,000 6,100,000 860,000 66,100,000 64,200,000 12,300,000 142,460,000 3,600,000 12,000,000 16,600,000 626,000 6,200,000 2,900,000 8,726,000 1,416,363 7,263,712 2,000,000 11,866,463 22,646,628 666,907 642,639 443,392 416,631 1,967,369 3,600,000 11,200,000 13,200,000 7,400,000 36,300,000 12,600,000 2,600,000 3,600,000 2,600,000 21,100,000 20. 29,000,000 22,600,000 14,200,000 16,600,000 82,300,000 13,600,000 10,200,000 9,600,000 8,600,000 41,900,000 327,998 330,916 608,266 368,262 1,626,441 1,122,906 1,222,262 1,296,344 1,317,004 4,968,616 2,400,000 1,100,000 1,660,000 1,800,000 6,960,000 1,600,000 860,000 900,000 900,000 4,160,000 260,000 305,000 360,000 300,000 1,206,000 Add the following decimals and cheek the answer : 21. 21.61 36.21 36.17 20.32 28.30 18.91 12.42 5.96 20.96 14.66 15.86 6.00 3.17 19.07 11.02 22. 44.33 73.16 71.69 14.36 8.16 43.20 47.14 126.04 85.05 70.42 93.35 80.13 31.15 62.51 49.17 49.17 . 29.37 47.26 31.10 206.38 37.69 47.26 36.59 60.47 73.26 23. On the following page is an itemized list of invest- ments. What is the total amount of investments ? What is the average rate of interest ? Review Interest, page 60. 240 VOCATIONAL MATHEMATICS FOR GIRLS List of Investments Held by the Sinking Punds of Fall Biver^ Mass. January 1, 1913 Name Rate Maturity Amount City of Boston Bonds H July 1, 1939 $16,000 City of Cambridge Bonds H Nov. 1, 1941 26,000 City of Chicago Bonds 4 Jan. 1, 1921 27,600 City of Chicago Bonds 4 Jan. 1, 1922 100,000 City of Los Angeles Bonds 4J June 1, 1930 60,000 City of So. Norwalk Bonds 4 July 1, 1930 23,000 City of So. Norwalk Bonds 4 Sept. 1, 1930 22,000 City of Taunton Bonds 4 June 1, 1919 39,000 Town of Revere Note 4.36 disc. Aug. 13, 1913 10,000 Boston & Albany R. R. Bonds 4 May 1, 1933 67,000 Boston & Albany R. R. Bonds 4 May 1, 1934 67,000 Boston Elevated R. R. Bonds 4 May 1, 1936 50,000 Boston Elevated R. R. Bonds 4i Oct. 1, 1937 68,000 Boston Elevated R. R. Bonds 4i Nov. 1, 1941 60,000 Boston & Lowell R. R. Bonds 4 April 1, 1932 16,000 Boston & Maine R. R. Bonds 4i Jan. 1, 1944 160,000 Boston & Maine R. R. Bonds 4 June 10, 1913 20,000 C. B. & Q. R. R. Bonds (111. Div.) 4 July 1, 1949 60,000 C. B. & Q. R. R. Bonds (111. Div.) H July 1, 1949 65,000 Chi. & N. W. R. R. Bonds 7 Feb. 1, 1915 92,000 Chi. & St. P., M. & 0.. R. R. Bonds 6 June 1, 1930 20,000 Cleveland & Pittsburg R. R. Bonds 4i Jan. 1, 1942 36,000 Cleveland & Pittsburg R. R. Bonds ^ Oct. 1, 1942 10,000 Fitchburg R. R. Bonds H Oct. 1, 1920 60,000 Fitchburg R. R. Bonds 3i Oct. 1, 1921 20,000 Fitchburg R. R. Bonds 4i May 1, 1928 60,000 Fre. Elk. & Mo. Val. R. R. Bonds 6 Oct. 1, 1933 85,000 Great Northern R. R. Bonds ^ July 1, 1961 . 25,000 Housatonic R. R. Bonds 5 Nov. 1, 1937 46,000 Louis. & Nash. R. R. Bonds (N. 0. & M.) 6 Jan. 1, 1930 20,000 Louis. & Nash. R. R. Bonds (St. L. Div.) 6 March 1, 1921 6,000 Louis. & Nai^h. R. R. Bonds (N. & M.) H Sept. 1, 1945 10,000 Louis. & Nai^h. R. R. Bonds 5 Nov. 1, 1931 36,000 Mich. Cent. R. R. Bonds 6 March 1, 1931 37,000 Mich. Cent. R.R. Bonds (Kal. & S. H.) 2 Nov. 1, 1939 60,000 ARITHMETIC FOR OFFICE ASSISTANTS 241 24. What is total amount of the following water bonds ? What is the average rate of interest ? Water Bonds of Fall Biver, Mass. Datk 01 I* Ibbub Rate TSBM Maturity Amount June 1. ,1893 ^L 30 years June 1, , 1923 $76,000 May 1 , 1894 30 years May 1, 1924 26,000 Nov. 1 , 1894 29 years Nov. 1, 1923 25,000 Nov. 1, , 1894 30 years Nov. 1, , 1924 26,000 May 1, 1896 30 years May 1. , 1926 25,000 June 1 , 1895 30 years June 1, , 1925 50,000 Nov. 1, , 1896 30 years Nov. 1 , 1925 25,000 May 1 ,1896 30 years May 1, , 1926 26,000 Nov. 1, , 1896 30 years Nov. 1, , 1926 26,000 April 1, , 1897 30 years April 1 , 1927 26,000 Nov. 1 , 1897 30 years Nov. 1, 1927 26,000 April 1. ,1898 30 years April 1. , 1928 26,000 Nov. 1, ,1898 30 years Nov. 1, , 1828 26,000 May 1, , 1899 30 years May 1, , 1929 60,000 Aug. 1, , 1899 30 years Aug. 1 , 1929 160,000 Nov. 1. , 1899 H 80 years Nov. 1, , 1929 176,000 Feb. 1, , 1900 H 30 years Feb. 1. , 1930 100,000 May 1, , 1900 H 30 years May 1, , 1930 20,000 April 1, 1901 H 30 years April 1. ,1931 20,000 April 1, 1902 H 30 years April 1. , 1932 20,000 April 1, , 1902 H 30 years April 1, 1932 60,000 Dec. 1. , 1902 H 30 years Dec. 1, , 1932 50,000 April 1, , 1903 H 30 years April 1 ,1933 20,000 Feb. 1 ,1904 3i 30 years Feb. 1. ,1934 176,000 May 2 , 1904 4 30 years May 2, 1934 20,000 1 SUBTRACTION Drill Exercise 1. 33 2, 35 3. 37 4. 38 5. 36 6. 32 7. 26 7 9 8 9 7 4 9 8. 42 9. 49 10. 46 U. 43 12. 41 17 18 19 16 15 242 VOCATIONAL MATHEMATICS FOR GIRLS 13. 45 14. 44 15. 364 16. 468 17. 566 17 17 126 329 328 18. 661 19. 363 20. 465 21. 362 324 127 228 129 22. 865,900 23. 891,000 24. 200,000 25. 30,071 697,148 597,119 121,314 28,002 26. 581,300 27. 481,111 28. 681,900 29. 868,434 391,111 310,010 537,349 399,638 iS _^_^,^_^_^^^_^^^ V.^__^-^_H^_^i^H ^_M^.^_^^_^^^^^^^ 30. 753,829 31. 394,287 32. 567,397 33. 487,196 537,297 277,469 297,719 311,076 34. 291,903 35. $835.00 36. $1100.44 37. $2881.44 187,147 119.00 835.11 1901.33 38. $3884.59 39. $4110.59 40. $2883.40 41. $3717.17 1500.45 1744.43 1918.17 1999.18 42. $1911.84 43. $2837.73 44. $5887.93 1294.95 1949.94 4999.99 MULTIPLICATION Drill Exercise By inspection, multiply the following numbers : 1. 1600x900. 11. 80x11. 2. 800 X 740. 12. 79 x 11. 3. 360 X 400. 13. 187 x 11. 4. 590x800. 14. 2100x11. 5. 1700x1100. 15. 2855x11. 6. 1900x700. 16. 84x25. 7. 788,000x600. 17. 116x50. 8. 49,009 X 400. 18. 288 x 25. 9. 318,000x4000. 19. 198x25. 10. 988,000 X 50,000. 20. 3884 x 25. Keview rules on multiplication, pages 8-9. ARITHMETIC FOR OFFICE ASSISTANTS 243 BILLS (Invoices) When a merchant sells goods (called merchandise), he sends a bill (called an invoice) to the customer unless payment is made at the time of the sale. This invoice contains an itemized list of the merchandise sold and also the following : The place and date of the sale. The terms of the sale (usually in small type) — cash or a number of days' credit. Sometimes a small discount is given if the bill is paid within a definite period. The quantity, name, and price of each item is placed on the same line. The entire amount of each item, called the exten- sion, is placed in a column at the right of the item. Discounts are deducted from the bill, if promised. Extra charges, such as cartage or freight, are added after taking off the discount. Make all Checks payable to We handle only highest grades Union Coal Company of Anthracite and Bitu- of Boston minous Coals UNION COAL COMPANY 40 Center Street BRANCH EXCHANGE TELEPHONE CONNECTING ALL WHARVES AND OFFICES SOLD TO L. T. Jones, 5 Whitney St. , Mattapan, Mass. BOSTOW, Sept. 3 , 1914. 6000 lb. Stove Coal 7.00 $21.00 4000 " Nut 7.25 14.50 35.50 REC'D PAYMENT SEPT. 28, 1914 UNION COAL CO. 244 VOCATIONAL MATHEMATICS FOR GIRIJS When the amount of the bill or invoice is paid, the invoice is marked. Received payment^ Name of firm. Per name of authorized person. This is called receipting a bill. Ledger Whenever an invoice is sent to a customer, a record of the transaction is made in a book called a ledger. The pages of this book are divided into two parts by means of red or double lines. The left side is called the debit and the right side the credit side. At the top of each ledger page the name of a person or firm that purchases merchandise is recorded. The record on this page is called the account of the person or firm. When the person or firm purchases merchandise, it is recorded on the debit side. When merchandise or cash is received, it is recorded on the credit side. The date, the amount, and the word Mdse. or cash is usually written. We debit an account when it receives value, and credit an account when it delivers value. E. D. REDINGTON 1917 1 1917 Jan, 2 Cash 109 1000 I Jail, 1 Ac&t to Perkins 114 810 5S Note, 60 ds. 114 1500 2 Mdse. 100 3057 50 9 Pagers Order 115 575 10 (i 100 575 25 Cash 109 500 22 Order to Jenness 115 375 27 Mdse. 93 157 50 688.05 1*818 08 31 Browne's Ace, 115 397 U130 53 03 Specimen Lbdobb Page ARITHMETIC FOR OFFICE ASSISTANTS 245 A summary of the debits and credits of an account is called a statement. The difference between the debits and credits represents the standing of the account. If the debits are greater than the credits, the customer named on the account owes the merchant. If the credits are greater than the debits, then the merchant owes the customer. EXAMPLES Balance the following accounts : BLANEY, BROWN & CO. 1917 Jan. 14 28 Cons't #1 ** Co.il 1917 177 669 98 Jan. 6 179 386 25 30 179 1200 75 Mdse, D/t. favor Button 171 180 1303 900 75 LUDWIG & LONG 1917 1917 Jan. 6 ConsU #2 177 1939 50 Jan. 6 Cash 172 1000 20 ** #^ St7.60 177 1327 50 15 28 n 172 172 939 1000 50 CHARLES N. BUTTON 1917 Jan. 7 Mdse. 168 651 12 Cash 173 1000 20 << 173 2000 29 ShipH Co. #1 179 795 30 D/t. on Blaneyt B. 180 900 1 1917 88 \jan. 9 1 ^^ 37 Ship't Co. §2 ConsH §2 t08.Bi 177 176 856 4699 67 09 246 VOCATIONAL MATHEMATICS FOR GIRLS D. K. REED & SON 1917 1917 Jan, 8 ComH #J 177 625 42 Jan, 8 Note at 60 ds. 180 625 42 17 Mdse, 170 202 50 17 Cash 172 202 60 26 Cons't SI " Co. #i 177 179 243 206 75 PROFIT AND LOSS (Review Percentage on pages 60-66) A merchant must sell merchandise at a higher price than he paid for it in order to have sufficient funds at the end of the transaction to pay for clerk hire, rent, etc. Any amount above the purchasing price and its attendant expenses is called profit; any amount below purchasing price is called loss, A merchant must be careful in figuring his profit. He must have a set of books so arranged as to show what caused either an increase or reduction in the profits. There are certain special terms used in considering profit and loss. The first cost of goods is called the net or prime cost. After the goods have been received and unpacked, and the freight, cartage, storage, commission, etc. paid, the cost has been increased to what is called gross or full cost. The total amount received from the sale of goods is called gross selling price. The sum of expenses connected with the sale of goods subtracted from the gross selling price is called the net selling price. A merchant gains or loses according as the net selling price is above or below the gross cost. There are two methods of computing gain or loss, each based on the rules of percentage. In the first method, the gross cost is the base, the per cent of gain or loss the rate, the gain or loss the percentage. The second method considers the selling price the base and will be explained in detail later. ARITHMETIC FOR OFFICE ASSISTANTS 247 EXAMPLES 1. Make extensions after deducting discounts and give total : CndH not allowed on goods returned wUbout our permission PETTINGELL-ANDREWS COMPANY ELECTRICAL MERCHANDISE General Offices and Warerooms 166 to 160 PEARL STREET and 401 to 611 ATLANTIO AVBNUB Terms : 30 Days Net NEW YORK, Nov 17 1911 SOLD TO City of Lowell School Dept, Lowell, Mass. SHIPPED TO Same Lowell Industrial School, Lowell, Mass. SHIPPED BY B &. L 11/14/11 * OUR REG. NO. 3786 ORDER REC'D 11/13/11 REELS COILS BUNDLES CASES BBLS. &-D ^7 : ' 3 "S II Order No. 78158 Reg. No. 52108 PRICE CO O'in 1 1 #4Comealong#ll293 Ea 4 00 15% ■ 1 1 #14492 16" Extension Bit Ea 2 00 50% 36 36 2 oz cans Nokorode Soldering Paste Doz 2 00 50% 15 15 #8020 Cutouts Ea 36 40% 2 2 #322 H & H Snap Sws Ea 76 30% 125 125 #9395 Pore Sockets Ea 30 45% 125 125 # 1999 Fuseless Rosettes Ea 08 45% 100 100 C Ball Adjusters for Lp Cord M 7 00 , 50 50 J" Skt Bushings C 50 200 200 Pr #43031 Std#328#l Single Wire Cleats M Pr 26 68 40% 200 200 Pr #43033 Single Wire Cleats MPr 40 00 40% 2 2 Lb White Exemplar Tape Lb 45 248 VOCATIONAL MATHEMATICS FOR GIRLS 2. Make extensions on the following items and give total : Gooda are Charged for the Convenience of Cuatomeri and Accounta are Rendered Monthly R A. Mc"Whirr Co. DEPARTMENT STORE FALL RIVER, MASS. A. A. MILLS. Pret't&Treas. J. H. MAHONEY, Supt. R. S. THOMPSON, Sec'y. Purchases for Fall River Technical High School September, 1913 City No. Ordei r Number 719 • Datb Ahticlbs Amounts Daily Total Credits Sept 4 2 Doz C Hangers 2 " Skirt " 90 45 5 120 Long Cloth 34S Cambric 15 5J 6 522 B Cambric 100 B Nainsook 24 Doz Kerr L Twist 8 Doz Tape Measures 84 •• W Thread 1 10/12 Doz Tape 18 16 120 25 51 25 9 1 Gro Tambo Cotton i Doz Bone Stillettos J '• Steel 40 Paper Needles 20 " " 520 46 46 3} 3} 2 Doz M Plyers 600 • 2 Boxes Edge Wire 125 12 '* Even Tie Wire 180 24 " Brace 225 2 •• Lace 160 2 Pk Ribbon 125 2 Roils Buckram 90 48 Yd Cape Net 15 13 100 Crinoline 125 5 5 ARITHMETIC FOR OFFICE ASSISTANTS 249 3. Make extensions on the following items and give total ; Goods are Charged for the Convenience of Customera and Accounts are Rendered Monthly R. A. Mc Whirr Co. DEPARTMENT STORE FALiLi RIVER, MASS. A. A. MILLS, PretH & Treat. J. H. MAHONEY, Vice-Pre»'t. R. S. THOMPSON, Sec'y. Purchases for September, 1913 Fall River Public Buildings City No. For Technical H igh School Date Articlxs Amounts Daily Total Cbbdits Sept 4 1 Dinner Set 1700 100 Knives 9 100 Forks 9 100 D Spoons 10 100 Tea Spoons 09 1 Doz Glasses 90 8i Doz Tumblers 45 8i " Bowls 96 54 Crash \\\ 7J " 11} 50 •• 3i } Doz Napkins 270 J " " 415 2 Table Cloths 360 12 120 Crash II } 15 2 Stock Pots 325 1 Lemon Squeezer 14 1 Doz Teaspoons 500 1 Butter Spreader 75 } Doz Forks 625 250 VOCATIONAL MATHEMATICS FOR GIRLS Example. — A real estate dealer buys a house for $4990 and sells it to gain $ 50. What is the per cent of gain over cost ? Solution. ^ x 100 = — = lAs%. Arts. 4990 499 ^'^ '^ Drill Exercisb Find per cent of gain or loss : Cost Oain Cost LoM 1. $1660 $175 6. $6110 $112 2. $1845 $135 7. $5880 $ 65 3. $1997.75 $412.50 8. $3181.10 $108.75 4. $2222.50 $319.75 9. $7181.49 $213.60 5. $3880.11 $610.03 10. $3333.19 $ 28.90 EXAMPLES 1. A dealer buys wheat at 91 cents a bushel and sells to gain 26 cents. What is the per cent of gain? 2. A farmer sold a bushel of potatoes for 86 cents, and gained 20 cents over the cost. What was the per cent of gain ? 3. Real estate was sold for $ 19,880 at a profit of $ 3650. What was the per cent of gain ? 4. A provision dealer buys smoked hams at 19 cents a pound and sells them at 31 cents a pound. What is the per cent of gain? 5. A grocer buys eggs at 28 cents a dozen and sells them at 35 cents a dozen. What is the per cent gain ? 6. A dealer buys sewing machines at $22 each and sells them at $ 40. What is the per cent gain ? 7. A dealer buys an automobile for $ 972 and sells it for $ 1472 and pays $ 73.50 freight. What is the per cent gain ? ARITHMETIC FOR OFFICE ASSISTANTS 251 Drill Exercise Find the per cent gain or loss on both cost and selling price : Cost Selling Price Cottt Selling Price 1. $1200 $1500 6. $2475 $2360 2. $1670 $1975 7. $1650 $1490 3. $2325 $2980 8. $4111.50 $2880.80 4. $4250.50 $5875.75 9. $4335.50 $4660.60 5. $3888.80 $4371.71 10. $2880.17 $2551.60 REVIEW EXAMPLES 1. A dealer buys 46 gross of spools of. cotton at $11.12. He sells them at 5 cents each. What is his profit ? What is the per cent of gain on cost ? on selling price ? 2. Hardware supplies were bought at $ 119.75 and sold for $ 208.16. What is the per cent of gain on cost and on selling price ? 3. A grocer pays $ 840 f .o.b. Detroit for an automobile truck. The freight costs him $ 61.75. What is the total cost of automobile truck ? What per cent of the total cost is freight ? 4. A dry goods firm buys 900 yards of calico at 5 cents a yard, and sells it at 9 cents. What is the profit ? What per cent of cost and selling price ? 5. A grocer buys a can (8J qt.) of milk for 55 cents and sells it for 9 cents a quart. What is the per cent of gain ? EXAMPLES 1. A dealer sold a piano at a profit of $ 115, thereby gaining 18 % on cost. What was the selling price ? Solution. — If $ 115 = 18 % of cost, which is 100 %, J % = j^5 = 6.3889 100% =.$638.89 cost Addin g 115.00 profit $753.89 selling price. 252 VOCATIONAL MATHEMATICS FOR GIRLS 2. A dealer sold furniture at a profit of $ 98. What was the cost of the furniture, if he sold to gain 35 % ? 3. A coal dealer buys coal at the wharf and sells it to gain $ 2 per ton. What is the cost per ton if he gains 31 % ? 4. A shoe jobber buys a lot of shoes for $ 1265 and sells to gain 26 %. What is the selling price ? 5. An electrician buys a motor for $ 48 and sells it to gain 18 %. What is the selling price ? 6. A pair of shoes was sold to gain 26 %, giving the shoe dealer a profit of 97 cents. What was the cost price ? What was the selling price ? FORMULAS Gain or loss = Cost x rate of gain or loss Gain or loss C08t = Rate of gain or rate of loss Selling Price = Cost (100 % + rate of gain) or (100 % - rate of loss) Cost ^ Selling Price Selling Price "" 100% + rate of gain ^' 100 9{> - rate of loss Drill Exercise Find the selling price in each of the following problems : Sold to Lose Cost Sold to Gain Cost 1. i&i% $96 6. 37% $250 2. 20% $115 7. 33%^ $644.50 3. 30% $48 8. 41% $ 841.75 4. 19% $ 112.50 9. 29% $ 108.19 5. 20^% $ 187.75 10. 22^% $ 237.75 COMPUTING PROFIT AND LOSS Second Method, — Many merchants find that it is better busi- ness practice to figure per cost profit on the selling price rather than on the cost price. Many failures in business can be ARITHMETIC FOR OFFICE ASSISTANTS 253 traced to the practice of basing profits on cost. We must bear in mind that no comparison can be made between per cents of profit or cost until they have been reduced to terms of the same unit value or to per cents of the same base. To illustrate: It costs $100 to manufacture a certain article. The expenses of selling are 22%. For what must it sell to make a net profit of 10%? Most students would calculate $132, taking the first cost as the basis of estimating cost of sales and net profit. The average business man would say that the expenses of selling and cost should be quoted on the basis of the selling price. Solution. — Expenses of selling = 22 % Profit = 10 % 32 % on selling price. . •. Cost on $ 100 = 68 % selling price. 100 % = $ 147 selling price. Example 1. — An article costs $ 5 and sells for $ 6. What is the percentage of profit? .dns, 16 J %. Process. — Six dollars minus $6 leaves $1, the profit. One dollar divided by $6, decimally, gives the correct answer, 16J%. Example 2: — An article costs $ 3.75. What must it sell for to show a profit of 25 % ? Ans, $ 5. Process. — Deduct 25 from 100. This will give you a remainder of 76, the percentage of the cost. If $3.76 is 75%, 1% would be $3.75 divided by 76 or 6 cents, and 100 % would be $ 5. Now, if you marked your goods, as too many do, by adding 26 % to the cost, you would ob- tain a selling price of about $ 4.69, or 31 cents less than by the former method. EXAMPLES 1. What is the percentage of profit, if an article costs $ 8.50 and sells for $ 10 ? 2. What is the percentage of profit on an automobile that cost $ 810 and sold for $ 1215 ? 3. An article costs $ 840. What must I sell it for to gain 30%? 254 VOCATIONAL MATHEMATICS FOR GIRLS 4. A case of shoes is bought for $ 30. For what must I sell them to gain 26 % ? Table for Finding the Selling Price of any Article COBT Net Per Cent Profit Desired TO PO Business ^^ - - 1 - t 1 84 88 3 82 4 81 6 80 6 79 7 78 8 77 9 76 10 76 11 74 IS 78 18 72 14 71 16 70 SO 66 S6 60 80 66 88 60 40 46 60 15% 35 16% 88 82 81 80 79 78 77 76 76 74 78 72 71 70 69 64 69 54 49 44 84 1T% 82 81 80 79 78 77 76 76 74 78 72 71 70 69 68 68 68 68 48 48 88 18% 19% 20% 21% 81 80 79 78 77 76 76 74 78 72 71 70 69 68 67 62 67 52 47 42 82 80 79 78 77 76 76 74 78 72 71 70 69 68 67 66 61 66 51 46 41 81 79 78 77 76 76 74 78 72 71 70 69 68 67 66 66 60 56 50 45 40 80 78 77 76 75 74 73 72 71 70 69 68 67 66 66 64 69 54 49 44 89 29 22% 28% 77 76 75 74 78 72 71 70 69 68 67 66 66 64 68 68 58 48 48 88 28 76 75 74 78 72 71 70 69 68 67 66 66 64 63 62 67 62 47 42 87 27 24% 75 74 78 72 71 70 69 68 67 66 66 64 68 62 61 66 51 46 41 86 26 26% 74 78 72 71 70 t9 68 67 66 66 64 68 62 61 60 66 50 45 40 36 25 The percentage of cost of doing business and the profit are figured on the selling price. Rule Divide the cost (invoice price with freight added) by the figure in the column of " net rate per cent profit desired " on the line with per cent it cost you to do business. Example. — If a wagon cost $ 60.00 Freight 1.20 $ 61.20 You desire to make a net profit of 6 per cent It costs you to do business 19 per cent Take the figure in column 5 on line 19, which is 76. 76|$61.2000 |$80.52, the selling price. 608 400 880 200 162 ARITHMETIC FOR OFFICE ASSISTANTS 255 Solve the following examples by table : 1. I bought a wagon for $84.00 f.o.b. New York City. Freight cost $ 1.05. I desire to sell to gain 8 %. If the cost to do business is 18 %, what should be the selling price? 2. I buy goods at $97 and desire a net profit of 7%. It costs 16 % to do business. What should be my selling price ? 3. Hardware supplies are purchased for $489.75. If it costs 23 % to do the business, and I desire to make a net profit of 11 %, for what must I sell the goods ? EXAMPLES 1. I bought 15 cuts of cloth containing 40^ yd. each, at 7 cents a yd., and sold it for 9 cents a yd. What was the gain? 2. A furniture dealer sold a table for $ 14.50, a couch for $ 45.80, a desk for $ 11.75, and some chairs for $ 27.30. Find the amount of his sales. 3. Goods were sold for $367.75 at a loss of $125. Find the cost of the goods. 4. Goods costing $145.25 were sold at a profit of $ 28.50. For how much were they sold ? 5. A woman bought 4^ yards of silk at $ 1.80 per yard, and gave in payment a $ 10 bill. What change did she receive.? 