?rJ?'i'-»'.f.-«j«r.< RIIHME ?»*^ h^i' ■wffwWfp^swin^i^+WM'iMs-tH* i.B10NDS.^NDJ0KI;,S Ml*f|i** APPROVED BT THE STATE BOAKD OF EDUCATION SACRAMENTO Friend Wm. Richardson, Supt. State Printing. Copyright, 1910, by THE PEOPLE OF THE STATE OF CALIFORNIA. Copyright, 1907, by J. W. McCLYMONDS and J). E. JONES. lyt iiiUi^^'nim .i?^^ Jn the compilation of thin book certain matter from an " EmentiaU of Arithmetic'''' by J. W. McClymondt and I). It. Jones has been used. All such matter it protected by the copyright entries noted above. 12th Ed.— 25.000— Oct., 1914. PREFACE This text is designed for use in the grammar grades, following the completion of the Elementary Arithmetic of the same series. In the preparation of this text the authors have aimed (a) to secure skill in numerical com- putations and (6) to develop the power necessary to the solution of any practical problem that may arise in the common experiences of life. The following are some of the distinguishing features of this text: 1. The text contains an unusually large number of exercises that are designed to give facility in numerical computations. 2. In the presentation of each topic an effort has been made to stimulate thought and to develop self-reliance on the part of the pupils. Whenever the nature of the work admits, it calls for action on the part of the pupils, as in making measurements, engaging in business relations with others in the class, etc. 3. The scope of the work is restricted to the needs of the majority of persons in the common experiences of life. Traditional materials that make no contribution to the mastery of the essentials of arithmetic have been carefully eliminated. All of the work prescribed in the text proper is easily within the capacity of pupils in the grammar grades. Certain topics that are prescribed in some courses of study but purposely ^j^yo^^fipm other courses have 4 PREFACE been presented in an Appendix, so that they may be used or omitted, as desired in each case, without destroying the continuity of the other work. 4. The problems of the text have been drawn from the common field of everyday experience. The necessary arithmetical training is had from dealing with practical problems within the experience of the pupils. No unreal problems, or problems dealing with artificial situations, or problems treating of situations remote from the experi- ences of the average pupil in the grammar grades, are introduced. The text aims to teach arithmetic only. 5. The text contains an unusual amount of oral work, including oral problems under every topic treated. The oral problems are everywhere related to the written work. No additional text in " mental " arithmetic need be used m conjunction with this text. 6. The methods of the text are those commonly em- ployed in business life. 7. The work in fractions and compound numbers is limited to the practical needs of life. Special attention is given in fractions to the use of those fractions which pupils must handle later on as the fractional equivalents of certain per cents. Commission, Taxes, Insurance, etc., are made part of the work in Percentage and are not treated as separate topics. The work in Interest has been considerably reduced, and but one method of finding interest is recommended. 8. A constant review of all previous work is maintained throughout the text. Finally, the aim of the authors has been to present a course in arithmetic that will secure a thorough knowl- edge of the essentials of this subject. CONTENTS PART I Review of Integers and Decimals PAGES The Decimal System — Notation and Nuix sration — Addition — Subtraction — Multiplication — Bills and Accounts — Divi- sion by Measurement and Partitio i — Comparison — Meas- urements — Divisibility of Numbers 7-89 PART II Fractions Objective Fractions — Ratio — Reduction — Addition — Sub- traction — Multiplication — Division — Scale Drawing — Aliquot Parts — Measurements 90-165 PART III Percentage Percentage — Profit and Loss — Commission — Insurance — Taxes — Customs and Duties — Trade Discount — Interest — Promissory Notes — Partial Payments — Compound Inter- est—Bank Discount— Present Worth . . . 166-220 PART IV Form^ and Measurements Lines — Angles — Surfaces — Solids — Longitude^ and Time — Ratio r . . 221-243 5 6 CONTENTS PART V Powers and Roots PAOX8 Powers — Square Root — Right-angled Triangles — Similar Sur- faces and Solids 244-255 PART YI Appendix Corporations, Stocks, and Bonds — Commission and Brokerage — Trade Discount — Partial Payments — Interest Table - Exact Interest — State and Local Taxes — Customs and Internal Revenue — Banking — Life Insurance — The Equa- tion — Proportion — Surfaces and Solids — Measurement of Public Lands — IMetric System — Tables of Denominate Measures — Table of Compound Interest . . . 256-320 Index 321-324 ESSENTIALS OF ARITIJMETiq,:^ PART I ..^i' -'^'-'•'' REVIEW OF INTEGERS AND DECIMALS 1, The Decimal System. 1. A unit is a single thing, or a group of things regarded as a single thing, as a book, an apple, a box of apples, etc. A unit is represented by the least whole number, one (1). 2. Point to several units of the same thing in your schoolroom. Can you think of a way by which you could tell your parents how many children there are in your room without using number? 3. Any definite quantity used to measure quantity of the same kind is called a unit of measure. The unit of 6 is 1 ; of 6 cows is 1 cow ; of 9 ft. is 1 ft. The inch, foot, yard, rod, and mile are units used to meas- ure length or distance. Name the units used to measure areas. What is the unit of 10? of 1^10? In finding the number of hats at $2 each that can be bought for $10, the unit of measure is 82. What is the unit of measure in finding the number of 4-ft. shelves that can be made from a board 12 ft. long? 4. Name the units used to measure liquids; time; weight. 8 REVIEW OF INTEGERS AND DECIMALS 5. Ih the number 111, the 1 at the right denotes some unit, and the 1 next toward the left denotes a unit ten times as great, and the 1 at the left denotes a unit ten times the second unit, or one hundred times the first unit. This aaay-be shp\Yij thus : one hundreds' uhit one tens' unit one unit 100 10 6. In 236, the 6 represents 6 units ; the 3 represents 3 units, each of which is ten times each of the units repre- sented by 6 ; and the 2 represents 2 units, each of which is ten times each of the units represented by 3, or one hundred times each of the units represented by 6. 7. Tell what each figure represents in 125, 47, 352. 8. In 30, the shows that there are no units of ones ; and the 3 represents 3 units of tens. What does each figure represent in 60, 600, 405, 530, 203, 478, 700, 520? 9. In 324, the units represented by 4 are called units of the first orders or of unM order ; the units represented by 2 are called units of the second order^ or of tens'* order ; and the units represented by 3 are called units of the third order ^ or of hundreds* order, 10. Our number system is a decimal system. Decimal means tens, A decimal system is one in which ten units of one order are equal to one unit of the next higher order. The decimal system is believed to have had its origin in the prac- tice of using the fingers for counting. DECIMAL SYSTEM 9 11. Beginning at the left of 111, the 1 in the third order represents some unit ; the 1 in the second order represents a unit one tenth as great ; and the 1 in the first order represents a unit one tenth as great as that represented by a unit of the second order. A unit one tenth as great as that represented by the 1 in the first order may be represented by 1 written to the right of a decimal point (.) placed to the right of units' order, thus : .1 (111.1). A unit one tenth as great as this last unit may be represented by 1 written in the second place to the right of the decimal point, thus : .01 (111.11). 12. .1 is read one tenth; .01 is read one hundredth ; .11 is read eleven hundredths ; 1.1 is read one and one tenth; A is read/(92^r tenths. Read 6.7; 8.05; 56.25. 13. The decimal point is placed after the figure that represents whole units. The figures to the right of the decimal point represent decimal parts of units. The parts thus represented are tenths, hundredtlis, thousandths, etc. ; and are called decimals. 14. A whole number is called an integer. Write an integer. On which side of the decimal point are integers written ? 15. What is the meaning of the word decimal? Why is our number system called a decimal system? 16. What does each 2 in 222.222 represent? 17. Write the following so that units of the same order are below one another: 45.5, 214.25, 347, 4.315, 17. 18. Compare the value of 2 in 24 with the value of 2 in 240 ; with the value of 2 in .24. 19. Is the system of United States money a decimal system? Explain your answer. 10 REVIEW OF INTEGERS AND DECIMALS NOTATION AND NUMERATION OF INTEGERS AND DECIMALS 2. 1. Numbers are commonly expressed by means of figures (or digits) as 5, 10, etc. ; by means of words, a,sfive, ten, etc. ; and by means of letters, as V, X, etc. The art of writing numbers by means of symbols is called notation. The word digit means finger. Why were the figures called digits ? 2. The figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, are called Arabic numerals, as they were introduced into Europe by the Arabs, who borrowed most of them from the Hindus. The system of denoting numbers by means of figures is called Arabic notation. 3. The figure is called naught, cipher, or zero. It has no value. It is used to fill out places that are not oc- cupied by other figures. Using figures, write six; six tens ; six hundreds. 4. The art of reading numbers is called numeration. 5. Integers of more than three places are read more easily when the figures are separated by commas into groups of three each, beginning at the right. The groups are called periods, and each period is named after the order of the right-hand figure in the group. 6. The names of the first four periods, and the orders in each, are as follows: 'S -s % •s Oedbes : 1 S ^ 1 CO a a Tens Units s Tens Units 7_ 8 9, 6 5, 8 4 2, 3 1 Peeiods : Billions Millions Thousands Units a b c 1. 3,625* 35,205 825,380 2. 8,017 20,007 308,016 3. 9,008 45,500 950,025 4. 6,303 12,012 404,040 5. What is the ) name of the first perio( . NOTATION AND NUMERATION 11 3. Reading Integers. To read an iiiteger of more than three figures^ begin at the right of the number and point off periods of three places each. Read the part occupying the left-hand period as though it stood alone^ and add the name of the period ; then read the part occupying the next period as though it stood alone, and add the name of the period. Continue until units' period is reached ; there omit the name of the period. Read the following : d 7,125,380 6,000,150 8,040,075 5,505,050 ? of the second period ? of the third period ? of the fourth period ? 6. How many periods are there in the numbers in column a? hi c? d? 7. How many places do the numbers in column c occupy ? in column d ? 8. What is the name of the left-hand period in the numbers in column a? 9. Kead the left-hand period of the first number in column d. Read the middle period of the same number. Read the number. 10. Read the third period of the fourth number in column d. Read the second period. Read the number. 11. When a number consists of three periods, how many places must there be in the lirst period ? in the second? How many places may there be in the third period ? * 625 is read six hundred twenty-five. 12 REVIEW OF INTEGERS AND DECIMALS 4. Writing Integers. To write a number in figures^ begin with the higheg, period and write it as though it stood alone^ and add a comma; then write the next highest period as though it stood alone, and add a comma; continue until units' period lias been written, thus: 5,807,050. Write in figures : 1. Three thousand, two hundred four. 2. One hundred two thousand, eight hundred ninety. 3. Twelve million, eight hundred seven thousand, eighty-four. 4. Seven hundred two million, sixteen thousand. 5. Write numbers dictated by your teacher. Read aloud the following statements : 6. The area of Rhode Island is 1,250 sq. mi. ; of Mas- sachusetts is 8,315 sq. mi.; of Illinois is 56,650 sq. mi.; of California is 158,360 sq. mi. ; of Texas is 265,780 sq. mi. 7. In 1900 the population of Rhode Island was 428,556 ; of Massachusetts was 2,805,346 ; of Illinois was 4,821,550 ; of California was 1,485,053; of Texas was 3,048,710. 8. The total number of votes cast for president in 1900 was 13,964,812. The five states polling the largest num- ber of votes were : New York, 1,548,042 ; Pennsylvania, 1,173,210; Illinois, 1,131,894; Ohio, 1,040,073, and Mis- souri, 683,656. 9. The grain production of the United States in 1902 in measured bushels was as follows: Indian corn, 2,523,648,312; wheat, 670,063,008; oats, 987,842,712; barley, 134,954,023 ; rye, 83,630,592 ; buckwheat, 14,629,770. ROMAN NOTATION 18 ROMAN NOTATION 5, 1. The letters used in Roman notation are: I V X L C DM 1 5 10 50 100 500 1000 2. The above letters are called Roman numerals. Other numbers are represented by combinations, thus : a. Repeating a numeral repeats its value. XXX de- notes 30, CCC denotes 300. The numerals V, L, and D are not repeated. Why ? h. If a numeral is followed by another of less value, the sum of their values is denoted. XXVI denotes the sum of 10, 10, 5, and 1. c. If a numeral is followed by another of greater value, the difference of their values is denoted. XC denotes the difference of 10 and 100, or 90; CD is 500-100, or 400. d. A bar glaced over a numeral increases its value 1000 times. V denotes 5000 ; IX denotes 9000. 3. Read the following and tell which of the above rules is illustrated in each: LX, XL, CIX, MDCC, IV, MDCCCCVI, LXXIV, MMDXL, MDCXX, XLII. 4. Write 1776 in Roman numerals. Model : 1776 may be divided into the parts 1000 — 700 — 70 — 6. These parts expressed in order, beginning at the left, are M — DCC — LXX — VI. 1776 is written MDCCLXXYI. 5. Write in Roman numerals : 18, 27, 68, 1492, 1907. 6. Write in Arabic figures XCVI, XLVII, XIX, LXXIV, MDCCCXII. Roman numerals are frequently used to designate chapter numbers in books, the hours on the clock face, dates on monuments and pub- lic buildings, etc. The M is used to designate a thousand feet of lumber. 14 REVIEW OF INTEGERS AND DECIMALS • UNITED STATES MONEY 6. 1. The units of United States money are decimal units. The standard unit of value is the dollar. The other units are derived from it. The dime is one tenth part of the dollar, and the units that represent dimes are there- fore written in the first place to the right of the decimal point. The cent is one hundredth part of the dollar, and the units that represent cents are therefore written in the second place to the right of the decimal point. 2. Dimes are written as cents. Two dollars and four dimes is written thus: 12.40. This is read two dollars and forty cents. 3. The unit one dollar is written $1. The unit one dime is written $.10. The unit one cent is written $.01. The unit one mill is written f.OOl. 7. Reading United States Money. Read the following : 1. 1425.15 5. $30,755 9. $8340.05 2. $301.08 6. $ 7.057 10. $9015.807 3. 1220.20 7. $10,105 11. $7200.50 4. $100.10 8. $ 4.005 12. $1306.065 8. Writing United States Money. Write the following in columns : 1. Six dollars and seventy-five cents. 2. Twenty-five dollars and fifty cents. 3. Eighty-five dollars and six cents. 4. Three hundred forty dollars and eighty cents. 5. One hundred dollars and fifty-two cents. 6. Eight cents. 7. Thirty-five cents and eight mills. READING AND WRITING DECIMALS 16 9. Reading Decimals. | •a 00 +3 -^ w "1" 2 Okders: '^wiS :S'«s^'«.2 9 13.452 876 Integers Decimals 1. Memorize the number of decimal places required for each of the first six orders. Tenths (first) . .' . . . . . . .5 Hundredths (second) 45 Thousandths (third) 367 Ten-thousandths (fourth) 6745 Hundred-thousandths (fifth) 62789 Million ths (sixth) 346329 To read a decimal, read the number without reference to the decimal point, and add the name of the order of the right-hand figure. 2. .375 is read three hundred seventy-five thousandths. 3.08 is read three and eight hundredths. Read: .125, .875, 4.625, 37.075, 670.005, 3.1416, 2150.42, .7854. 10. Writing Decimals. 1. Write sixty-two thousandths. As thousandths is the name of the third order to the right of the decimal point, three figures will be required in writing the num- ber. Two figures are necessary to denote sixty-two ; so one cipher must be supplied. To write sixty-two thou- sandths, first write the decimal point, then write 0, and then write 62 (.062). 2. Write the following: Sixty-nine ten-thousandths^ forty-eight hundred-thousandths ; thirteen thousandths. 16 REVIEW OF INTEGERS AND DECIMALS ADDITION OF INTEGERS AND DECIMALS 11, 1. A number that is not applied to any particular thing, as 6, 43, etc., is called an abstract number. 2. A number that is applied to some particular thing, as 6 ft., 43 lb., etc., is called a concrete number. 3. Quantities that are expressed in the same unit of measure, as 3 lb. and 6 lb., are called like quantities. 4. Quantities that are expressed in different units of measure, as 5 lb. and 4 hr., are called unlike quantities. 5. Write two abstract numbers; two concrete num- bers; two unlike quantities. Like quantities can be combined and expressed as a single quantity. 3 ft. and 2 ft. may be combined and expressed as 5 ft. Can the unlike quantities 5 lb. and 4 hr. be combined and ex- pressed as a single quantity? 6. Units of the same order may be combined and expressed as single numbers. 3 tens and 2 tens are 5 tens. 7. When two or more numbers are combined and ex- pressed as a single number, this number is called their sum, or amount. 8. The process of finding the sum of two or more numbers is called addition. The numbers that are added are called addends. 