6. I bought 25 yards of carpet at $2.75 per yard, and 6 chairs at $ 4.50 each, and gave in payment a $ 100 bill. What change should I receive ? TIME SHEETS AND PAY ROLLS Office assistants must tabulate the time of the different em- ployees and compute the individual amount due each week. In addition, they must know the number of coins and bills of different denominations required so as to be able to place the exact amount in each envelope. This may be done by making out the following j)ay roll form. 256 VOCATIONAL MATHEMATICS FOR GIRLS FoBM Used to Determine the Number of Different Denominations Noi Persons Amt. Rec'd $10 2 2 $5 3 2 9 $2 2 6 /6 $1 2 2 50 j^ 2 3 5 25^ 3 7 10^' 3 8 2 /3 5^ 2 2 \f 2 /d.60 8 ^.86 f 7.^8 /2 2 ^./8 6 Total Number Coins /8 Time Card Week Ending. No. NAME .191-- MORNING AFTERNOON LOST OR OVERTIME ^ Toe Wed Tim FrI tet Sun U OCT II OUT n OIT ? Tot»l Time Hr$. Rate Total Wage* for Week $ Form Used to Send to the Bank FOR THE Monet for Pat Koll MEMORANDUM OF CASH FOR PAY ROLL WANTED BY J9— Twenties Tens Fives Twos Ones Halves Quarters Dimes . Nickels Pennies Total ARITHMETIC FOR OFFICE ASSISTANTS 257 TABLE OF WAGES 1 To find the amount due at any rate from 30 cents to. 56 cents per hour, look at the column containing the number of hours and the amoimt will be shown. Time and a half is counted for overtime on regular working days, and double time for Sundays and holidays. * • • 1 P4 a s II a S s m S 1 g s 1 1 S • § i 1 • n s s s n M M « o Q (4 M ^ Q $0 45 & o » %••• $0 80 $0 15 $0 22^ $0 80 $0 82^ $0 16i $0 24| $0 32) $0 22) $0 88f $0 46 1... 80 80 46 60 82^ 82^ 48| 66 46 45 67i 90 2... 30 60 90 1 20 82^ 66 97^ 1 80 46 90 1 85 1 80 Oi • • 80 90 185 1 80 82^ 97^ 1 46i 1 96 45 135 2 02) 2 70 4... 80 1 20 180 240 82^ 1 80 1 95 2 60 45 1 80 2 70 360 6... 80 1 60 2 25 8 00 82^ 1 62i 248| 8 25 45 225 8 87i 460 o< • • 80 1 80 2 70 8 60 82J 196 2 92^ 3 90 45 2 70 405 5 40 7... 80 2 10 8 15 4 20 82i 2 27i 8 41i 466 45 8 15 4 72) 6 80 vl • • • 80 2 40 8 eo 4 80 32^ 2 60 3 90 620 45 3 60 540 7 20 9... 80 2 70 405 640 32i 2 92i 4 88| 586 45 405 6 07) 8 10 10... 80 8 06 460 6 00 82i 8 25 4 87i 660 45 450 6 75 9 00 %••• $0 47i $0 28| $0 86| $0 47i 10 60 $0 25 $0 87i $0 50 $0 65 $0 271 $0 41J $0 56 1... 47i 47i 7U 96 60 60 76 1 00 65 66 82) 110 2... 47i 95 1 42| 1 90 60 1 00 1 50 2 00 55 1 10 1 65 2 20 o* • • 4H 1 42| 2 1df 285 60 1 60 225 8 00 56 1 65 2 47i 8 80 4... 47i 1 90 2 86 8 80 50 2 00 8 00 4 00 . 55 2 20 8 80 440 o< .* 47i 2 87i 8 56i 4 76 50 2 60 8 75 6 00 55 2 75 4 12) 560 o« . * 47i 2 85 4 27i 5 70 60 8 00 450 6 00 65 8 80 4 96 6 60 7... 47i 8 82i 4 98| 665 60 8 50 525 7 00 55 3 86 6 77) 7 70 o* • • 47i 880 6 70 7 60 50 4 00 6 00 8 00 55 440 6 60 880 9... 47i 4 27i 6 411 8 65 60 450 6 75 9 00 55 4 96 7 42) 9 90 10... 47i 4 76 7 12J 9 50 50 500 7 60 10 00 56 560 825 11 00 EXAMPLES 1. Find the amount due a carpenter who has worked 8 hours regular time and 2 hours overtime at 55 cents per hour. ^ Similar tables may be constracted for other rates. 258 VOCATIONAL MATHEMATICS FOR GIRLS 2. A plasterer worked on Sunday from 8 to 11 o'clock. If his regular wages are 45 cents per hour, how much will he receive ? 3. A machinist's regular wage is 55 cents an hour. How much money is due him for working July 4th from 8-12 a.m. and 1^.30 p.m. ? 4. A plumber works six days in the week; every morning from 7.30 to 12 m. ; three afternoons from 1 to 4.30 p.m. ; two afternoons from 1 to 5.30 ; and one from 1 until 6 p.m. What will he receive for his week's wages at 50 cents per hour? Wages op Employees Superintendent $1,200.00 per annum Matron 700.00 per annum Nurses, 2 at 45.00 per month Nurses, 1 at 40.00 per month Nurses, 3 at 35.00 per month Attendant 6.00 per week Cook 12.00 per week Assistant cook 1.00 per day Kitchen maid 6.00 per week Ward maids, 4 at 6.00 per week Waitresses, 2 at 6.00 per week Laundress 8.00 per week Washwomen, 2 at 6.00 per week Janitors, 1 day and 1 night 16.00 per week Barber . . . • 6.00 per week 5. Find the total of coins and bills of all different denomi- nations necessary to make up the weekly pay roll (52 weeks = a year) of the above. Assume full time for a week. Make out the currency memorandum for baiik. 6. Find the total of coins and bills of the different denomi- nations necessary to make up the following pay roll ; 47^ hours, at 30 cents. 48 hours, at 45 cents. 48 hours, at 47 J cents. 46 hours, at 32^ cents. ARITHMETIC FOR OFFICE ASSISTANTS 259 7. Make a pay roll memorandum for the following pay roll: 48at42i, 39 at 45, 46iat48i. TEMPORARY LOANS The following is a statement of the temporary loans of a New England city negotiated during the year, — amount, time, rates. Datr Amount of Loan TiMB Rate of Intbbbst Amount of JL^J^ A JM — ^» Intbbbst Months Imys Feb. 28 350,000 243 2.76 Feb. 28 26,000 243 2.76 Feb. 28 26,000 243 2.76 June 6 100,000 6 3.26 June 19 26,000 126 3.52 June 19 26,000 126 3.52 June 19 25,000 126 3.62 June 19 26,000 126 3.62 July 3 25,000 124 3.66 July 3 26,000 124 3.55 July 8 26,000 124 3.56 July 3 26,000 124 3.66 July 3 26,000 124 3.55 July 3 25,000 124 3.65 Aug. 14 26,000 2 4. Aug. 20 26,000 80 4.07 Aug. 20 26,000 80 4.07 Aug. 20 25,000 80 4.07 Aug. 20 26,000 80 4.07 Sept. 4 25,000 40 4. Sept. 4 25,000 40 4. Sept. 4 16,000 40 4. Write in a column after each loan, as suggested above, the amount of interest on each loan for the time and at the rate. CHAPTER XII ARITHMETIC FOR SALESGIRLS AND CASHIERS • The majority of employees in a department store are sales- girls. It may be well to describe briefly the method of operation of such a store and to indicate what part a salesgirl has in it. A department store is a combination of a number of distinct stores or departments under one roof and general manage- ment. It is organized in this way for the purpose of economy. Each department is conducted as a separate store, and is in charge of a buyer, who both buys and plans the sales for his department. His department is charged for rent, according to its location, and must also pay for overhead charges. The buyer in charge of each department has under him salesgirls or saleswomen, who sell the goods. Each salesgirl has a book containing sales slips in duplicate and a card to show the amount of sales. The sales slip shows the name and address of the purchaser if the merchandise is to be sent to the customer's home. In the case of a charge accoimt a special form of sales slip is used. The name and quality of the article purchased are written in large space and the amoimt extended to th^ right. The amount of money received from the purchaser is placed at the top of the sales slip. A carbon copy of each sales slip is made. The carbon copy is given to the customer and the original is sent with the money to the cashier. It is then used to tabulate data in regard to sales, etc. 260 ARITHMETIC FOR SALESGIRLS AND CASHIERS 261 8606 ^^^^^*' ^^^'^ ^^-- Name Address — , 80LD BY... Pur. by Ha- 1^1 oo'S lis- D E P .T 1 AMT REC'D. Am't Rec'd Sold by Am't of Sile 8606 1 8AI«K8BIAK>8 VOVCHBR In Gate of Error Ploaso Rotum Goods ind Bill J. n. BmBRSOBi CO. 8606 ^^^^^*' ^^^-^ ^^-- Name- Addrei 80LI BY.. iS D E 3 P AMT T REC'D Pur. by Oustamerit loiU please report any failure to deliver bill vdth goods Tkis Slip miut go in CDRtAiner'i Ptreel. ViolfttioB of tiis Knle is cane for Initaiit DlimiiMl 1 Department. SaIjESMAN- Date Cash Sales Charge Sales Cash Sales Charge Sales 1 Forward 2 10 3 11 4 12 5 13 6 14 7 15 8 16 • 9 17 262 VOCATIONAL MATHEMATICS FOR GIRLS Salesgirls should be able to do a great many calculations at sight. This ability comes only through practice. EXAMPLES Find the amount of the following : 1. 10 yd. percale at 12^ cents. 2. 12 yd. voile at 16 1 cents. 3. 27 yd. silesia at 33^ cents. 4. 60 yd. serge at $ 1.50. 5. 28 yd. mohair at $ 1.25. 6. 48 yd. organdie at 37^ cents. 7. 91|^ yd. gingham at 10 cents. a 112 yd. calico at 4^ cents. 9. 36 yd. galatea at 16 cents. 10. 11 yd. lawn at 19 cents. 11. 64 yd. dotted muslin at 62^ cents. 12. 24 yd. gabardine at $ 1.75. 13. 18 yd. poplin at 29 cents. 14. 16 yd. hamburg at 15 cents. 15. 12 yd. lace at 87^ cents. 16. 19 yd. val lace at 9 cents. 17. 26 yd. braid at 26 cents. 18. 48 dz. hooks and eyes at 12 centSo 19. 19 yd. cambric at 15 cents. 20. 18 pc. binding at 16 cents. 21. 6 yd. canvas at 24 cents. 22. 56 yd. linen at 62^ cents. 23. 18 yd. albatross at $ 1.60. 24. 22 yd. silk at $ 2.25. ARITHMETIC FOR SALESGIRLS AND CASHIERS 263 PROBLEMS 1. I bought cotton cloth valued at $ 6.25, silk at $ 13.75, handkerchiefs for $ 2.50, and hose for $ 2.75. What was the whole cost ? ' • • 2. Ruth saved $ 15.20 one month, $ 20.75 a second month, and the third month $4.05 more than the first and second months together. How much did she* save in the three months ? 3. Goods were sold for $ 367.75, at a loss of $ 125. Find the cost of the stock. 4. Goods costing $ 145.25 were sold at a profit of $ 28.50. For how much were they sold ? ' 5. A butcher sold 8f pounds of meat to one customer, 9^ pounds to a second, to the third as much as the first plus 1^1^ pounds, to a fourth as much as to the second. How many- pounds did he sell ? 6. Edith paid $ 42.75 for a dress, one-half as much for a cloak, and $ 7.25 for a hat. How much did she pay for all ? 7. A merchant sold four pieces of cloth; the first piece contained 24 yards, the second 32 yards, the third 16 yards, and the fourth five-eighths as many yards as the sum of the other three. How many yards were* sold? 8. From a piece of cloth containing 65f yards, there were sold 23;J^ yards. How many yards remained ? 9. A merchant sold goods for $ 528.40 and gained $ 29.50. Find the cost. 10. From 11 yards of cloth, 3f were cut, for a coat, and 6^ yards for a suit. How many yards remained ? 11. I bought 15 cuts of cloth containing 40^ yards each at 7 cents a yard and sold it for 9 cents a yard. What was the gain? 264 VOCATIONAL MATHEMATICS FOR GIRLS 12. What is the cost of 13| yards of silk at $ 3.76 per yard ? 13. What is the cost of 16^ yards of broadcloth at $ 2.25 per yard ? 14. What is the cost of 3 pieces of cloth containing 12f , 14 J, and 15^ yards at 12^ cents per yard ? 15. What will 6| yards of velvet cost at $ 2.75 per yard ? 16. What is the cost of three-fourths of a yard of crgpe de chine at $ 1.75 per yard ? 17. A saleslady is paid $1.00 per day for services and a bonus of 2 % on all sales over $ 50 per week. If the sales amount to $ 175 per week, what will be her salary ? 18. At $ 1.33^ a yard, how much will 15 yards of lace cost ? 19. At $ 1.16| a yard, how much will 9 yards of silk cost ? 20. At $ 1.12^ per yard, how much will 6 yards of velvet cost ? 21. At 33^ cents each, find the cost of 101 handkerchiefs. 22. A salesgirl sold 14|^ yards of gingham at 25 cents, 9 yards of cotton at 12^ cents, 10^ yards of Madras at 35 cents. Amount received, $ 10. How much change will be given to the customer ? 23. Sold 6^ yards of cheviot at $ 1.10, 5f yards of silk at $ 1.25, 9^ yards of velveteen at 98 cents. Amount received, $ 25.00. How much change will be given to the customer ? 24. Sold 11^ yards of Persian lawn at $ 1.95, 6f yards of dimity at 25 cents, 12|^ yards of linen suiting at 75 cents. Amount received, $ 40. How much change will be given to the customer ? 25. Sold 9^ yards of Persian lawn at $ 1.37^, 5\ yards of cheviot at $ 1.25, 15 yards of cotton at 12|^ cents. Amount received, $30. How much change will be given to the cus- tomer ? ARITHMETIC FOR SALESGIRLS AND CASHIERS 265 26. Sold 7 yards of muslin at 25 cents, 12^ yards of lining at 11 cents, 6f yards of lawn at $ 1.50, 7 yards of suiting at 75 cents. Amount received, $ 20. How imucli change will be given to the customer ? 27. Sold 16 yards of velvet at $ 2.25, 14^ yards of suiting at 48 cents, 23 yards of cotton at 15 cents, 6f yards of dimity at 24 cents, 7^ yards of ribbon at 25 cents. Amount received, $ 50. How much change will be given to the customer ? 2a At 12^ cents a yard, what will 8f yards of ribbon cost ? 29. At $ 2.50 a yard, what will 2.8 yards of velvet cost ? 30. If it takes 5^ yards of cloth for a coat, 3J yards for a jacket, and ^ a yard for a vest, how many yards will it take for all ? 31. I gave $ 16.50 for 33 yards of cloth. How much did one yard cost ? 32. Mary went shopping. She had a $ 20 bill. She bought a dress for $ 9.50, a pair of gloves for $ .75, a fan for $ .87, two handkerchiefs for $ .37 each, and a hat for $ 4.50. How much money had she left ? 33. Emma's dress cost $ 11.25, and Mary's cost | as much. How much did Mary's cost? How much did both cost ? 34. What is the cost of 16f yards of silk at $ 2.75 a yard ? 35. What is the cost of 14^ yards of cambric at 42 cents a yard ? 36. If 5| yards of calico cost 33 cents, how much must be paid for 14| yards ? 37. One yard of sheeting cost 22| cents. How many yards can be bought for $ 15.15 ? 38. From a piece of calico containing 33|^ yards there have been sold at different times llf , 7f , and 1^ yards. How many yards remain ? 266 VOCATIONAL MATHEMATICS FOR GIRLS 39. I bought 16 J yards of cloth for $ 3J per yard, and sold it for $ 4^ per yard. What was the gain ? 40. A merchant has three pieces of cloth containing, respec- tively, 28|, 35^, and 41 1 yards. After selling several yards from each piece, he finds that he Has left in the three pieces 67 yards. How many yards has he sold ? * ARITHMETIC FOR CASHIER How to Make Change. — Every efficient cashier or saleslady makes change by adding to the amount of the sale or purchase enough change to make the sum equal to amount presented. The change should be returned in the largest denominations possible. To illustrate : A young lady buys dry goods to the amount of $1.52. She gives the saleslady a $5 bill. What change should she receive ? iy will say: $1.52, $1.65, $1.66, $1.75, $2.00, $4.00, «6.00. That is, $1.62 + $.0.3 = $1.55; $1.65 -h $.10 = $ 1.65 ; $1.66 ^5; $1.76 + $.26 = $2.00; $2.00 + $2.00 = $4.00 ; $4.00 The saleslady » 5.00. That is, ^ i.oz -\- ^ .u.> = ^ i.oo ; ^ i.oo + + $.10 = $1.75; $1.76 + $.26 = $2.00; $2.00 + + $1.00 = $5.00. EXAMPLES 1. What change should be given for a dollar bill, if the following purchases were made ? a. $.87 c. $.43 e. $.20 6. $.39 d. $.51 /. $.23 2. What change should be given for a two-dollar bill, if the following purchases were made ? a. $1.19 d. $1.57 g* $ .63 5. $.89 e. $1.42 h, $ .78 c. $1.73 /. $1.12 i, $.27 ARITHMETIC FOR SALESGIRLS AND CASHIERS 267 3. What change should be given for a five-dollar bill, if the following pui'chases were made ? a. $3.87 d. $2.81 g. $1.93 b. $2.53 e. $3.74 h, $.17 c. $4.19 /. $4.29 I $.47 4. What change should be given for a ten-dollar bill,, if the following purchases were made ? a. $8.66 d. $6.23 g. $3.16 b. $9.31 e. $5.29 h. $2.29 c. $ 7.42 /. $ 4.18 i. $ 1.74 5. What change should be given for a twenty-dollar bill, if the following purchases were made ? a. $18.46 c. $17.09 e. $8.01 b. $19.23 d $12.03 /. $6.27 CHAPTER XIII CIVIL SERVICE Almost every government position open to women has to be obtained through an examination. In most cases Arithmetic is one of the subjects tested. It is wise to know not only the subject, but also the standards of marking, and for this reason some general rules on this subject follow. Marking Arithmetic — Civil Service Papers 1. On questions of addition, where sums are added across and the totals added, for each error deduct 16} ^o, 2. For each error in questions containing simple multiplication or division, as a single process, deduct 50 9^ ; as a double process, deduct 3. In questions involving fractions and problems other than simple computation, mark as follows : (a) Wrong process leading to incorrect result, credit 0. (6) For inconvenient or complex statement, process, or method, giving right result, deduct from 6 to 26 9^?. (c) If the answer is correct but no work is shown, credit 0. (d) If the answer is correct and the process is clearly indicated, but not written in full, deduct 26 ^o, (e) If no attempt is made to answer, credit 0. (/) If the operation is incomplete, credit in proportion to the work done. (g) For the omission of the dollar sign (|) in final result or answer, deduct 6. (h) In long division examples, to be solved by decimals, if the answer is given as a mixed number, deduct 26. 4. For questions on bookkeeping and accounts, mark as follows : (a) For omission of total heading, deduct 25 ; for partial omission, a commensurate deduction. (b) For every misplacement of credits or debits, deduct 25. 268 CIVIL SERVICE 269 (c) For every omission of date or item, deduct 10. (d) For omissions or misplacement of balance, deduct 26. NoTB. — Hard and fast rales are not always applicable because the impor- tance of certain mistakes differs with the type of example. Before, a set of examples is marked, the deductions to be made for various sorts of errors are decided upon by the examiners. In general, examples in arithmetic for high-grade positions are marked on practically the same basis as clerical arithmetic. Arithmetic in lower-grade examinations, such as police and fire service and the like, is marked about 60% easier than clerical. CIVIL SERVICE EXAMPLES (Give the work in full in each example.) 1. Multiply 83,849,619 by 11,079. 2. Subtract 16,389,110 from 48,901,001. 3. Divide 18,617.03 by .717. 4. At $0.37 per dozen, how many dozen eggs can be bought for $ 33.67 ? 5. What would 372 pounds of com meal cost if 4 lb. cost 12 cents ? 6. If a man bought 394 cows for $ 23,640 and sold 210 for $ 14,700, what was the profit on each cow ? 7. What is the net amount of a bill for $ 93.70, subject to a discount of 37 J % ? 8. How many pints in a measure containing 14,784 cubic inches ? 9. What number exceeds the sum of its fourth, fifth, and sixth by 23 ? 10. If a man's yearly income is $ 1600, and he spends $ 25 a week, how much can he save in a year ? 11. What will 16 J pounds of butter cost at 34 cents a pound ? 12. How many hogs can be bought for $ 1340 if each hog averages 160 pounds and costs 9 cents a pound ? 13. How many tons of coal can be bought for $446.25, if each ton costs $ 8.76 ? 270 VOCATIONAL MATHEMATICS FOR GIRLS 14. A young lady can separate 38 letters per minute. If a letter averages 6^ ounces, how many pounds of mail does she handle in an hour ? 15. Multiply 53| by 9f and divide the product by 2^. (Solve decimally.) ' 16. Roll matting costs 73 J cents per sq. yd. What will be the cost of 47 rolls, each roll 60 yd. long and 36 in. wide ? 17. A man paid $ 5123.25 for 27 mules and sold them for $ 6500. How much did he gain by the transaction ? 18. A wheel measures 3' 7" in diameter. What is the dis- tance around the tire ? 19. A bricklayer earns 70 cents an hour. If he works 129 days, 8 hours a day, and spends $ 50 a month, how much does he save a year ? 20. A rectangular courtyard is 48' 5" long and 23' 7" wide. How many square yards is it in area ? 21. How many days will it take a ship to cross the Atlantic Ocean, 2970 miles, if the vessel sails at the rate of 21 miles an hour? 22. Eleven men bought 7 tracts of land with 22 acres in each tract. How many acres will each man have ? 23. A. merchant sends his agent $10,228 to buy goods. What is the value of the goods, after paying $ 28 for freight and giving the agent 2 ^o for liis commission ? 24. If milk costs 6 cents a quart, and you sold it for 9 cents a quart, and your profit for the milk was $48, how many quarts of milk did you sell? 25. A traveler travels llf miles a day for 8 days. How many more miles has he yet to travel if the journey is 134 miles ? 26. What is the net amount of a bill for $ 29.85, subject to a discount of 16| % ? CIVIL SERVICE 271 27. Add across, placing the totals in the spaces indicated ; then add the totals and check : Totals 8,431 • 17,694 18,630 91 707 5,912 305 3,777 871 8,901 6,801 29,006 5,891 30 16,717 5,008 10,008 7,771 144 9,001 13,709 10,999 39 1,113 3,444 28. Divide 37,818.009 by .0391. 29. A pile of wood is 136 ft. long, 8 ft. wide, and 6 ft. high, and is sold for $ 4.85 per cord, which is 20 % more than the cost. What is the cost of the pile ? 30. Add the following column and from the sum subtract 81,376,019 : 80,614,304 68,815,519 32,910,833 54,489,605 96,315,809 75,029,034 21,201,511 31. A man bought 128 gal. cider at 23 cents a gallon ; he sold it for 28 cents a gallon. How much did he make ? 32. A laborer has $48 in the bank. He is taken sick and his expenses are $ 7.75 a day. How many days will his fund last? 33. In paving a street If mi. long and 54 ft. wide, how many bricks 9 in. long and 4 in. wide will be required ? 34. Find the simple interest on $ 841.37 for 2 yr. 3 mo. 17 da. at 5%. 35. Find the simple interest on $ 367.49 for 1 yr. 7 mo. 19 da. at 4 %. 272 VOCATIONAL MATHEMATICS FOR GIRLS SPECDfEN ARITHMETIC PAPERS Clerks, Messengers, etc. Rapid Computation 1. Add these across, placing the totals in the spaces in- dicated ; then add the totals : 16,863 3,176 368 61,461 36,196 Totals 27,368 7,242 82,463 24,176 62,837 3,724 61,493 68,317 68,417 41,682 4,738 16,837 6,281 62,683 26,364 73,642 26,164 42,626 70,463 1,476 18,672 7,368 16,726 71,394 62,968 2. Multiply 82,473,659 by 9874. Give the work in full. 3. From 68,515,100 subtract 24,884,574. Give the work in full. 4. Divide 29,379.7 by .47. Give the work in full. 5. What is the net amount of a bill for $19.20, subject to a discount of 16| % ? Give the work in full. Arithmetic 1. How many times must 720 be added to 522 to make 987,642 ? Give the work in full. 2. If the shadow of an up- right pole 9 ft. high is 8^ ft. long, what is the height of a church spire which casts a shadow 221 ft. long ? Give the work in full. 3. How many sods, each 8 in. square, will be required to sod a yard 24 feet long and 10 feet 8 inches wide ? Give the work in full. 4. A retired merchant has an income of $ 25 a day, his property being invested at 6 %. What is he worth ? Give the work in full. 5. Find the principal that will yield $ 38.40 in 1 yr. 6 mo. at 4 % simple interest. Give the work in full. 6. If the time past noon increased by 90 minutes equals ^ of the time from noon to midnight, what time is it ? Give the work in full. 7. A merchant deducts 20 % from the marked price of his goods and still makes a profit of 16 % . At what CIVIL SERVICE 273 advance on the cost are the goods marked? Give the work in full. 8. If a grocer sells a tub of butter at 22 cents a pound, he will gain 168 cents, but if he sells it at 17 cents a pound, he will lose 112 cents. Find (a) the weight of the butter and (6) the cost per pound. Give the work in full. 9. The product of four factors is 432. Two of the factors are 3 and 4. The other two factors are equal. What are the equal factors ? Give the work in full. Stenographer-Typewriter 1. From what number can 857 be subtracted 307 times and leave a remainder of 49 ? Give the work in full. 2. What number exceeds the sum of its fourth, fifth, sixth, and seventh parts by 101 ? Give the work in full. 3. A sells to B at 10'% profit; B sells to C at 5 % profit; if C paid $ 5336.10, what did the goods cost A ? Give the work in full. 4. Find the simple interest of $ 297.60 for 3 yr. 1 mo. 15 da. at 6 %. Give the work in full. 5. A man sold \ of his farm to B, ^ of the remainder to C, and the remaining 60 acres to D. How many acres were in the farm at first ? Give the work in full. « Sealers op Weights and Measures (Keview Weights and Measures, pages 43, 276) 1. A measure under test is found to have a capacity of 332.0625 cu. in. What is its capacity in gallons, quarts, etc. ? Give the work in full. 2. How many quarts, dry measure, would the above meas- ure hold ? Give the work in full, carrying the answer to four decimal places, 3. What is the equivalent of 175 lb. troy in pounds avoir- dupois ? Give the work in full. 1 av. lb. = 7000 grains.* 274 VOCATIONAL MATHEMATICS FOR GIRLS 4. How many grains in 12 lb. 15 oz. avoirdupois ? Give the work in full. 5. Reduce 15 lb. 10 oz. 20 grains avoirdupois to grains troy weight. Give the work in full. 6. What part of a bushel is 2 pecks and 3 pints ? Give the work in full and' the answer both as a decimal and as a common fraction. 7. What will 10 bushels 3 pecks and 4 quarts of seed cost at $ 2.10 per bushel ? Give the work in full. 8. What part of a troy pound is 50 grains, expressed both decimally and in the form of a common fraction ? 9. A strawberry basket was found to be 65.2 cubic inches in capacity, (a) How many cubic inches short was it? (b) What percentage of a fidl quart did it contain ? Give the work in full. 10. In testing a spring scale it was found that in weighing 22 lb. of correct test weights on same, the scale indicated 22 lb. 10^ oz. What was the percentage of error in this scale at this weight ? Give the work in full. Visitor 1. A certain "home" had at the beginning of the year $ 693.07, and received during the year donations amoimting to $ 1322.48. The expenses for the year were : salaries, $387.25 ; printing, etc., $175 ; supplies, $651.15 ; rent, $104.25 heat, etc., $ 75 ; interest, $ 100 ; miscellaneous, $ 72.83. Find the cash on hand at the end of the year. Give the work in fulL - 2. Of the 72,700 persons relieved in a certain state at public expense in the year ending March 31, 1912, 76 % were aided locally, and the remainder by the state. Find the number relieved by the state. Give the work in full. CIVIL SERVICE 275 3. There was spent in state, city, and town public poor relief in Massachusetts in one year the sum of $3,539,036. The number of beneficiaries was 72,700. What was the average sum spent per person ? Give the work in full. 4. Of the 72,900 persons aided by public charity in this state in a certain year -j^ were classed as sane. Of the re- mainder, ^ were classed as insane, J as idiotic, and the rest as epileptic. How many epileptics received public aid?. Give, the work in full. PART V — ARITHMETIC FOR JfURSES CHAPTER XIV A NURSE should be familiar with the weights and measures used in dispensing medicines. There are two systems used — the English, based on the grain, and the Metric system, based on the meter. Apothecaries^ Weight (Troy Weight) 20 grains (gr.)= 1 scruple (3) 8 3 =ldram (3)=60gr. 8 3 =1 ounce ( 3 ) = 24 3 = 480 gr. 12 3 =1 pound (»))= 96 3 = 288 3 = 5760 gr. The number of units is often expressed by Roman numerals written after the symbols. (See Roman Numerals, p. 2.) EXAMPLES 1. How many grains in iv scruples ? 2. How many grains in iii drams ? 3. How many grains in iv ounces ? 4. How many scruples in lb i ? 5. How many grains in lb iii ? 6. How many drams in lb iv ? 7. How many grains in 3 ii ? 8. How many scruples in 5 v ? 9. How many drams in 5 vii ? 10. How many ounces in lb viii ? 276 ARITHMETIC FOR NURSES 277 U. Salt S i will make how many quarts of saline solution, gr. xc to qt. 1 ? 12. How many drams of sodium carbonate in 10 powders of Seidlitz Powder ? Each powder contains gr. xl. Apothecaries^ Fluid Measure 60 minims (m) = 1 fluid dram = (f 3 ). 8f 3 =1 fluid ounce (f 3). 16 f 3 =1 pint (O) = 128 f 3 = 7680 m. 8 =1 gaUbn (C) = 128 f 3 =1024 f 3 . EXAMPLES 1. How many minims in f 3 iv ? 2. How many minims in f 5 iii ? 3. How many fluid drams in 1 ? 4. How many minims in 5 pints ? 5. How many pints in 8 gallons ? 6. How many fluid drams in ii ? 7. How many minims in f 5 viii ? 8. How many fluid drams in C vii ? 9. How many pints in C v ? 10. How many minims in f 5 ix ? 11. If the dose of a solution is m xxx and each dose contains ^^ gr. strychnine, how much of the drug is contained in f 5 i of the solution ? 12. 3 ii of a solution contains gr. i of cocaine. How much cocaine is given when a doctor orders m x of the solution ? Approximate Measures of Fluids (With Household Measures) * An ordinary teaspoonful is supposed to hold 60 minims of pure water and is approximately equal to a fluid dram. A 278 VOCATIONAL MATHEMATICS FOB GIRI^ drop is ordinarily considered equiTalent to a minim, but this is only approxi- mately true in the case of water. The specific gravity, shape, and surface ftom which the drop is poured influence the size. In preparing medicines to be taken internally, minima should never be measured out as drops. There are minim droppers and measures for this purpose. A level teaspoonful of either a fluid or solid preparation is equal to a dram. Level spoonfuls are always considered A Qkaddatb. 1 teaspoonful = 1 fluid dram. 1 dessertspoonful = 2 fluid drHjns. I tablespoonful = 4 fluid drams or } fluid ouDce. 1 wineglassful = 2 fluid ounces. 1 teacupful — fluid ounces. 1 tumblerful = 8 fluid ounces. EXAMPLES 1. How many dessertspoonfuls in 8 fluid ounces? 2. How many wineglassfuls in 2 tumblerfuls ? 3. How many tablespoonfuls in 3 fluid drams ? 4. How many teaspoonfuls in 6 fluid ounces ? 5. How many teacupfuls in 4 fluid drams ? 6. How many dessertspoonfuls in 6 fluid drams ? 7. How many teaspoonfuls in 1 gallon ? 8. How many drops of water in 1 quart ? 9. How many teaspoonfuls in 3 ounces ? 10. How many minims in 3 pints ? ARITHMETIC FOR NURSES 279 U. What household measure would you use to make a solu- tion, 3 i to a pint ? 12. Read the following apothecaries' measurements and give their equivalents : a, 3 iv. /. 3 ss.^ 6. gr. V. g. iv. c. ii. h, 3 ii. d. 5 ii« *• 5 iv. e. 5 ij. j. 5 ss. Metric System of Weights and Measures (Review Metric System in Appendix.) The metric system of weights and measures is used to a great extent in medicine. The advantage of this system over the English is that, in preparing solutions, it is easy to change weights to volumes and volumes to weights without the use of common fractions. In medicine the gramme (so written in prescriptions to avoid confusion with the dram) and the milligramme are the chief weights used. 1 gramme = wt. of 1 cubic centimeter (cc. ) of water at 4° c. 1000 grammes = 1 kilogram or ** kilo.** 1 kilogram of water = 1000 cc. = 1 liter. Conversion Factobs 1 gramme = 16.4 or approx. 16 grains. 1 grain = 0.064 gramme. 1 cubic centimeter = 16 minims. 1 minim = 0.06 cc. 1 liter = 1 quart (approx.). The liter and cubic centimeter are the principal units of volume used in medicine. ^ ss means one-half. 280 VOCATIONAL MATHEMATICS FOR GIRLS A micro-millimeter is used in measuring microscopical dis- tances. It is j^ mm. and is indicated by the Greek letter /lu To convert cc. into minims multiply by 15. To convert grammes into drams divide by 4. To convert cc. into ounces divide by 30. To convert minims into cc^ divide by 16. To convert grains into grammes divide by 15. To convert fluid drams into cc. multiply by 4. To conveH drams into grammes multiply by 4. 1 grain = 0.064 gramme. 2 grains = 0.1 gramme. 5 grains = 0.3 gramme. 8 grains =0.5 gramme. 10 grains = 0.6 gramme. 15 grains = 1 gramme. 1 milligramme = 0.01^ grain. Review Troy (apothecary) and avoirdupois weights, pages 43 and 276. EXAMPLES 1. A red corpuscle is 8 ft in diameter. Give the diameter in a fraction of an inch. 2. A microbe is ^5^00 ^^^^ ^ diameter. What part of a millimeter is it ? 3. Another form of microbe is ^^^^^ of an inch in diameter. What part of a millimeter is it ? 4. A bottle holds 48 cc. What is the weight of water in the bottle when it is filled ? 5. How many liters of water in a vessel containing 4831 grams of water ? 6. Give the approximate equivalent in English of the following : a. 48 grammes d, 8 kilos 6. 3.6 kilograms e. 3:9 grammes c. 3.5 liters /. 53 milligrammes ARITHMETIC FOR NURSES 281 7. Give the approximate equivalents in the metric system of the following : tt. 39 grains ^ e. 13 quarts b. 4 drams /. 2 gallons c. 7 fluid drams g, 39 minims d. 47 lb h, 8321 grains Approximate Equivalents between Metric and Household Measures • 1 teaspoonful = 4 cc. or 4 grams of water. 1 dessertspoonful = 8 cc. or 8 grams of water. 1 tablespoonful = 16+ cc. or 15+ grams of water. 1 wineglassful = 60 cc. or 60 grams of water. 1 teacupful = 180 cc. or 180 grams of water. 1 glassful = 240 cc. or 240 grams of water. EXAMPLES (Give approximate answers.) 1. What is the weight of two glassfuls of water in the metric system ? 2. What is the weight of a gallon of water in the metric system ? 3. What is the weight of three liters of water in the English system ? 4. What is the volume of four ounces of water in the metric system ? 5. What is the volume of twelve cubic centimeters of water in the English system ? 6. What is the volume of f 3 iii in the metric system ? 7. What is the volume of eighty grammes of water ? • 8. What is the weight of 360.1 cc. of water ? 9. What is the volume of 4 kilos of water ? 10. What is the weight of 6.1 liters of water ? 11. With ordinary household measures how would you obtain the following : 5 gm., m xv, 1.5 L., 25 cc, S ii, f 5 ss ? 282 VOCATIONAL MATHEMATICS FOR GIRLS METRIC SYSTEM EXAMPLES 1. Change the following to milligrammes : 8 gm., 17 dg., 13 gm. 2. Change the following to grammes : 13 mg., 29 dg., 7 dg., 21 mg. 3. Add the following : 11 mg., 18 dg., 21 gm., 4.2 gm. Express answer in grammes. 4. Add the following : 25 mg., 1.7 gm., 9.8 dg., 21 mg. Express answer in milligrammes. 5. The dose of atropine is 0.4 mg. What fraction of a gramme is necessary to make 25 cc. of a solution in which 1 cc. contains the dose ? 6. Give the equivalent in the metric system of the following doses : a. Extract of gentian, gr. ^. 6. Tincture of quassia, 3 i. c. Tincture of capsicum, m iii. d. Spirits of peppermint, 3 L e. Cinnamon spirit, m x. /. Oil of cajuput, m xv. g. Extract of cascara sagrada, gr. v. h. Eluid extract of senna, 3 ii. i. Agar agar, 5 ss. 7. Give the equivalent in the English system of the follow- ing doses : a. Ether, 1 cc. b. Syrup of ipecac, 4 cc. c. Compound syrup of hypophosphites, 4 cc. ARITHMETIC FOR NURSES 283 d, Pancreatin, 0.3 gm. e. Zinc sulphate, 2 gm. /. Copper sulphate, 0.2 gm. gr. Castor oil, 30 ee. h. Extract of rhubarb, 0.6 gm. i. Purified aloes, 0.5 gm. DOSES Since all drugs are harmful or poisonous in sufficiently large quantities, it is necessary to know the least amount needed to produce the desired change in the body — the minimum dose. This has been ascertained by careful and prolonged experiments. Similar experiments have told us the largest amount of drug that one can take without producing dangerous effect — the maximum dose. On the average, children under 12 years of age require smaller doses than adults. To determine the fraction of an adult dose of a drug to give to a child, let the child's age be the numer- ator, and the sum of the child's age plus twelve be the denomina- tor of the fraction. For infants under one year, multiply the adult dose by the fraction ^ge in months , •^ 150 To illustrate: How much of a dose should be given to a child of four ? Age of child = 4. Age of child + 12 = 16. Fraction of dose ^ = J. Ans. J of a dose. EXAMPLES 1. What is the fraction of a dose to give to a child of 8 ? 2. What is the fraction of a dose to give to a child of 6 ? 3. What is the fraction of a dose to give to a child of 3 ? 4. What is the fraction of a dose to give to a child of 10 ? 284 VOCATIONAL MATHEMATICS FOR GIRLS 5. If the normal adult dose of aromatic spirits of ammonia is 1 dram, what is the dose for a child of 7 ? 6. If the normal adult dose of castor oil is one-half ounce, what is the dose for a child of 6 ? 7. If the normal adult dose of epsom salts is 4 drams, what is the dose for a child of 4 ? 8. If the normal adult dose of strychnine sulphate is ^^^ gr., what is the dose for a child of 8 ? 9. If the normal adult dose of ipecac is 15 grains, what is the dose for a child of 11 ? 10. If the normal adult dose of aromatic spirits of ammonia is 4 grammes, what is the dose for a child of 5 m'onths ? U. If the normal adult dose of ipecac is 1 gramme, what is the dose for a child 10 months old ? 12, The normal adult dose of strychnine sulphate is 3.2 mg. How much should be given to a child 2 years old ? STRENGTH OF SOLUTIONS A nurse should know about the strength of substances used in treating the sick. Most of these substances are drugs which are prepared according to formulas given in a book called a Pharmacopoeia, Preparations made according to this standard are called official preparations, and often have the letters U. S. P. written after them to distinguish them from patented preparations prepared from unknown formulas. Drugs are applied in the following forms : solutions, lini- ments, oleates, cerates, powders, lozenges, plasters, ointments, etc. An infusion is a liquid preparation of the drug made by extracting the drug with boiling water. The strength of an infusion is 5% of the drug, unless otherwise ordered by the physician. ARITHMETIC FOR NURSES . 285 The strength of a solution may be written as per cent or in the form of a ratio. A 10% solution means that in every 100 parts by weight of water or the solvent there are 10 parts by weight of the substance. This may be written in form of a fraction — ^^^ or ^. In other words, for every ten parts of solvent there is one part of substance. Since a fraction may be written as a ratio, it may be called a solution of one to ten, written thus, 1 : 10. EXAMPLES 1. Express the following per cents as ratios i 5%, 20%, 2%, 0.1%, 0.01%. Since per cent represents so many parts per hundred, a ratio may be changed to per cent by putting it in the form of a fraction and multiplying by 100. The quotient is the per cent. 2. Express the following in per cents : 1 : 4, 1 ; 3, 1 ; 6, 1 : 15, 1 : 25, 1 : 40. 3. Arrange the following solutions in the order of their strength : 3 %, 8 %, 24 %, 6 %, 1 : 10, 1 ; 14, 1 : 50, 40 %, 1 : 45, 50%. 4. Express the strength of the following solutions as per cents, and in ratios. a. 80 ounces of dilute alcohol contains 40 ounces of absolute alcohol. 6. 6 pints of dilute alcohol contains two pints of absolute alcohol. 5. Change the following ratios into per cents : 1 : 18, 1 : 20, 1:5, 1 : 35, 1 : 100. Arrange in order, beginning with the highest. 6. Change the following per cents to ratios: 33%, 12%, 15%, .5%, 1%. 7. Is it possible to make an 8 % solution from 4 % ? Ex- plain. 286 VOCATIONAL MATHEMATICS FOR GIRLS 8. Express the following strengths in terms of ratio : a. 25 CO. of alcohol in 100 cc. solution. b. 5 pints of alcohol in 3 qts. c. f S i contains f 3 iii. 9. Express the following strengths in terms of per cent : a. 50 cc. of , solution containing 5 cc. of peroxide of hydrogen. 6. f S iii of dilute alcohol containing ^ ii of pure alcohol. How to Make Solutions of Different Strengths from Crude Drugs or Tablets of Known Strengths Exact Method Illustrative Example. — How much water will be neces- sary to dissolve 5 gr. of powdered bichloride of mercury to make a solution of 1 part to 2000 ? Since the whole powder ia dissolved, 1 part is 5 gr. 2000 parts = 10,000 grains. 480 gr. = 1 oz. 82 oz. = 1 qt. ^m^ = 20{. Approz. 21 oz. or 1^ pints of water should be used to dissolve it. The above example may be solved by proportion, when x = no. oz. of water necessary to dissolve powder ; then wt. of powder : drug ::x: water. ,f ^ : 1 : : ac : 2000. ^ ^ 5j<J000 ^ 125 ^ 20| oz. Approx. 21 oz. 480 6 ' EXAMPLES Solve the following examples by analysis and proportion : 1. How much water will be required to dissolve 5 gr. of powdered corrosive sublimate to make a solution of 1 part to 1000? ARITHMETIC FOR NURSES 287 2. How much water will be required to dissolve a TJ-grain tablet of corrosive sublimate to make a solution 1 part to 2000 ? Illustrative Example. — How much of a 40 % solution of formaldehyde should be used to make a pint of 1 : 500 solution ? 480 minims = 1 oz. 7680 minims = 1 pint. ^^ = I62V iiiinims = amt. of pure formaldehyde necessary to make a pint of 1 : 600. Since the strength of the solution is 40 9^, 15/^ minims represents but ^ or f of the actual amount necessary. Therefore, the full amount of 40 % solution is obtained by dividing by |. 192 ^ X 2 = 1??= 38| minims to apmt. 6 To Determine the Amount of Orvde Drug tfecessary to Make a Certain Quantity of a Solution of a Given Strength To illustrate : To make a gallon of 1 : 20 carbolic acid solu- tion, how much crude carbolic acid is necessary ? 1 : 20 : : X : 1 gal. 1 : 20 : : 05 : 8 pints or 128 ounces. 20 X = 128 ounces. a; = 6| ounces crude carbolic acid. EXAMPLES 1. How much crude boric acid is necessary to make 6 pints of 5 % boric acid ? 5 : 100 : : a : 6 pts. 5 : 100 ::x: 576 drams. 100 x = 2880. X = 28.8 drams. 2. How much crude boric acid is necessary to make 2 quarts of 1 : 18 boric acid ? 288 VOCATIONAL MATHEMATICS FOR GIRLS 3. How much crude drug is necessary to make f S iii of 2 % cocaine? 4. How many T^grain tablets are necessary to make 2 gal- lons of 1 : 1000 bichloride of mercury ? ^ 5. How much crude drug is necessary to make O vi of 1 : 20 phenol solution ? 6. How much crude drug is necessary to make vii of 1 : 500 bichloride of mercury ? 7. How much crude drug is necessary to make iii of 1 : 10 chlorinated lime ? Hypodermic Doses Standard strong solutions and pills are kept on hand in a hospital and from these weaker solutions are made as required by the nurse for hypodermic use. This is done by finding out what part the required dose is of the tablet or sol.ution on hand. The hypodermic dose is not administered in more than 25 or less than 10 minims. The standard pill or solution is dissolved or diluted in about 20 minims and the fractional part, corresponding to the dose, is used for injection. To illustrate : A nurse is asked to give a patient ^kif S^^ strychnine. She finds that the only tablet on hand is -^ gr. How will she give the required dose ? TOTT "^ TIT == ^iir X 30 = T^. The required dose is ^^ of the stock pill. Therefore she dissolves the pill in 80 minims of water and administers 12 minims. The reason for dissolving in 80 rather than in 20 minims is to have the hypodermic dose not less than 10 minims. EXAMPLES 1. Express the dose, in the illustrative example, in the metric system. ^ Hospitals usually use 1 tablet for a pint of water to make 1 : 1000 solation. ARITHMETIC FOR NURSES 289 2. How would you give a dose -^ gr. strychnine sulphate from stock tablet -^ gr,? 3. How would you give gr. ^^y if only ^ grain were on hand? 4. How would you give gr. ^, if only ^grain tablets were on hand? 5. How would you give gr. ^, if only ^grain tablets were on hand? 6. How would you give gr. -^^^ if only y^grain tablets were on hand ? 7. How would you give gr. y^ ^^ atropine sulphate, if only y^grain tablets were on hand ? a How would you give gr. -^ of apomorphine hydrochloride if only iVgrain tablets were on hand ? To Estimate a Dose of a Different Fractional Part of a Grain from the Prepared Solution Nurses are often required to give a dose of medicine of a different fractional part of a grain from the drug they have. To illustrate : Give a dose of -^ gr. of strychnine when the only solution on hand is one containing -^ gr. in every 10 minims. Since ^ grain is contained in 10 minims, 1 grain or 30 x ^ grain is contained in 300 minims. Then, ^ of a grain is^of300 = |/5x300 = 12m. EXAMPLES 1. What dose of a solution of 60 minims containing -^ gr. will be given to get y^ gr. ? 2. Reckon quickly and accurately how much of a tablet gr. ^ should be given to have the patient obtain a dose gr. ^. 290 VOCATIONAL MATHEMATICS FOR GIRLS 3. What dose of a solution of m x containing gr. ^ morphine sulphate will be given to give gr. ^ ? 