9. The sign of addition is + and is read plus. 10. This sign = is the sign of equality, and when placed between two numbers is read equals or is equal to^ thus: 6 = 4 + 2 means that 6 is equal to the sum of 4 and 2. ADDITION 17 12. Oral Exercises.* To each number in Exs. 1-4, add in succession 3, 2, 7, 6, 9, 4, 8, 5. 1. 23 35 84 69 26 88 82 57 47 60 2. 39 76 48 87 65 74 33 22 81 30 3. 52 86 49 61 73 95 40 18 67 94 4. 90 66 38 17 41 93 55 74 12 99 13. Add each column as written. Add each column, increasing the number at the bottom of the column by 10, by 20, etc., to 90 ; thus for column a, having increased the number at the bottom of the column by 20: 22, 25, 29, etc. a b c defghijklmn 98958797778896 3 7 8 8 7 9 3 3 7 9 9 9 5 8 6 5 9 2 5 7 9 6 8 8 9 5 6 4 4 6 3 7 4 6 5 8 6 6 5 9 4 9 3 8 9 6 8 9 5 2 4 9 1 8 8 8 8 9 7 5 9 6 8 7 8 7 7 7 3 9 7 7 6 4 5 8 5 5 6 8 1 9 4 2 7 7 3 5 7 6 9 9 4 4 8 7 3 7 2 4 5 9 8 8 7 8 5 9 4 6 9 6 3 7 2 3 7 9 6 5 8 8 2 8 5 5 4 3 9 5 6 4 9 7 9 7 6 3 7 7 3 8 9 2 9 6 6 7 8 5 8 2 7 7 2 6 8 4 4 5 7 9 6 2 8 9 7 4 * If the pupils require a more extended drill upon addition than is pro- vided in the above exercises, the method indicated in the elementary text should be followed. AR. — 2 18 REVIEW OF INTEGERS AND DECIMALS 14. Written Exercises. Numbers to be added or subtracted must be written so that units of the same order are directly below one another, units under units, tens under tens, and tenths under tenths, etc. Why ? When numbers are written so that the decimal points are directly below one another, units of the same order are directly below one another. Explain. Add: 1. 2. 3. $ 345.67 $ 58.06 $ 9.045 84.075 275.936 590. 650. 83.07 5.15 70.004 342.457 69.075 572.806 34.08 610.75 6.605 8.125 57.246 852.451 64. . 540.375 ^ ^ W " Kead aiduii 6ach of ihe aqove.' ^ 7. Write the above from dictation. 8. Add 74.06 mi., 6.8 mi., 320.45 mi., 17.04 mi. 9. Add 64.5 A., 79.14 A., 160.75 A., 321.15 A. 10. Add 60.5 cu. in., 352.24 cu. in., 80.125 cu. in. 11. Add 168.05, 1107.98, 1730.04, f 9.75, 1894, $80, 1740.40, 1375.15, f486.75, 1836.95, i.95. 12. Add six and nine hundredths, thirty-seven and six tenths, eighty-five thousandths, seven hundredths. 13. Find the sum of nine hundred eighty and five tenths, seventy and seven hundredths, one hundred and five thousandths, six hundred twenty-five. 14. Write five addition exercises similar to Exs. 1-5 above and add each. Read each answer. 4. 5. $405.27 $ 68. 73.435 125.87 487.50 45.369 50.258 845.075 250.50 8.75 .375 100. 62.50 58.268 ADDlTIOiSr 19 15. Oral Exercises . Add a h c d e / 9 h i 1. 40 60 130 120 60 150 90 140 80 50 20 90 30 140 40 30 60 20 2. TO 40 140 50 60 80 70 20 70 25 54 63 139 42 59 96 192 .36 3. 29 55 79 56 54 89 46 92 m 90 80 70 90 80 70 80 90 60 4. 23* 43 64 36 59 54 68 94 39 89 52 95 . 94 43 46 43 36 27 5. Frank weighs 95 lb. and his little brother weighs 34 lb. How much do they both together weigh ? 6. A farmer has 46 sheep and his neighbor has 54 sheep. How many have both together? 7. A man paid f 94 for a wagon and 836 for a harness. How much did both cost him ? 8. Mr. White had 23 head of cattle and bought 39 more. How many had he then ? 9. A girl spent 50 ^ for cloth and 45/ for lace. How much did she spend for both ? 10. A boy placed 60/ into his bank one week and 46/ the next week. How much did he put into the bank in the two weeks? 11. A girl spent 20/ for stamps, 25/ for some meat, and 50 / for sugar. How much did she spend for all ? 12. Make and solve ten oral problems in addition. ♦ Add : 89, 109, 112. 20 REVIEW OF INTEGERS AND DECIMALS SUBTRACTION OF INTEGERS AND DECIMALS 16. 1. Like quantities, such as 5 marbles and 9 marbles, may be compared, and the difference between them found, thus: 9 marbles: ••••••••• 5 marbles: • • • • • 2. If there is added to 5 marbles a quantity that will make it equal to 9 marbles, how much is added ? This amount is the difference between the two quantities. 3. If that part of 9 marbles that is equal to 5 marbles is taken from 9 marbles, liow many will remain ? This remainder is the difference between the two quantities. 4. How does the difference as found in Ex. 3 compare with the difference as found in Ex. 2 ? 5. The difference between the two quantities may be found by answering either of the following questions: a, 5 marbles and how many marbles are 9 marbles ? h. 5 marbles from 9 marbles leaves how many marbles ? In either case, the answer is known by recalling that the sum of 5 marbles and 4 marbles is 9 marbles. 6. The difference between two numbers is the number which when added to one number makes the other number. 7. The process of finding the difference between two numbers is called subtraction. 8. The number to which the difference is added is called the subtrahend. 9. The sum of the subtrahend and difference is called the minuend. Or the subtrahend is the number which is subtracted, and the minuend is the number from which the subtrahend is taken. SUBTRACTION 21 17. Oral Exercises. a b c 1. 6 and — are 11 8 and — are 12 9 and — are 16 2. 9 and — are 14 7 and — are 13 8 and — are 14 3. 8 and — are 11 4 and — are 11 7 and — are 11 4. 7 and — are 12 9 and — are 15 5 and — are 14 5. 5 and — are 13 5 and — are 11 6 and — are 15 6. 4 and — are 12 3 and — are 12 9 and — are 13 7. 8 and — are 15 5 and — are 12 8 and — are 16 8. 7 and — are 16 6 and — are 14 7 and — are 14 9. 9 and — are 17 8 and — are 13 8 and — are 17 10. 6 and — are 12 7 and — are 15 9 and — are 11 11. 9 and — are 12 9 and — are 18 6 and — are 13 9 12. _ r is read 5 and how many are 9? Or, ^ from 9 leaver how many? Use the form with which you are familiar. 13. The sign of subtraction is ~, and is called minus. It indicates that the number that follows it is to be sub- tracted from the number that precedes it. 7 — 4 is read seven minus four, 18. Explanation of Subtraction. 1. Find the missing addend. (one addend) The other addend may be 2874 (one addend) ^^^"^ ^7 a^d^^S *° *^« g^^^^ rnor« y £ ^ i i i n addeiid the number that will ozdo (sum 01 two addends) . ^, ^, ^ j « ^ ^ give the sum, thus : 4 and 2 are 6 ; 7 and 6 are 13 ; carry 1 to 8, making it 9 ; 9 and 3 are 12 ; carry 1 to 2, making it 3; 3 and 2 are 5. Missing addend, 2362. 22 REVIEW OF INTEGERS AND DECIMALS 2. From 5236 subtract 2874. Model a : 5236 -A-fld to the subtrahend the number that will 2874 give the minuend, thus : 4 and 2 are 6 ; 7 and 23(32 6 are 13 ; carry 1 to 8 as in addition, making it 9; 9 and 3 are 12; carry 1 to 2 as in addition, making it 3; 3 and 2 are 5. AVrite the answer as in the model. This is known as the Austrian, or additive, method. Model b : Subtract thus : 4 from 6 leaves 2 ; as 7 tens cannot be taken from 3 tens, 1 hundred is " borrowed " from 2 hundreds and called 10 tens; 10 tens and 3 tens are 13 tens; 7 tens from 13 tens leaves 6 tens ; as 1 hundred was borrowed from 2 hundreds, there is left 1 hundred; as 8 hundreds cannot be taken from 1 hundred, 1 thousand is borrowed from 5 thousands and called 10 hundreds ; adding 10 hun- dreds to 1 hundred gives 11 hundreds ; 8 hundreds from 11 hundreds leaves 3 hundreds ; as 1 thousand was taken from 5 thousands, there are left 4 thousands; 2 thousands from 4 thousands leaves 2 thousands. Model c : If the same number is added to both the minuend and the subtrahend, the difference remains unchanged. Subtract thus: 4 from 6 leaves 2 ; as 7 tens cannot be taken from 3 tens, add 10 tens to 3 tens, making 13 tens ; 7 tens from 13 tens leaves 6 tens ; as 10 tens were added to the minuend, the same number must be added to the subtrahend, so 1 hundred (10 tens) is added to 8 hundreds, making 9 hundreds; as 9 hundreds cannot be taken from 2 hundreds, 10 hundreds are added to 2 hundreds, making 12 hundreds; 9 hundreds from 12 hundreds leaves 3 hundreds; as 10 hundreds were added to the minuend, the same number must be added to the subtrahend, so 1 thousand (10 hundreds) is added to 2 thousands, making 3 thousands ; 3 thousands from 5 thousands leaves 2 thousands. 19. Written Exercises. Solve : 1. 38,256-21,359 6. 1,106,800-289,060 2. 40,175-19,688 7. 4,083,453-613,757 3. 85,430-41,856 8. 3,256,845-465,868 4. 93,950-17,275 9. 4,741,242-572,847 5. 97,204-57,240 lO. 2,814,004-935,940 SUBTRACTION 23 20. Oral Exercises. Subtract : a h c d e f g hi 1. 40 140 150 100 120 150 120 90 110 20 30 20 40 30 60 90 40 50 2. 95 40 126 90 83 50 142 60 149 80 155 70 153 20 129 40 124 80 3. 124* 92 109 44 138 85 96 13 75 24 139 68 136 45 88 16 99 44 4. 75 1 38 142 96 34 19 57 29 83 68 74 18 42 27 36 19 52 29 5. Harry bought 120 yd. of string and used 85 yd. for a kite string and gave the rest to George. How many yards did he give to George ? 6. A farmer had 52 head of cattle and sold 29. How- many had he left ? 7. Mary read 87 pages in a book that contained 124 pages. How many more pages must she read to complete the book ? 8. There are 38 pupils in Room A and 47 in Room B. How many pupils are there in both rooms ? How many more pupils are there in Room B than in Room A? 9. The frontage of a certain city lot is 40 ft. and its depth is 135 ft. Find the difference between the depth and frontage of the lot. * Suggestion. The difference iDetween 92 and 124 Is 30 and 2, or 32. t Suggestion. The difference between 38 and 75 is 80 (38 to 68) and 7 (68 to 76), or 37 ; or 40 less 3, or 37. 24 REVIEW OF INTEGERS AND DECIMALS 21, Before solving, represent each by a diagram. 1. Two boys started from the same place. One boy rode east 32 mi. and the other boy rode west 24 mi. How far apart were they then? W. 2^ mi. S 32 mL From S. to E. is 32 mi. and from S. to W. is 24 mi. From E. to W. is the sum of 32 mi. and 24 mi., or ^Q mi. 2. Two boys started from the same place. One rode east 32 mi. and the other rode east 24 mi. How far apart were they then ? 3. How far apart are two places, if one is 40 mi. north of the center of a certain city, and the other is 65 mi. south of the center of the same city ? 4. Mary lives 8 blocks east of the schoolhouse, and Ethel lives 14 blocks west of the schoolhouse. How far apart do the girls live ? 5. Two trains left a certain station at the same time, going in opposite directions. How far apart were they at the end of 2 hours, if one traveled at the average rate of 42 mi. an hour, and the other at the average rate of 36 mi. an hour ? 6. How far apart would the trains mentioned in Prob. 5 be at the end of 2 hours, if both traveled in the same direction ? 7. In a bicycle race Frank and Henry rode around a park 400 ft. long and 200 ft. wide. When Frank had ridden once around tlie park, Henry hud gained 200 ft. on him. At tlie same rate of gain, how many times will Frank ride around the park before Henry overtakes him ? SUBTRACTION 25 22. United States Money. Write units of tlie same kind below one another. Do not supply unnecessary O's. 1. Subtract: a. $12.75 from 137.25; h. $12 from 137.25; c. 112.75 from $37. Model a: $37.25 Model &: $37.25 Model c: $37. 12.75 12. 12.75 $21.50 $25.25 $24.25 Solve : 2. $307.57 -$200.69 6. 120.375-93 3. $925.07 -$570.80 7. 690.125-209 4. $700.40 -$180.05 8. 542-45.78 5. $860.455 -$280 9. 640-70.65 10. Read aloud each of the above amounts. 11. Write the above amounts from dictation. 23. Decimals. Subtract : 1. 2. 3. 4. 320.564 450.125 35.7 600. 206.7 86.75 6.875 57.375 5. A man owned 158.15 acres of land. He sold 79.5 acres. How many acres had he left ? 6. If it is 844.7 mi. from San Francisco to Ogden and 1004.7 mi. from Ogden to Omaha, how far is it from San Francisco to Omaha? How much farther is it from Ogden to Omaha than from San Francisco to Ogden ? 7. A cubic foot of rain water weighs 62.5 lb. and a cubic foot of petroleum weighs 54.875 lb. How much heavier is a cubic foot of rain water than a cubic foot of petroleum (kerosene) ? 26 REVIEW OF INTEGERS AND DECIMALS 24. 1. Show the effect, if any, upon the difference : (a) of adding the same number to both minuend and subtrahend; (6) of subtracting the same number from both minuend and subtrahend. Illustrate each with several exercises. 2. Write ten exercises in subtraction of decimals and solve each. 25. Oral Exercises. 1. Name five combinations whose sums are 10. When these combinations occur in a column, they should be treated as 10. Exercise a below may be added : 15, 25, 32, 42, 48, 58, 66, 16. Add the following exercises in a similar manner : abcdefgh i j k I m 57841574 5 87 89 5326.9526573 98 8697648567712 r5 3841347638 97 15 726976348246 68798866766 64 r5 7269782145 6 b l5 384132898 559 75586997723 21 r5 3269541379 85 15 7841569891 79 66789999977 83 989786 5 29688^4 2. Write ten columns, in each of which some of the five combinations wliose sums are 10 occur several times. Add these columns. SUBTRACTION 26. Written Exercises. 27 Nbw England States Area in Sy. MiLBs Gkbat Lakes Area in Sq. Miles 33,040 9,305 9,565 8,315 1,250 4,990 Superior Huron Michigan Erie Ontario 31,200 23,800 22,450 9,960 7,240 New Hampshire Vermont . . . Massachusetts . Rhode Island . Connecticut . . 1. Find the combined area of the New England states; of the Great Lakes. 2. Find the difference between the combined area of the New England states and of the Great Lakes. 3. Compare the area of Vermont with the combined area of Massachusetts and Rhode Island. 4. Find the difference between the area of Lake Ontario and the combined area of Rhode Island and Connecticut. 5. Find the difference between the area of Lake Su- perior and the combined area of Lakes Huron, Erie, and Ontario. 6. Compare the area of Maine with the area of Lake Superior. 7. Find the difference between the area of Maine and the combined area of the other five New England states. 8. The area of Missouri is 69,415 sq. mi. Compare the area of Missouri with the combined area of the New England states. 28 REVIEW OF INTEGERS AND DECIMALS 27. Written Exercises. In solving a problem, follow these steps in the order given : a. Read the problem carefully^ if convenient, aloud. h. State what facts are given in the problem and what fact you are asked to find. c. Determine what relation the given facts have to one another, and state what operation you must use in finding the facts that are asked for, — whether you must add or subtract, etc. d. Make an estimate of the answer. When you have found the answer, compare it with this estimate. 1. A man bought a house for $2400 and sold it for $3000. Find the amount gained. 2. A farmer sold his farm for $7500, which was $1800 more than it cost him. How much did he pay for the farm ? 3. A farmer bought a farm for $6250 and sold it at a gain of $1200. How much did he get for the farm? 4. By selling a farm for $4500, a farmer received $900 less than it cost him. How much did it cost him? 5. A room is 24 ft. long and 18 ft. wide. Find how many feet of picture molding it will require for the room. 6. After drawing out $2300 from a bank, a merchant had $760 left in the bank. How much had he on deposit in the bank? 7. A merchant had $1600 on deposit in a bank on Jan. 1, 1907. On Jan. 2 he drew out $200. On Jan. 5 he deposited $750. On Jan. 15 he drew out $2000. How much had he left in the bank? SUBTRACTION 29 28. Written Exercise*. 1. The total production of corn in the United States in 1899 was 2,666,440,279 bu. In 1889 tiie total production was 2,122,327,547 bu. How much had the production increased during the decade (10 years)? 2. In 1899 the production of corn in Illinois was 398,149,144 bu. The production in 1889 was 289,697,256 bu. Find the increase in production during the decade. 3. From the amounts given in Frobs. 1 and 2, find the total number of bushels produced in all states other than Illinois in 1899. 4. The total production of rice in the United States in 1899 was as follows : Louisiana . South Carolina Hawaii Georgia North Carolina Texas Florida Alabama . Mississippi Arkansas . Virginia 172,732,430 lb. 47,360,128 lb. 83,442,400 lb. 11,174,562 lb. 7,892,580 lb. 7,186,863 lb. 2,254,492 lb. 926,946 lb. 739,222 lb. 8,630 lb. 4,374 lb. a. Read aloud the above quantities. h. Write the above from dictation. c. Find the total number of pounds produced. d. Was the amount produced by Louisiana more or less than that produced by all others combined, and how much ? e. Compare the amount produced in South Carolina with the total amount produced in Hawaii and Georgia. 30 REVIEW OF INTEGERS AND DECIMALS 29. Written Exercises. 1. Ill 1890 the population of San Francisco was 298,997, and in 1900 it was 342,782. Find the increase in popula- tion between 1890 and 1900. 2. In 1898 the population of London was 4,504,766, and' in 1900 the population of New York was 3,437,202 and of Chicago was 1,698,575. Find the difference be- tween the population of London and the combined popu- lation of New York and Chicago. 3. The area of the earth's surface is about 196,940,000 sq. mi. Of this, 141,486,000 sq. mi. is covered with water. Find the area of the land. 4. A cattle dealer bought some cattle, for which he paid $ 380. He paid out f 67 for feed and care of the cattle. He then sold them for ef 500. How much was his net profit, that is, the profit after deducting all expenses ? 5. A real estate dealer bought a city lot for 11750. He built a house on it that cost $ 3275 and then sold the property for $ 6000. Find the amount of his gain or loss. 6. The total area under broom-corn cultivation in the United States in 1899 was 178,584 acres. In 1889 it was 93,425 acres. How much was the increase in the area under cultivation during the decade ? 7. The appropriation for the maintenance of the navy for 1907 was $ 98,773,692, for the military $ 72,305,270, for pensions $ 143,746,106. How much was appropriated for these three purposes ? How much more were the combined appropriations for the navy and military than for pensions ? 8. The total appropriations of the government for 1907 amounted to $5 701,551,566. Find the appropriations for all purposes other than the navy, military, and pensions. MULTIPLICATION 31 MULTIPLICATION OF INTEGERS AND DECIMALS 30, 1. Find the sum of a column of four 2's. 2. The sum of a column of four 2's is . In this column the addend 2 is repeated 4 times. Four times 2 are . 