4. What dose of a solution of m xx containing gr. -^ strych- nine sulphate will be given to give gr. ^ ? 5. What dose of a solution of 1 cc. containing 0.1 cc. of the fluid extract of nux vomica will be given to give 0.06 cc. ? To Obtain a Definite Dose from a Stock Solution of Definite Strength To illustrate: To give a patient a ^grain dose when the stock solution has a strength of 1%. 1 ^0 solution means that each drop of the solution contains j^ part or —^ of strychnine. 100 ^ gr. is contained in as many drops as y^ is contained in it. A-T*7r=J^xlOO = 4. Therefore 4 drops of the 1 % solution contains '^ gr. EXAMPLES 1. To give ^ gr. strychnine from 2 % solution. 2. To give ^j gr. strychnine from solution containing in ten minims t^^ gr. 3. To give 3 gr. of caffeinic sodium benzoate from a 25 % solution. 4. To give Y^ir S^* ^^ atropine from 1 % solution. 5. To give j^ gr. of strychnine from ^ % solution. 6. To give -^jf gr. atropine from solution containing in ten minims -^ gr. ARITHMETIC FOR NURSES 291 n Temperature The temperature of the body is due to the combined activity of all its various systems but is regulated chiefly by the skin and circulatory system. It remains very nearly constant in the nor- mal person, in spite of the variations of the outdoor temperature. A variation of more than one degree from the normal tempera- ture, that is, above 99^° F. or below 97^® F., may be regarded as a sign of a disease. The temperature is obtained by means of a small thermometer — called a clinical thermometer. See Appendix, page 337, for descrip- tion of the different thermometers. Temperature readings are usually expressed in the Fahrenheit scale, but scientific data gathered in laboratories are expressed according to the Centi- grade scale. Therefore, we should be able to change readings from one scale to another. Fahrenheit readings may be obtained by adding 32® to f of the Centigrade reading. This rule may be abbreviated into a formula as follows ; i^=|C+32°, where F = Fahrenheit reading, C = Centigrade reading. • Centigrade readings may be obtained by sub tracting 32° from the Fahrenheit and taking ^ of the remainder. This may be abbreviated into a formula as follows : HTr 0=1(2^-32°). Clinical Thbbmom- BTEB EXAMPLES 1. Albumin is coagulated by heat at 165** F. What is the degree Centigrade ? 292 VOCATIONAL MATHEMATICS FOR GIRLS 2. When milk is heated above 170® F., the albumin coagulates and forms a scum on the milk. To what degree on Centigrade scale does this correspond ? 3. Egg albumin (white of egg) coagulates at 138** F. At what degree on the Centigrade scale ? 4. Milk is pasteurized by bringing milk in the bottle to a temperature of 165® F. To what degree on the Centigrade scale? 5. " Gentle heat " is a term used to denote the temperature between 32® to 38° C. What are the corresponding degrees on the Fahrenheit scale ? Baths (Change the following temperatures to Centigrade scale.) A bath with a temperature between 33® and 65® F. is known as a cold bath. A bath with a temperature between 65® and 75® F. is known as a cool bath. A bath with a temperature between 75® and 85® F. is known as a temperate bath. A bath with a temperature between 85^ and 92® F. is known as a t^id bath. A bath with a temperature between 92® and 98® F. is known as a warm bath. A bath with a temperature between 98® and 112® F. is known as a hot bath. Medical Chart (Graph) (See Graphs in the Appendix.) In order to follow the condition of a patient from day to day, the temperature, the pulse beats, and respirations are recorded morning and night on a special ruled chart. The name of the patient is placed on each chart. ARITHMETIC FOR NURSES 293 NAMtf.. WARD.. OATS.. 'Oav«« 1 VffMty 41* 40* |3r| ■ 'T - ■ lor . . TOP « * >' i IT i K k! z i ISl i I0»» ^ " ■'" • IT * •r 180 140 ISO 110 110 MA - '. - - * — t 4... 1 1 • < WW to ■ . TO 00 M ' I n-i 1 40 40 » ■ , _ . ■ 1 . m m n . — i ' ■ ■ 294 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES Chart the following case of pneumonia : Morning Evening 2 day 102° 104° 3 day 102.6° 106° 4 day 102.4° 104.2° 5 day 102.4° 103.6° 6 day 102.4° 104° 7 day 102.4° 104.4° 8 day 101.8° 103° 9 day 102.9° 104° 10 day 102° 102.8° 11 day 98.4° 98.5° 12 day 97.4° 98.2° 13 day 97.4° 98.2° 14 day 98.2° 98.4° PROBLEM^ IN HOUSEHOLD CHEMISTRY Bacteria are low forms of vegetable and animal life, and some are capable of producing disease. Chemicals that are employed to destroy bacteria are known as germicides. Those which limit the growth or destructive power of bacteria are called antiseptics. Deodorants remove or neutralize unpleasant odors. 1. Bacteria multiply in all temperatures between 2® and 70° C. What are the temperatures in the Fahrenheit scale within which bacteria will grow ? 2. Creolin is used as a germicide and deodorant for offen- sive wounds in solutions of from 2 to 5 %. The creolin must never be added to water over 98° C, as its strength is impaired. What is the corresponding temperature on the Fahrenheit scale ? 3. The most important medium or preparation for growing bacteria is nutrient bouillon. It is made of the following : ARITHMETIC FOR NURSES 295 Meat extract 5 grams Peptone 10 grams Salt 6 grams Water inter Wliat per cent of each ? 4. A sugar bouillon culture is used for artificially cultivat- ing bacteria. It is made by adding 1 % of glucose to nutrient bouillon. How many grams of glucose to a liter of solution ? 5. Carbolic acid is bought by hospitals in a 95 ^J solution and diluted as required. A solution of carbolic acid 1 : 20 is used to destroy germs. How much 95 ^o solution will be required to make 5 gallons 1 : 20 ? 6. 1 : 1000 solution means how many grams to the gallon ? 7. A normal salt solution is made by dissolving 9 grams of salt to the quart. How many teaspoonfuls to the quart ? How many grains to the quart ? 8. What is the ratio of a pure drug ? What is the per- centage of purity of a pure drug ? 9. If I desire to make a lotion of 1 : 1000 corrosive subli- mate, how much of the substance would be added and how much water used ? 10. How much water and corrosive sublimate are required for a gallon of the following strengths? a. 1:2000. d 1:20,000. 6. 1:4000. e. 1:100,000. c. 1:10,000. /. 1:150,000. 11. A saturated solution of boric acid may be* made by dis- solving 3 V to pint (0 i) of water. What is the per cent of the saturated solution ? 12. A saturated solution of KMn04 may be made by dis- solving I i to i ? What is the per cent ? 13. How much of the saturated solution should be added to water i to make 1 % solution ? 296 VOCATIONAL MATHEMATICS FOR GIRLS Water Analysis Every nurse should be able to interpret a biological and chemical analysis of water. Terms used in Chemical and Bacteriological Reports The following brief explanation of the terms used in chemi- cal and bacteriological examinations of water is given in order that the reports of analyses of samples may be clearly imder- stood. As the quantities to be obtained by analyses are usu- ally very small, they are ordinarily expressed in parts per million (p. p. m.), and always by weight. Turbidity of water is caused by fine particles such as clay, silt, and microscopic organisms. Sediment is self-explanatory. The amount and nature of the sediment are usually noted. Color is measured by comparing the sample with artificial standards made by dissolving certain salts in distilled water, or sometimes with colored gla^s disks. The color of large lakes is usually below 0.10. Odor, This requires no explanation. Residue on Evaporation^ or Total Solids, indicates the total solid matter, both organic and inorganic, in 1,000,000 parts of water. The determination is made by placing about 100 grams of water in a platinum dish and weighing the whole accurately. The water is then evaporated to dryness by mod- erate heat and the dish again weighed ; the difference between this and the weight of the empty dish gives the total solids in the water. The dish is then heated red hot, to bum out the organic matter, when the weight of the remaining ash gives the inorganic or fixed solids. The loss on ignition, sometimes reported, is a measure of the organic solids. Ammonia. Ammonia in water indicates the presence of organic matter in an advanced stage of decay, and although ARITHMETIC FOR NURSES 297 the amount is small, it affords a valuable indication of what is going on in the water. It is determined in two forms, called "free" and "albuminoid." Free Ammonia is that which has actually been set free in the water in the process of decay of organic matter, while Albuminoid Ammonia is that which has not yet been set free, but which is liable to be freed under the action of the oxygen in the water. The sum of the two gives an indication of the total amount of organic matter in the water. Water which has 0.05 p. p. m. of free ammonia is probably pure, while if it has more than 0.1 p. p. m., it is perhaps dan- gerous. A low figure for albuminoid ammonia is 0.06 p. p. m., and a high one is 0.60. Chlorine in water usually represents sodium chloride, or common salt. It may be due to sewage pollution or to near- ness to the ocean. It is always found in natural waters, the normal amount decreasing from the seacoast inland. If the amount exceeds 20 p. p. m., it may cause corrosion in boilers and plumbing fixtures. Properly interpreted, the chlorine content is one of the most useful indexes of the extent of sewage pollution. Nitrogen is usually determined in the form of nitrates and nitrites, the former being the final result of decomposition, while the latter is the incomplete result of the same action. If an analysis shows the ammonia to be low and the nitrates high, it indicates that the water has become completely puri- fied, while the reverse indicates that the decaying process is going on and the water is dangerous. In good drinking water the nitrates may be as high as 1 or 2 p. p. m., while the nitrites, if present, are practically always a sign of pol- lution. Oxygen Consumed. This is the amount of oxygen absorbed by the water from potassium permanganate. As the oxygen is absorbed by the organic matter present in the water, the amount consumed gives a measure of the amount of impurities 298 VOCATIONAL MATHEMATICS FOR GIRLS contained in it. Less than 1 p. p. m. indicates probable pur- ity, while as high as 4 or 5 p. p. m. indicates danger in drink- ing water. Hardness, A water is said to be " hard " when it contains in solution the carbonates and sulphates of calcium or mag- nesium. When a hard water is used for washing, these salts have to be decomposed by soap before a lather can be formed. In boilers, a hard water forms scale. Hardness is expressed by the number of parts of calcium carbonate in 1,000,000 parts of water. Rain water has a hardness of about 5, and river waters of from 50 to 100. Iron may be troublesome in a water used for domestic pur- poses if it is present in quantities greater than 0.3 to 0.5 p. p. m. Alkalinity or Temporary Hardness is that part of the total hardness which is due to carbonates removable by boiling, thus causing the formation of scale. For purposes of softening water for boiler use, it is necessary to know both the total hardness and the alkalinity. Bacteria, While it is obvious that the quality of a water of turbid appearance and unpleasant odor is suspicious, it does not follow that it is dangerous, nor is a water which is entirely free from color and odor necessarily a safe drinking water, for epidemics of typhoid have been caused by such. The bacteri- ological examination of water, by which the number of bacteria present in one cubic centimeter (1 cc.) is determined, is there- fore an important part of an analysis. As a general statement, it may be said that fresh water con- taining less than 100 bacteria per cc. is pure, that water contain- ing 500 bacteria per cc. should be viewed with suspicion, and that water containing 1000 bacteria per cc. is undoubtedly con- taminated. In considering these figures with relation to a water supply, it must be remembered that all natural surface waters contain some bacteria and that, except where there is pollution, the greater part of them are absolutely harmless. ARITHMETIC FOR NURSES 299 The bacteria are so small that they may be seen only with the aid of a high-powered microscope. In order to count them a culture jelly of gelatine, albumin, and extract of beef is prepared and 1 cc. of the water is thoroughly mixed with 10 cc. of the culture jelly, a small measured portion of this mixture then being poured in a thin layer on a sterilized plate to harden. Each bacterium eats and multiplies to such an extent that in about forty-eight hours a visible colony is produced. From a count of these colonies within a measured area of the plate the number of bacteria in the original 1 cc. of water is determined. Different species of bacteria may be detected by the use of different media for development, or they may be found by further examination with the microscope. The well-known colon bacillus (B. coli)y which, although harmless itself, is an indication of sewage pollution, is detected by the gas which it produces in a closed tube. As B. coli are found in practically all warm-blooded animals and sometimes in fish and elsewhere, the finding of a few in large samples of water, or their occa- sional discovery in small samples, is of no special significance ; but if they are found in a larger proportion in small samples and in considerable numbers in larger ones, sewage pollution is indicated. EXAMPLES 1. If a bacterium multiplies tenfold every half hour in a person's mouth, how many will be produced in twenty-four hours ? 2. A sample of water contains 0.24 parts per million of free ammonia. How many parts per 100,000 ? 3. A sample of water contains 1.1 parts per million of iron. How many parts per 100,000 ? 4. A sample of water contains 10 parts per million of lime. How many parts per 10,000 ? 300 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES ON ANALYSES OP WATER (Parts in 100,000) Bbsidub on Evaporation Ammonia H 1 n o .72 NiTBOGEN AS • OXYGBN •-* Consumed < n 1.3 Total Loss on Igni- tion Fixed Free Albuminoid CO 1 00 1 Total In Solu- tion In Sus- pen- sion 1 a. 4.00 1.65 2.35 .0026 .0190 .0156 .0034 .0030 .0001 .0160 h. 4.65 2.00 2.65 .0028 .0172 .0148 .0024 .68 .0030 .0000 .28 1.3 .0080 c. 3.85 1.15 2.70 .0014 .0148 .0130 .0018 .68 .0000 .0000 .31 1.1 .0080 d. 4.20 1.50 2.70 .0062 .0140 .0128 .0012 .71 .0000 .0000 .32 1.1 .0050 e. 4.15 1.35 2.80 .0018 .0170 .0152 .0018 .71 .0000 .0000 .26 1.0 .0080 /. 5.00 1.75 3.25 .0014 .0162 .0142 .0020 .73 .0000 .0000 .36 1.0 .0120 ff- 4.36 1.60 2.75 .0020 .0178 .0160 .0028 .70 .0010 .0001 .24 1.3 .0080 h. 4.10 1.15 2.95 .0018 .0162 .0136 .0026 .71 .0010 .0001 .24 1.3 .0100 1. Give the number of parts of free ammonia in 10,000 in a. 2. Give the number of parts of nitrates in 10,000 in b, 3. Give the number of parts of nitrites in 10,000 in d. REVIEW EXAMPLES 1. Give the number of cubic centimeters of water you would measure out to get the following : a. 70 gm. b, 11 kg. c. 0.4 gm. d. 61 mg. 2. How much would the following amounts of water weigh ? a. 9 1. b, 4.7 cc. c. ^ 1. d. 48 cc. 3. If the dose of aromatic spirits of ammonia is 30 minims, what is the dose for a child 6 years old ? 4. Give the approximate equivalents in household measures of the following : a. 7 drams c. 4 ounces e, 12 fluid ounces 6. 36 grams d, 90 minims /. 3 fluid drams ARITHMETIC FOR NURSES 301 5. Give the approximate equivalents in household measures of the following : a. 1500 cc. c. 3 liters e. 1 gramme b, 11 CO. d, 0.003 grain /. 0.008 gramme 6. How many grammes in 3 ounces of 1 % solution ? 7. How many drams in 1 gallon of 1 ; 50 solution ? 8. How many grammes in a liter of 10 % solution ? 9. How many grammes in 5 liters of 1 : 25 solution ? 10. How many teaspoonf uls of pure carbolic acid in a gallon of 1 % solution ? 11. How many drops (minims) of carbolic acid in a quart of 1 : 1000 solution ? 12. A basin of rain water has a temperature of 94® F. Give the equivalent on the Centigrade scale. 13. A cool bath registers a temperature of 26® C. Give the equivalent on the Fahrenheit scale. 14. A dose of ipecac is 20 to 30 grammes. What is the dose for a child of seven years ? 15. A dose of 1 : 500 solution means how many grammes to a quart ? 16. Given a 5 % solution of- silver nitrate, how would you make a gallon of 1 : 5000 solution ? 17. How would you make a gallon of 3 ^J? solution of acetic acid from the pure acid ? 18. How would you make two quarts of 5 % solution of car- bolic acid from pure acid ? (Consider pure acid 95 % .) 19. A 1 : 50 solution is used for disinfecting wounds. How would you make a gallon of this fluid from standard solution ? (Consider standard strength about 40 %.) 302 VOCATIONAL MATHEMATICS FOR GIRLS 20. A 2^0 solution of boric acid is used for eye and ear irrigations. How much boric acid will be necessary to make a quart of the solution ? 21. Give the approximate equivalents of metric and apothecaries' measures of the following : a. 31 cc. d. 50 minims b. y^^ gram e, 5 pints c. 1.6 gram /. 101 cc. 22. If the pharmacy nurse buys 3 oz. of trional, how many powders of 10 grains each can she make ? 23. The dose of the tincture of opium is 0.5 cc. ; 10 cc. of the tincture contains 1 gm. of opium ; 12 % of opium is morphine. How many milligrams of morphine in one dose of the tincture ? 24. a. Convert the following to milligrams : 5 dg. and 0.27 gm. h. Convert the following to grams : 483 dg. and 7 mg. 25. How much alcohol (15 % strength) will be necessary to make a quart of alcohol containing 80 % volume of absolute alcohol? 66%? 37%? 75%? 26. If lactic acid is composed of 75 % of absolute acid, how much absolute acid in a pound of the official preparation ? 27. Diluted alcohol contains 41.5 % absolute alcohol. How much absolute alcohol in a gallon of dilute alcohol ? 28. The dose of morphine sulphate is 0.008 gm. What is the dose for a baby 7 months old ? 29. The dose of camphorated tincture of opium (paregoric) is f 3 i. What is the dose for a baby 6 months old ? 30. Hands and arms are often disinfected by washing in a solution of permanganate of potash (two ounces to four quarts of water) followed by immersion in a solution of oxalic acid (eight ounces to four quarts of water). What is the percentage of each ? ARITHMETIC FOR NURSES 303 31. Adhesive iodoform gauze is made by saturating sterilized gauze in the following solution : Iodoform 22 grams Resin 10 grams Glycerine 6 cc. Alcohol 26 cc. (Consider specific gravity of alcohol and glycerine as 1.) What per cent of each ? Give quantity in English system. 32. How much of each ingredient should be used in prepar- ing a pound of the following mass ? Zinc oxide 6 parts Gelatine 5 parts Glycerine 12 parts Water 10 parts 33. What per cent of the following solution is atropine sulphate? Atropine sulphate 1^ gr. Water J fluid ounce 34. What amount of carbolic acid crystals is used to make 4 oz. of 3 % carbolized petrolatum ? 35. What per cent of the following solution is boric acid ? Boric acid 18 gr. Water 1 oz. 36. How much bichloride of mercury is required to make 1 qt. of a 1 : 25,000 solution ? 37. How much potassium permanganate will be necessary to make a pint of a 1 : 1000 solution ? PART VI — PROBLEMS ON THE FARM CHAPTER XV Every young person who lives on the farm has more or less to do with the bookkeeping and the arithmetic connected with the selling of the eggs, milk, and other products. Very few of the men on the farm have the time or the inclination to do this work, and it is usually performed by the wife or daughter. EXAMPLES 1. I sold 16 dozen eggs at 30 cents a dozen and took my pay in butter at 40 cents a pound. How many pounds did I receive ? 2. A dealer bought 16 cords of wood at $ 4 a cord and sold it for $ 96. Find the gain. 3. Three men bought a farm. Henry paid $ 1135.75, Philip $2400.25, and Carl as much as Henry and Philip. Find the value of the farm. 4. A farmer divided his farm as follows : to his elder son he gave 257f acres, to his younger son 200^^ acres, and to his wife as many acres as to his two sons. How many acres in the farm ? 5. One farm contains 287f acres and another 244J acres. Find the difference between them. 6. One bin contains 165^ bushels of grain and the other bin 184y^Tj^ bushels. How many bushels more does the larger bin contain than the smaller ? 304 PROBLEMS ON THE FARM 305 7. From a farm of 375^ acres, 84^^ acres were sold. How many acres remained ? 8. A farmer owning 67f acres of land sold 28^ acres and afterwards bought 14J acres. How many acres did he then own? 9. A farm contained 132 acres, one-eighth of which is woodland, one-sixth is pasture, and the remainder is culti- vated. What part of the farm is cultivated? How many acres are cultivated ? 10. From four trees, 14| barrels of apples were gathered. One man bought 5^ bbl., another S\ bbl. How many barrels remained ? 11. I owned two-fifths of a farm and sold three-fourths of my share for $ 1360. Find value of the whole farm. 12. I bought 5 loads of potatoes containing 33^ bushels, 27f bushels, 40^ bushels, 35^ bushels, and 29J bushels. I sold 12f bushels to each of three men, and 25^ bushels to each of four men. How many bushels were left ? 13. If two-thirds of a farm costs $2480, what is the cost of the farm ? 14. Mr. Thomas bought 168 sheep at $5.50 a head. He sold three-sevenths of them at $ 6 a head, and the remainder at $ 7 a head. Find the gain. 15. A farm is divided into four lots. The first contains 30^ acres, the second 42|^ acres, the third 35^ acres, the fourth 28f acres. How many acres in the farm ? 16. A farmer sold sheep for $ 62.50, cattle for $ 102.60, a horse for $ 125.75, and a plow for $ 18.25. How much did he receive ? 17. Farmer Blake raised 114 bushels of apples and 73f bushels of pears. How many more bushels of apples than pears did he raise ? 306 VOCATIONAL MATHEMATICS FOR GIRLS 18. A farmer paid $ 78 for a cow, $ 165 for a horse. How much more did the horse cost than the cow? 19. Mr. Borden has 450^2^ acres of woodland and sells 304f acres. How much has he left ? 20. Mr. Sherman bought ten acres of land at $ 65 an acre and sold it for $ 24.60 an acre. How much did he lose ? 21. A's farm contains 265f acres, B's 43^ acres. What is the difference in the size of their farms ? 22. Mr. Grover had 110 acres of land, and sold 7^ acres. How many acres had he left ? 23. Mr. Dean sold one-third of his farm to one man, one- fourth to another, and one-eighth to another. What part had he left ? 24. I paid $ 365.75 for a horse, and sold him for four-fifths of what he cost. What was the loss ? 25. How many bushels of grain can be put into 16 bags, if they hold 2| bushels each ? 26. A farmer carries 35 bushels of apples to market. What is half this load worth at 75 cents a bushel ? 27. I paid $76.50 for 18 sheep. What was the average price ? 28. Mr. Piatt gave 435 acres of land to his sons, giving each 72^ acres. How many sons had he ? 29. If 4^ bushels of potatoes were bought for $ 3.60, how many bushels can be bought for $ 10.80 at the same price per bushel ? 30. Mr. White paid $ 16.25 for 2^ cords of wood. How many cords could he buy for $ 74.75 at the same price per cord? 31. A father divided 183 acres of land equally among his sons, giving to each 45f acres. How many sons had he ? PROBLEMS ON THE FARM 307 FARM MEASURES (Review Mensuration and Table of Measures.) 1. If a bushel of shelled corn contains IJ cubic feet, how many bushels in a bin 8' x 4' x 2' 6" ? 2. A bushel of ear corn contains 2^ cubic feet. How many bushels in a crib 10' x 4' 3" x 2' 4" ? 3. A ton of tame hay contains 512 cubic feet. How many tons in a space 14' x 12' x 13' ? 4. A ton of wild hay contains 343 cubic feet. How many tons in a space 28' 6" x 18' 9" x 13' 5" ? 5. A bushel of potatoes contains 1^ cubic feet. How many bushels in a bin 8' 6" x V 5" x 9' 3" filled with potatoes ? 6. How many bushels of com on the ear in a pointed heap 12' X 8' and 6' high ? 7. How many bushels of com in a circular crib with a diameter 12' 6" and a height 8' ? 8. How many gallons of water in a rectangular trough 6' 3" X 2' 6" X 3' 4" ? (Consider a gallon | cubic foot.) 9. How many acres in 694 sq. rods ? 10. A 60-acre piece of land, half a mile across, is 6' 8" higher on one side than the other. How much of a fall (grade) to the rod ? 11. How many bushels of corn in a rectangular crib with sloping sides 16' long, 7' high and 4' 6" wide at the bottom and 6' 8" wide at the top ? ENSILAC^E PROBLEMS 1. A farmer with the purpose of filling his silo with com began the preparation of one acre of land for planting : 8 loads of stable fertilizer were used in dressing the land. What is the average number of square rods a load will fertilize ? 308 VOCATIONAL MATHEMATICS FOR GIRLS 2. The field was plowed in a day. Mr. A receives, when working for others, 20 cents per hour for his horses and 15 cents per hour for his own work. How much is his time worth for the day of 10 hours ? 3. Mr. A paid $ 12 for his plow and two extra points. The regular price without extras was $ 10.50. Mr. A broke a plow point on a rock. How much was the loss ? 4. It took three-fifths as long to harrow the field (see ex- ample 2) as to plow it. If the work was begun at 7 o'clock in the morning, at what time would the com piece be harrowed ? (Noon hour from 12 m. to 1 p.m.) • 5. Mr. A bought seed corn at $1.25 per bushel. What did the seed cost, 12 quarts being the amount used ? 6. He bought 4 one-hundred-pound bags of fertilizer at $ 1.40 per hundred. How much did the fertilizer cost ? 7. Mr. A is agent for Bradley fertilizers and receives a commission of 10 % on what he sells for the company. How- much must he sell to receive a commission equal to the cost of fertilizer used on his own corn piece, and also the expense of hauling from the railroad station, which amounted to $ 2.50 ? 8. Mr. A hires a man to plant his corn with a horse planter. He pays $ 2 for the planting, which is at the rate of 30 ^ per hour. How long did it take ? 9. Mr. A cultivated his com three times, each time requir- ing about 8 hours. Besides this he and his hired man spent 3 days hoeing the com once. Which was more expensive, the hoeing or the cultivating ? How much ? 10. The corn was planted June 1st. It was ready for cut- ting September 1st. Some of the stalks had grown to a height of 6 ft. What was the average weekly growth ? 11. Mr. A's silo is rectangular, 10 ft. long, 10 ft. wide, and 20 ft. deep. The floor is cemented. How many sq. yd. of cement in the floor ? PROBLEMS ON THE FARM 309 12. If the lumber is 1 inch thick, how many board feet in one thickness of the walls ? 13. How many cubic feet of ensilage will the silo hold? How many cubic feet below the level of the barn floor, which is 5 ft. higher than the cemented floor of the silo ? 14. How many bushels of the cut and compressed corn stalks must hare been produced on the acre of land to fill the space ? 15. On September 1st a gang of men helped Mr. A fill the silo. Two men worked in the field cutting down the stalks at $ 1.50 per day each. Two men hauled to the barn with teams at $ 3 per day each. Two men, a cutting machine, and horses for power cost $ 7. One man leveled corn in the silo at $ 1.50 per day. What did Mr. A pay these men for the work of the day ? The next day the men with the cutting machine, one man with a team, and the man for the silo worked two hours to finish the work. Add this expense to that of the previous day. 16. A week later the ensilage had settled 8 feet and Mr. A filled the space with surplus corn. He and a helper hoisted it with a pulley in a two-bushel basket. How many times must he fill the basket ? 17. The mass was left to the fermenting process for two months. When Mr. A opens the silo, he begins feeding regu- larly to his 10 cows, giving each one-half bushel twice a day. At this rate when will the silo be emptied ? When should the ensilage be even with the barn floor ? 18. It is estimated that 1 ton of ensilage is equal in value to one-third of a ton of hay. If ensilage weighs 50 pounds per bushel, how many pounds of hay equal a feed of ensilage ? 19. How many tons of hay is the ensilage worth ? What is the value at $ 15 per ton ? 310 VOCATIONAL MATHEMATICS FOR GIRLS 20. Does it pay the farmer to raise ensilage ? Note. — Ensilage could not be used as a substitute for hay, but is ex- cellent as a milk producer when fed in moderate quantities. Cows like it better than hay. DAIRY PRODUCE Milk is graded according to the amount of cream (fat) in it. In addition to cream, it contains casein (cheese), milk sugar, and -about 84 % water. Milk is usually sold by the farmer by weight and the per cent cream. To illustrate : A sample of milk from a large can weighing 50 lb. contains 4 % cream. The large can contains 2 lb. of butter fat. EXAMPLES 1. A sample of milk from a cooler containing 48^ lb. tested 3f % butter fat. How much butter fat in the cooler ? 2. A cow gives 3J gallons of milk per day. If a gallon weighs 8f lb., what is the weight of milk per day ? per week ? per month ? 3. If the milk in example 2 contains 4.7 per cent of cream, how much butter fat does it yield per week ? per month ? 4. If butter fat is worth 26 cents a pound, how much is ob- tained per week from the butter fat in example 3 ? How much per month ? 5. Another cow gives 3^ gallons of milk a day that tests 4.6 % butter fat. Is it more profitable to keep this cow or the one in examples 2 and 3, and by how much ? 6. Skim milk from the butter fat is usually sold to feed the pigs at 5\ cents a gallon. Is it cheaper to sell milk at 5^ cents a quart or to make butter and sell it at 26 cents a pound and give the skim milk to the pigs ? PROBLEMS ON THE FARM 311 PROBLEMS ON EQUIPPING A COOPERATIVE CHEESE FACTORY Seventy-three farmers came together, and after the election of officers it was decided that a stock company of seventy-three shares should be formed, and each member bought a share at the rate of $ 75. Part of the money was used to erect a cheese factory, and the rest was deposited in a bank and drawn out as it was needed to run the business until the sale of the products should be sufficient to supply money for carrying on the business and paying a small per cent on the money invested by each man. The following are the items of expense : Half acre of land at $0.03 a square foot. The building cost $ 2000 for material and work. Three large vats, $ 60 each ; 4 % discount. A Babcock tester $30 ; 2| </o discount. A small engine, belts, etc., $63.86. Whey trough and leads, $ 64 ; 4 9^ discount. Cheese press, $ 28 ; S^o discount. Rennet, salt, coloring, wood, cheesecloth, boiler, and piping, $15. Boxes, acid for test, etc., $67 ; S^o commission. Scales, weights, and weighing can, $27.86. A year's salary to the cheese maker, $620. The money left after these expenses was put at 3 9^ interest. PROBLEMS 1. How much money was put into the business ? 2. How much did the cheese maker average a week ? 3. What was the cost of the land ? 4. The man who sold the boiler and piping and also the engine and belt to the company received 6 % commission. What did he receive for his sales ? 5. The man who bought the whey trough, the leads, and the large vats received the discount as his commission. How much did he receive ? - 6. How much did the company that sold the whey trough, leads, and vats receive ? 312 VOCATIONAL MATHEMATICS FOR GIRLS 7. How much did the buying company pay out for the whey trough, leads, and vats ? a An agent sold the tester ; his commission was 6 %. How much did he receive ? 9. How much did the tester cost the company ? 10. The man who bought the press and material received 3 % commission. How much did the press and materials cost the company ? 11. How much did it cost to buy the land, build the fac- tory, and equip the plant ? 12. How much was left at interest ? 13. How much would the interest be for 3 years 6 months and 16 days ? 14. What is the interest for one year ? 15. What per cent of the whole investment is this interest ? 16. What per cent of the whole was left at interest ? PROBLEMS ON POULTRY One hen has to have five square feet of room in the house. It costs about ten cents a month to feed one hen. One dozen eggs sell on the average for 30 cents. One hen lays about 100 eggs per year. Broilers are sold at 25 cents a pound. Hens are sold for 15 cents a pound. An incubator costing $ 20 holds 150 eggs. Setting eggs cost $ 1.00 a dozen. Brooders cost $7.50. A small chicken coop costs $ 8.00. One hen costs 60 cents. Little chickens 1 week old cost 10 cents. Chicken wire, 6 ft. wide, costs 4^ cents per foot. PROBLEMS ON THE FARM 313 PROBLEMS 1. How large would the floor of my poultry house have to be for 30 chickens ? for 50 ? 80 ? 200 ? 2. If I have a poultry house the floor of which is 30 ft. by 50 ft., how many chickens can I put in it? 3. How much will it cost to keep the chickens one month ? one year ? 4. If I have one hen, how much does it cost me to feed her one year ? Suppose she lays 90 eggs, how much will I receive for them ? Does it pay me to keep the hen ? 5. Suppose I sold 25 broilers, 12 weighing 3 lb., 5 weigh- ing 5 lb., and the rest an average of 4 lb. How much would I receive for them ? If I had kept them 14 weeks, how much would they have cost me ? Would I gain or lose in keeping them ? How much ? 6. If I bought an incubator for $ 20, a brooder for $ 7.50, a chicken coop for $ 8, and 150 eggs to put into the incubator, how much did I pay in all ? 7. If from the 150 eggs only 139 were hatched and lived, how much would I receive for the little chickens when I sold them? 8. Suppose I had a chicken yard 100 by 250 ft. How many feet of wire would I need to fence it in ? How much would it cost me to put wire around it ? 9. How many eggs would I receive from 60 hens in one year ? If I sold all from 40 hens, how much would I receive for them ? 10. If I bought 50 little chickens, kept them 16 weeks, and then sold them, each weighing on the average 3 lb., how much profit did I make on them ? 314 VOCATIONAL MATHEMATICS FOR GIRLS POULTRY RAISING 1. A man wishes to build and stock a henhouse for $ 125. If he has $ 75, how much will he have to borrow ? How much interest will he have to pay for 1 year at 5 % ? 2. If he pays $ 20 for labor, three times as much for mate- rial, one-fourth as much for . apparatus as for material, how much will he have left ? How many hens could he buy with the remainder if each hen cost 50 cents ? 3. If it cost $ 1 per year to keep one hen, how much would it cost to keep all of his hens for 1 year ? for 5 years ? 4. If each hen lays 100 eggs a year, how many eggs would they yield in one year ? how many dozen ? 5. If he sold 400 dozen at 25 cents per dozen, how much would he receive for them ? 6. If he sold the remaining dozen " for setting " at 50 cents a dozen, how much would he receive for these ? How much did he receive for all his eggs ? 7. If it cost him the above amount to keep the hens for a year, how much did he gain from selling his eggs ? 8. If, from 100 dozen eggs sold for " setting," 9 chickens were hatched from each dozen, how many chickens were hatched in all ? 9. If it cost him 27 cents to raise oiie broiler,' how much would it cost to raise all the chickens for broilers ? 10. If for each pair of broilers he received $ 1.50, how much would his entire stock net him ? 11. After considering the cost of raising the broilers and the price received for them, what was his profit ? 12. After he had paid his interest, what was his net profit ? PROBLEMS ON THE FARM 315 REVIEW EXAMPLES 1. A crib of com is 12' wide, 34' long, and has an average depth of 11' of corn in it. How many bushels ? 2. How many bushels of oats in a bin 12' wide, 12' long, 18' deep ? 3. A freight car is 8' x 32' x 11'. If it is filled 3|' deep with apples for the cider mill, how many bushels in the car ? 4. At 26 cents a barrel, what is the car of apples worth ? {21 bu. = 1 bbl.) 5. A field of hay is 88 rods long and 64 rods wide. How much is it worth at $ 98 an acre ? 6. A cow gives 3f gallons of milk a day. The milk tests 4.2 % butter fat. At 27 cents a pound for butter, and 5 cents a quart for skim milk, how much is obtained a week from this cow ? 7. A flock of 200 hens averaged 135 eggs a year, and at the end of four years were sold for 10^ cents a pound, the average weight being 6^ lb. If the cost of feed for a year is $ 27.05 for the whole flock, what is the average gain per hen ? 8. A man receives S 35 a month. How much per hour, if the month contains 26 working days of 10 hours a day ? 9. What is the cost of 963^ bushels of oats at 47 cents per bushel ? 10. If I buy 125 bushels of com at 41| cents per bushel and sell it at 62^ cents a bushel, how much do I gain ? APPENDIX METRIC SYSTEM The metric system is used in nearly all the countries of Continental Europe and among scientific men as the standard system of weights and measures. It is based on the meter as the unit of length. The meter is supposed to be one ten- millionth part of the length of the meridian passing from the equator to the poles. It is equal to about 39.37 inches. The unit of weight is the gram^ which is equal to about one thirtieth of an ounce. The unit of volume is the liter, which is a little larger than a quart. Measures of Length 10 millimeters (mm.) . . . . =1 centimeter cm. 10 centimeters =1 decimeter dm; 10 decimeters =1 meter m. 10 meters =1 dekameter Dm. 10 dek3,meters =1 hektometer Hm. 10 hektometers =1 kilometer Em. Measures of Surface {not Land) 100 square millimeters (mm.) . . = 1 square centimeter . . sq. cm. 100 square centimeters . . . . = 1 square decimeter . . sq. dm. 100 square decimeters . . . . = 1 square meter sq. m. . Measures of Volume 1000 cubic millimeters (mm.) . . = 1 cu. centimeter . . . cu. cm. 1000 cubic centimeters . . . . = 1 cubic decimeter . . . cu. dm. 1000 cubic decimeters . . . . = 1 cubic meter cu. m. I The gram is the weight of one cubic centimeter of pure distilled water at a temperature of 39.2^ F. ; the kilogram is the weight of 1 liter of water ; the metric ton is the weight of 1 cubic meter of water. 817 318 VOCATIONAL MATHEMATICS FOR GIRLS Measures of Capacity 10 miliaiters (ml.) =1 centiliter 10 centiliters =1 deciliter 10 deciliters =1 liter i . 10 liters =1 dekaliter 10 dekaliters =1 hektoliter 10 hektoliters . . . . ,» . . =1 kiloliter el. dl. L Dl. HI. Kl. Measures of Weight 10 milligrams (mg.) 10 centigrams . . 10 decigrams . . 10 grams . . . 10 dekagrams . . 10 hektograms . . 1000 kilograms = 1 centigram eg. = 1 decigram dg. = 1 gram g. = 1 dekagram ...... Dg. = 1 hektogram Hg. = 1 kilogram Kg. = 1 ton T. 1 meter 1 centimeter 1 millimeter 1 kilometer Ifoot 1 inch Metric Equivalent Measures Measures of Length - 39.87 in. = 3.28083 ft.= 1.0936 yd. = .8937 inch = .03937 inch, or ^ inch nearly = .62137 mile = .3048 meter = 2.64 centimeters = 26.4 millimeters 1 square meter 1 square centimeter 1 square millimeter 1 square yard 1 square foot 1 square inch Measures of Surface 10.764 sq. ft. = 1.196 sq. yd. .166 sq. in. .00165 sq. in. .836 square meter .0929 square meter 6.462 square centimeters =646. 2 square millimeters Measures of Volume and Capacity 1 cubic meter = 36.314 cu. ft. = 1.308 cu. yd. = 264.2 gal. 1 cubic decimeter = 61.023 cu. in. = .0363 cu. ft. 1 cubic centimeter = .061 cu. in. 1 The liter is equal to the volume occupied by 1 cubic decimeter. METRIC SYSTEM 319 1 liter = 1 cubic decimeter = 61.023 cu. in. = .0353 cu. ft. = 1.0667 quarts (U. S.)=.2642 gallon (U. S.) = 2.202 lb. of water at 62° F. 1 cubic yard = .7645 cubic meter 1 cubic foot = .02832 cubic meter = 28.317 cubic decimeters = 28.317 Uters 1 cubic inch = 16.387 cubic centimeters 1 gallon (British) = 4.543 liters 1 gallon (U. S.) = 3.785 liters Measures of Weight 1 gram = 15.432 grains 1 kilogram = 2.2045 pounds 1 metric ton = .9842 ton of 2240 lb. = 19.68 c\\i;. = 2204.6 lb. 1 grain = .0648 gram 1 ounce avoirdupois = 28.35 grams 1 pound = .4536 kilogram 1 ton of 2240 lb. = 1.016 metric tons = 1016 kilograms Miscellaneous 1 kilogram per meter = .6720 pound per foot 1 gram per square millimeter = 1.422 pounds per square inch 1 kilogram per square meter = .2084 pound per square foot 1 kilogram per cubic meter = .0624 pound per cubic foot 1 degree centigrade = 1.8 degrees Fahrenheit 1 pound per foot = 1.488 kilograms per meter 1 pound per square foot = 4.882 kilograms per square meter 1 pound per cubic foot = 16.02 kilograms per cubic meter 1 degree Fahrenheit = .5556 degree centigrade 1 Calorie (French Thermal Unit) = 3.968 B. T. U. (British Thermal Unit) 1 horse power = 33,000 foot pounds per minute = 746 watts 1 watt (Unit of Electrical Power) = .00134 horse power = 44.24 foot pounds per minute 1 kilowatt = 1000 watts = 1.34 horse power =44,240 foot pounds per minute Table of Metric Conversion To change meters to feet multiply by 3.28083 feet to meters multiply by .3048 square feet to square metera . . . multiply by .0929 square meters to square feet . . . multiply by 10.764 320 VOCATIONAL MATHEMATICS FOR GIRLS To change square centimeters to square inches . multiply by .156 square inches to square centimeters . multiply by 6.452 inches to centimeters multiply by 2.54 centimeters to inches multiply by .8937 grams to grains multiply by 15.43 grains to grams multiply by .0648 grams to ounces multiply by .0853 ounces to grams multiply by 28.35 pounds to kilograms multiply by .4536 kilograms to pounds multiply by 2.2045 liters to quarts multiply by 1.0567 liters to gallons multiply by .2642 gallons to liters multiply by 8.78543 liters to cubic inches multiply by 61.023 cubic inches to cubic centimeters . . multiply by 16.387 cubic centimeters to cubic inches . . multiply by .061 cubic feet to cubic decimeters or liters multiply by 28.317 kilowatts to horse power multiply by 1.34 calories to British Thermal Units . . multiply by 3.068 EXAMPLES 1. Change 8 m. to centimeters ; to kilometers. 2. Reduce 4 Km., 6 m., and 2 m. to centimeters. 3. How many square meters of carpet will cover a floor which is 25.5 feet long and 24 feet wide ? 4. (a) Change 6.5 centimeters into inches. (b) Change 48.3 square centimeters into square inches. 5. A cellar 18 m. x 37 m. x 2 m. is to be excavated ; what will it cost at 13 cents per cubic meter to do the work ? 6. How many liters of capacity has a tank containing 5.2 cu. m.? 7. What is the weight in grams of 31 cc. of water ? 8. Give the approximate value of 36 millimeters in inches. 9. Change 84.9 square meters into square feet. 10. Change 23.6 liters to cubic inches. METRIC SYSTEM 321 11. Change 7.3 m. to millimeters; to centimeters; to kilometers. 12. Reduce 9.8 m. to kilometers ; to centimeters ; to milli- meters. 13. What is the difference in millimeters between 2.7 m. and 48.1 mm. ? 14. What part of a kilometer is 1.8 mm. ? 15. What part of a meter is 1.3 cm. ? 16. How many square centimeters are there in 26 square kilometers ? 17. How many square meters in 4 rectangular gardens, 3.4 Dm. long and 85.7 dm. wide ? la How many cubic meters in a wall 43 m. long, 8.4 dm. high, and 69 cm. wide ? 19. Reduce 869.7 eg. to milligrams ; to kilograms ; to grams. 