2 X 4 3. Four 2's are 8 may be indicated thus : - • Here 2 is taken 4 times, or is multiplied hy 4. The 2 is the addend that is repeated, and the 4 tells the number of times this addend is repeated. % 82 4. The sum $32 taken 4 times may be ^ oo found by addition, thus: ^ g2 It may also be found by ^ ^^ multiplication, thus : ^Toq $128 Since four 2's are 8 and four 3's are 12, four |32's are $128. 5. Find the cost of 3 cows at $48 each by addition; by multiplication. Which method is the shorter? 6. A man paid the following amounts for three horses : $120, $85, and $100. Can the cost of the three horses be found by multiplication ? Give a reason for your answer. 7. Multiplication is the process of taking one number as many times as there are units in another. 8. The number that is multiplied is called the multi- plicand; the number by which we o^^c^oa i^- t j u- 1 • ^^ A ^x. 1.- 1- ^284, multiplicand, multiply IS called the multiplier; « if r and the result obtained is called ^^t^ , . ,, , ^ $702, product, the product. ^284 . . ^^.^. 9. Express « as an exercise in addition. 32 .REVIEW OF INTEGERS AND DECIMALS 10. Regard the multiplicand as a repeated addend and tlie multiplier as the number of times the addend is repeated. 11. The product and the multiplicand are always like quantities. Why ? 12. The multiplier is always an abstract number. It tells how many times the multiplicand is to be taken, or how many times the addend is to be repeated. 13. The sign of multiplication is x . It indicates that the number before it is to be multiplied by the number after it. $3 x 4 is read 83 multiplied by 4. The sign ( X ) is sometimes used in place of the word times in such an expression as 2 times f 5. 14. Express each of the following in the form of addi- tion : $4x6; 5 lb. X 3; 4 times 6 yd. ; 7 in. x 8; 9x4. 31. Law of Commutation. 1. In the following diagram there are 3 rows of squares, with 4 squares in each row. Or, there are 4 rows of squares, with 3 squares in each row. There are in all 12 squares. We see that 3 times 4 squares and 4 times 3 squares are the same number of squares. 2. Find the sum of three 4's and of four 3's. Since the sum of four 3's is the same as the sum of three 4's, the product of 3 and 4 is the same, without regard to which is multiplier and which is mul- tiplicand. 3. Show by the addition of columns that the sum of five 6*s equals the sum of six 5's. Show by a diagram that 5 times 6 squares equals 6 times 5 squares. MULTIPLICATION 33 32. Remembering that the multiplicand is the same as the repeated addend, answer the following : 1. Can you multiply 16 by 3? 3 by |6? 6 ft. by 3 ft.? 2. Can you find 8 lb. X 2 ? 2x8? 2 ft. x 3 ft. ? 3. When the multiplicand is some number of yards, what is the product? 4. Can the multiplier ever be concrete ? Why ? 5. Which is more, |6 x 3 or |3 x 6? 33. Table of Products and Quotients. For reference only. 1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 C 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 "28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 6Q 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 Note. If the pupils are not thoroughly familiar with the facts of multiplication and division, a thorough mastery of these facts should precede the attempt to use them in the exercises that follow. For a systematic method of teaching these facts, see Elementary Arith- metic of this series. AR.— 3 84 REVIEW OF INTEGERS AND DECIMALS 34. Oral Exercises. Supply products for x^ and add to each product the numbers above the column, as in a (products 12, 16, 35, etc.); adding 2: 14, 18, 37, etc. ; adding 6: 18, 22, 41, etc. a h c d ^x2 = x 3, 5 9x3 = 2; 4, 7 7x7 = 2; 8, 9 7 x4 = 2; 4x4 = 2; 8x5 = 2; 8x9 = 2; 7x5 = 2; 5 X 1 = x 4x9 = 2; 6x6 = 2; 4x5 = 2; 8x4 = 2: 6x7 = 2; 7x9 = 2; 9x8 = 2; 8x7 = 2; 9x5 = 2; 6 x4 = 2; 5 x5 = 2; 9 x4 = 2; 6 x8 = 2; 9x6 = 2; 4x8 = 2; 5x6 = 2; 9x9 = 2; 8x8 = 2; 3x9 = 2; 9x3 = 2; 7 X 8 = 2; 8x7 = a; 8x6 = a; 35. Written Exercises. 1. Multiply 25,435 by 304. Model: 25435 Explanation: 25435 304 304 101740 101740 = 4 times 25435 76305 7630500 = 300 times 25435 7732240 7732240 = 304 times 25435 Multipl}^ by 4; multiply by 3 (that is, 300), placing the product of 3 and 5 directly below the 3, or in hundreds' place. Add. Solve : 2. 435,450x504 7. $386.50x527* 3. 978,689x450 8. $868.75x689 4. 230,302 X 800 9. $768.30 x 843 5. 967,843 X 769 10. $364.97 x 107 6. 845,397x896 11. $426.87x489 ♦ Point off two places for cents in the answer. MULTIPLICATION 35 36. Oral Exercises.* 1. How much will 4 sheep cost at $5 each? Model for oral recitation : Since 1 sheep costs $5, 4 sheep will cost 4 times $5, or 120. 2. How much will 4 chairs cost at $3 each ? 3. At f 2 each, how much will 6 books cost? 4. There are 4 quarts in a gallon. How many quarts are there in 5 gallons ? 5. Make and solve ten similar problems in multiplication. 37. Written Exercises. 1. Find the cost of 32 acres of land at $75 an acre. Model ; Statement : 1 75 X 32 = a; Work: $75, cost of 1 acre. 32 150 225 $2400, cost of 32 acres. 2. If a boy attends school 180 days each year for 12 years, how many days will he attend in 12 years ? 3. Find the cost of 25 cows at $45 each. 4. If a man earns $1.75 a day, how much will he earn in 24 days ? 5. Make and solve ten problems in multiplication simi- lar to those a clerk in a grocery store has to solve. * Drill should be given upon these and similar problems until the pupils are familiar with the language forms used in the analysis. The written form should be taken up after the oral form has been mastered. Apply this form of analysis (or some suitable form) to similar problems on the succeeding pages of the text. When tlie form has been mastered, the pupils should be permitted to reply briefly, thus for Prob. 1 : Four times five dollars, or twenty dollars. 86 REVIEW OF INTEGERS AND DECIMALS REVIEW — FARM PROBLEMS 38. 1. A man bought a farm of 160 acres at i 75 an acre. Find the cost of the farm. 2. During the first year he expended the following sums for improvements : repairing fences, $165.80; dig- ging a well, $95; building a carriage house, $640; re- shingling the barn, $124.35; draining a marsh, $60. Find the cost of the improvements. 3. The farm was divided into 5 fields of 20 acres each, 3 fields of 10 acres each, 10 acres of orchard, 5 acres for yards and garden, and the rest was timber land. How many acres of timber were on the farm ? 4. To stock up the farm, the farmer bought 14 head of cattle at an average of $ 37 apiece, 5 horses at an average of $120 apiece, 24 sheep at $4.50 apiece, 6 hogs at $5.25 apiece, and 30 chickens at $.35 apiece. Find the cost of all. 5. The following amounts were received from the sale of milk for one year: Jan., $60.80; Feb., $68.17; March, $70.30; April, $71.90; May, $79.25; June, $80; July, $72.35; Aug., $66.10; Sept., $63.28; Oct., $59.37; Nov., $50.40; Dec, $54.30. Find the amount received during the year. 6. The farmer employed one man for 8 months, paying him $35 a month, and another man for 3 months, paying him $ 38 a month. Find the amount expended in wages. 7. Two of the 20-acre fields were sown in oats, and the yield was 45 bu. to the acre. If oats were worth 34^ per bushel, find the value of the crop. 8. During the month of April the farmer sold $15.75 worth of eggs. At the same rate, how much would the sale of eggs amount to in one year? MULTIPLICATION 37 39. Multiplication and Division by 10, 100, etc. 1. Compare the value of 2 in 20 and in 2; in 200 and in 20 ; in 2000 and in 200. 2. What effect upon the value of a figure has (a) mov- ing it one place to the left ? (J) moving it two places to the left ? (c) moving it three places to the left ? 3. Annexing a cipher to an integer has the effect of moving the digits each one place to the left. This mul- tiplies the number by 10. Annexing two ciphers has what effect upon the places occupied by the digits ? State a short method of multiplying an integer by 100 ; by 1000. 4. Using the short method, multiply each by 10; by 100; by 1000: 6, 47, 390, 20, 475, 8, 600, 72, 25, 64, 640. 5. State how an integer may be multiplied by 10; by 100; by 1000. 6. Moving a figure one place to the right has what effect upon its value ? Compare the value of 6 in 60 and in 6; in 600 and in 6; in 6000 and in 6. 7. Dropping the cipher at the right of 60 changes the number to 6. What change does this make in the value of the number ? State a short method of dividing an in- teger ending in a cipher by 10. 8. What change is made in the value of 400 by drop- ping the two ciphers ? State a short method of dividing a number ending in two ciphers by 100. 9. Divide each by 10 ; by 100: 4500,700,400, 3700. 10. An integer that does not end in a cipher may be divided by 10 by placing a decimal point to the left of the right-hand figure, thus: 87 divided by 10 is 8.7. Is this the same as moving each digit one place to the right ? 88 REVIEW OF INTEGERS AND DECIMALS 11. An integer that does not end in two ciphers may be divided by 100 by placing a decimal point at the left of the figure in tens' place, thus: 475 divided by 100 is 4.75. 12. Give the quotient of each divided by 10; by 100: 325, 560, 4582, 4500, 48, 4, 2, 10, 5, 50. 13. Write integers and divide each by 10 ; by 100. * 40, Short Methods. ( To be used in subsequent work. ) 1. What part of 100 is 25 ? Compare 25 times a num- ber with 100 times the number. 2. To multiply by 25, multiply by 100 and divide by 4. Multiply $489 by 25. Model : $ 489 Explanation : Write as in multiplication. 25 Mentally multiply $489 by 100. Divide the $ 12225 product by 4. 3. Multiply by 25: 1680,11225, 5280 ft., 231 mi., 187.56,1247.82. 4. Multiply 7865 by 369. Model: 7865 Explanation : First multiply by 9. As 70785 36 is 4 times i), multiply 70786 by 4, writ- 283140 ^"S ^^® product as in the model. Add. 2902185 5. In multiplying by 84, first multiply by 4 ; then mul- tiply this product by 2, writing the first figure of the prod- uct in tens' place. State how you would multiply by 63; by 126 ; by 246 ; by 729; by 279. Illustrate each. 6. Multiply 6840 by 248 ; by 328 ; by 648 ; by 168. 7. Multiply 1840 by 287 ; by 147 ; by 637 ; by 639. MULTIPLICATION 39 4:1. Multiplication of Decimals. 1. Find the sum of 5.2 mi., 5.2 mi., and 5.2 mic This sum is the same as the product of 5.2 mi. x 3. How many decimal places are there in this product? Why? 2. Find the sum of 5.2 mi., 5.2 mi., 5.2 mi., 5.2 mi., and 5.2 mi. Multiply 5.2 mi. by 5. How many decimal places are there in the product? Why? 3. Write 6.08 x 3 in the form of addition and find the sum. Multiply 6.08 by 3. How many decimal places are there in the product? Why? 4. Write 6.08 X 5 in the form of addition and find the sum. Multiply 6.08 by 5. Compare the results. How many decimal places are there in the result? Why? 5. State a short method of multiplying an integer by 10. 6.25 may be multiplied by 10 by moving the decimal point one place to the right. 6.25 x 10 is 62.5. Has moving the decimal point one place to the right the same effect as moving the digits one place to the left? Compare 2.2 with 22. Compare 75.25 with 752.5. 6. State a short method of multiplying a decimal by 100. How many places to the right must the decimal point be moved to multiply by 10? by 100? by 1000? 7. Compare 25 with 2.5. Here the digits have been moved one place to the right. Compare 2.5 with .25. Here the digits have been moved another place to the right. This has been done by moving the decimal point one place to the left. Moving the decimal point one place to the left has what effect upon the value of a decimal? upon the value of an integer? 8. State a quick way of dividing a decimal by 10 ; by 100. Illustrate with integers and with decimals. 40 REVIEW OF INTEGERS AND DECIMALS 42. Oral Exercises. 1. Divide each by 10 and by 100: 450, 25.74, 45, .4, 4.5, 346.2. 2. What is 6 times 4? 1 times 4? J of 4 ? ^ of 4 ? 3. What is meant by 4x6? 4x1? 4xJ? 4x.l? 4. 4 X .1 is the same as 4 divided by v^^hat number? 4 X. 1 = 37. 4x.2 = a:. 4x.3 = a;. 5. Divide 4 by 100. What is meant by 4 x .01 ? 4 x .01 = x. 4 X ,06 = x. 6. What is meant by .4 x 2? .4 x 13? .4 x .1? .4 x .1 is the same as .4 divided by 10. .4 -s- 10 = a;. .4 x .1 = a;. .4 X .2 — x. .4 X .5 = x, 7. Multiply 82.30 by 10. Multiply 12.36 by 100. Multiply 12.30 by .1. Multiply $24.50 by .01; by .1. Multiply 145.75 by 10 ; by .1 ; by 100 ; by .01. \J 8. To multiply by .1 is the same as to divide by 10. What change made in the place of the decimal point in the multiplicand divides it by 10 ? n 43. Multiply each by 10 ; by .1 ; by 100 ; by .01. 1. $37.50 5. 625 ft. 9. 2240 lb. 13. 3.1416 2. $2500 6. 1726 yd. 10. 2000 lb. 14. 24 cwt. 3. 14.525 7. 5280 ft. 11. 625 lb. 15. 4.75 cvvt. 4. 17500 8. 2150.42 12. 630 lb. 16. 20 T. ^ 17. 624 X .001 = a:. 2000 lb. x .001 = a:. 18. Divide by 100: 4632 lb.; 3000 lb. ; 2160 mi.; 37.40 mi.; $234.50; .425 mi.; .03 mi.; 3.1416 ft.; $60. 19. At $5 per hundredweight, how much is sugar worth per pound ? at $4.75 per cwt. ? at $4.50 per cwt. ? 20. Multiply each in the shortest way : 5280 ft. X 25 ; 1728 X 25 ; 987,647 by 648 ; 7,389,675 by 369. MULTIPLICATION 41 44, Written Exercises. 1. Multiply 6.23 by 4.2. Model : Explanation : 6.23 6.23 First multiply 6.23 by .2. This is equiva- . ^ . ^ l^iit to dividing 6.23 by 10 and multiplying -^ ^ ^ the quotient by 2. 6.23 -- 10 = .628 ; .623 1246 1.246 ^2 = 1.246. Next, multiply 6.23 by 4. 2492 24.92 6.23 x 4 = 24.92. Add the products. 26.166 26.166 Notice that in the above the number of decimal places in the product is the same as the sum of the number of decimal places in the multiplicand and multiplier. As this is always true, the following method may be employed : To multiple/ decimals^ multiple/ as in integers and point off in the product as many decimal places as there are in both multiplicand and multiplier. Solve. Estima^ each result b efore mu ltiplying: 2. 59.786 X8.97 r- .0056 X 385.07 3. 487.69 53.008 x.479 X 7.086 \- 7.0758 X 67.09 4. 9. .07854 X 8.0065 5. .69387 X 6.9075 10. 46,897x4.008 6. 13.006 X 3.1416 11. 785.06 X 6300 45. Written Exercises. 1. If a train travels at an average rate of 43.5 mi. an hour, how far will it travel in 24 hours ? 2. The circumference of a circle is 3.1416 times its diameter. Find the circumference of a circle that is 9.5 in. in diameter. 3. If it costs a boy $.50 a week to keep a pony, how much will it cost to keep it for 1 year (52 wk.)? 42 REVIEW OF INTEGERS AND DECIMALS BILLS AND ACCOUNTS 46. A Receipted Bill. Los Angeles, Cal., May 31, 1907. Mr. James J. Davies, 2217 Vine St. In account with S. D. James & Co. May 3 Z lb. coffee $.40 2 lb. tea M 6 bars soap .05 $1 1 20 30 30 (4 5 3 cans corn .08 1 doz. lemons .15 24 15 (( 6 10 lb. sugar .06 Received Payment^ 60 $3 79 S. D. James ^ Co. 1. The person who buys on account is called the debtor, and the person who sells on account is called the creditor. 2. Each purchase, or payment, is an item. How many items of debit are there in this bill ? of credit ? 3. A bill must show the date of each transaction. "When was the above bill made out, or rendered? 4. A bill must also name the debtor and the creditor and the several items of debit and credit. Who is the debtor named in the above bill ? Who is the creditor ? 5. When a bill is paid, the creditor writes " Paid " or " Received payment " on the bill and signs his name below. Has the above bill been paid? 6. Should a person make a practice of keeping receipted bills ? Why ? MULTIPLICATION 43 47. Written Exercises. Make out and receipt the following bills. Supply all necessary data not contained in the problems: 1. Mr. J. S. White, residing at 234 First Street, bought of the grocery firm of Allen and Baker the following : May 25, 1907, 2 lb. tea @ 60^; 3 lb. coffee @ 45/; 2 lb. bacon @ 20 /; 1 lb. butter @ 30 /. On May 31, 4 cans tomatoes @ 8/; 2 doz. eggs @ 18j^; 1 lb. cheese @ 20 j^. 2. Mrs. Harry Smith, residing at 1450 Jackson Street, bought of Cole Bros, the following : April 24, 1907, 9 yd. silk @ $1.50 ; 2 yd. dress lining @ 25^; 4 spools silk @ 10/; 1 bolt skirt binding, 15/; dress trimmings, $1.50. The bill was rendered April 30. 3. Insert your own name as purchaser of the following bill of hardware: 1 garden rake, 45/; 1 shovel, 60 /; 3 lb. nails @ 6/; 12 yd. wire netting @ 30 /; 2 lb. staples @ 5/; 1 lawn mower, 12.50. 4. Dr. C. L. Ward employed a schoolboy to take care of his horses, for which he agreed to pay him $6 per month, with extra pay for additional services. During the month of May the boy mowed the lawn twice, for which he was to receive 50 / each time ; and he also worked 12 hr. in the garden, at 10 /. per hour. At the end of the month Dr. Ward asked the boy to render his bill for services during the month. Make out the above bill, using your own name or the name of some boy in your school as the creditor. - 5. Make out a bill for 8 music lessons at $1.50 each. 6. Make out a bill for purchases at a furniture store. 7. Make out a bill for purchases at a meat market. 8. Make out a bill for purchases at a dry goods store. k 44 REVIEW OF INTEGERS AND DECIMALS RECEIPTS 48. 1. Explain the meaning of the following : Oakland, Cal., May 1, 1907. Received of Mr. C. W. Smith twenty-five dollars ($ 25) in full for rent of house at 704 Logan Avenue for the month of May, 1907. D. S. Stone. 2. E. M. Day bought a sewing machine of P. Orr for |;30. Write a receipt for the payment of this amount. 3. Mr. J. E. Thomas rented a farm of D. R. James for $ 300 per year. Write out a receipt for the payment of rent for the year beginning March 1, 1905. 4. The manager of an athletic club received $7.50 from the treasurer of the club. Write the receipt. 49. Oral Exercises. Whenever the partial sum is 10, 20, etc., take the next two numbers together, as in <2, 10, 19, 26, 30, 39, etc. a b c. d e / 9 h i J A; Z 6 8 4 6 8 7 7 9 5 4 7 8 2 3 6 7 8 6 5 9 8 9 8 5 3 3 3 9 5 7 8 5 4 7 9 4 8 7 9' 8 9 6 5 7 3 8 7 3 4 7 6' 3 8 7 7 9 8 5 5 8 5 3 6 8 8 4 8 9 5 7 8 9 4 3 8' 9 4 3 5 2 7 6 6 8 7 4 6 3 8 6 7 9 5 7 2 3 4 3 6 8 8 7 5 9 8 8 2 8 5 3 4 9 6 9 8 7 4 6 9 6 4 2 7' 7 7 6 8 8 9 8 4 7 6 8 9 3 7 5 9 5 7 8 7 \) ^ DIVISION 45 DIVISION OF INTEGERS AND DECIMALS ^ 50. Factors and Multiples. 1. 4 times 3 are 12. 4 and 8 are each a factor of 12. Name two other factors of 12; two factors of 14; of 20. 2. 2 and 8 are each a factor of . 3 is a factor of . 5 and are each a factor of 10. 3. The factors of a number are the integers which, when multiplied, make the number. A number may have sev- eral pairs of factors, thus : the pairs of factors of 24 are 4 and 6, 3 and 8, 2 and 12. Some numbers have only a single pair of factors, thus : the factors of 15 are 8 and 5. 4. Name all the pairs of factors of each of the follow- ing : 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30. 5. As 3 times 5 are 15, 15 is a multiple of both 3 and 5. Name a multiple of both 4 and 5; of both 4 and 7. 6. The number obtained by multiplying together two integers is called a multiple of either integer. Name a multiple of 4; of both 6 and 5. 7. In multiplication, two factors are given to find their product. In division, the product and one of two factors are given to find the other factor. 61. 1. The process of finding the other factor, when the product and one of two factors are given, is called division. 2. Since three 5's are 15, the number of 5's in 15 is 3. 3 This may be indicated thus : 5)15 Here 5 is the given factor, 15 is the given product, and 3 is the other factor. 3. The given factor is called the divisor, the given product is called the dividend, and the factor found is called the quotient. 46 REVIEW OF INTEGERS AND DECIMALS 62. 1. Name all the multiples of 3 to 30 ; of 4 to 40 ; of 5 to 50; of 6 to 60; of 7 to 70; ofStoSO; of9to90. 2. In each of the following, name the higliest multiple : a, Of 2 in 17, 11, 13, 9, 5, 7, 15, 19, 3, 10, 14, 21, 17, 19. h. Of 3 in 10, 8, 14, 19, 57, 16, 20, 29, 23, 26, 17, 22. c. Of 4 in 10, 17, 7, 15, 38, 31, 22, 27, 35, 19, 25, 30. d. Of 5 in 17, 23, 7, 12, 28, 34, 42, 39, 48, 27, 33, 19. e. Of 6 in 16, 10, 29, 22, 38, 45, 51, 34, 57, 43, 53, 41. /. Of 7 in 52, 46, 17, 33, 25, 68, 57, 39, 13, 30, 44, 53. ff. Of 8 in 52, 15, 28, 20, 35, 46, 69, 75, 58, 30, 60, 44. 7i. Of 9 in 20, 78, 34, 42, 47, 37, 59, 68, 11, 29, 53, 16. 3. Repeat Ex. 2, naming the highest multiple and the difference between it and the given number, thus for g : 48 and 4 ; 8 and 7 ; 24 and 4, etc. 4. Repeat Ex. 2, giving the quotients and the remain- ders, thus for g : 6 and 4 over ; 1 and 7 over ; etc. The sign of division is -^. It indicates that the num- ber before it is to be divided by the number after it. 6 ^ 3 is read 6 divided hy 3. Division may also be indi- cated thus : 3)^; or thus : |. 53. Oral Exercises. Supply the value of x in each of the following : a h c d 24-^6 = a: 58^8 = a; 67^9 = a; 76-f-8 = a? 30-j-8 = a: 76^9 = 2: 69-^8 = a; 61^-9 = a; 42-^9 = ic 40-^7 = 2; 60-j-7=.r 45-!- 6 = a; 18-r-7=a; 23^9 = a: 52 - 6 = a: 39^ 7 = a: 25-5-7=2; 70-8 = 2: 43-^9 = 2; 26-^3 = 2: 36-*- 8 = 2; 52-7 = 2: 59-r-6 = 2? 22-6 = 2: 51-1-6 = 2; 28^6 = 2: 39^-5 = 2: 50-5-8 = 2: DIVISION 47 54, Written Exercises. Use as successive divisors the numbers above the columns. Solve by short division. a h c d* 6,T,8 9,5,4 3,8,7 2,9,6 1. 672,458 327,459 836,594 ^ $4387.24 2. 237,400 574,063 683,127 $3058.26 3. 946,305 508,342 427,060 i $9576.30 4. 375,268 970,654 738,967 i $4256.75 5. 834,008 624,307 520,380 $5687.50 6. 463,925 207,193 946,425 $7495.38 7. 927,384 423,075 315,017 $8250.15 55. Written Exercises. Use as multipliers the numbers above the columns : a 789 6 465 c 305 d 890 1. $8796.50 7568.93 975.864 432.501 2. $4578.69 835.769 586.097 34.2056 3. $9760.50 58.7964 4376.89 Q^Q4.M 4. $6389.75 95,687.3 605.008 5046.32 5. $4975.86 2456.78 789.645 265.423 6. $7869.45 697.583 83.7956 123.456 7. $3698.70 4309.58 3945.78 45.0635 8. $8970.56 37.6895 687.905 5643.06 9. $6875.09 709.608 '201.003 326.504 .0. $3204.56 854.076 4567.98 Q^MbQ * Place a decimal point in the answer above the decimal point in the dividend. 48 REVIEW OF INTEGERS AND DECIMALS 56. Measurement and Partition. 1. All division is either measurement or partition. 2. 6 ft. 6 ft. _4 6 ft. 24 ft. when expressed as addition is 6 ft. 6 ft. 24 ft. 3. The quantity 24 ft. may be measured by the quantity 6 ft. The measure 6 ft. is contained in 24 ft. 4 times. This may be indicated thus : ^ „ .r—- ^ 6 ft.)24 It. 4. The process of finding how many times one number or quantity is contained in another is called division by measurement. 5. In Ex. 2 the quantity 6 ft. is taken 4 times to give the quantity 24 ft. Therefore, one fourth of 24 ft. is 6 ft. (K f '■ This may be expressed thus : A^^^oTTr 6. The process of finding one of the equal parts of a number or quantity is called division by partition. 7. In every problem in division you will be required to find either (a) how many times some unit of measure is contained in a quantity to be measured, or (5) to find a given part of some quantity to be divided. 67. Measurement. 1. The number of 4 cubes in 12 cubes may be found by using the unit 4 cubes as a measure to measure the quantity 12 cubes. 4 cubes, the unit of measure, is con- tained in 12 cubes, the quantity to be measured, exactly 3 3 times. This may be indicated thus: 4 cubes) 12 cubes DIVISION 49 2. Show with objects or by a diagram how many 3 apples there are in 12 apples. Indicate this in the form of division. Write the quotient. 3. Show by a diagram or by actual measurement how many 2 ft. there are in 12 ft. Indicate this in the form of division. Write the quotient. 4. Show with objects or by a diagram what is meant by each: 2 books)10 books; 5 pencils)15 pencils; 4 boys) 12 boys; 6 books) 12 books; 3 ft.)9lt7; 5^)20/. Write a problem for each. 5. In finding the number of 4 hr. there are in 24 hr., what is the unit of measure ? What is the quantity to be measured? 6. Show by using books (a) how many 3 books there are in 9 books ; (6) how many 3 books there are in 10 books; (? J57? What name is given to each part of the line in jB? in (7? in i>? in ^? In each case, how many of the parts does it take to equal the entire line? 5. The length of the line is represented in turn by 1? f ^ \'> f 1 and \\. 1 of the line = | = | = ^^g of the line. 6. I of the line = f = ^^^ of the line. 7. \ of the line + \ of the line = f of the line. \ of the line — J of the line = f of the line. 8. What is the sum of \ of the line and \ of the line? What is the difference between J of the line and J of the line? \ of the line and | of the line? 00 FRACTIONS 91 9. I of the line is longer than f of the line. -^^ of the line is longer than | of the line, f of the line is longer than I of the line. 10. Using 8 objects, show that |^ of 8 objects is the same as | of 8 objects, and that | of 8 objects is the same as "I of 8 objects. 34. Show by dividing circles that | of a circle is equal to I of a circle ; that | of a circle is equal to | of a circle ; that J of a circle plus |^ of a circle is equal to |- of a circle ; that f of a circle is equal to ^| of a circle. 12. Show by dividing rectangles that |^, |, -|, -j^, and -^ of a rectangle are equivalent parts. 13. Using objects, show that ^ of 12 objects is the same as I of 12 objects ; that J of 12 objects is the same as | of 12 objects. 14. Show by folding paper that i = |== | =« ^g- ; that J =- 1 ^ 3 _ _4_ 6 9 ~ 12* 116. Ratio. A S . c . 1. Line A is what part of line B ? what part of line 0? 2. If B is called 1, what isA?0? If (7 is called 1, what is ^ ? ^ ? If (7 is called 6, what is ^ ? J5 ? 3. If A is called 3, what is ^? (7? If J. is called J, what is J5? 0? 4. The ratio of line A to line B is ^, What is the ratio of line A to line (7? of line B to line ^ ? of line B to line a? of line O to line B ? of line O to line A ? FRACTIONS n R 116. 1. The surface A is what part of the surface Bl oiC^ oiD? oiU? 2. The ratio of ^ to ^ is ; of A to (7 is ; of -A to 2> is ; oi A to U is . 3. The surface B is what part of the surface C? of D ? of ^? The ratio of B to (7 is ; of J5 to i) is ; of 5 to ^ is . 4. What is the ratio of C to U? If (7 represents 40 A, what does U represent ? 5. What is the ratio of ^ to ^ ? oi Oto A? of i> to A7 of ^ to A? of toB? oiEto (7? of i>to^? of ^ to 5? 6. The ratio of (7 to D is |, or § ; oi D to C is f, or |. 7. What is the ratio oiDtoEl of ^ to i>? oiUtoC? 8. If A represents 10 acres, what does B represent ? C? B? B? 9. If B represents 40 acres, what does A represent ? C? B? JEJ? 10. If the cost of the land represented by B is f 100, what is the cost of the land represented hy A? C? B? B? 11. If the area represented by B is 640 acres, what is the area represented hy 0? B? A? B? 12. Draw two lines such that the ratio of one to the other is -J; |; 2; 5. 13. Draw oblongs such that the ratio of one to the other FRACTIONS 93 117. 1. The unit of 3 is 1 ; of 3 da. is 1 da.; of 3 mi. is 1 mi. 2. The unit 1 mi. may be regarded as composed of equal parts, as of 2 half miles, of 4 quarter miles, of 8 eighth miles, etc. If the unit 1 mi. is regarded as com- posed of 4 equal parts, each part is expressed as ^ mi. ; 3 such parts are expressed as | mi. A unit may be re- garded as composed of 2 or more equal parts. 3. A fraction is one or more of the equal parts of a unit, as |, |, etc. 4. In the fraction |, 4 is the denominator. It shows the number of equal parts into which the unit has been divided. It names the equal parts. 3 is the numerator. It shows the number of the equal parts of the unit that have been taken to make the fraction |. | denotes 3 of the 4 equal parts of the unit 1. 5. When a unit is divided into two or more equal parts, each of these parts becomes in turn a unit. Such a unit is called a fractional unit. \, \, ^, etc., are frac- tional units. The unit of ^ is \, What is the unit of each of the following : |, |, |, |- yd., -f^ yr. ? 6. Draw a line 1 ft. long. Divide it into 4 equal parts. Show the part that is expressed by \ ft. ; by | ft. ; by ^ ft. The ratio of 1 part of the line to the whole line is J. What is the ratio of 2 parts of the line to the whole line ? of 3 parts ? What is the ratio of the line to 1 part ? to 2 parts ? to 3 parts ? 7. Draw a line 8 in. long. Let it represent 1 mi. Show the part that represents f mi.; | mi.; | mi. Show the part whose ratio to the whole line is \^ ■^, |^, |, |. Show the part to which the ratio of the whole line is 2; 8; 4; j; j; f 94 FRACTIONS 118. 1. f f wk., f yd., f gal., IJ yr., |, J;^ lb., f, I mi., -Jj^, 3%. «. Read aloud each of the above fractions. h. Tell into how many parts the unit in each has been divided. c. Name the unit in which each is expressed. d. Tell how many of these parts are expressed in each fraction. e. Read the denominator of each fraction. /. Read the numerator of each fraction. g. Draw a line to represent the unit. Mark on this line the parts expressed in each fraction. 2. The numerator and denominator are called the terms of the fraction. 3. A fraction whose numerator is less than the denomi- nator is called a proper fraction, as |, ^, etc. Name ten proper fractions. 4. A fraction whose numerator is equal to or greater than the denominator is called an improper fraction, as J, |, etc. Name ten improper fractions. 5. When a number is composed of an integer and a fraction, it is called a mixed number. Q\ is a mixed num- ber. Its value is expressed in two different units. The 6 is expressed in units of ones; the -J is expressed in units of one jifths. Name ten mixed numbers. 6. The value of 1, expressed in the fractional unit ^, is f ; of 2 is ^ ; of 3 is f ; of 4 is f ; of 5 is f ; of 8 is f. 7. What kind of a number is 5|? In what unit is 5 expressed? In what unit is | expressed? The value of 6| may be expressed in the fractional unit J. There are I in 1. In 5 there are 5 times |, or ^, ^ and | are ^. What kind of a fraction is ^? REDUCTION 96 REDUCTION 119. Changing Mixed Numbers to Improper Fractions. Change 4|- to an improper fraction. Model : 5 times 4 is 20 ; 20 and 3 are 23 ; write 23 over the de- nominator, thus : \^. To change a mixed number to an improper fraction^ mul- tiply the integer hy the denominator of the fraction^ add the 7iumerator, and write the sum over the denominator of the fraction. 120. Oral Exercises. Change the following to improper fractions : * a b c d e / ff ^ * J ifc 1. H n 8f H 5| 4f H 2| 1| 8i 8f 2. 9| 5| 7f ^ n 9* H 2| H H 7i 3. 5f. 5i If ^ 2f 8* H 7f 4| 6f 5f 4. 2f 9f 7^ 41 n 5i 6f 3f 8f 2f 9^ 5. 2| 9f 7f 6t H 7| 6f 4f 2A 7f s* 6. 6* 2*31 ^ n 3i'j 9t% 7t^ 5^ 8i«J 6* 7. 4i^ 8fi 9i\ 3f 7| 6t\ 7i\ 5A 7A 4^ 5A 8. Write ten mixed numbers and change them to im- proper fractions. 9. Express the value of the following integers in the fractional unit ^: 3, 5, 7, 6, 9, 2, 8, 10, 12. 10. Write ten proper fractions. State what the frac- tional unit is in each. U. Change to improper fractions : 3| yd., 4| in., 8|- mi. * This exercise contains practically all the combinations in addition and multiplication. It should be used frequently as a review exercise. 96 FRACTIONS 121. Changing Improper Fractions to Whole or Mixed Numbers. 1. What kind of a fraction is ^? What is the unit in which its value is expressed? How many of these frac- tional units does it take to make the unit 1 ? How many units of 1 are there in ^? in ^? in J^? in -1^3.? in J^? 2. What does the denominator of a fraction show? Which term of the fraction tells the number of the frac- tional units it takes to make a unit? To change an improper fraction to a whole or a mixed number, divide the numerator hy the denominator, 122. Oral Exercises. Change the following to whole or mixed numbers : * a b c d e / 9 A i y A 1. ¥ V- V Y ¥ ¥ ¥ ¥ ¥ ¥ ¥ 2. ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ 3. ¥ ¥ h^ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ 4. V ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ 5. Y ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ 6. ¥ \l ¥ ¥ ¥ M ¥^ \l ^ ¥i^ If 7. M W ¥i^ ¥ ¥ a !! \\ !f !i \{ 8. Write ten improper fractions and change them to whole or mixed numbers. 9. Write ten mixed numbers and change them to im- proper fractions. ♦ This exercise contains nearly all of the facts of division and subtrac- tion. It should be used frequently as a review exercise. ADDITION AND SUBTRACTION 97 ADDITION AND SUBTRACTION OF FRACTIONS 123. Oral Exercises. 1. What is the sum of 2 books and 3 books and 1 book ? Are these quantities expressed in the same unit of measure ? 2. Name three quantities that are not expressed in the same unit of measure. Can their sum be found ? 3. Are the following fractions expressed in the same unit of measure : fi f ? i ? Fractions that are expressed in the same unit of measure are said to be similar fractions. Similar fractions have the same denominator. Only fractions that are expressed in the same unit of measure can be added. 4. The sum of ^, |, |, and | is |, which is equal to 2^. 5. Add 2| ft., 5^ ft., and 6J ft. Modi !l: ^ ft. Add the fractions first : J ft. and J ft. and 1 ft. 5J ft. are % ft., which are equa! 1 toli ft. Write } ft. 6i 14^ ft. below the column of fractions, and carry : Lft.to ft. the column of whole numbers. 6. Add the following : a h G d e / ff h 2 6i 3 H 4f H H 6| 7^ 5t% 8 7* 9f 9f U 6 n 5 6if 3f ^ 4 ^ H H H 6^ 8t% H 4* 7i 2 H 4f ^ 4t^ 7x^ 5i n 8* 8+ 6i 2i ^ 3^ 3t^ 4| 6| 5i 1\ 5J 7f 9| 8iV m 8 2* 6| ^ »i n 6| 7* 9^ 98 FRACTIONS 124:, Oral Exercises. 1. } ft. - J ft. = f ft. I mi. - I mi. = f -A yj*- = 1% y^' f da. -I da. = f da. 2. 6f ft. -4 ft. = ft. i8|-$5 = $- mi. Hyr- mi. 2mi. - mi. 3. Subtract > the fractions first and then the whole numbers : •• a 8f wk. If wk. h HI s/yd. 3| yd. d ^ yr. 6|lb. 4flb. / 8f wk. 5 wk. 3-A in. in. h t9| 24| yd. 17i yd. 29| yd. 19| yd. 4. Find, the sum of each of the above. 5. Subtract 3J ft. from 6 ft. Since there is no fractional part in the minu- Model : 6 ft. end, the suna of the fraction of the subtrahend SI ft. and the fraction of the difference is : I ft. i ft. 2|ft. and I ft. are 1 ft. Carry 1 ft. to 3 ft. and sub- tract tht ) integers. 4 ft. and 2 ft. are ( 3 ft. 6. Subtract : 8 hr. 1 da. 19 16 5 ft. 8 lb. 9 mi. 4| hr. 2J da. $H $2| If ft. 311b. 6|mi. 7. Subtract: 7fyd. gjyr. 8 A. 6|mi. 5f wk. 6 yd. 8|hr. 2Jvd. 6 yr. 4J A. 3|mi. 3f wk. 3fvd. 4 hr. 8. If a boy attended school 3| da. in a certain week, how many days was he absent ? ADDITION AND SUBTKACTION 99 9. A girl who was taking lessons on the piano prac- ticed as follows during one week : Monday, 1 J hr. ; Tuesday morning, 1 hr. ; Tuesday afternoon, | hr. ; Wednesday, Ifhr.; Thursday, I hr. ; Friday, 1^ hr. ; Saturday morn- ing, If hr. ; Saturday afternoon, | hr. How many hours did she practice during the week ? 10. A boy had 10 mi. to travel. If he traveled 3| mi. on foot and rode the remainder of the distance, how far did he ride ? 125. Oral Exercises. 1. Subtract 6| from 9f . Model : 91 The sum of | and the fraction of the difference p4 is 1§. Find what must be added to f to make 1 and 93 add it to I . f and ^ are 1. | and f are f. Carry 1 to 6. 7 and 2 are 9. The nnmerator of the fraction in the difference may be found by subtracting 4 (the numerator of the fraction in the subtrahend) from 5 (the denominator of the fraction in the minuend^, and adding 2 (the numerator of the fraction in the minuend). Explain why this method will give the correct result. Use this method* in subtracting. 2. Subtract without the use of a pencil : a b c d e / ff h I 6| n n. H 9H Vj m 6if n 21 6f 3i H 4if sji m ii! ^ 6t^ H n m ^ 8i^ 9t^ 6t^ m ?i M M m H 2H 14 4^^ m 3. Find the sum of each of the above exercises. 