20. What is the weight in grams of 48.7 cc. of water ? What is the weight in kilograms of 43.9 1. of water? 21. Mercury weighs 13.6 times as much as water ; what is the weight of 87.5 cc. of mercury ? Of 5 1. of mercury ? 22. A tank is 7.9 m. by 4.3 m. by 3.1 m. How many grams of water will it hold ? 23. What is the weight of 874 cc. of copper, the density of which is 89 g. per cubic centimeter ? 24. What is the capacity of a bottle that holds 5 kg. of alcohol, the density of which is 0.8 g. per cubic centimeter ? 25. What is the weight in grams of 56.8 cc. of alcohol? What is the weight in kilograms of 7 1. of alcohol ? 26. What part of a liter is 1.7 cc. ? ORAPHS A SHEET of paper, ruled with horizontal and vertical lines that are equally distant from each other, is called a sheet of cross-section, or coordinate, paper. Every tenth line is very distinct so that it is easy for one to measure off the horizontal and vertical distances without the aid of a ruler. Ruled or ORAFB SHOWIHa TUB VaBUTIOH in PrICB OT COTTON Y&lUr FOB A Skbibs of Yeabs coordinate paper is used to record the rise and fall of the price of any commodity, or the rise and fall of the barometer or thermometer. Trade papers and reports frequently make use of coordinate paper to show the results of the changes in the price of com- modities. In this way one can see at a glance the changes GRAPHS 323 and condition of a certain commodity, and can compare these with the results of years or months ago. He also can see from the slope of the curve the rate of rise or fall in price. If similar commodities are plotted on the same sheet, the effect of one on the other can be noted. Often experts are able to prophesy with some certainty the price of a commodity for a month in advance. The two quantities which must be employed in this comparison are time and value, or terms corresponding to them. The lower left-hand corner of the squared paper is generally used as an initial point, or origin, and is marked 0, although any other corner may be used. The horizontal line from this corner, taken as a line of reference or axis, is called the db- scissa. The vertical line from this corner is the other axis, and is called the ordinate. Equal distances on the abscissa (horizontal line) represent definite units of time (hours, days, months, years, etc.), while equal distances along the ordinate (vertical line) represent certain units of value (cost, degrees of heat, etc.). By plotting, or placing points which correspond to a certain value on each axis and connecting these points, a line is ob- tained that shows at every point the relationship of the line to the axis. EXAMPLES 1. Show the rise and fall of temperature in a day from 8 A.M. to 8 P.M., taking readings every hour. 2. Show the rise and fall of temperature at noon every day for a week. 3. Obtain stock quotation sheets and plot the rise and fall of cotton for a week. 4. Show the rise and fall of the price of potatoes for two months. . 5. Show a curve giving the amount of coal used each day for a week. FORMULAS Most technical books and magazines contain many formulas. The reason for this is evident when we remember that rules are often long and their true meaning not comprehended until they have been reread several times. The attempt to abbre- viate the length and emphasize the meaning results in the formula, in which whole clauses of the written rule are ex- pressed by one letter, that letter being understood to have throughout the discussion the same meaning with which it started. To illustrate : One of the fundamental laws of electricity is that the quantity of electricity flowing through a circuit (flow of electricity) is equal to the quotient (expressed in amperes) obtained by dividing the electric motive force (pressure, or expressed in volts, voltage) of the current by the resistance (expressed in ohms). One unfamiliar with electricity is obliged to read this rule over several times before the relations between the different parts are clear. To show how the rule may be abbreviated, Let / = quantity of electricity through a wire (amperes) E = pressure of the current (volts) B = resistance of the current (ohms) Then /= jrH-i?=:? B It is customary to allow the flrst letter of the quantity to represent it in the formula, but in this case I is used because the letter C is used in an- other formula with which this might be confused. Translating Rules into Formulas The area of a trapezoid is equal to the sum of the two parallel sides multiplied by one half the perpendicular distance between them. 824 FORMULAS 325 We may abbreviate this rule by letting A = area of trapezoid L = length of longest parallel side 3f = length of shortest parallel side JV= length of perpendicular distance between them Then A= (^L + M) x—, or The area of a circle is equal to the square of the radius multiplied by 3.1416. When a number is used in the formula it is called a constant, and is sometimes represented by a letter. In this case 3.1416 is represented by the Greek letter ir (pi). Let A = area of circle B. = radius of circle Then A — kx i^, or (the multiplication sign is usually left out between letters) Thus we see that a formula is a short and simple way of stating a rule. Any formula may be written or expressed in words and is then called a rule. The knowledge of formulas and of their use is necessary for nearly every one engaged in the higher forms of mechanical or technical work. * When two or more quantities are to be multiplied or divided or other- wise operated upon by the same quantity, they are often grouped together by means of parentheses ( ) or braces { }, or brackets [ ] . Any number or letter placed before or after one of these parentheses, with no other sign between, is to multiply all that is grouped within the parentheses. In the trapezoid case above, — is to multiply the sum of L and Jf, hence the parentheses. To prevent confusion, different signs of aggregation may be used for different combinations in the same problem. For instance, r= iir^r5(ra + r'2) + :^1 which equals 326 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES Abbreviate the following rules into formulas : > 1. One electrical horse power is equal to 746 watts. 2. One kilowatt is equal to 1000 watts. 3. The number of watts consumed in a given electrical circuit, such as a lamp, is obtained by multiplying the volts by the amperes. 4. The number of volts equals the watts divided by the amperes. 5. Number of amperes equals the watts divided by the volts. 6. The horse power of an electric machine is found by mul- tiplying the number of volts by the number of amperes and dividing the product by 746. 7. The speed at which a body travels is equal to the ratio between the distance traveled and the time which is required. 8. To find the pressure in pounds per square inch of a column of water, multiply the height of the column in feet by 0.434. 9. The amount of gain in a business transaction is equal to the cost multiplied by the rate of gain. 10. The selling price of a commodity is equal to the cost multiplied by the quantity 100 % plus the rate of gain. 11. The selling price of a commodity is equal to the cost multiplied by the quantity 100 % minus the rate of loss. 12. The interest on a sum of money is equal to the product of the principal, time (expressed as years), and the rate (ex- pressed as hundredths). FORMULAS 327 13. The amount of a sum of money may be obtained by adding the principal to the quantity obtained by multi- plying the principal, the time (as years), and the rate (as himdredths). 14. To find the length of an arc of a circle: Multiply the diameter of the circle by the number of degrees in the arc and this product by .0087266. 15. To. find the area of a sector of a circle : Multiply the number of degrees in the arc of the sector by the square of the radius and by .008727; or, multiply the arc of the sector by half its radius. Translating Formulas into Rules In order to understand a formula, it is necessary to be able to express it in simple language. 1. Oue of the simplest formulas is that for finding the area of a circle, A^^ir B? Here A stands for the area of a circle, B for the radius of the circle. IT is a constant quantity and is the ratio of the circumference of a circle to its diameter. The exact value cannot be expressed in figures, but for ordinary purposes is called 3.1416 or 3f . Therefore, the formula reads, the area of a circle is equal to the square of the radius multiplied by 3.1416. 2. The formula for finding the area of a rectangle is ^ = Xx W Here A = area of a rectangle L = length of rectangle W = width of rectangle The area of a rectangle, therefore, is found by multiplying the length by the width. 328 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES Express the facts of the following formulas as rules : 1. Electromotive force or voltage of electricity delivered by a current, when current and resistance are given: E = RI 2. For the circumference of a circle, when the length of the radius is given : C-2irR0VirD 3. For the area of an equilateral triangle, when the length of one side is given: ^ a*V3 4. For the volume of a circular pillar, when the radius and height are given: 5. For the volume of a square pyramid, when the height and one side of the base are given : 3 6. For the volume of a sphere, when the diameter is given : F= irD" 6 7. For the diagonal of a rectangle, when the length and breadth are given : 8. For the average diameter of a tree, when the average girth is known : n _ ^ IT 9. For the diameter of a ball, when the volume of it is known. 8/6« FORMULAS 329 10. The diameter of a circle may be obtained from the area by the following formula : D = 1.1283 X VA U. The number of miles in a given length, expressed in feet, may be obtained from the formula M= .00019 X F 12. The number of cubic feet in a given volume expressed in gallons may be obtained from the formula C = .13367 X G 13. Contractors express excavations in cubic yards; the number of bushels in a given excavation expressed in yards may be obtained from the formula C = .0495 X Y 14. The circumference of a circle may be obtained from the area by the formula (7=3.5446 xV3 15. The area of the surface of a cylinder may be expressed by the formula u4 = (0 X X) + 2 a When C = circumference L = length a = area of one end 16. The surface of a sphere may be expressed by the formula S^D'x 3.1416 17. The solidity of a sphere may be obtained from the formula S^D'x .5236 18. The side of an inscribed cube of a sphere may be ob- tained from the formula S = Bx 1.1547, where S = length of side, B = radius of sphere. 330 VOCATIONAL MATHEMATICS FOR GIRLS 19. The solidity or contents of a pyramid may be expressed by the formula F S = Ax—, where A = area of base, F = height of pyramid. 20. The length of an are of a circle may be obtained from the formula L= Nx .017453 R, where L = length of arc, N= number of degrees, B = radius of circle. 21. The loss in a transaction may be expressed by the formula X = c X r, where L = loss, c = cost, r = rate of loss. « 22. The rate of loss in a transaction may be expressed by the formula — = r. c 23. The cost of a commodity may be expressed by the formula c = -— , where S = selling price, 100 + r' . ' c = cost, r = rate. 24. The volume of a sphere when the circumference of a great circle is known may be determined by the formula 25. The diameter of a circle the circumference of which is known may be found by the formula FORMULAS 331 26. The area of a circle the circumference of which is known may be found by the formula Coefficients and Similar Terms When a quantity may be separated into two factors, one of these is called the coefficient of the other ; but by the coefficient of a term is generally meant its numerical factor. Thus, 4 & is a quantity comi>osed of two factors 4 and & ; 4 is a coef- ficient of 6. Similar terms are those that have as factors the same letters with the same exponents. Thus, in the expression, 6 a, 4 &, 2 a, 5 a&, 5 a, 2 &. 6 a, 2 a, 5 a are similar terms ; 4 &, 2 & are similar terms ; 5 ah and 6 a are not similar terms because they do not have the same letters as factors. 3 a&, 5 a&, 1 a6, 8 ah are similar terms. They may be united or added by simply adding the letters to the numerical sum, 17 ah. In the following, 86, 56, 8 a6, 4 a, a6, and 2 a, 8 6 and 56 are similar terms ; 3 a6 and a6 are similar terms ; 4 a and 2 a are similar terms ; 8 6, 8 a6, and 4 a are dissimilar terms. In addition the numerical coefficients are algebraically added ; in subtraction the numerical coefficients are algebraically sub- tracted ; in multiplication the numerical coefficients are alge- braically multiplied ; in division the numerial coefficients are algebraically divided. EXAMPLES State the similar terms in the following expressions : 1. 5 0?, 8 aa;, 3 aj, 2 ax. 6. 16 abCy 2 ahCy 4 abc, 2 ah, 2. 8a6c, 7c, 2a6, 3c, 8a6, 3a6. 9a6c. 7. 8a;, 6a?, 13a;y, 5aj, 7y. 3. 2 j9g, 5 j9, 8 g, 2p, 3 g, 5^. a 7y,2y,2xy,Sy,2xy. 4. 4:y, 5 yz, 2 y, 15 z, 5 z, 2 yz. '^ ^ o^ 5. 18 mn, 6 m, 5 w, 4 mn, 2 m, ^ 332 VOCATIONAL MATHEMATICS FOR GIRLS Equations A statement that two quantities are equal may be expressed mathematically by placing one quantity on the left and the other on the right of the equality sign (=). The statement in this form is called an equation. The quantity on the left hand of the equation is called the left-hand member and the quantity on the right hand of the equation is called the right-hand member. An equation may be considered as a balance. If a balance is in equilibrium, we may add or subtract or multiply or divide the weight on each side of the balance by the same weight and the equilibrium will still exist. So in an equation we may perform the following operations on each member without changing the value of the equation : We may add an equal quantity or equal quantities to each memr her of the equation. We may subtract an equal quantity or equal quantities from each member of the equation. We may multiply each member of the equation by the same or equal quantities. We may divide each member of the equation by the same or equal qvxintities. We may extract the square root of each member of the equation. We may raise each member of the equation to the same power. The expression, A = irB? is an equation. Why ? If we desire to obtain the value of B instead of A we may do so by the process of transformation according to the above rules. To obtain the value of B, means that a series of opersr tions must be performed on the equation so that B will be left on one side of the equation. (1) A = irB^ (2) - = J?2 (Dividing equation (1) by the coefficient of iP.) TT (8) ^— = B (Extracting tlie square root of each side of the equation.) FORMULAS 333 Methods of Representing Operations Multiplication The multiplication sign ( X ) is used in most cases. It should not be used in operations where the letter (x) is also to be em- ployed. Another method is as follows ; 2.3 a. 6 2a. 36 4aj.6a This method is very convenient, especially where a number of small terms are employed. Keep the dot above the line, otherwise it is a decimal point. Where parentheses, etc., are used, multiplication signs may be omitted. For instance, (a + 6) x (a — b) and (a + b){a — b) are identical ; also, 2'{x — y) and 2(x —y). The multiplication sign is very often omitted in order to simplify work. To illustrate, 2 a means 2 times a ; 5 xyz means 5 • a; • y • « ; x(a — b) means x times (a — 6), etc. A number written to the right of, and above, another (ic*) is a sign indicating the special kind of multiplication known as involution. In multiplication we add exponents of similar terms. Thus, iB* . aj» = aj2+' = a^ dbc • ab • d?h = a*b*c The multiplication of dissimilar terms may be indicated. Thus, a-6»c«a?«y-2; = abcxyz. Division The division sign (-^) is used in most cases. In many cases, however, it is best to employ a horizontal line to indicate division. To illustrate, — i— means the same as (a + b) -i- x-y (x — y) in simpler form. The division sign is never omitted. 334 VOCATIONAL MATHEMATICS FOR GIRLS A root or radical sign (V^> \/^ is a sign indicating the special form of division known as evolution. In division, we subtract exponents of similar terms. Thus, a^-5.a^ = ^ = a»-2^a? The division of dissimilar terms may be indicated. dbc Thus, (abc) -i- xyz = Qcyz Substituting and Transposing A formula is usually written in the form of an equation. The left-hand member contains only one quantity, which is the quantity that we desire to find. The right-hand member contains the letters representing the quantity and numbers whose values we are given eithjer directly or indirectly. To find the value of the formula we must (1) substitute for every letter in the right-hand member its exact numerical value, (2) carry out the various operations indicated, remem- bering to perform all the operations of multiplication and division before those of addition and subtraction, (3) if there are any parentheses, these should be removed, one pair at a time, inner parentheses first. A minus sign before a parenthesis means that when the parenthesis is removed, all the signs of the terms included in the parenthesis must be changed. Find the value of the expression 3a + 6(2a — 6 + 18), wherie a = 5, 6 = 3. Substitute the value of each letter. Then perform all addition or subtraction in the parentheses. 3x5 + 3(10 -3 + 18) 15 + 3(28-3) 15 + 3(26) 15 + 75 = 90 FORMULAS 335 EXAMPLES Find the value of the following expressions : 1. 2u4x(2 + 3^)x8, when^ = 10. 2. 8 a X (6 — 2 a) X 7, when a = 7. 3. 86 + 3c + 2a(a + & + c)-8,whena = 9; 6 = 11; c=:.13. 4. 8(aj + y), when a? = 9; y = ll. 5. 13 (a?— y), when a? = 27 ; 2^ = 9. 6. 24y + 82;(2 + y)-3y, when 2/ = 8; 2 = 11. 7. Q(6J»f+3iV^)4-2 0, when JW=4, JV^=5, Q = 6, = 8. a Find the value of X in the formula X = ^(^•^+^\ when JW= 11, JV^= 9, P = 28. 9. 0? = ?^^:^, when 71 = 5, m = 6,P=8, Q = 7. 10. Find the value of T in the equation y^S(^4-y)+7(a:-hy) ^^ ^^^ ^g U. 3a+4(6 — 2a + 3c)-c, when a = 4, 6 = 6, c = 2. 12. 5 j9 — 8 5 (|9 + r — /S) — g, when p = 5, g = 7, r = 9, /S = 11 13. ^ + ^_p2--3(/S' + * + 2>), whenp = 5, /S=8, « = 9. 14. a'-6'+c2, when a = 9, 6 = 6, c = 4. 15. (a + 6) (a + 6 — c), when a = 2, 6 = 3, c = 4. 16. (a^ - 6') (a* + 6^), when a = 8, 6 = 4. 17. (c»+^)(c'-- (i*), c = 9, d = 5. 18. Va« + 2a6 + 6^ when a = 7, 6 = 8. 19. Vc* — 61, when c^6. 336 VOCATIONAL MATHEMATICS FOR GIRLS PROBLEMS Solve the following problems by first writing the formula from the rule on page 326, and then substituting for the answer. 1. How many electrical horse power in 4389 watts ? 2. How many kilowatts in 2389 watts ? 3. (a) Give the number of watts in a circuit of 110 volts and 25 amperes. (b) How many electrical horse power ? 4. What is the voltage of a circuit if the horse power is 2740 watts and the quantity of electricity delivered is 25 amperes ? 5. What is the resistance of a circuit if the voltage is 110 and the quantity of electricity is 25 amperes ? 6. What is the pressure per square inch of water 87 feet high? 7. What is the capacity of a cylinder with a base of 16 square inches and 6 inches high? (Capacity in gallons is equal to cubical contents obtained by multiplying base by the height and dividing by 231 cubic inches.) 8. What is the length of a 30° arc of a circle with 16" diameter ? 9. What is the area of a sector which contains an arc of 40° in a circle of diameter 18" ? 10. What is the amount of $ 800 at the end of 5 years at 5 %? 11. What is the amount of gain in a transaction, when a man buys a house for $ 5000 and gains 10 % ? 12. What is the selling price of an automobile that cost $ 896, if the salesman gained 33 % ? 13. What is the capacity of a pail 14" (diameter of top), 11" (diameter of bottom), and 16" in height ? 14. What is the area of an ellipse with the greatest length 16" and the greatest breadth 10" ? FORMULAS 337 Interpretation of Negative Quantities The quantity or number — 12 has no meaning to us according to our knowledge of simple arithmetic, but in a great many problems in practical work the minus sign before a number assists us in understanding the different solutions. To illustrate : Fahrinhbit Tbebmombtbb Boiling point of water Freezing point of ■ water 212" a o a 82» 3 i«« OB i 0" Cbntiobadb Thbbmomxtba Boiling point of winter Freeadng point of • water lOO* SB. o a m cr On the Centigrade scale the freezing point of water is marked 0**. Below the freezing point of water on the Centigrade scale all readings are expressed as minus readings. — 30° C means thirty degrees below the freezing point. In other words, all readings, in the direction below zero are expressed as — , and all readings above zero are called -h. Terms are quantities connected by a j/lus or minus sign. Those preceded by a plus sign (when no sign precedes a quan- tity plus is understood) are called positive quantities, while those connected by a minus sign are called negative quantities. 338 VOCATIONAL MATHEMATICS FOR GIRLS Let us try some problems involving negative quantities. Find the corresponding reading on the Fahrenheit scale cor- responding to — 18® C. F = I C + 32° F = |(-18°)H-82° Notice that a minus quantity is placed in parenthesis when it is to be multiplied by another quantity. F =- H29 4. 32" = _ 82f° H- 32° ; F =- i°. The value — J° is explained by saying it is | of a degree below zero point on Fahrenheit scale. Let us consider another problem. Find the reading on the Centi- grade scale corresponding to — 40° F. Substituting in the formula, we have C = f (- 40° - 82°) = K- 72°) = - 40°. Since subtracting a negative number is equivalent to adding a positive number of the same value, and subtracting a posi- tive number is equivalent to adding a negative number of the same value, the rule for subtracting may be expressed as fol- lows : Change the sign of the subtrahend and proceed as in addition. For example, 40 minus — 28 equals 40 plus 28, or 68. 