4. A dressmaker had two pieces of cloth containing 8| yd. and 6| yd., respectively. She used lOJ yd. in making a dress. How much cloth was left? 100 FRACTIONS REDUCTION 126. Changing to Higher and Lower Terms. w. « M » s ^ ^ 1. If A represents a unit divided into 2 equal parts, what does ^ represent ? C? i>? ^? J'? 2. The fractional unit of ^ is \\ what is the fractional unit oi CI Bl El Fl What part of the fractional unit of A is the fractional unit of ^? (7? i>? ^? ^? How many of the fractional units of B does it take to make one of the fractional units of -4. ? How many of (7? of 2>? of El of Fl 3. The fractional unit \ is what part of the fractional unit J ? I = |. I is what part of J ? 4. The denominator of the fractional unit ^ is 2 times the denominator of the fractional unit \. It shows that the unit has been divided into twice as many equal parts. It will therefore take 2 of the fractional units sixteenths to make one of the fractional units eighths. i%=|- 5. The fractions J^ f ^ f ^ ^ are the same in value. They differ in form. Changing the form of a fraction without changing its value is called reduction. 6. The fraction | is equal to the fraction ■^. Compare their numerators. 8 is times 4. Compare their de- nominators. 16 is times 8. How may ^ be derived from I ? How may } be derived from ^ ? REDUCTIOlSr-''' • " • ' ' 101 ' •'• * 1 •.* ' ''*»? * V, '. A 7. Compare in a similar way tlie tet'nis of the' fractions I and ^2 5 \ ^"^ 1^5 f ^^^ A* What effect upon the value of a fraction has multiplying both terms by the same number ? 8. A fraction is an indicated division. | is the same as 6}^. The denominator of the fraction is the divisor, and the numerator is the dividend. What effect upon the quotient has multiplying both the dividend and the divisor by the same number ? Is multipl3dng both the numerator and the denominator of a fraction by the same number the same as multiplying both the dividend and the divi- sor by the same number ? | = | = ^^ = j^. 9. The fraction \^ is equal to the fraction |-. Compare their numerators. 5 is what part of 10 ? Compare their denominators. 8 is what part of 16 ? Compare in a sim- ilar way -5^2 with |; -^^ with |. What effect upon the value of a fraction has dividing both terms by the same number? 3-^ = | = |=f. 10. What effect upon the quotient has dividing both dividend and divisor by the same number ? Is dividing both numerator and denominator of a fraction by the same number the same as dividing both dividend and divisor by the same number ? Multiplying or dividing both terms of a fraction hy the same number does not alter the value of the fraction. 11. Change the form of the following without changing their value : f , ^, f , if, ^, If, |^, ^. 12. By what must the terms of the fraction ^ be multi- plied to reduce the fraction to lOths ? to 15ths ? to 20ths ? 13. How many 12ths are there in 1 ? in ^ ? in | ? in|^ 102 FRACTIONS 127. Written Exercises. 1. Change | to 12ths. As the denominator 12 is 4 times the denominator of |, the numerator of the required fraction must be 4 times the numerator of |. Model: ^^f^- 3 is contained in 12 four times. 4 times 2 is 8. | = -j^. Another method. 1 = H i i = i of [f , or ^^2 ; | = 2 tunes ^^^ or ^. 2. State how you would find the number that 3 must be multiplied by to change | to 20ths. 3. Change to 12ths 4. Change to 18ths 5. Change to 24ths 6. Change to 20ths 7. Change to 36ths 8. Change to 30ths 2' 3' t' 6' 3' I' h 6' f • 2' 3' 6' 9' 6' 9' 6' 9' "S* f' I' '2' 6' 8' 1^' t' A* 2' I' h A' I' 10' h tV- I' h i I h {h tV h i' iV ^u^ h h h h \h h A- 9. Write eight fractions and change them to 48ths. 10. When the terms of a fraction have been made larger by reduction, the fraction is said to have been reduced to higher terms. 128. Oral Exercises. 1. Express each in a different form without changing the value: f, 8|, ^ J, 2, 2^, |, f, 7^, J^, |f, 1. 2. Find the sum of 3J, 2|, and 7. 3. Find the difference: gf^?! |?| S| 4. Show that J of 2 yd. is equal to | of 1 yd.; that J of 8 ft. is the same as | of 1 ft. REDUCTION 103 129. Oral Exercises. 1. Show by a diagram that f of a line is equivalent to f of the line ; that J f t. = f ft. ; that | ft. + J ft. = | ft., or 11 ft. 2. Show with objects that | of 12 objects = 3^2 of 12 objects; that 1 of 12 objects = ^^2 ^^ ^^ objects; that f of 12 objects + i of 12 objects = -^^ of 12 objects + -^^ of 12 objects, or \^ of 12 objects. 3. Why is it necessary to change | and | to 12ths be- fore finding their sum ? 4. Show by a diagram that the difference between J ft. and \ ft. is ^^2 ^^' 5. If I of a group of objects contains 6 objects, show the number of objects in | of the group. 6. Show with objects that if | of a group of objects is 6 objects, the whole group contains 9 objects. 7. Show by a diagram that if | of the length of a line is 4 ft., the entire length of the line is 10 ft. 8. If 6 objects represent | of the number of books on a certain shelf, represent by objects ^ of the number of books on the shelf. Represent all the books. 9. What is the least number of boys that may be separated either into groups of 3 boys or of 4 boys ? 10. What is the least number of equal parts into which a rectangle can be divided so that either ^ or -J of the rectangle may be shown ? 11. What is the least number of girls that can be sepa- rated into groups containing as many girls as are indicated in the denominator of any one of the following : J, f , f , f ? 104 FRACTIONS 130. Common Factors. 1. The exact measures of 12 ft. are 1 ft., 2 ft., 3 ft., 4 ft., and 6 ft. Name the exact measures of 18 ft. Which are exact measures of both 12 ft. and 18 ft.? 1 ft., 2 ft., 3 ft., and 6 ft. are common measures of 12 ft. and 18 ft. 2. A number that is a factor of two or more numbers is called a common factor, or a common measure, of the numbers. 3. Name the common factors of 18 and 24. Name the largest common factor of 18 and 24. 4. The greatest common factor of two or more numbers is the largest number that will exactly divide each of them. This is also called the greatest common measure^ or the greatest common divisor^ of the numbers. 5. Draw a line 12 in. long and another line 18 in. long. What is the longest measure that can be applied without a remainder in measuring both of these lines ? What is the greatest common measure of 12 ft. and 18 ft.? 6. Show with objects or by a diagram all the common measures of 8 objects and 12 objects. 131. Name the exact measures of : 1. 10 ft. 6. 30 mi. 11. 50 rd. 16. $80 2. 16 gal. 7. 36 yd. 12. 27 pt. 17. 190 3. 20 da. 8. 40 yr. 13. 28 da. 18. $60 4. 18 hr. 9. 481b. 14. 64 ft. 19. 175 5. 24 in. 10. 45 qt. 15. 72 mi. 20. $84 132. Name the greatest common factor of : 1. 6 and 8 4. 12 and 18 7. 12 and 48 10. 20 and 60 2. 8 and 12 5. 24 and 30 8. 15 and 30 11. 18 and 36 3. 9 and 12 6. 24 and 36 9. 30 and 36 12. 14 and 28 REDUCTION 105 133. Oral Exercises. 1. The fractional unit J must be repeated how many times to equal the fractional unit ^ ? The fractional unit ^^ must be repeated how many times to equal the frac- tional unit -^^ ? 2. By what must the terms of the fraction |^| be divided to give the fraction ^2 • i M M ^ 2 ^ 3. When the terms of a fraction have been made smaller by reduction, the fraction is said to have been reduced to lower terms. A fraction is in its lowest terms when the terms have no common factor. 4. The fraction \^ is not in its lowest terms, as both terms are exactly divisible by 5. Which of the following are in their lowest terms : |, f, |, |, 1|, if, IJ. |f ? 5. Dividing both terms of a fraction by a common fac- tor is called canceling the common factor. In reducing a fraction to its lowest terms, cancel in turn the largest factors that are seen to be common to both teims. Canceling the greatest common factor of both terms reduces the fraction to its low- est terms . 134. Oral Exercises. Reduce to lowest terms ; 1. 1^2' tI' iV ^2' A' it 9. II. If. If. If. M. 11 2. A' A' ^¥^ A' A' A 10. A. ^. If. M. 1 2' H 3. M^if^Maili^il 11. If. f I. ^i. M. If. ft 4. /o^ A' -io' A' ro' A 12. fi.fMi.if.ii.H 5. il ih il' ih ih u 13. if. 1!. ih If. If. If 6. ih /r B^6^ A' ih Jl 14. if. Ih ih f«. If. iV\ 7. U^ M^ 1% iV A' A 15. it 1%. If. li. II. H 8- M. H. if. if. H- It 16. if, if if H, T%, ,vk 106 FRACTIONS 135. Multiples. 1. 2 is a factor of 4, 6, 8, 10, etc. Each of these num- bers is a multiple of 2. Name the multiples of 3 to 27. 2. A number that is exactly divisible by a given num- ber is called a multiple of the given number. Name a mul- tiple of 6 ; of 8 ; of 7. 3. Write the multiples of 3 to 27 and of 4 to 36. Which of the numbers written are multiples of both 3 and 4 ? These numbers are common multiples of 3 and 4. Which is the least multiple common to 3 and 4 ? 4. A number that is a multiple of each of two or more numbers is called a common multiple, and the least number that is a common multiple of each of two or more numbers is called the least common multiple of the numbers. 5. Write all the multiples of 4 to 36 and of 6 to 54. Name the multiples common to 4 and 6. Which of these is the least common multiple of 4 and 6 ? 6. Name the least common multiple of 3 and 4 ; of 2, 3, and 4. Since 4 is a multiple of 2, the least common multiple of 2, 3, and 4 is the same as the least common multiple of 3 and 4. Tlie least common multiple of 2, 3, 4, and 6 is the same as the least common multiple of 4 and 6. Why? 7. Write four numbers such that the least common multiple of the numbers is the same as the least common multiple of some two of the numbers. 8. In finding the least common multiple of 2, 3, 4, and 9, which numbers need not be considered, and why ? 9. Find the least common multiple of 3, 5, and 7. Show by several illustrations that the least common multiple of two or more prime numbers is their product. MULTIPLES 107 136. Name the least common multiple of : 1. 4 and 5 7. 8 and 12 13. 10 and 12 8. 6 and 9 14. 12 and 9 9. 4 and 10 15. 3 and 7 10. 5 and 10 16. 2, 3, and 6 11. 10 and 4 17. 3, 4, and 8 12. 5 and 8 18. 4, 5, and 15 2. 4 and 6 3. 3 and 4 4. 6 and 8 5. 5 and 7 6. 4 and 8 137. 1. Find and 48. Model : ^ 48 Find the least common multiple of 8, 10, 18, As 48 is a multiple of 8, cancel 8. As the factor 2 is common to 10 and to 48, cancel this factor of 10, leaving the factor 5. As 6 is a factor common to 18 and 48, cancel this factor of 18, leaving the factor 3. 5 x 3 x 48, or 720, is the least common multiple of 8, 10, 18, and 48. 2. In finding the least common multiple of 3, 4, 6, 9, and 12, which numbers may be canceled? From which number may a factor be canceled? 3. In finding the least common multiple of 4, 12, 7, and 35, which number may be canceled because it is a factor of 12 ? Which may be canceled because it is a factor of 35 ? 138. Find by inspection the least common multiple of: 7. 10, 15, 25, 40 13. 7, 35, 45, 90, 70 8. 36, 48, 60, 72 14. 5, 14, 42, 60 9. 12, 18, 24, 36 15. 6, 7, 8, 9, 84 10. 3, 5, 30, 45 16. 20, 24, 30, 100 11. 4,18,27, 72 17. 4,9,20,54 12. 8, 12, 15, 60 18. 6, 15, 24, 86 1. 4, 5, 8, 24 2. 3, 12, 15, 30 3. 5, 8, 25, 40 4. 2, 6, 15, 45 5. 7, 21, 49, 84 6. 9, 15, 36, 60 108 FRACTIONS ADDITION AND SUBTRACTION 139. Written Exercises. 1. Change to 12ths and add |, |, f . Model : l=* 2. 3. 4. 5. 6. 7. l = A I 6 1 f i* § A S = M i 1 i i i f !J = 2^J=2J i 6* 1 i 8. Add the fractions in Exs. 3-8, Sec. 127. 9. Change to 24ths and add 3f, 4 J, 6l|, 7|. Model: 3f = 31f 41 __ 4_4 The sum of the fractions is f |, which f\ll _ r2 2 reduces to 2|. Write f as the frac- 12 ~ ^21 tional part of the answer. Carry 2 to *t ~ ^2ir the column of integers. 22f 10. Change to 12ths and add 4f, 3f, 4f , 5J, 6^\. 11. Change to 18ths and add 8f, 7|, 9J, 6^, 51^. 12. Change to 24ths and add 7|, 6^^, 9J, 8|, 4J. 13. Change to 36ths and add 311 7i|, sj, 51 9f . 14. Change to 48ths and add 7 J, 9f, 3^, 5||, 8^^. 15. Change to 72ds and add 8^, 9f, 7^, 18^^, 3|. 16. Reduce to lowest terms: ||, ||, -||, |-|. 17. Change to improper fractions: 7|, 9|, 8|, 7|. 18. Change to mixed numbers : -^, -j^, -^1, -^. 19. Add 4f , 6f 3f 8f 9f , 7f 20. Change 4 to 12ths ; 3 to 18ths ; 5 to 20ths. 21. Write ten fractions and reduce them to lower terms. ADDITION 109 140. Oral Exercises. 1. What is the least common multiple of 2, 3, and 4 ? of 3, 5, and 6 ? of 4, 5, and 6 ? of 4, 6, and 8 ? of 3, 6, and 9 ? of 5, 8, and 12 ? 2. Can you add the following fractions without first re- ducing them : |^, |^, |-? Are they expressed in the same fractional unit ? 3. Can you add the following fractions without first re- ducing them : |, |, and J ? Are they expressed in the same fractional unit ? Only fractions that are expressed in the same fractional unit can be added. 4. What is the unit of measure in f ft. ? | ft. ? -^ ft. ? These fractions may be expressed in the same unit of measure, -^ ft. | ft. = f^ ft. f ft. = f^ ft. 5. Can the following be expressed in the same unit of measure : -J ft., | da., and ^ gal.? Can the following : 1 ft., f ft., and iV ft.? 6. What is the least common multiple of the denomi- nators of the fractions |^, f , |, and ^ ? The least common multiple of the denominators of two or more fractions is called their least common denominator. 141. Reduce the fractions to fractions having the least common denominator, and add ; a h c d e / 9 h i H n H 3A 4iJ 3^ 5! ^ii 6f 6f H 6f 4| n 8| 8A m 8f n ^ 7^1 n 6f 6-11 ^ n 4i^ 4A 6il 8i H n ^ 5i H 6« n n 5i 7f 5A 6J n 8f n 110 FRACTIONS u 2. Written Exercises.* 1. From 5f subtract a§. Model : 5| = 8| = 5-5^ Reduce 3^ the least 2^1^ tract. Subtract : a b i 2. 6* ^ 8| 5f 6f 7f 8fl Reduce the fractions to fractions having the least common denominator, and sub- e f g h 2^ 9| 23^ 43f ■"""' "~~" """■ 3. 9| 6| 4| 8i 6f 6i li 5| 2f 6| fi 4. 5i 4f 6f 4t^ 7^ 2| 9* 18? 7| 19| 9| 12i 8t\ 15i 8| B. 21f 10| 80| 9 20 9! 26f 4| 43| 7 16 101 18| 81 29| 9^ 6. 87^ 64f 79^ 90A 3TA 68J, 19t\ 74| 20 13^f 20 9H 4f 8£ 143. 1. Add each exercise in Sec. 142. 2. The lengths of the bhicl^boards in a certain school- room are 12J ft., 14J ft., 8| ft., and GJ ft., respectively. Find the combined length of the four blackboards. 3. Find the difference in the weight of two turkeys, if one of the turkeys weighs 22J lb. and the other 17| lb. « See Sec. 125. ADDITION AND SUBTKACTION 111 U4. Review Sec. 134. Reduce to lowest terms : 1- Ih If ih il Ih If 5- If. t\\' AV 1%' m 2- 100' TOO"' TOO' T0T7' Too ®- T2' "50' yS' ■g'O' QO"' 80 3 70 JLO__ _9JL JL5_ _4 5_ 7 100 JTS _2_5_ _5JL __4 0_ **• 100' 100' 100' 100' 100 '' 150' 150' 150' 200' 200 *. 1^%' ^¥t' 1^5' T^A' ilt 8. ^, ^%' Hi if^' ii 145. Review Sec. 120. Reduce to improper fractions : 1. 14|, 30J, 161, 331 66f 4. 17f mi., 51 yd., 8f wk. 2. 16f, 37f , 111, 9_i_, 28f 5. 231 ft., 8f lb., 9f wk. 3. 38^^, 5f, 16f , 25|, 671 g. 6^^ yd., 4J mi., 811 A. 146. Review Sec. 122. Reduce to integers or mixed numbers : 1. 101 iQjQ., loii^ i.Qii, loii 5. ^ yd., ■^24 50 72 80 Find the least common multi- pie of 24, 50, 72, 80. Cancel 24 as it is a factor of 72. Select a prime number that is a factor of two or more of the remaining 5 9 2 numbers. Divide the multiples of this number by the number used as a divisor, and write the quotients and the numbers that are not exactly divisible as shown in the model. Continue the division until no two numbers bro\ight down have a common factor. The product of the several divisors and numbers remaining is the least common multiple of the denominators. 1. C. m.=2x2x 2x 5x Find the least common multi- R w Q vx o oc*c\f\ pl® of the same numbers by the method explained m Sec. 137. 2) 25 36 40 2) 25 18 20 5) 25 9 10 MULTIPLICATION AND DIVISION 113 MULTIPLICATION AND DIVISION OF FRACTIONS 151. Multiplying a Fraction by an Integer. 1. In the fraction |, which term tells the number of equal parts into which the unit has been divided ? How many of the equal parts are expressed in the fraction ? Write a fraction expressing twice as many equal parts. 2. Draw a diagram to show what part of a mile is ex- pressed in I mi. Show the part that represents f mi. Compare the part | mi. with the part | mi. 3. What is the sum of |, |, and | ? 3 times | = f • 4. Write I four times as an addend and find the sum. State how the sum was found. 5. If I is written five times as an addend, what is the sum ? If f is multiplied by 5, what is the product ? 6. State how a fraction may be multiplied by a whole number. Compare the results thus obtained with the re- sults obtained by addition. 7. Multiply. Reduce all products to their simplest forms: | by 5; | by 3 ; f by 6 ; f by 7 ; | yd. by 4. 8. If I is multiplied by 3 by multiplying the numera- tor by 3, the result will not be in its lowest terms. Why? I may be multiplied by 3 by dividing the denominator 9 by 3. |x3 = f 9. Dividing the denominator of a fraction by a whole number has what effect upon the value of the fraction ? 10. Multiply I by 12. As the factor 4 is common to both 12 and 8, it is canceled before mul- tiplying. Canceling 4 in 12 leaves 3 ; canceling 4 in 8 leaves 2. 3 times | AR. — 8 114 FRACTIONS 152. Oral Exercises. Solve each in the shortest way : a h c d e 1. 1 x5 Hx5 f i X 10 i| X 9 Hx6 2. 1 x4 i%x3 I|x7 -Hx8 i|xl2 3. f x3 Ax7 11 X 8 If X 3 Bx6 4. 1 x5 i|x4 «x6 11x7 Mx4 5. 1 x20 1 xl2 41x24 1 xl6 If x48 6. 1 x24 i^.x7 ^^ X 30 f X 42 Iix3 7. 11x18 f x5 t\x8 ||x16 }fx28 8. If XIO i\x5 ft X 75 If x 24 Hx60 9. ^x36 H X 28 Mx7 /,x8 fix 17 153. Written Exercises. 1. Multiply ■43|by 8. Model: 43| o ^ First, multiply | by 8. Next, multiply 43 by 8. Add the products. 344 350 Solve. Perform the cancellation and reductions with- out the use of a pencil : a h c d e 2. 47f x5 64f x7 82fxl4 749i-Jx5 708f x 30 3. 68fxT 74f x9 65-1 x 15 896i| x 7 580f J X 15 4. 96fx3 59f x6 94f X 12 780i| X 3 496Jix48 5. 78fx9 76^x9 70f X 18 973f X 9 573| X 25 6. 56fx9 38|-x8 27f X 24 587| x 2 609f X 42 7. Write ten mixed numbers and multiply them by in- tegers. MULTIPLICATION AND DIVISION 115 154:, Multiplying an Integer by a Fraction. 1. What is the meaning of 4 ft. x 2? of 4 ft. x 1? of 4 ft. X I? Name the multiplicand and the multiplier in each, and tell what each shows. 