40 minus + 28 equals 40 plus — 28, or 12. — 40 minus + 32 equals — 40 plus — 82 = - 72. (Notice that a positive quantity multiplied by a negative quantity or a negative quantity multiplied by a positive quantity always gives a negative product. Two positive quantities multiplied together will give a positive product, and two negative quantities multiplied together will give a positive product.) To illustrate : 5 times 6 = 5 x 6 = 26 5 times - 6 = ^ X (- 6) =- 25 (-5) times (-5) = + 25 In adding positive and negative quantities, first add all the positive quantities and then add all the negative quantities FORMULAS 339 together. Subtract the smaller from the larger and prefix the same sign before the remainder as is before the larger number. For example, add : 2a, 5a, —6a, 8-a, —2a 2a + 5a + 8a = 15a; — 6a-2a=-8a 15 a — 8a = 7a EXAMPLES Add the following terms : 1. 3 a?, — 05, 7 a?, 4 a;, — 2 a?. 2. 6y,2y,9y, -Ty. 3. 9 ab, 2dbf 6ab, — 4 ab, 7 ab, — 6 ab. Multiplication of Algebraic Expressions Each term of an algebraic expression is composed of one or more factors, as, for example, 2 ab contains the factors ^, a, and 6. The factors of a term have, either expressed or understood, a small letter or number in the upper right-hand corner, which states how many times the quantity is to be used as a factor. For instance, ab\ The factor a has the exponent 1 understood and the factor b has the exponent 2 expressed, meaning that a is to be used once and b twice as a factor. oJb^ means, then, a X 6 X 6. The rule of algebraic multiplication by terms is as follows: Add the exponents of all like letters in the terms multiplied and use the result as exponent of that letter in the product. Multiplication of unlike letters may be expressed by placing the letters side by side in the product. For example : 2 aft x 3 62 = 6 ^fts 4a X 36 = 12a6 Algebraic or literal expressions of more than one term are multiplied in the following way : begin with the first term to the left in the multiplier and multiply every term in the multi- plicand, placing the partial products underneath the line. Then 340 VOCATIONAL MATHEMATICS FOR GIRLS repeat the same operation, using the second term in the multi- plier. Place similar products of the same factors and degree (same exponents) in same column. Add the partial products. Thus, a-\- b multiplied by a — &. a + b a— b a2 4. a6 - 62 ~ ab Notice the product of the sum and difference of the quantities is equal to the difference of their squares. EXAMPLES 1. Multiply a -f 6 by a -f 6. State what the square of the sum of the quantities equals. 2. Multiply 05 — y by 05 — y. State what the square of the difference of the quantities equals. 3. Multiply (i) + g)(p — g). 7. Multiply (a? — y)(aj — y). 4. Multiply (i) + g)(i> + g). a (a? + y)«=? 5. Multiply (r + «)(r - «). 9. {x - y)« = ? 6. Multiply (a ± b){a ± b). 10. {x -f y)(x -y) = ? USEFUL MECHANICAL INFORBiATION There are certain mechanical terms and laws that every girl should know and be able to apply to the labor-saving devices and machines that are used in the home to-day. Time and Speed Two important terms are time and speed. Speed is the name given to the time-rate of change of position. That is, S-oeed — C/hange of position or distance ^ Time taken EXAMPLES 1. A train takes 120 seconds to go one mile; what is its speed in miles per hour ? One hour contains 60 minutes, 1 minute contains 60 seconds, then 1 hour contains 60 X 60 = 3600 seconds. If the train goes one mile in 120 seconds, in one second it will go y^ of a mile and in 3600 seconds it will go 3600 X xl^ = 30 miles per hour. Ans, 2. At the rate of 80 seconds per mile, how fast is a train moving in miles per hour ? In a second it will move ^ of a mile ; in 3600 seconds it will move 3600 times as much. 3. At the rate of 55 miles an hour, how many seconds will it require to travel between mile-posts ? 4. A watch shows 55 seconds between mile-posts ; what is the speed in miles per hour ? 341 342 VOCATIONAL MATHEMATICS FOR GIRLS 5. What number of seconds between mile-posts will corre- spond to a speed of 40 miles an hour ? 6. The rim of a fly-wheel is moving at the rate of one mile a minute. How many feet does it move in a second ? 7. If a train continues to travel at the rate of 44 feet a second, how many miles will it travel in an hour ? 8. If a train travels at the rate of 3.87 miles in 6 minutes, how many miles an hour is it traveling ? Motion and Momentum Many interesting facts about the motion of bodies can be understood by the aid of a knowledge of the laws of motion and momentum. A body acted upon by some force,^ such as steam or elec- tricity, starts slowly, increasing its speed under the action of the force. To illustrate: — when an electric car starts, we often experience a heavy jarring ; this is due to the fact that the seat starts before our body, and the seat pushes us along. There is a tendency of bodies to remain in a state of rest or motion, which is called inertia, that is, the inability of a body itself to change its position, to stop itself if moving, or to start if at rest. The momentum of a body is defined as the quantity of motion in a body, and is the product of the mass 2 and the velocity in feet per second (speed). Example. To find the momentum of a body 9 pounds in weight, when moving with the velocity of 76 feet per second. If the mass of the body upon which the force acts is given in pounds, and the velocity in seconds, the force will be given in foot-pounds. Mass Velooitt Momentum 9 X 76 = 676 foot-pounds. 1 Force is that which tends to produce motion. 3 Mass is the quantity of matter in a body. USEFUL MECHANICAL INFORMATION 343 We may abbreviate this rule by allowing letters to stand for quantities. Let the mass be represented by M and the veloc- ity by F. EXAMPLES 1. What is the momentum of a car weighing 16 tons, mov- ing 12 miles per hour ? 2. What is the momentum of a motor-car weighing 3 tons, moving 26 miles per hour ? 3. What is the momentum of a person weighing 135 pounds, moving 5 miles per hour ? 4. A truck weighing 4 tons has a momentum of 620,000 foot- pounds. At what speed is it moving ? Work and Energy Work is the overcoming of resistance of any kind. Energy is the ability to do work. Work is measured in a unit called a foot-pound. It is the work done in raising one pound one foot in one second. One horse power is 33,000 foot-pounds in one minute. EXAMPLES 1. A woman lifts a package weighing 15 lb. from the floor to a shelf 6 ft. above the floor in two seconds. How many foot-pounds of force does she use ? 2. How much work does a woman weighing 130 pounds do in climbing a 13-story building in 20 minutes ? Each story is 16' high. 3. If an engine is rated at 5 H. P.,^ how much work will it do in 8 seconds ? in 3 minutes ? ^ Remember that 1 H. P. means 33,000 ft.-lb. in one minute. 344 VOCATIONAL MATHEMATICS FOR GIRLS 4. Find the horse power developed by a locomotive when it draws at the rate of 31 miles per hour a train offering a resist- ance of 130,000 lb. Machines Experience shows that it is often possible to use our strength to better advantage by means of a contrivance called a machine. Every home-maker is interested in labor-saving devices. The mechanical principles of all simple machines may be resolved into those of the lever, including the wheel and axle and pulley y and the inclined plane, to which belong the wedge and acreio. In all machines there is more or less friction} The work done by the acting force always exceeds the actual work accomplished by the amount that is transformed into heat. The ratio of the useful work to the total work done by the acting force is called the efficiency of the machine. Fffio'p o — ^^^^^^ work accomplished Total work expended • Levers. — The efficiency of simple levers is very nearly 100 (fo because the friction is so small as to be disregarded. Inclined Planes. — In the inclined plane the friction is greater than in the lever, because there is more surface with which the two bodies come in contact ; the efficiency is some- where between 90 % and 100 %. Pulleys. — The efficiency of the commercial block and tackle with several movable pulleys varies from 40 % to 60 %. Jack Screw. — In the use of the jack screw there is neces- sarily a very large amount of friction so that the efficiency is often as low as 25 %. ^ Friction is the resistance which every material surface offers to the slid- ing or moving of any other surface upon it. USEFUL MECHANICAL INFORMATION 345 EXAMPLES 1. Mention some instances in which friction is of advantage. 2. If 472 foot-pounds of work are expended by a dredge in raising a load, and only 398 pounds of useful work are accom- plished, what is the efficiency of the dredge ? 3. If 260 foot-pounds of work are expended at one end of a lever, and 249 pounds of useful work are accomplished, what is the efficiency of the lever ? 4. If 689 foot-pounds of work are expended in raising a body on an inclined plane, and only 684 pounds of useful work are accomplished, what is the efficiency of the inclined plane ? 5. If 844 foot-pounds of work are expended in raising a body by means of pulleys and only 612 pounds of useful work are accomplished, what is the efficiency of the pulley ? Water Supply The question of the water supply of a city or a town is very important. Water is usually obtained from lakes and rivers which drain the surrounding country. If a lake is located in a section of the surrounding country higher than the city (which is often located in a valley), the water may be obtained from the lake, and the pressure of the water in the lake may be sufficient to force it through the pipes into the houses. But in most cases a reservoir is built at an elevation as high as the highest portion of the town or city, and the water is pumped into it. Since the reservoir is as high as the highest point of the town, the water will flow from it to any part of the town. If houses are built on the same hill with the reservoir, a stand- pipe, which is a steel tank, is erected on this hill and the water is pumped into it. Water is conveyed from the reservoir to the house by means 346 VOCATIONAL MATHEMATICS FOR GIRLS of iron pipes of various sizes. It is distributed to the differ- ent parts of the house by small lead, iron, or brass pipes. Since water exerts considerable pressure, it is necessary to know how to calculate the exact pressure in order to have pipes of proper size and strength. Water Supply "^mmmp^ The distribution of water in a city during 1912 is as follows : Months January February March April May June July August September October . November December 167,866,290 147,692,464 146,933,054 143,066,067 161,177,486 176,479,364 189,063,260 179,379,666 169,394,758 176,067,671 153,484,712 161,976,208 H 0. OD >< < 5,092,461 6,092,844 4,739,776 4,768,869 5,199,274 6,882,645 6,098,816 6,786,438 5,646,492 6,679,699 5,116,167 4,902,468 H -4 09 gs i: "•WW « -. * S IB ^ as 09 0. S fe. Pki ^ W o ^Q 1-t 1-t UO o CO -^ ■^ ^0 1-t r^ 1-t r^ ^ V M S? "< s « ^ u ^ ^ S 3 5 » What is the number of gallons per day for each consumer ? What is the number of gallons per day for each inhabitant ? PLUMBING AND HYDRAULICS 347 EXAMPLES 1. Water is measured by means of a meter. If a water meter measures for five hours 760 cubic feet, how many gal- lons does it indicate ? Note. — 231 cubic inches = 1 gallon. 2. If a water meter registered 1845 cubic feet for 3 days, how many gallons were used ? 3. A tank holds exactly 12,852 gallons ; what is the capacity of the tank in cubic feet ? 4. A tank holds 3841 gallons and measures 4 feet square on the bottom ; how high is .the tank ? Rectangular Tanks. — To find the contents in gallons of a square or rectangular tank, multiply together the length, breadth, and height in feet; multiply the result by 7.48. I = length of tank in feet b = breadth of tank in feet h = height of tank in feet Contents = Ibh cubic feet x 7.48 = 7.48 Ibh gallons (Note. — 1 cu. ft. = 7.48 gallons.) If the dimensions of the tank are in inches, multiply the length, breadth, and height together, and the result by .004329. 5. Find the contents in gallons of a rectangular tank having in- side dimensions (a) 12' x 8' x 8'; (b) 15" x 11" X 6"; (c) 3' 4" X 2' 8" X 8"; (d) 5' 8" x 4' 3" x 3' 5"; (e) 3' 8" x 3' 9" x 2' 5", Cylindrical Tank. — To find the contents of a cylin- drical tank, square the diameter in inches, multiply this by the height in inches, and the result by .0034. d = diameter of cylinder h = height of cylinder Contents = d^h cubic inches x .0034 = d^h .0034 gallons 6. Find the capacity in gallons of a cylindrical tank (a) 14" in diameter and 8' high; (b) 6" in diameter and 6' high; 348 VOCATIONAL MATHEMATICS FOR GIRLS (c) 15" in diameter and 4' high; (d) V 8" in diameter and 6' 4" high ; (e) 2' 2" in diameter and 6' 7" high. Inside Area of Tanks. — To find the area, for lining purposes, of a square or rectangular tank, add together the widths of the four sides of the tank, and multiply the result by the height. Then add to the above the area of the bottom. Since the top is usually open, we do not line it. In the following problems find the area of the sides and bottom. 7. Find the amount of zinc necessary to line a tank whose inside dimensions are 2' 4" x 10" x 10". 8. Find the amount of copper necessary to line a tank whose inside dimensions are 1^9" x 11" x 10", no allowance made for overlapping. 9. Find the amount of copper necessary to line a tank whose inside dimensions are 3' 4" x 1' 2" x 11", no allowance for overlapping. 10. Find the amount of zinc necessary to line a tank 2' 11" X 1' 4" X 10". Capacity of Pipes Law of Squares. — The areas of similar figures vary as the squares of their corresponding dimensions. Pipes are cylindrical in shape and are, therefore, similar figures. The areas of any two pipes are to each other a& the squares of the diameters. Example. — If one pipe is 4" in diameter and another is 2" in diameter, their ratio is J^, and the area of the larger one is, therefore, 4 times the smaller one. EXAMPLES 1. How much larger is a section of 5" pipe than a section of 2" pipe ? 2. How much larger is a section of 2y pipe than a section of 1" pipe ? 3. How much larger is a section of 5" pipe than a section of 3" pipe ? PLUMBING AND HYDRAULICS 349 Atmospheric Pressure The atmosphere has weight and exerts .pressure. The pres- sure is greatest at sea level, because here the depth of the atmosphere is greatest. In mathematics the pressure at sea level is taken as the standard. Men have learned to make use of the principles of atmospheric pressure in such devices as the pump, the barometer, the vacuum, etc. Atmospheric pressure is often expressed as a certain number of "atmospheres." The pressure of one " atmosphere " is the weight of a column of air, one square inch in area. At sea level the average pressure of the atmosphere is approximately 15 pounds per square inch. The pressure of the air is measured by an instrument called a barometer. The barometer con- sists of a glass tube, about 3H inches long, which has been entirely filled with mercury (thus removing all air from the tube) and inverted in a vessel of mercury. The space at the top of the column of mercury varies as the air pressure on the surface of the mercury in the vessel increases or decreases. The Basomstbb pressure is read from a graduated scale which indi- Barombteb Tube 350 VOCATIONAL MATHEMATICS FOR GIRLS cates the distance from the sor&ce of the mercuiy in the vessel to the top of the mercoiy oolomn in the tube. QUESnOHS 1. Four atmospheres would mean how many poonds ? 2. Give in pounds the following pressures: 1 atmosphere; ^ atmosphere ; f atmosphere. 3. If the air, on the average, will support a column of mercury 30 inches high with a base of 1 square inch, what is the pressure of the air ? (One cubic foot of mercury weighs 849 pounds.) Water Pressure When water is stored in a tank, it exerts pressure against the sides, whether the sides are vertical, oblique, or horizontal. The force is exerted perpendicularly to the surface on which it acts. In other words, every pound of water in a tank, at a height above the point where the water is to be used, possesses a certain amount of energy due to its position. It is often necessary to estimate the energy in the tank at the top of a house or in the reservoir of a town or city, so as to secure the needed water pressure for use in case of fire. In such problems one must know the perpendicular height from the water level in the reservoir to the point of discharge. This perpendicular height is caUed the head. Pressure per Square Inch. — To find the pressure per square inch exerted by a column of water, multiply the head of water in feet by 0.434. The result will be the pressure in pounds. The pressure per square inch is due to the weight of a column of water 1 square inch in area and the height of the column. Therefore, the pressure, or weight per square inch, is equal to the weight of a foot of water with a base of 1 square inch multiplied by the height in feet. Since the weight of a column of water 1 foot high and having a base of 1 square inch is 0.434 lb., we obtain the pressure per square inch by multiply- ing the height in feet by 0.434. PLUMBING AND HYDRAULICS 351 EXAMPLES What is the pressure per square inch of a column of water (a) 8' high? (6) 15' 8" high? (c) 30' 4'' high? (d) 18' 9'' high ? (e) 41' 3" high ? Head. — To find the head of water in feet, if the pressure (weight) per square inch is known, multiply the pressure by 2.31. Let p = pressure h = height in feet Then p=:h x 0.434 lb. per sq. in. 0.434 0.434 Xp=2.Slp EXAMPLES Find the head of water, if the pressure is (a) 49 lb. per sq. in.; (b) 88 lb. per sq. in.; (c) 46 lb. per sq. in.; (d) 28 lb. per sq. in. ; (e) 64 lb. per sq. in. Lateral Pressure. — To find the lateral (sideways) pressure of water upon the sides of a tank, multiply the area of the submerged side, in inches, by the pressure due to one half the depth. Example. — A tank 18" long and 12" deep is full of water. What is the lateral pressure on one side ? length depth 18" X 12" =216 square inches = area of side depth 1' X 0.434 = .434 lb. pressure at the bottom of the tank = pressure at top 2) .434 lb. .217 lb. average pressure due to one half the depth of the tank .217 X 216 = 46.872 pounds = pressure on one side of the tank — Preanire is zero ^— — — — PreMure is half that at base Lateral Pj^essubb 352 VOCATIONAL MATHEMATICS FOR GIRLS Water Traps The question of disposing of the waste water, called sewage, is of great importance. Various devices may be used to prevent odors from the sewage entering the house. In order to prevent the escape of gas B T Water Traps from the outlet of the sewer in the basement of a house or building, a device called a trap is used. This trap consists of a vessel of water placed in the waste pipe of the plumbing fixtures. It allows the free pas- a3ge of waste material, but prevents sewer gases or foul odors from enter- ing the living rooms. The vessels holding the water have different forms ; (see illustration). These traps may be emptied by back pressure or by siphon. It is a good plan to have sufficient water in the trap so that it will never be empty. All these problems belong to the plumber and in- volve more or less arithmetic. To determine the pressure which the seal of a trap will resist : Example. — What pressure will a l|-inch trap resist ? If one arm of the trap has a seal of 1} inches, both arms will make a column twice as high, or 8 inches. Since a column of water 28 inches in height is equivalent to a pressure of 1 pound, or 16 ounces, a column of water 1 inch in height is equivalent to a pressure of |^ of a pound, or ^} = ^ ounces, and a column of water 8 inches in height is equivalent to 8 X ^ = "V* = 1.7 ounces. Therefore, a IJ-inch trap will resist 1.7 ounces of pressure. PLUMBING AND HYDRAULICS 353 EXAMPLES 1. What back pressure will a |-iiich seal trap resist ? 2. What back pressure will a 2-inch seal trap resist ? 3. What back pressure will a 2^inch seal trap resist ? 4. What back pressure will a 4J-inch seal trap resist ? 5. What b^ck pressure will a 5-inch seal trap resist ? Water Power When water flows from one level to another, it exerts a certain amount of energy, which is the capacity for doing work. Consequently, water may be utilized to create power by the use of such means as the water wheel, the turbine, and the hydraulic ram. Friction, which must be considered when one speaks of water power, is the resistance which a substance encounters when moving through or over another substance. The amount of friction depends upon the pressure between the surfaces in contact. When work is done a part of the energy which is put into it is naturally lost. In the case of water this is due to the friction. All the power which the water has cannot be used to advantage, and efficiency is the ratio of the useful work done by the water to the total work done by it. Efficiency. — To find the work done upon the water when a pump lifts or forces it to a height, multiply the weight of the water by the height through which it is raised. Since friction must be taken into consideration, the useful work done upon the water when the same power* is exerted will equal the weight of the water multiplied by the height through which it is raised, multiplied by the efficiency of the pump. Example. — Find the power required to raise half a ton 354 VOCATIONAL MATHEMATICS FOR GIRLS (long ton, or 2240 lb.) of water to a height of 40 feet, when the efficiency of the pump is 75 %. Total work done = weight x height x efficiency counter 1120 X 40 X W = 59,733.8 ft. lb. H. P. required = 5M§M = 1.8. Ans. ^ 33000 EXAMPLES 1. Find the power required to raise a cubic foot of water 28', if the pump has 80 % efficiency.