2. 4 ft. X I is the same as J of 4 ft. How may |^ of a number be found? How may ^ oi sl number be found? When you know what J of a number is, how can you find J of the number? 3. Show with objects what is meant by | of 9 things ; of 12 things ; of 6 things. 4. I of 24 yd. means 5 of the 6 equal parts of 24 yd. Draw a line to represent 24 yd. Divide it into 6 equal parts. Show the part that represents f of 24 yd. 5. Show by a diagram what is meant by | of 12 in. ; by I of 1 mi. ; by | of 6 mi. ; by | of 10 mi. 6. How many thirds of 18 are equivalent to 18? Are I of 18 more or less than 18? If 18 is multiplied by |, will the answer be more or less than 18? Why? 7. Read each of the following, name the multiplicand and the multiplier in each, and tell what each shows : 120 X f ; 16 yr. X f ; 25 mi. x f ; 18 mo. x | ; 24 lb. x f . 8. Compare | of 120 with ^ of 3 times 1 20. Compare ■| of 25 mi. with ^ of 4 times 25 mi. 9. f of 8 ft. is the same as J of ft. | of $5 is the same as ^ of $ . 10. Show by a diagram that | of 1 yd. is the same as i of 3 yd. 11. 5 divided by 7 may be indicated IJS, or ^. Indi- cate ^ of 3 ; J of 2 ft. ; J of 5 mi. ; | of 5 mi. 12. The products of -^ x 18 and of 18 x | are the same. 116 FEACTIONS 155. Written Exercises. 1. Multiply 36 by J|. 3 11 Model : ^^ x ^ = .^ = 16 J. 12 is a factor common to 36 and 24. Cancel the con*- mon factor. 3 times 11 is 33; -8^=16i. Solve : a h c rf e 2. 30 x| Sxif 25 xf 144 xf 100 xf 3. 48xiJ Txf 30x^5^ 54 X J 100 xf 4. 36 X ^ 8xi\ 27 X il 60 Xt^ 100 x| 5. 21 xf 9xi| 45 xf 16xJ| 100 xf 6. 4xf 6xf 72x|J 86 x| 100 xf 156. Written Exercises. 1. Multiply 845 by 4f . Model : 845 f of 845 is 120f ; ^ of 845 are 3 times 120f,oP ^T 362f 4 times 845 are 3380. Add the products. 120f (I of 845) 3621 (f of 845) 3380 (4 times 845) 3742| Multiply : a h c d 2. 60x8f 64 X 45f 827 X 47f 801 X 84f 3. 36x9f 81x47^2^ 459 X 75f 153 X 46f 4. 55 X 8^^ 72 X 67f 693 X 68| 360x48| 5. 27x6| 96 X 87-1 745 X 47| 578x96f 6. 33x4| 48 X 541 J 584 X 37| 609x24f 7. 45x8f 75 X 49^ 144 X 35{j. 586 x 27| MULTIPLICATION AND DIVISION 117 157. Dividing a Fraction by an Integer. 1. Divide 12 by 3. Find ^ of 12. -l of a number may be found by dividing the number by 3. State how |^ of a number may be found. 2. Draw a line 12 inches long. Show -^^ of the line. Show \ of -j^ of the line. Divide -f^ of the line into 3 equal parts. How does each of these parts compare with I of -^^ of the line? J of -^^ is the same as -^^ -f- 3. 3. What is \ of I? of I? of if? of 1^? 4. State how a fraction may be divided by an integer, when the numerator is exactly divisible by the integer. 5. Solve: f^-^3; l|-^7; lf-^6; ||-^8; ^f-^5. 6. Draw a line 12 in. long. Divide it into 12 equal parts. Each part is -^^ of the whole line. Divide each part into 3 equal parts. Each of these smaller parts is what part of the entire line ? To divide -^ of the line into 3 equal parts, each of the 8 parts must be divided into 3 equal parts and ^ of these taken. Show that J of -^ of the line is ^ of the line. This result may be found by multiplying the denominator of ^^2 t)y 3. 7. Divide \^ by 4 by dividing the numerator by 4 ; by multiplying the denominator by 4, Compare the results. 158. Oral Exercises. Divide, using the shortest method for each : 1. -rV^yS 5. if by 2 9. |T. by2 2. f^bylO 6. If by 6 10. $tby7 3. I by 6 7. if by 7 11. 1| mi. by 5 4- M by 8 8. 2§ ft. by 3 12. f da. by 3 118 FRACTIONS 159. Written Exercises. 1. Divide 65| bj 8. TVT . ^S^ ^ ^^ contained in 65 eight times, with 1 J.M.I ^^^^- 8)65| over ; ^ of If is ^ of h or ij. 2. 37^^ 7 5 325|^4 c 1321 H- 5 3. 62|-j-9 423f^6 836f^4 4. 87f-f-8 7561^8 456^ ^ 2 5. 46|-^9 637|-f-7 3871-*- 3 6. 90-1- ^ 8 436^^9 7261^14 7. Divide 645f by 24 ; 1645| by 3^ ; ; 195| by 27. 8. Divide 347f by 46; 73|byl4; 4721 by ^c>. 160, Written Exercises. 1. A carpenter sawed a board 9 ft. 4 in. (9 J ft.) long into 4 equal parts to make shelves for a bookcase. How long was each shelf? 2 ft. 4 in. Or, divide 9 ft. 4 in. by 4 thus: 4)9 ft. 4 in. ^ of 8 ft. is 2 ft.; I of 16 in. (1 ft. 4 in.) is 4 in. Ans. : 2 ft. 4 in. 2. The perimeter of a square flower bed is 14 ft. 8 in. (14| ft.). How long is each side? 3. If a train travels at an average rate of 45 mi. per hour, how far will it travel in 4 hr. 45 min. (4| hr.) ? 4. At 5 ^ per pound, how much will 11| lb. of sugar cost ? 5. In a magazine of 160 pages, 45 pages were devoted to advertisements. What part of the magazine was de- voted to advertisements? 6. Two boys caught 8 fish. The combined weight of the fish was lOJ lb. What was their average weight? MULTIPLICATION AND DIVISION 119 161. Multiplying a Fraction by a Fraction. 1. Show by a diagram or with objects what is meant by f of 6 ft. ; by f of 1 ft. ; by f of i ft. | of ^ ft. is what part of 1 ft. ? 2. Draw a line and divide it into 5 equal parts. What is each part called ? Show ^ of one of these parts. | of ^ of the line is what part of the line ? 3. Draw a line and divide it into 4 equal parts. Show ■J of one of these parts. |^ of J of a line is what part of the line ? 4. Draw a line and divide it into 3 equal parts. Show ^ of one of these parts. Show f of one of these parts. -J^ of |- of the line is what part of the entire line ? 5. How much is I of 6 da. ? J of 6 sevenths ? 6. How much is -J of ^g- ? of -^j ? of 1| ? 7. When you know what |^ of a number is, how can you find | of the number ? When you know what ^ of a fraction is, how can you find | of the fraction ? 8. State how you would find :| of i| ; J of 1| ; J of if. 9. State how you would find J of f ; | of | ; ^ of | ; I off. 10. Divide rectangles to show | of | ; | of | ; | of |^ ; f of ih 11. Show by a diagram that J of |- is ■^. Since | of J is ^2' f o^ i ^s ^^w many times -^^ ^ 12. Divide rectangles to show that ^ of J is equivalent to J off 13. Divide rectangles to show | of |^ and |^ of | ; f of f and f off; I of 1 audi off. 120 FRACTIONS 14. Draw a line. Show ^ of the line. Show | of the line. Show J of ^ of the line. What part of the line is J of I of the line ? Show J of | of the line. J of f of the line is what part of the line ? Show | of | of the line, f of |- of the line is what part of the line? l—J . 1 . . 1 U—i \ . . I . ■ I J of ^ of the line = ^ of the line. L-j—j I I I I I I I I I \ ■ ■ 1 I of the line = -^ of the line. I of f of the line = ^^ of the line. ^ of f = ^^. 1 1 I ^ ' ■ \ » * \ t I I ' ^ V > I of I of the line = -^ of the line, f of | = i^* 15. Draw a line. Show | of the line. Show | of the line. Show J of ^ of the line. J of J of the line is what part of the line ? Show J of f of the line. | of f of the line is what part of the line ? Show | of | of the line. I of I of the line is what part of the line ? 16. Finding |- of a number is the same as multiplying the number by J. Finding f of a number is the same as multiplying the number by |. Examine the illustrations under Ex. 14, and tell how J of J of a number is found; J of I of a number ; |- of | of a number. 17. Since J of ^ is ^, J of f is ^^ and | of f is ^. In each case the product of the numerators is the numer- ator of the answer, and the product of the denominators is the denominator of the answer. To multiply a fraction hy a fraction, multiply the numera- tors for the numerator of the product and the denominators for the denominator of the product. Before midtiplyitig^ cancel factors common to both terms. MULTIPLICATION AND DIVISION 121 162. Written Exercises. 1. Multiply fl by f . Model : 3 1 8 is a common factor of 24 and 8. 5 is a com- ?^,^_3 nion factor of 5 and 25. Cancel these common n^"^ factors and multiply. 5 1 Solve. Before multiplying, cancel common factors : a b c d e 2- fxf f|xf II X 11 h^xi Mxl! 3- ixf II X il f>'l |x| Ifxi| 4. |X| \\ X \ l>-f Jxii ^Xt^j 5. fxf if X 1 ¥xf ^x^ Wx| 163. Written Exercises. Change the mixed numbers to improper fractions and solve : a 6 c d 1. Hx^ fx2i 2|x4f l|x3i 2. 4ix| |x6|. 6|x4| 4Jx2J 3. Sfxf |x4| 7|x4| 6|x4| 4. 5^x1 T%x5i 9fx6| 8|x2J 164. Written Exercises. Review Sees. 152, 153, and 156. 1. Multiply 45f by |. Model : 45| ^ of 45§ is 7ii; f of 45f = 5 times 7H.. or 5 38^1,. 6 1{1 (^of45|) 5 38 J^ (iof45|) 122 FRACTIONS Solve without reducing the mixed numbers to improper fractions : a b c d e 2. 34|x| 84fxf 546| x| 654f xf 840f x f 3. ISfxf 55|xf 385| xf 235| xf 468f x | 4. 72fx| 40|x| 463J xf 900,9^ x | 479| x | 5. 48fx| 38Jx| 847A x| 783f x J 673f x f 6. 96fxf 94^ X I 170^\x| 680^ x ^^ 574| x | 7. 481 x^ 63Jx| 4311 x| 598f x^ 650J x | 165. Written Exercises. 1. Multiply 349f by 3f , Model : 349| First multiply 349f by f . Next multiply 349| ga by 3. Add the products. mi 3 (I times 349f) 209t 1049 12581 (1 times 349f) (3 times 349f) Solve without reducing the mi^ed numbers to improper fractions : abed 2. 645fx6f 584^^ x4f 963J x 7J 642f x 7^ 3. 867fx5f 982f x8f 333ix6§ 789^ X 7f 4. 694fx7i 648| x5J 78Hx9f 537^ X 6 J 5. 748^ x5f 457f x7| 450f x 7f 521^ x2| 6. 384fx4f 926| x2f 467J X 9J 830| x4f 7. 412|x6| 726| x8J 940^ X 3f 590^ X 7J 8. 240|x4f 948| xl^ 640^ X 7f 810f x3f MULTIPLICATION AND DIVISION 123 166. Review Questions. 1. What is a proper fraction? an improper fraction? a mixed number ? 2. Write 5 proper fractions; 5 improper fractions; 5 mixed numbers. 3. What is a fractional unit? How many fractional units are expressed in |? 4. What is meant by a factor of a number? Illustrate. 5. What is meant by a multiple of a number ? Illustrate. 6. When is a fraction said to be in its lowest terms? Write five fractions that are in their lowest terms. 7. How may a fraction be reduced to its lowest terms ? Write five fractions and reduce them to their lowest terms. 8. How may an improper fraction be reduced to a whole or a mixed number? Write five improper fractions and reduce them to whole or mixed numbers. 9. How may a mixed number be changed to an im- proper fraction? Write five mixed numbers and change them to improper fractions. 10. What effect upon the value of a fraction has multi- plying or dividing both terms of the fraction by the same number? Illustrate. 11. What is cancellation? Illustrate. 12. State two ways in which a fraction may be multi- plied by an integer. Illustrate. When a proper fraction is multiplied by an integer, is the product greater or less than the multiplicand? Why? Is the product greater or less than the multiplier? Why? Illustrate. 13. When an integer is multiplied by a proper fraction, is the product greater or less than tlie multiplicand? than the multiplier? Why? Illustrate. 124 FRACTIONS 167. Dividing by a Fraction, 1. Draw a line 4 ft. long. Make a measure J ft. long. Apply this measure to the line. How many times must the measure J ft. be applied to measure a 4-ft. line? 2. After finding how many times the measure | ft. must be applied to measure 1 ft., how may you find, with- out performing the actual measurement, how many times the measure must be applied to measure a 4-ft. line ? 3. Repeat Exs. 1 and 2 above, using a J-ft. measure to measure a 6-ft. line. 4. As the measure ^ ft. must be applied 2 times to measure 1 ft., to measure any given number of feet it must be applied as many times 2 as the number of feet to be measured. To measure 24 ft., it must be applied 24 times 2, or 48 times. How many times must the measure J ft. be applied to measure 6 ft. ? 10 ft. ? 12 ft. ? 5. To measure a line J ft. long, the measure J ft. must be applied J times 2, or J times. That is, one half of the measure must be applied. 6. The expression J ft. -4- J ft. indicates that a line J ft. in length is to be measured by a measure J ft. in length. What is meant by each of the expressions: 6ft.^Jft.? |ft.+Jft.? Jjft.-i-Jft.? 7. Draw a line J ft. long. Determine how many times each of the following measures must be applied to measure it: J ft., J ft., ^ ft. 8. The measure | ft. must be applied how many times to measure a 1-ft. line? 1J= J. It must be applied | times. How many times must it be applied to measure a 4-ft. line? a 6-ft. line? a 15-ft. line? What is meant by the expression 18 ft. -*- f ft. ? MULTIPLICATION AND DIVISION 125 9. If a |-ft. measure must be applied | times to measure a 1-ft. line, to measure a |-ft. line it must be ap- plied I times |, or | times. 10. To measure a 1-ft. line, a 2-ft. measure must be ap- plied I time. That is, one half of the measure must be applied. How many times must the measure 3 ft. be ap- plied to measure a 1-ft. line? 11. Determine how many times the measure in each must be applied to measure 1 ft., and solve each: 5 ft.-^^ ft.; 3 ft.^J ft.; | ft. -^ 2 ft.; J ft. -i- 3 ft.; 4 ft. -^ 1 ft. 12. To measure a 1-ft. line, the measure ^ ft. must be used 2 times. 2 is the reciprocal of J. To measure a 1-ft. line, the measure 2 ft. must applied J times. \ is the reciprocal of 2. 13. The reciprocal of 4 is | ; of 3 is J ; of ^ is 3 ; of f is |; of 1^ is |. If I is used as a measure to measure 1, the quotient is |. Multiply | by |. The product is 1. 14. When the product of two numbers is 1, the num- bers are said to be reciprocals of each other. 15. The reciprocal of the number used as divisor shows the number of times the divisor is contained in a unit, thus : The divisor ^ is contained in 1 three times. The divisor J is contained in 1 f times. Hence the following rule : To divide hy a fraction^ miultiply the reciprocal of the di- visor hy the dividend. 16. What is the reciprocal of each: |? ^? |^? y^^? ^? 17. Compare the terms of a fraction with the terms of the reciprocal of the fraction. When the terms of a frac- tion are interchanged, the fraction is said to be inverted. 56 Fl 168. Written Exercises MCI IONS 1. Divide 6| by f . 3 Model: 6| -f- ^ = ?I > 6 i ^5 Solve : 2 2. 3| yd. -1- f yd. 10. 6f^6f 18. H^f 3. 5iyd. ^|yd. 11. 4| ^ 6| 19. 3J^2J 4. 6f wk. -i- f wk. 12. 5|-2,V 20. 81J^9^ 5. 8| yd. H- 1 yd. 13. 7f^8J 21. m^^^ 6. |7f-i-|| 14. f^l 22. 16|^14f 7. 3y\ in. ^ 1 in. 15. 1%-^T% 23. 6|^12| a 8|-f-4| 16. A^l 24. 7J-^6|i 9. 7^^5f 17. A^? 25. 24|^8 26. Divide 100 by 331 ; by 66f ; by 371; by 87 J. 27. Divide if by 4 ; if by 5 ; 316 J by 8 ; 435| by 27. 28. Multiply 635f by 8; 315f by 8; 80f by 9. 29. Add 8f, 4|, 31 6^, 8^. 30. Take 32f from 96|. From 80| take 19f . 31. Divide 8.125 by .04; 180.40 by .05; 725 by 1.25. 32. Multiply 3.1416 by 4|; .7854 by6J; 2150.42 by 60|. 169. Oral Exercises. 1. Divide each by 100 : |43, 3.14, 60.75, .9, 2000. 2. Add J and \ ; J and \ ; \ and \. State a short method of getting the sum of two fractions whose nu- merators are 1 and whose denominators are prime to each other. MULTIPLICATION AND DIVISION 127 170. 1. How many strips of carpet 1 yd. wide will it take to cover a room 7 yd. wide ? 2. How many strips of carpet | yd. wide will it take to cover a room 9 yd. wide ? 6 yd. wide ? 3 yd. wide ? 3. Draw a diagram of a room 24 ft. long and 18 ft. wide. Show on the diagram the number of strips of carpet | yd. wide that are needed to cover the floor, the strips running lengthwise of the room, 4. What is the length in yards of each strip (Prob. 3)? How many yards of carpet are needed to cover the room, making no allowance for matching the strips ? 5. At 75^ per yard, how much will it cost for carpet for a room 28 ft. long and 18 ft. wide, the carpet being 27 in. wide, the strips running lengthwise of the room ? 6. How many ribbons each -| yd. long can be made of 8 yd. of ribbon? of 12 yd.? of 18 yd.? of 6 yd.? 7. At 5/ a pound, how many pounds of sugar can be bought for 40^? 8. How many pounds of sugar can be bought for 1 5, at 4^^ a pound ? at 4|^ a pound ? at 5| ^ a pound ? 9. How many strips of matting 42 in. (IJ yd.) wide will it take to cover a room 21 ft. (7 yd.) wide? 10. If a certain lamp consumes | pt. of oil each even- ing, how long will a gallon of oil last ? 5 gal. ? 11. What part of 1 yd. is 1 ft.? 30 in. ? 32 in. ? 27 in. ? 171, 1. How many times must 2| T. be written as an addend so the sum of the column will be 24 T. ? 2. If a boy earns . 16 rd. erd 111. City Lots. 1. The figure on p. 135 is a diagram of a city block. The dimensions are expressed in feet. Using your ruler, determine the scale used in making this diagram. Find the width of East Avenue. SCALE DRAWING 135 _j Grove St I 0) 1. too ^ o ^ -^ ^ l\) Co Cr> Co Co •li^ Oi 6 7 8 9 /O ^5 60 40 30 30 40 East Av. ~^ 186 FRACTIONS 2. Find the length and the width of the block. 3. Find the area of each of the lots. 4. Lot 7 was sold at $35 per front foot. Find the selling price of the lot. 5. At the same price per front foot, what is the value of each of the other lots ? 6. At $45 per front foot, what is the value of Lot 6 (60 ft. front) ? 7. Lot 1 was sold for $2250. How much was this per front foot? 8. Lot 10 was sold for $ 1600. How much was this per front foot ? 9. The selling price of Lot 5 was $ 750. How much was this a front foot ? 10. From its location in the block, which should be worth the more per front foot. Lot 7 or Lot 10 ? Lot 4 or Lot 5 ? 11. Find the cost of laying a 6-ft. cement sidewalk in front of Lot 9 at 12 ^ per square foot. 12. Find the cost of laying a 6-ft. cement sidewalk in front of Lot 2 at 13 ^ per square foot. 13. Mr. Thomas bought Lot 8 at $35 per front foot. He sold it for $1200. Did he gain or lose, and how much? 14. Mr. Brown paid $ 35 per front foot for Lots 3 and 4. He sold both lots for $2500. Did he gain or lose, and how much ? 15. Mr. Newton paid $35 per front foot for Lot 7. He had a 6-ft. cement sidewalk laid in front of the lot, cost- ing 12^ per square foot. He afterward sold the lot for $1500. Did he gain or lose, and how much ? REVIEW 137 REVIEW 178. Written Exercises. 1. Add 34.125, 4.36, 180.006, .67, 3.1416, 10.07, 2. Solve: 326.87-83.65; 9.82785 - 4.003 ; 346.85 - 184. 3. Multiply: 32.064 by .045 ; $465.73 by .08 ; $2456 by .06. 4. Divide: 13.046 by 1.8; 143.78 by.06; $120.78 by 1.06. 5. Write five improper fractions, and change each to a whole or a mixed number. 6. Write five mixed numbers, and change each to an improper fraction. 7. Write five proper fractions that are not in their lowest terms, and change each to lowest terms. 8. Add 6f , 4|, 71, 21, 14^3_. 9. From 28^\ subtract 9y\. From 834f take 186f 10. Multiply 8| by 6| ; 683| by 3| ; 31 1 by 18. 11. Divide 654| by 9 ; 195J by 7 ; 200 by 3J. 12. Divide f by I ; 6f by 71; 8| by 7|. ^ n 1 A ' vf 12x14x21 X 9 13. Cancel and simplity : -— — -- — -— — — - • ^ ^ 16 X 20 X 15 X 30 14. Reduce to lowest terms : |^, -5^, ||f , l^\, ^^* T. A 4. 1 w 75 66^ 331 40 12J 15. Reduce to lowest terms: — , ^, ^, — , ^, 100' 100 ■ $8 FRACTIONS 179. Oral Exercises. Solve : 1. 12 xj 10. 1 off 25 19. f of 21 da. 2. 24 x| 11. 1 of 140 20. ■^ of 84 yd. 3. 15 X J 12. 1 of 21 ft. 21. 1 of 48 mo 4. 10 Xf 13. f of 64 mi. 22. Jf of 30 da. 5. 18 x| 14. 1 of 36 in. 23. f of 120 6. 14 Xf 15. 1 of 18 in. 24. 1 of $120 7. 30 x| 16. f of 35 yd. 25. J of 160 8. 16 xf 17. f of 24 lb. 26. f of 170 9. 27 X 1 18. 1 of 45 lb. 27. f of $100 180. Oral Exercises. Find the quotient of : 1. 1 ^4 S. 1^6 9. 1,\ -r-4 13. 2J-i-5 2. lf-^6 6. 1^10 10. 6f ^3 14. 31-*. 8 3. {i^7 7. 1^2 11. m -^20 15. 4j-*-4 4. if^5 8. 1^8 12. If -4-10 16. 9f-i-4 181. Oral Exercises. Find the product of : • 1. ioii '• f off 13 ■ *xi 2- ioii 8- f of| 14. |x| 3. foff 9- 1 off 15. ?xi 4- fofj 10. J, Of H 16. |xf 5. -|0fA u. 1 off 17. *xi . 6. fofi 12. 1 off 18. JxH REVIEW 139 182, Oral Exercises. 1. If I of a ton of coal costs $6, what is the cost of a ton? 2. If f of the cost of a farm is f 2400, what is the cost of the farm ? What is | of the cost of the farm ? 3. If I of the cost of a carriage is $80, what is the cost of the carriage ? What is ^ of the cost of the carriage ? 4. A farmer sold -| of his crop of oats for f 160. At the same rate, how much was the entire crop worth ? ^ of the crop ? 5. Some men entered into partnership. One man contributed $800, which was f of the capital invested. How much capital was invested ? How much was con- tributed by one of the partners who furnished J of the capital ? 6. A man sold | of his land for f 1200. At this rate, what was the value of all his land ? 7. A poultry dealer sold 80 turkeys and then had ^ of his stock left. What part of his stock of turkeys did he sell ? How many turkeys had he at first ? 8. After spending 'S 18 for an overcoat, a man had $6 left. What part of his money did he spend ? What part of his money did he have left ? 9. After traveling 24 miles, a man still had f of his journey to travel. Find the length of the entire journey. 10. Mary had | as much money as Ethel. If Mary had 60 ^, how much did Ethel have ? 140 FRACTIONS 11. If George has f as many books as Walter, and George has 12 books, how many books has Walter ? 12. 120 is f of ^ of $20 = 13. $35 is ^ of fof$35 = 14. After increasing his farm by buying | as many acres as his farm contained, a farmer owned 120 acres. How many acres did he own before making the purchase ? 15. Ethel weighs J more than Edna. Ethel's weight is 105 lb. What is Edna's weight ? 16. Thomas solved | more problems than Henry. He solved 6 more problems than Henry. How many problems did Henry solve ? How many did Thomas solve ? 17. $80 is If (f ) times what amount ? 18. $120 is 2| times what amount ? 19. $200isfofic. $60 is IJ of a;. 20. What amount less | of itself equals $100 ? 21. What amount less | of itself equals $60 ? 22. $1200 is 2| times x, ^ of some amount is $160. What is the amount ? 23. After gaining -J of his capital, a merchant had $14,000. Find the amount of his capital at first. 24. After buying 3 books, a girl had 8 books. The number of books bought was what part of the number she previously had ? 25. $80 is f of . $120 is J of . $90 is} of . 26. $60 is 1 J times $150 is 1 J times . 27. $6 -4- $.75. $9-<-$1.50. $50 + $1.25. flEVIEW 141 183. Oral Exercises. 1. If I of the cost of a pair of skates is 60 ^, the cost of the pair of skates is how many times 60 ^ ? 2. If J of the cost of a desk is $3, the cost of the desk is how many times $3 ? 3. Compare | with |. Show by a diagram that | is IJ times |, or ^ times |. 4. If I of the cost of a table is $9, the cost of the table is how many times $9 ? 5. I of the cost of a clock is $S, In finding the cost of the clock we may find ^ of its cost, and then J of its cost. Show that multiplying $8 by J is the same as find- ing first ^ of the cost, and then f of its cost. 6. If I of the value of a horse is $60, what is its value? 7. If 30 sacks of oats is | of the yield per acre, what is the yield per acre ? Find the answer in two ways. S. A man sold | of his crop of apples for $120. At the same rate, what was the value of his entire crop ? 9. If I c' a ton of coal costs 86, what is the cost per ton ? Are these two solutions identical in character ? A. 3)16, 'B. $2 f 2, cost of { ton. ^^ >^ i = S8 __4 ^ $8, cost of 1 ton. 1 10. If I of the cost of a farm is f 6000, what is the cost of the farm ? What is ^ the cost of the farm ? 11. After selling | of his sheep, a farmer had 60 sheep left. How many had he at first ? 12. A boy sold 16 papers, which was J of all he had. How many papers had he at first ? 142 FRACTIONS 13. After solving 8 problems, a girl had | of her prob- lems yet to solve. How many problems had she to solve at first ? 14. By selling an article for 45^, a merchant gained J of the cost. Find the cost of the article. 15. By selling a horse for '1^90, a man lost J^ of its cost. For what part of the cost did he sell the horse ? Find the cost of the horse. 16. By selling a book for 60^, a boy lost | of its cost. For what part of its cost did he sell the book ? Find the cost of the book. 17. Two boys bought a sled in partnership, one paying I of its cost and the other paying | of its cost. The boy who paid | of its cost paid 70^. Find the cost of the sled. 18. After having his salary increased by J, a boy re- ceived $)20 a month. What was his salary before it was raised ? 19. A dealer advertised second-hand books at f of their ordinary price. At what price does he sell a book that costs 60^ when new? What is the price when new of a book which he sells for 90^? 184. Written Exercises. Find the whole when the part is given : 1. 112 is 1 7. 160 A. is { 13. 3| mi. is J 2. 120 is 1 8. 320 rd. is | 14. H gal. is A 3. 75 mi. is | 9. $42.50 is f 15. 81 ft. is 5^ 4. lis} 10. 36 yd. is 1 16. $1.20 is 1 5. fT.isf 11. 90 ft. is f 17. 3|isA 6. $6400 is f 12. 5280 is 1 18. ^44 is f ' f of the ; 1 of the t¥o of the REVIEW 143 185. Oral Exercises. 1. Find the whole amount when | of the amount is $60; is $18; is $54; is $90; is $240; is $1500. 2. Find the whole amount when $ 36 is | of the amount; | of the amount; -f^ of the amount; -^^ of the amount; | of the amount. 3. Find the whole amount when $120 is 1^ times the amount; IJ times the amount; 1^ times the amount; IJ times the amount; If times the amount. 4. Find the whole amount when $240 is amount; f of the amount; | of the amount amount; |^ of the amount; ^ of the amount. 5. Find the whole amount when $600 is amount; 1|^ of the amount; ^|gof the amount; -^-^^ of the amount; -^^^ of the amount; \^ of the amount. 6. A boy walked 2 blocks, which was ^ of the distance from his home to the schoolhouse. How many blocks must he walk in going to and coming from school each day, if he goes home for lunch ? 7. Charles weighs -^ less than Albert. The difference in their weight is 11 lb. How much does each weigh? 8. Margaret wrote 7 more words than Emma, which was ^ more words than Emma wrote. How many words did each write ? How many did both together write ? 9. A collector charged ^ of the amount of a certain bill for collecting it. Find the amount of the bill, if the creditor received $24. 10. After selling 60 acres, a farmer had | as much land left. How many acres had he before making the sale ? 11. $80 is I of . 90 mi. is f of 144 FRACTIONS REVIEW 186. Written Exercises. 1. 47f-14| 9. 87 -66f 17. 2|x3| 2. 93|-52f 10. 47|+62f is. 41- f 3. 48fx84f 11. 19fx38| 19. 3f-2| 4. 9|-4J 12. 2fx6f 20. 7f+8J 5. 324|x| 13. 96|x7| 21. | = 2% 6. 453|-h5 14. 30Jx45| 22. 87 = f 7. 526|^f 15. 897|4-6 23. |x|xf 8. 736fx5| 16. 7301 x I 24. 3^x4| 187. Written Exercises. 1. Find the value of | of a farm of 160 A. at f 85 per acre. 2. A man sold | of his farm for $4800. At the same rate, what was the value of his entire farm? 3. If 3| yd. of cloth cost #2.25, what is the cost per yard? 4. Find the cost of 8| yd. of silk at $1.14 per yard. 5. Find the cost of a roast of lamb weighing 4| lb. at IQj^ per pound. 6. A turkey weighing 9^ lb. was bought for f 1.90. Find the price paid per pound. 7. Express in cents and find the sum of the following : ^. $|, $i, I J, ^. li, $1 $i, H. ^h H^ ^h H' 8 Write ten improper fractions and change them to whole or mixed numbers. 9. Write ten mixed numbers and change them to im- proper fractions. NUMBER RELATIONS 145 NUMBER RELATIONS 188. Oral Exercises. Express all fractional parts in their lowest terms. 1. What part of 10 is 5 ? of 6 is 3 ? of 8 is 2 ? of 12 is 4 ? of 20 is 5 ? of 30 is 6 ? 2. What is the ratio of 5 to 15? of 6 to 12? of 8 to 24? of 9 to 81? of T to 56? of 20 to 4? of 28 to 7? of 42 to 6? 3. What part of 4 is 3 ? of 8 is 5 ? of 9 is 7 ? of 11 is 3? of 10 is 6? 4. What is the ratio of 3 to 5 ? of 5 to 3 ? of 3 to 11 ? of 11 to 3 ? of 7 to 9 ? of 9 to 7 ? 5. What is the ratio of 6 sacks of oats to 18 sacks of oats ? 6 T. of coal will cost what part of the cost of 18 T. ? 18 T. will cost how many times the cost of 6 T. ? 6. If 5 sacks of flour cost $7.50, how much will 10 sacks cost ? 7. If a boy earns i3 in 4 da., how much will he earn in 16 da. ? 8. If a boy rides at the rate of 7 mi. in 2 hr., how far at the same rate will he ride in 6 hr. ? 9. If 12 pads cost 1.60, how much will 36 pads cost? 10. If 3 T. of coal cost $24, how much will 9 T. cost ? 11. What number expresses the ratio of 4 lb. to 8 lb. ? of 5 lb. to 20 lb. ? of 15 yd. to 5 yd.? of 20 A. to 4 A.? of $20 to $30? of $24 to $36? of 18 bu. to 24 bu.? of 36 ft. to 21 ft.? of 48 mi. to 36 mi.? of $25 to $50? 12. What fraction expresses the ratio of 5^ to 25^? of 10^ to 50^? of 20^ to 100^? of 25^ to 100^? of 25^ to 75^? of 10^ to 40^? of 5^ to 45^? of 20)^ to 50^? iK. — 10 146 J'ractionS 13. What fraction expresses the ratio of 3 qt. to 4 qt.? of 5 mi. to 8 mi.? of 4 lb. to 6 lb.? of 8 bu. to 12 bu.? of 12 yd. to 9 yd. ? of 20 mi. to 15 mi.? of 2b^ to 40^? of 18 yr. to 12 yr.? of 6 mo. to 9 mo.? of 9 mo. to 12 mo.? 14. What part of $1 is 25^? 50^? 75^? 40^? 60^? 70^? 80^? 90^? 5/? lOj^? 20/? 30/? 15. What part of |1 is 12|/? 37|/? 621/? 87^/? 33-1-/? 662^? 162/? 831/? 142/9 8J/? 16. What part of 1 ft. is 2 in.? 3 in. ? 4 in. ? 5 in. ? 6 in. ? 7 in.? 8 in.? 9 in. ? 10 in.? 11 in. ? 17. What part of 1 yd. is 2 in.? 3 in.? 4 in.? 6 in.? 8 in.? 9in. ? 12 in.? 18 in.? 20 in. ? 24 in.? 30 in.? 13 in. ? 21 in. ? 18. Whatpartof lib. isJoz.?4oz.? 8oz.? 12 oz.? 7oz.? ' 19. What part of 1 yr. is 2 mo. ? 3 mo. ? 4 mo. ? 5 mo. ? 6 mo.? 7 mo.? 8 mo.? 9 mo.? 10 mo.? 11 mo.? 20. What part of 1 da. is 2 hr. ? 3 hr. ? 4 hr. ? 5 hr. ? 6 hr.? 8 hr.? 10 hr. ? 12 hr. ? 15 hr.? 16 hr. ? 18 hr.? 20 hr.? 21. What part of 1 hr. is 5 min. ? 10 min. ? 15 min. ? 20 min. ? 25 min. ? 30 min. ? 35 min. ? 40 min. ? 45 min. ? 50 min. ? ^^ min. ? 17 min. ? 22. What part of 1 mi. is 10 rd. ? 20 rd. ? 40 rd. ? 80 rd.? 60 rd.? 90 rd. ? 23. What part of 1 T. is 1000 lb. ? 500 lb. ? 250 lb. ? 200 lb. ? 100 lb. ? 24. What part of 1 section of land (1 sq. mi., or 640 A.) is 320 A.? 160 A.? 80 A. ? 40 A.? 20 A.? ALIQUOT PARTS 147 189. Aliquot Parts. 1. Name several amounts that are exactly contained in 136. 2. How many times is each of the following contained in$l: 50^? 25^? 12^^? 10)^? 20^? 5/? 4^? 2^? 3. A part of a number or a quantity that will divide it without a remainder is called an aliquot part. Name several aliquot parts of 100. 4. What part of $1 is each: 50/? 25^? 10^? 20)^? 6/? 121/? 331^? 16|/? Hi/? 6|/? 2/? 4/? 14^/? 5. If 40 sheep can be bought for $100, how many sheep can be bought for 120 Q of |100)? for $25? for $12.50? for 110? for $5? 6. If 100 sacks of potatoes cost $80, how much will 25 sacks cost? 50 sacks? 10 sacks? 5 sacks? 20 sacks? 7. How much will 30 yards of cloth cost at $1 a yard? at 25/ a yard? at 12^/ a yard? at 20/ a yard? at 16|/ a yard? at 33J/ a yard? at 8^/ a yard? 8. From the cost of any number of articles at $1 each how may the cost of the same number of articles at 25/ each be found? at 50/ each? at 20/ each? at 121/ each? at 33^/ each? 190. Memorize the following fractional parts of 1 : .50= i .20 = i ^^^ = i .40 =1 .25= I .83J= i .37^ = 1 .60 =1 •W=rV .75= 1 .66|= 1 .05 =^V •62J = f •87|=J •16f = J •14? = f From the above table construct a table showing the same fractional parts of $1 ; of 100 ; of $100 ; of 1000. 1. At 50 ^ each. 2. At 25 ^ each. 3. At 75 ^ each. 4. At 121^ each. 5. At 331^ each. 6. At 66|^each, 148 FRACTIONS 191. Oral Exercises. From the cost of 120 articles at f 1 each find the cost: 7. At 20 ^ each. 8. At37|^each. 9. At 621^ each. 10. At 40 ^ each. 11. At 60 ^ each. 12. At 80 / each. 192. Written Exercises. Solve eacli by the shortest method. 1. Find the cost of 24 yd. of cloth at 37^^ per yard. Suggestion : At $1 per yard the cloth would cost $24. 2. Find the cost of 24 yd. of cloth at 87 J i per yard. Suggestion : 87^^ per yard is \ less than $1 per yard. 3. Find the cost of 24 yd. of cloth at 66|^ per yard. 4. Find the cost of 16 articles at 1 25 each; at $250 each (1 of $1000) ; at $125 each (^ of $1000) ; at $75 each (J less than $100) ; at $37.50 each. Suggestion : At $100 each the 16 articles would cost % 1600. 5. How many articles can be bought for $48 at $1 each? at 25^ each? at 331^ each? at 66|j^ each? at Vl\^ each? at 20)^ each? at 37| j^^ each? 193. Short Methods. Solve each, using the shortest method: 1. $40x25 5. $2040x.l2J 9. 400 1b. x. 625 2. $120x25 6. 640A. x37J lo. $8.60x75 3. $80x250 7. 240 mi. X. 125 ii. $5.60x750 4. 60mi. x33i 8. 36 ft. x 125 12. $4.64x12.5 ALIQUOT PARTS 149 194. 1. Divide by 25: 12; $16; 640 A. To divide hy 25, divide hy 100 and multiply the quotient by 4:, 2. State a short method of dividing a number by 250; by 50 ; by SSJ ; by 66| ; by 371 ; by 12| ; by 375 ; by 75 ; by 125; by .25; by .125; by 12.5; by 2.5; by 62.5; by 625 ; by 500. ivide : 3. $400 by 25 10. 2240 lb. by .25 4. $ 300 by 250 11. 2000 lb. by 2.5 5. $600 by 50 12. 5280 ft. by 37.5 ft. 6. 320 rd. by .125 13. 1728 by 250 7. 640 mi. by 12.5 14. $400 by 87.5 8. 540 ft. by 331 15. $ 3200 by f 625 9. 120 yr. by 66f 16. $1500 by 12.50 195. 1. What is the cost of 24 yd. of cloth at 50 / per yard? at 12| / per yard? at 16|)^ per yard? at 75/ per yard ? at 87^ ^ per yard ? at 37| / per yard ? 2. How many yards of cloth can be purchased for $12 at 25 / a yard ? at 12| / a yard ? at 61 / a yard ? at 37^ / a yard ? at 33^ / a yard ? a\; 66 J / a yard ? at $1.50 a yard? at $1,331 a yard? at $1.20 a yard? 3. If 40 acres of land cost $ 2000, how much will 50 acres cost at the same rate ? 60 acres ? 100 acres ? 45 acres ? 55 acres (40 acres + | of 40 acres -4- J of -J of 40 acres) ? 4. George White paid Thomas Evans $ 12 for the loan of some money for 60 da. At the same rate, how much must he pay for the use of the same sum for 90 da. ? for 30 da. ? for 75 da. (60 da. + I of 60 da.) ? for 120 da. ? for 70 da.? for 50 da. ? for 20 da. ? for 80 da. ? 150 FRACTIONS REVIEW 196. 1. Draw a square and show the following parts of it : .50, .25, .75, .121 .37^, .871, .33J, .66|. 2. Draw a square, and divide it into as many equal parts as are necessary to show either J or J of the square. 3. What is the 1. c. m. of 2, 3, 9? of 2, 3, 4, 6? 4. Draw a square, and divide it into as many equal parts as are necessary to show all of the following parts : h h h h h S^C)w on the square the parts |, |, |, |. 5. Describe a cubic foot. Think of a box whose inside measure is 1 ft. by 1 ft. by 1 ft. How many bricks do you think the box will contain? How can you find the exact number it will contain? If this box is watertight, how many gallons will it contain? (1 gal. = 231 cu. in.) 6. A boy made a bookcase. The top of the lower shelf is Q" from the floor. The space between the lower shelf and the top of the case is 3' 6''. The case contains four shelves. The space between the two lower shelves is 12''. The other shelves are placed so that the distance between them is exactly equal. How far apart are they if the shelves are each ^" in thickness? The width of the case is 2' 6", and the depth 1'. Draw a diagram, using the scale 2'' = 1'. 197. 1. Divide each by 10 : 47; 138.40; f.80; 3.1416. 2. Divide by 100: 22001b. ; 5280 ft.; 1760 yd.; 1457.50. 3. State a short method for multiplying by 10 ; by 100; by 25; by 33^; by 66|; byT5; by 12|; by .25; by .87J; by 37|. Give several illustrations of each. 4. Explain what is meant by the reciprocal of any number. What is the reciprocal of |? of 8? of ^? of 5^? REVIEW 161 5. Divide f ft. by ^\ ft.; 6.2 by .02; if by 6 ; if by 3. 6. State how to multiply and how to divide a fraction by a fraction ; a whole number by a fraction ; a fraction by a whole number; a mixed number by a whole number or a mixed number. Illustrate each. 7. State how to determine the place of the decimal point in the quotient in division of decimals. Fix the decimal point in: .002)4.8368; 2.36)13:4; 34)4.275. 8. The question, How many square inches are there in 1 square foot? is answered by the number 144. Ask a similar question that is answered by each of the follow- ing: 3, 12, 9, 4, 2, 60, 24, 51 144, 7, 320, 231, 1728, 5280, 365, 640, 27, 52, 160, 128, 2150.42, 8, 32, 100, 16, 2000, 2240. 9. The question. What is ^ of 100? is answered by the number 121. Ask a question concerning a fractional part of 100 that is answered by each of the following : 20, 75, 25, 121, 871, 5, 621 qqi^ 871 331, 40, 10, 80, 16|. 10. The question. What is the ratio of 10^^ to |1? is answered by the number -^q. Ask a question concerning the ratio of some quantity to $1 that is answered by each- 1 -^ 4 4 1 4 ^ X 4 i ^ J- -1- 4 ^ 4^ 2 -5^ 4 Ctn^u . 