^ 2. Find the power required to raise 80 gallons of water 15', if the pump has 75 % efficiency. 3. Find the power required to raise 253 gallons of water 18', if the pump has 70 % efficiency. 4. Find the power required to raise a gallon of water 16', if the pump has 85 % efficiency. 5. Find the power required to raise a quart of water 25', if the pump has 70 % efficiency. Density of Water The mass of a unit volume of a substance is called its density. One cubic foot of pure water at 39.1° F. has a mass of 62.425 pounds ; therefore, its density at this temperature is 62.425, or approximately 62.5. At this temperature water has its greatest density. With a change of temperature, the density is also changed. With a rise of temperature, the density decreases until at 212° F., the boiling point of water, the weight of a cubic foot of fresh water is only 59.64 pounds. When the temperature falls below 39.1° F., the density of water decreases until we find the weight of a cubic foot of ice to be but 57.5 pounds. 1 Consider the time 1 minute in all power examples where the time is not giyen. PLUMBING AND HYDRAULICS 355 EXAMPLES 1. One cubic foot of fresh water at 62.6** F. weighs 62.365 lb., or approximately 62.4 lb. What is the weight of 1 cubic inch ? What is the weight of 1 gallon (231 cubic inches) ? 2. What is the weight of a gallon of water at 39.1** F. ? 3. What is the weight of a gallon of water at 212** F. ? 4. What is the weight of a volume of ice represented by a gallon of water ? 5. What is the volume of a pound of water at ordinary temperature, 62.5** F. ? Specific Gravity Some forms of matter are heavier than others, Le, lead is heavier than wood. It is often desirable to compare the weights of different forms of matter and, in order to do this, a common unit of comparison must be selected. Water is taken as the standard. Specific Gravity is the ratio of the mass of any volume of a substance to the mass of the same volume of pure water at 4** C. or 39.1** F. It is found by dividing the weight of a known volume of a substance in liqui(J by the weight of an equal volume of water. Example. — A cubic foot of wrought iron weighs about 480 pounds. Find its specific gravity. Note. — 1 cu. ft. of water weighs 62.425 lb. Weight of 1 cu. ft. of iron __. 480 __ 7 7 Ang Weight of 1 cu. ft. of water ~ 62.426 ~ * * To find Specific Gravity. — To find the specific gravity of a solid, weigh it in air and then in water. Find the difference between its weight in air and its weight in water, which will be the buoyant force on the body, or the weight of an equal volume of water. Divide the weight of the solid in air by its buoyant force, or the weight of an equal volume of water, and the quotient will be the specific gravity of the solid. 356 VOCATIONAL MATHEMATICS FOR GIRLS Tables have been compiled giyjng the specific gravity of different solids, so it is seldom necessary to compute it. The specific gravity of liquids is very often used in the industrial world, as it means the " strength " of a liquid. In the carbonization of raw wool, the wool must be soaked in sulphuric acid of a certain strength. This acid cannot be bought except in its concentrated form (sp. gr. 1.84), and it must be diluted with water until it is of the required strength. Tbe simplest way to determine the specific gravity of a liquid is with a hydrometer. This instrument consists of a closed glass tube, with a bulb at the lower end filled with mercury. This bulb of mer- cury keeps the hydrometer upright when it is immersed in a liquid. The hydrometer has a scale on the tube which can be read when tbe instrument is placed in a graduate of the liquid whose specific gravity is to be determined. But not all instruments have the specific gravity recorded on the stem. Those most commonly in use are graduated with an impartial scale. In England, Twaddell^s scale is commonly employed, and since most of the textile mill workers are English, we find the same scale in use in this country. The Twaddell scale bears a marked relation tQ specific gravity and can be easily converted into it. Another scale of the hydrometer is the Beaume, but these readings cannot be converted into specific gravity without the use of a complicated formula or reference to a table. Htdrombtkr Scale Formula for Gontbrtino into S. G. 1. Specific gravity hydrometer Gives direct reading 2. Twaddell S. G. = O^xyiOO 100 8. Beaume S. G. = ^^'^ 146.8 - N iV= the particular degree which is to be converted. Example. — Change 168 degrees (Tw.) into S. G. (.6x168) + 100 ^13^ ^^ 100 PLUMBING AND HYDRAULICS 357 Another formula for changing degrees Twaddell scale into specific gravity is : (^ ^ ^^ ^^^ = specific gravity. In Twaddell^s scale, P specific gravity = 1.006 2° specific gravity = 1.010 8° specific gravity = 1.016 and so on by a regular increase of .005 for each degree. To find the degrees Twaddell v^rhen the specific gravity is given, multi- ply the specific gravity by 1000, subtract 1000, and divide by 5. Formula : (S. G. X IWO) - 1000 ^ j^g^ TwaddeU ExAMPi.a. — Change 1.84 specific gravity into degrees Twad- deU, (1.M X 1000) - 1000 ^ jgg j^g^ TwaddeU 5 EXAMPLES 1. What is the specific gravity of sulphuric acid of 116** Tw.? 2. What is the specific gravity of acetic acid of 86** Tw. ? 3. What is the specific gravity of a liquid of 164** Be. ? 4. What is the specific gravity of a liquid of 108** Be.? 5. What is the specific gravity of a liquid of 142** Tw.? Heat Heat Units. — The unit of heat used in the industries and shops of America and England is the British Thermal Unit (B, T. U.) and is defined as the quantity of heat required to raise one pound of water through a temperature of one degree Fahrenheit. Thus the heat required to raise 5 lb. of water through 15 degrees F. equals 6 X 16 = 75 British Thermal Units (B. T. U.) Similarly, to raise 86 lb. of water through J° F. requires 86 X i = 43 B. T. U. The unit used on the Continent and by scientists in America is the metric system unit, a calorie. This is the amount of heat necessary to raise 1 gram of water 1 degree Centigrade. 358 VOCATIONAL MATHEMATICS FOR GIRLS EXAMPLES 1. How many units (B. T. U.) will be required to raise 4823 lb. of water 62 degrees ? 2. How many B. T, U. of heat are required to change 365 cubic feet of water from 66° F. to 208° F.? 3. How many units (B. T. U.) will be required to raise 785 lb. of water from 74° F. to 208° F.? (Consider one cubic foot of water equal to 62^ lb.) 4. How many B. T. U. of heat are required to change 1825 cu. ft. of water from 118° to 211° ? 5. How many heat units will it take to raise 484 gallons of water 12 degrees ? 6. How many heat units will it take to raise 5116 gallons of water from 66° F. to 198° F.? f{ f( 100 213' Temperature The ordinary instruments used to measure temperature are called thermometers. There are two kinds — Fahren- heit and Centigrade. The Fahrenheit ther- mometer consists of a cylindrical tube filled with mercury with the position of the mercury at the boiling point of water marked 212, and the position of mercury at the freezing point of water 32. The intervening space is divided into 180 divisions. The Centigrade thermometer has the position of the boiling point of water 100 and the freezing point 0. The intervening space is divided into 100 spaces. It is often necessary to convert the Centigrade scale into the Fah- renheit scale, and Fahrenheit into Centigrade. nTo convert F. into C, subtract 32 from the F. degrees and multiply by f , or divide by 1.8, or C. = (F. - 32°) I, where C. = Centigrade reading Thermometers and F. = Fahrenheit reading. 17^ I 32' 0' HEAT AND TEMPERATURE 359 To convert C. to F., multiply C. degrees by f or 1.8 and add 32°. 5 Example. — Convert 212 degrees F. to C. reading 5(212^ - 32-) ^ 5(180°) ^ 900'^ ^ ^^p ^ 9 9 9 * Example. — Convert 100 degrees C. to F. reading. ^ ^ ^^° H- 32° = ?52! + 32° = 180° + 32° = 212° F. 6 6 If the temperature is below the freezing point, it is usually written with a minus sign before it : thus, 15 degrees below the freezing point is written — 15°. ^n changing — 15° C. into F. we must bear in mind the minus sign. . Thus, \p = —-1-32° j?T_ - 15° X 9 ^ 3^o ^_27° + 32° =6° 6 6 Example. — Change — 22° F. to C. C. = 5 (F. - 32) C. =1 (-22°-32°) =5 (-64°) =-30° EXAMPLES 1. Change 36° F. to C. 6. Change 225° C. to F. 2. Change 89° F. to C. 7. Change 380° C. to F. 3. Change 289° F. to C. 8. Change 415° C. to F. 4. Change 350° F. to C. 9. Change 580° C. to F. ^ 5. Change 119° C. to F. Latent Heat By latent heat of water is meant that heat which water ab- sorbs in passing from the liquid to the gaseous state, or that heat which water discharges in passing from the liquid to the 360 VOCATIONAL MATHEMATICS FOR GIRLS solid state, without affecting its own temperature. Thus, the temperature of boiling water at atmospheric pressure never rises above 212 degrees F., because the steam absorbs the excess of heat which is necessary for its gaseous state. Latent heat of steam is the quantity of heat necessary to convert a pound of water into steam of the same temperature as the steam in question. COMMERCIAL ELECTRICITY Amperes. — What electricity is no one knows. Its action, however, is so like that of flowing water that the comparison is helpful. A current of water in a pipe is measured by the amount which flows through the pipe in a second of time, as one gallon per second. So a current of electricity is measured Water Analogy of Fall op Potbntdll by the amount which flows along a wire in a second, as one coulomb per second, — a coulomb being a unit of measurement of electricity, just as a gallon is a unit of measurement of water. The rate of flow of one coulomb per second is called one ampere. The rate of flow of five coulombs per second is five amperes. Volts. — The quantity of water which flows through a pipe depends to a large extent upon the pressure under which it flows. The number of amperes of electricity which flow along a wire depends in the same way upon the pressure behind it. COMMERCIAL ELECTRICITY 361 The electrical unit of pressure is the voU. In a stream of water there is a difference in pressure between a point on the surface of the stream and a point near the bottom. This is called the difference or drop in level between the two points. It is also spoken of as the pressure head, " head '' meaning the difference in intensity of pressure between two points in a body of water, as well as the intensity of pressure at any point. Similarly the pressure (or voltage) between two points in an electric circuit is called the difference or drop in pressure or the potential. The amperes represent the amount of electricity flowing through a circuit, and the volts the pressure causing the flow. Ohms. — Besides the pressure the resistance of the wire helps to determine the amount of the current : — the greater the resistance, the less the current flowing under the same pressure.. The electrical unit of resistance is called an ohm. A wire has a resistance of one ohm when a pressure of one volt can force no more than a current of one ampere through it. Ohm's Law. — The relation between current (amperes), pressure (volts), and resistance (ohms) is expressed by a law known as Ohm^s Law, This is the fundamental law of the study of electricity and may be stated as follows : An electric current flowing along a conductor is equal to the pressure divided by the resistance. Currera (amperes) = ^e^^"'-^ ijo^^) Resistance (ohms) Letting /= amperes, E = volts, R = ohms, J=^-f-i?or/ = =| R E==IR "-1 Example. — If a pressure of 110 volts is applied to a re- sistance of 220 ohms, what current will flow ? 362 VOCATIONAL MATHEMATICS FOR GIRLS Example. — A current of 2 amperes flows Ina circuit the resist- ance of which is 300 ohmB. What is the voltage of the circuit ? ExAMPLR. — If a current of 12 amperes flows in a circuit and the voltage applied to the circuit is 240 volts, find the resistance of the circuit. ^ = B — =20oUms. Am. I 12 Ammeter and Voltmeter. — Ohm's Law may be applied to a circuit as a whole or to any part of it. It is often desirable to know how much current is flowing in a circuit without calculating it by Ohm's Law. An instrument called an ammeter is used to measure the current. This instrument has a low resistance so that it will not cause a drop in pressure. A volt- Toeter is used to measure the voltaga This instrument has high resistance so that a very small current will flow through it, and is always placed in shunt, or parallel (see p. 235) with that part of the circuit the voltage of which is to be found. Example. — What is the resistance of wires that are carry- ing 100 amperes from a generator to a motor, if the drop or loss of potential equals 12 volts ? Drop in voltage — IE / = 100 amperes Drop Id volU =12 11= ? obms It=- B= — = 0.12ohai. Atu. I 100 Example. — A circuit made up of incandescent lamps and conducting wires is supplied under a pressure of 115 volts. COMMERCIAL ELECTRICITY 363 The lamps require a pressure of 110 volts at their terminals and take a current of 10 amperes. What should be the resist- ance of the conducting wires in order that the necessary cur- rent may flow ? Drop in conducting wires = 115 — 110 = 6 volts Current through wires = 10 amperes E 6 jB= — = — = 0.6 ohm resistance. -4ns. / 10 EXAMPLES 1. How much current will flow through an electromagnet of 140 ohms' resistance when placed across a 100-volt circuit ? 2. How many amperes will flow through a 110-volt lamp which has a resistance of 120 ohms ? 3. What will be the resistance of an arc lamp burning upon a 110-volt circuit, if the current is 5 amperes ? 4. If the lamp in Example 3 were to be put upon a 150-volt circuit, how much additional resistance would have to be put into it in order that it might not take more than 5 amperes ? Motor Electric Road System 5. In a series motor used to drive a street car the resistance of the field equals 1.06 ohms ; the current going through equals 30 amperes. What would a voltmeter indicate if placed across the field terminals ? 6. If the load upon the motor in Example 5 were increased so that 45 amperes were flowing through the field coils, what would the voltmeter then indicate ? i^ INDEX Addition, 3 Compound numbers, 46 Decimals, 33 Fractions, 21 Aliquot parts, 39 Alkalinity of water, 298 Ammeter, 362 Ammonia, 296 Amount, 53 Amperes, 169, 360 Angles, 66 Complementary, 66 Right, 66 Straight, 66 Supplementary, 66 Annuity, 192 Antecedent, 37 Apothecary's weights, 276 Apothem, 72 Approximate equivalents between metric and household measures, 281 Approximate measures of fluids, 277 Arc, 64 Area of a ring, 65 Area of a triangle, 69 Atmospheric pressure, 349 Avoirdupois weight, 43 Bacteria, 294, 298 Banks, 178 Cooperative, 179 National, 178 Savings, 179 Baths, 292 Bed linen, 161 Beef, 118 Bills, 243 Blue print reading, 80 Board measure, 131 Bonds, 187 Brickwork, 134 Building materials, 133 Buying Christmas gifts, 94 Cotton, 229 Rags, 229 . Wool, 229 Yarn, 229 Cancellation, 13 Capacity of pipes, 348 Carbohydrates, 102 Cement, 136 Chlorine, 297 Circle, 64 Circumference, 64 Civil Service, 268 Claims, 196 Clapboards, 138 Clothing, 91 Coefficients, 331 Color of water, 296 Common denominator, 20 Fractions, 17 Multiple, 15 Comparative costs of digestible nutrients and energy in different food materials at average prices, 114, 115 Compound numbers, 42, 46 Addition, 46 Division, 47 Multiplication, 47 Subtraction, 46 Computing profit and loss, 252 Cone, 75 Consequent, 57 Construction of a house, 128 Cooperative banks, 179 Cost of food, 105 Cost of furnishing a house, 146 Cost of subsistence, 91 Cotton, 217 Yarns, 223 365 4 366 INDEX Counting, 44 Credit account, 244 Cube, 61 Cube Root, 61 Cubic measure, 42 Cuts of Beef, 120 Mutton, 122 Pork, 121 Cylindrical tank, 347 Dairy Products, 310 Debit, 244 Decimal Fractions, 30 Addition, 33 Division, 36 Mixed, 31 Multiplication, 35 Reduction, 32 Subtraction, 34 Denominate fraction, 45 Number, 42, 45 Denomination, 42 Denominator, 17 Density of water, 354 Deodorants, 294 Diameter, 64 Distribution of income, 89 Division, 9 Compound numbers, 47 Fractions, 25 Income, 92 Drawing to scale, 85 Dressmaking, 198 Dry measure, 43 Economical marketing, 125 Uses of Meats, 117 Economy of space, 130 Efficiency of water, 353 Electricity, 360 EUipse, 72 En^ish system, 276 Ensilage problems, 307 Equations, 332 Substituting, 334 Transposing, 334 Equiangular triangle, 68 Equilateral triangle, 68 Estimating distances, 86 Weights, 87 Evolution, 61 Exchange, 193 Expense account book, 95 Factors, 13 Farm measures, 307 Problems, 305 Filling, 217 Flax, 217 Flooring, 139 Fluid measure, 277 Food, 100 Values, 110 Formulas, 327 For computing profit and loss, 253 Fractions, 17 Addition, 21 Common, 17 Decimal, 30 Division, 25 Improper, 17 Multiplication, 24 Reduction, 17 Subtraction, 22 Frame and roof, 132 Free ammonia, 297 Frustum of a cone, 76 Furnishing a bedroom, 153, 154, 155 Dining room, 156 Hall, 146 Kitchen, 162 Living room. 149, 150. 152 Sewing room, 161 Germicides, 294 Graphs, 322 Greatest common divisor. 15 Hardness of water. 298 Heat, 357 And Ught, 167 Units, 357 Hem, 200 Hexagon, 72 Horizontal addition, 237 Household linens, 160 Measures, 277 How to make change, 266 Solutions of various strengths from crude drugs or tablets of known strength, 286 INDEX 367 How to read an electric meter, 169 Gas meter, 168 Hypodermic doses, 288 Improper fractions, 17 Inclined planes, 344 Income, 89 Inside area of tanks, 348 Insurance, 188 Fire, 188 Life, 189 Integer, 1, 17, 31, 45 Interest, 53 Compound, 56 Simple, 53 Interpretation of negative quantities, 337 Invoice bills, 243 Involution, 61 Iron in water, 298 Isosceles triangle, 68 Jack screw, 344 Kilowatt, 169 Kitchen weights and measures, 103 Latent heat, 359 Lateral pressure, 351 Lathing, 141 Law of squares, 348 Least common multiple, 15 Ledger, 244 Levers, 344 Linear measure, 42 Linen, 217 Yarns, 222 Liquid measure, 43 Lumber, 131 Machines, 344 Measure, of time, 43 Length, 317 Medical chart, 292 Mensuration, 64 Menus, making up, 113 Merchandise, 243 Methods of heating, 174 Methods of solving examples, 87, 88 Metric system, 276, 279, 282, 317-319 Millinery problems, 212 Mixed decimals, 31 Mohair, 217 Momentum, 342 Money orders, 194 Mortar, 133 Mortgages, 180 Motion, 342 Multiplication, 8, 242 Algebraic expressions, 339 Compound numbers, 47 Decimals, 31 Fractions, 24 Mutton, 122. 123 National banks, 178 Nitrogen, 297 Notation, 1 Notes, 181 Numerals, Roman, 2 Numeration, 1 Numerator, 17 Nurses, arithmetic for, 276-303 Nutritive ingredients of food, 101 Octagon, 72 Odor of water, 296 Ohm, 361 Ohm's Law, 361 Oxygen consumed, 297 Painting, 141 Papering, 142 Paper measure, 44 Pay rolls, 255 Pentagon, 72 Percentage, 50 Perimeter, 72 Plank, 131 Plastering, 133 Polygons, 72 Poultry problems, 312 Power, 30 Pressure, lateral, 351 Per square inch, 350 Water, 350 Principal, 53 Profit and loss, 246 Promissory notes, 182 Proper fractions, 17 Proportion, 57, 58, 59 368 INDEX Protractor, 67 Pulleys. 344 Pyramid, 75 Quadrilaterals, 71 Radius, 64 Rapid calculation, 233 Rate (per cent), 50 Ratio, 57 Raw silk yams, 222 Reading a blue print, 80 Rectangle, 71 Reduction, 42 Ascending, 42, 44 Descending, 42, 44 Right triangles, 68, 69 Roman numerals, 2 Root, cube, 62 Sqi^are, 61 Ruffles, 201 Rule of thumb methods, 88 Savings bank, 179 Interest tables, 56 Scalene triangle, 68 Sector, 64 Sediment in water, 296 Shingles, 137 Shoes, 219 Silk, 217 Similar figures, 77 Terms, 331 Simple interest, 53 Proportion, 59 Slate roofing, 137 Specific gravity, 355 Specimen arithmetic papers, 272 Sealers of Weights and Measures, 273 State visitors, 274 Stenographers, 273 Sphere, 76 Spun silk yarns, 223 Square measure, 42 Square root, 62 Stairs, 140 Steers and beef, 118 Stocks, 184 Stonework, 135 Strength of solutions, 224 Studding, 132 Substituting in equations, 334 Subtraction Compound numbers, 46 Decimals, 34 Fractions, 22 Supplement, 66 Table linen, 161 Table of metric conversion, 317 Table of wages, 257 Tanks, 347 Taxes, 143 Temperature, 290, 358 Temporary loans, 259 Terms used in chemical and bacterio- logical reports, 296 Time and speed, 341 Time sheets, 255 Trade discount, 52, 207 Transposing in equations, 334 Trap)ezium, 71 Trapezoid, 72 Triangles, 68 Equiangular, 68 Equilateral, 68 Isosceles, 68 Right, 68, 69 Scalene, 68 Trust companies, 179 Tucks, 199 Turbidity of water, 296 Two-ply yarns, 223 Unit, 1 United States revenue, 144 Useful mechanical information, 341 Use of tables, 88 Uses of nutrients in the body, 102 Value of coal to produce heat, 167 Volt, 169, 360 Voltmeter, 362 Volume, 74 Warp, 217 Water, alkalinity of, 298 Ammonia in, 296, 297 Analysis of, 296 Bacteria in, 298 Chlorine in, 297 INDEX 369 Water, ^ — continued. Color of, 296 Hardness of, 298 Iron in, 298 Nitrogen in, 297 Odor of, 296 Oxygen consumed by, 297 Power, 353 Pressure, 350 Residue on evaporation, 296 Sediment in, 296 Supply, 345 Traps, 352 Turbidity of. 296 Watt, 170 Wool, 211 Woolen yarns, 220 Work, 343 Worsted yarns, 219 Yarns, 217 Cotton, 223 Linen, 222 Raw silk, 222 Spun silk, 223 Two-ply, 223 Woolen, 220 Worsted, 219 1^-^ YB 0520C 1