2^ 3? ^1 ^5 6' 1' 5' 8' ^' ^' 5' 10' 2 0' 1' 2' 4' "^^ 3' F* 198, The first number is the product of two numbers. The second number is one factor. Find the other factor. 1. 36, 6 7. 1.5, .5 13. 816.40, 1 2. 30,1 8. $.15, 3 14. $20, .331 3. 15, .5 9. 1.15, 5^ 15. 96 ft., .121 4. l'2 10. 11.80, .06 16. 96 ft., 1 5. f'i 11. $30, .05 17. $34.40, .08 6. .5,10 12. 125.60, .08 18. $10, .04 152 FRACTIONS 199. Oral Exercises. Fill in the amounts omitted under each heading Cost Selling Price Gain Loss Part Gained Part Lost 1. 140 150 $10 i 2. 125 130 X X 3. 160 $50 X X 4. $80 X $20 X 5. X $100 $25 X 6. X $150 $50 X 7. $75 X $50 X 8. X $75 X \ 9. $150 X X \ 10 X $110 X tV 11. Express as hundredths the part gained or lost in each of the above exercises, as \ gain= .25 gain. 12. Write and solve ten exercises similar to Exs. 1-10 above. 13. Example 5 above may be stated as a problem: Illustration: A man sold a horse for $100 at a gain of $25. Find the cost of the horse and what part the gain is of the cost. 14. State Exs. 1-4 and 6-10 as problems. 200. 1. If .04 of some amount is $10, what is the amount? 2. By what must $240 be multiplied to produce $12? 3. Multiply: $600 by .06; $300 by .04; $80.50 by .07. 4. Find .06 of $360; of $4 ; of $24; of $30; of 80 mi. 5. If 4 times some amount is $16, what is the amount? REVIEW 153 201. Oral Exercises. 1. If 12 articles cost $ 36, how may the cost of 6 articles be found without finding the cost of 1 article ? 2. What part of the cost of 12 articles must be added to the cost of 12 articles to give the cost of 18 articles ? of 15 articles ? of 14 articles ? of 16 articles ? Illustrate. 3. What part of the cost of 6 articles must be added to the cost of 6 articles to give the cost of 9 articles ? of 8 articles ? of 7 articles ? Illustrate each. 4. What part of the cost of 6 articles must be sub- tracted from the cost of 6 articles to give the cost of 5 articles ? of 4 articles ? Illustrate each. 5. When the cost of 6 articles is known, how may the cost of 3 articles be found ? of 2 articles ? of 1 article ? of 12 articles ? Illustrate each. 6. When the cost of 50 articles is known, how may the ►st of 12J articles be articles ? of 75 articles ? cost of 12J articles be found ? of 37J articles ? of 62^ 202. 1. If .06 times some amount is $12, what is the amount? If .04 of an amount is $20, what is the amount ? If .03 of an amount is $ 24, what is the amount ? 2. 145 is .09 of what amount? 175 is .15 of what amount ? f 1.60 is .08 of what amount? 3. $ 15 is 1.25 of what amount ? is .20 of what amount ? 4. How much is 1.75 of $80? of $200? of 640 A.? 5. $ 40 is what part of $ 80 ? J = yf ^. 6. $20 is how many hundredths of $40? of $80? 7. .6 of 600 is .12 of what number? 8. .9 of 800 is .3 of what number ? 154 FRACTIONS 203. Written Exercises. Keep each result until all the problems have been solved. 1. A farmer rented a field 60 rd. long and 40 rd. wide. Find the number of acres in the field. 2. The annual rent of the field was $8.75 per acre. Find the rent of the field for 1 yr. 3. The farmer planted the field in broom corn, which yielded J T. to the acre. Find the total yield of broom corn. Each acre of broom corn yielded 1 T. of seed, valued at $ 16 per ton. Find the value of the seed. 4. The farmer sold the broom corn at $80 a ton. Find the value of the crop. 5. The farmer paid a commission merchant $4 a ton for selling the broom corn. Find the commission. 6. The commission merchant paid $2.50 per ton freight and f .75 per ton cartage on the broom corn. How much should he remit to the farmer, after deduct- ing these expenses and his commission ? 7. The expense of seed and of planting and harvest^ ing the crop amounted to $ 15 per acre. How much was the farmer's net profit per acre from the crop ? 8. If 25 lb. of broom corn are used in making 1 doz. brooms, how many dozen brooms can be made from the yield of 1 A.? 9. If the manufacturer sells the brooms for $2.50 per dozen, how much does he receive for the brooms made from the yield of 1 A.? 10. How much did the broom corn cost per pound at $80 per ton? '■^'^ 11. How much is the cost of the broom corn used in making 1 doz. brooms ? REVIEW 155 12. If the cost of labor and of the material other than the broom corn is f .80 for each dozen brooms, how much do the brooms cost the manufacturer per dozen ? 13. How much is the manufacturer's profit on each dozen brooms ? The manufacturer's profit on each dozen brooms is what part of the cost of a dozen ? 14. A wholesale merchant bought'the brooms from the manufacturer at f 2.50 a dozen and sold them to retail dealers at a profit of -^^ of the cost. What was the price of the brooms per dozen to the retail dealer? 15. The retail dealer sold the brooms to his customers at a profit of ^ of the cost to him. Find the price paid to the retail dealer for each broom. 16." If a retail dealer's net profit on each dozen brooms is ^ of the gross profit, how much was his net profit on the sale of 1 doz. brooms ? 17. Find the difference between the cost of 1 doz. brooms to the manufacturer and the cost to the consumer. 204. Written Exercises. 1. .06 of some amount is f 30. Find .03 of the same amount. Suggestion : .03 is one half of .06. Take one half of $ 30. 2. .06 of an amount is $30. Find .09 of the amount. Suggestion: .09 is one half more than .06. Follow the above suggestions in the solution of each : 3. Find .06 of 16400. From the answer find .02 of 16400; .03of|6400; .09of 16400; .08 (1 more than. 06) of 16400 ; .04 (i less than .06) of 16400 ; .05 (J less than .06) of $6400; .07 of $6400,- J.2 of ^6400. 156 FRACTIONS 205. Changing Decimal Fractions to Common Fractions. 1. What is the numerator in .375? What is the denominator? 2. How many decimal places are there in .375? Write .375 as a common fraction. Compare the number of decimal places in .375 with the number of O's in -f^. 3. Change .875 to a common fraction. Model: .875 = ^V^ = f To change a decimal fraction to a common fraction^ write the numerator of the fraction over the denominator of the fraction. Reduce to lowest terms. Change to common fractions : 4. .1, .2, .3, .4, .5, .6, .7, .8, .9. 5. .10, .20, .30, .40, .50, .60, .70, .80, .90. 6. .12, .15, .25, .35, .45, .55, Sb, .75, .85, .95. 7. .125,' .375, .625, .875, .025, .075, .04, .05, .02. 8. Reduce to mixed numbers : 4.25, 26.5, 8.75, 15.375, 45.125, 7.875, 12.625, 8.20, 35.60, 2.04, 5.40. 9. In the following the cents and mills are expressed decimally as fractions of a dollar. Write with the cents expressed as common fractions; thus, |56.40 = 'f 6|: $5.25, $8.20, $4.10, $3.50, $7.80, $2.75, $6.60, $4,125, $9,625, $8,375, $4,875, $25.30, $15.05, $4.01, $7.90. 10. What decimal is equivalent to J? i? |? i? |? |? F V f? I? 1^? A? T%? 11. Express each as an improper fraction: 1.25, 1.10, 1.20, 1.125, 2.40, 1.375, 1.625, 1.875, 3.6, 1.80. 12. Add as decimals: .25, .125, .4, .875, .2,4.75,6.07. Change to common fractions and add. Common to DfeciMAt 157 S06. Changing Common Fractions to Decimal Fractions. 1. Change | to a decimal fraction. 3 -7- 4 is the same as -I- A fraction is an indicated division, in which the numerator is the dividend and the denominator is the divisor. The division indicated by | may be performed by placing the decimal point after 3 and dividing, thus : .75 I = . ' . The fraction | has been reduced to a decimal fraction. 2. Perform the indicated division. Continue the division until there is no longer a remainder. I, |, |^, ^, h h h h h To change a common fraction to a decimal fraction^ divide the numerator hy the denominator. 3. The prime factors of 10 are 2 and 5. Name all the numbers to 30 which have no other prime factors than 2 or 5. Find by trial whether any fraction whose denom- inator has any prime factors other than 2 or 5 can be changed to an exact decimal. Which of the fractions in Sec. 207 can be reduced to exact decimals ? 207. Written Exercises. Change the following to decimals. Where the decimal is inexact, continue the division to three places. 1. f 7. 1 13. A 19. 2| 2S. $H 2. f 8. f 14. ^ 20. m 26. |7t 3. \ 9. 1 6 15. f 21. Hs 27. »8fV 4. i 10. \ 16. 1 22. H 28. I7f 5. f 11. f 17. i 23. 82V 29. f5i 6. i 12. 1 18. A 24. ^2\ 30. i6T-L 158 FRACTIONS 208. Changing Fractions to Hundredths. 1. Change | to a fraction whose denominator is 100. 4 X Model: r— ^j — 7" ^ i^ contained in 100 twenty times. ^ 1^^ Multiply both terms of i by 20. i = ^%. Since 1 is equivalent to {^^, ^ is equivalent to ^q%, and ^ is equiva- lent to j%. 2. Change to fractions whose denominators are 100 : 1' h h h h i ft,' sV- #5' h h h h h h h h T^- 3. Express as decimal fractions each of the fractions in Ex. 2. 4. Memorize the following : 1 = ?^ = .334 8 100 ^ 3 100 ^ 8 100 ' ? = ?li = .37i 8 100 ^ '^ = ^^ = .621 8 100 ^ 8 100 * 6 100 ^ l = lil = .14? 7 100 ^ 5. Express as common fractions in lowest terms: .25, .20, .40, .50, .60, .70, .75, .80, .05, .02, .04, .10, .90, .01. 6. Write with the fractional part expressed as a deci- mal : 7^, 4f , 6f , 8J, 9|, 4^, Z^, S^, 12^, 9f 7. Express as dollars and cents and add t8J, i3|, $9J, f4fj, $7^, *8^, il2|. |15f, IWjV, i72i5. 1 = .ioo_ 100 1 1_ 2 60 100 .50 1_ 4 25 'ioo~ .25 3_ 4 75 100 .75 1_ 5 20 100 .20 2 = 6 . 40 _ 100 .40 3_ 5 60 100 .60 4_ 5 80 _ 100 .80 1 _ 20" 5 100 .05 1 _ 25 4 100 .04 1 _ 50 2 100 .02 1 _ 12 100 .08^ 1 10 _ 10 _ "100 .10 3 _ 10 - 30 _ "100 .30 7 10 70 _ "100 .70 9 _ 10" . 00 _ "100 .90 REVIEW 159 209. Written Exercises. 1. There are 2000 lb. in a ton. How many tons of hay are there in 5400 lb. ? 2. At $8.50 per ton, how much will 6500 lb. of hay cost? 3. At f 10.50 per ton, how much will 950 lb. of coal cost? 4. How many hundredweight (100 lb.) are 575 lb. ? At $5.60 per hundredweight, how much will a farmer receive for some hogs weighing 3750 lb. ? 5. At $37.50 per ton, how much will a farmer receive for 12,400 lb. of wheat ? 6. At $1.25 each, how many hats can be bought for $20? 7. How much will 45.75 A. of land cost at $65.50 per acre ? 8. The circumference of a circle is 3.1416 times its diameter. Find the diameter of a tree, the circumference of which is 7.75 ft. 9. Find the circumference of a cylindrical tank, the diameter of which is 4 ft. 9 in. (4.75). 10. At $38.50 each, how much will 14 cows cost ? 11. The area of a rectangle is 42.625 sq. in. Its length is 7.75 in. How wide is the rectangle ? 12. If a train travels at an average rate of 46.75 mi. per hour, in how many hours will it travel 390.6 mi. ? 13. At 5J^ ($.0525) per pound, how many pounds of sugar can be bought for $4.20? 14. When hay is worth $7.50 per ton, how many tons can be bought for $90? 160 FRACTIONS LUMBER MEASURE 210. 1. The unit used in measuring lumber is the board foot, which is the equivalent of a piece of board 1 ft. long, 1 ft. wide, and 1 in. thick. A board 12 ft. long, 12 in. (1 ft.) wide, and 1 in. or less in thick- ness contains 12 times 1 board foot, or 12 board feet In measuring lumber, boards less than 1 in. thick are considered inch boards. The name board foot is generally shortened to " foot." The Roman jiumeral "M" is used to denote a thousand feet. 2. How many board feet are there in a piece of board 1 ft. long, 12 in. (1 ft.) wide, and 2 in. thick? 1 ft. long, 6 in. (^ ft.) wide, and 2 in. thick ? 3. What part of a board foot is there in a piece of board 1 ft. long, 6 in. Q ft.) wide, and 1 in. thick ? 1 ft. long, 8 in. (I ft.) wide, and 1 in. thick? 1 ft. long, 4 in. (1^ ft.) wide, and 1 in. thick? 4. A piece of board 1 ft. long, 6 in. wide, and 2 in. thick contains 2 times | board foot, or 1 board foot. Explain. How many board feet are there in a piece of board 1 ft. long, 6 in. wide, and 3 in. thick ? To find the number of hoard feet in a piece of lumber^ multiply the number of board feet in one foot of the length by the number of feet in the length of the piece, 5. Find the number of board feet in 16 pieces of 3" by 4^', each 12 ft. long. Model- ^^^^ either 3" or 4" as the width. Taking 4" as the 16 X ?;2 X 3 X 4 board feet = f^^.' *''t"T'f ' f ^"^ ^^ yp feet m 1 ft. of the length is 192 board feet. ^ ^ i^ ^^^^^ ^^®*5 and in 1 piece 12 ft. long, 12 x 3 x ^^ board feet ; and in 16 pieoes, 16xl2x3x^ board feet, or 192 board feet. LUMBER MEASURE 161 211. Oral Exercises. Find the number of feet in a piece of lumber of the following dimensions: 1. V^ X 12'', 10 ft. long 5. 4" X 4'^ 12 ft. long 2. 2'^ X 12'^ 1 ft. long 6. 4'^ X 8'', 15 ft. long 3. 2" X 6", 14 ft. long 7. 6'^ x 6'', 18 ft. long 4. 2'^ X 4'^ 16 ft. long 8. 1'^ X 16'^ 15 ft. long 212. Written Exercises. 1. Find the number of feet in 120 pieces of lumber, each 2" by ^" by 16'. 2. Measure various pieces of lumber. 3. Find the cost of lumber for a bridge 10 ft. long, if planks 3'' x 12'' x 14' are laid over four timbers 8" x 8" xl4'. Boards costing 118 per M; timbers 1 20 per M. 4. Find the cost of the lumber for a 5-board fence around an orchard 160 ft. by 240 ft. The boards used are 6 in. by 1 in. by 16 ft. and cost 1 14 per M. The posts, 8 ft. in length, are set 8 ft. apart, and are made of pieces 4 in. by 4 in. by 16 ft., costing f 16 per M. 213. Flooring. 1. When tongued and grooved^ a board 3 in. wide is 2^ in. wide when laid. The part of the board thus lost is ^ of the width covered by the board after it has been fitted. Explain. If 168 ft. of flooring 3 in. wide are needed for a certain floor, ^ as much must be added if tongued and grooved flooring 3 in. wide is used. Why? 2. Find the number of feet of flooring needed for a room 24 ft. wide and 30 ft. long, if tongued and grooved flooring 3 in. wide, | in. thick, and 12 ft. long is used. What is the cost of the flooring at $40 per IVf" di).. XB. — 11 162 FRACTIONS 3. What part of 5|- in. is J in.? of 2J- in, is J in.? Having found the number of feet of lumber needed to floor a certain room with boards 6 in. wide, how may the number of feet needed to floor the same room with boards 6 in. wide that have been tongued and grooved be found ? 214. Shingling. 1. The unit of shingling is a square, which is an area of 100 square feet. 2. When shingles have been laid, about 4 inches of their length is exposed to the weather. The average width of a shingle is about 4 inches. Consequently the exposed surface of one shingle is about 16 square inches, or about J square foot. It will thus take about 900 shingles to cover a square. Allowing for waste, 1000 shingles are estimated for a square. A hunch of shingles contains 250 shingles. How many bunches should be allowed to each square ? 3. Find the cost of the shingles necessary to cover both sides of a roof, if each side is 24' by 40', at f 2.25 per thousand shingles. Model : 24 x 40 x 2 x .01 x 1 2.25 = x. The number of square feet in both sides of the roof is 24 x 40 x 2, and the number of squares is .01 times this product. The cost of the shingles is ^2.25 multiplied by the number of squares. Why? 4. Find the cost of the shingles necessary to cover both sides of a roof, if each side is 36' by 48', and the shingles cost $2.50 a thousand. 5. Estimate the cost of shingles to cover the roof of your schoolhouse, at $2 a thousand. yr. 1907 mo. 1 da. 4 1893 5 26 DIFFERENCE BETWEEN DATES 163 215. Difference between Dates. 1. Walter Harris was born May 26, 1893. How old was he on January 4, 1907? Model : Write the later date as the minuend and the earlier date as the subtrahend. It is evident that some number of days added to 26 da. equal 1 mo. and 4 da. Subtract thus: 26 da. and 4 da. are 1 mo. 4 da. 13 7 8 and 4 da. (in the minuend) are 8 da. Carry 1 mo. to 5 mo., as in subtraction of integers. 6 mo. and 6 mo. are 12 mo. ; 6 mo. and 1 mo. are 7 mo. Carry 1 yr. tp 1893. Com- plete the subtraction. 2. Find your age by subtraction. 3. Find the time from the landing of Columbus in America to the date when the Declaration of Independence was signed. 4. Frank Thomas borrowed 1 750 of Charles Gray on Oct. 8, 1902, and paid it on July 2, 1903. How long did he have the money? 5. When the exact number of days between two dates that are less than a year apart is required, it is necessary to take account of the number of days in each month in- cluded, as in the following : Find the exact number of days from Jan. 4, 1907 to April 11, 1907. There are 27 full days left in January, 28 days in February, 31 days in March, and 11 days in April (including April 11), or 27 da. + 28 da. + 31 da. + 11 da., or 97 da. 6. Find the exact number of days from the Fourth of July to Christmas; from Christmas to May 1. 7. Mr. Jenkins borrowed a team of Mr. Slate on Aug. 21 and returned it on Nov. 15. At il.50 a day, how much did he owe for the use of the team? 164 FRACTIONS o 1} 216. Review Exercises. 1. Make a drawing to represent a city lot 40 ft. front and 120 ft. deep, using the scale 1 in. = 20 ft. Using the same scale, represent at the back of the lot the space occu- pied by a barn 20 ft. by 30 ft. 2. A man bought a tract of land 160 rd. long and SOi^ rd. wide. How many ' "^ acres did it contain ? The tract was di- vided as shown in the ^ , figure. Find the area of each field. Find the cost of fencing the tract as shown in the /60rd. 40ra/. 40rcf. eOrd. 1 40rd. 40rd 1 \ aord. 4/t figure at f 1.25 per rod. 3. A field containing 20 A. is 40 rd. wide. How long 4. Mr. James bought Lot 2 (p. 135) for 140 per front foot. After paying for a 6-ft. cement sidewalk costing -/ ^VS 12)^ per square foot, he sold the lot at a profit of $320. How much did he receive for it ? 5. After selling 60 acres a farmer had | of his land i / /; ^ left. Kow many acres had he before making the sale ? 6. If 80 A. of land cost 14000, how much at the same^/ .,^ price per acre will 320 A. cost? ,■ 7. If hay is worth $12 a ton, how much is 500 lb. of^jL hay worth ? How much is 400 lb. worth ? '^; • ' 8. If a boat traveled 120 mi. during the first 8 hr. after leaving port, how far at the same rate will it travel in 1 da.? in 2 da.? ^^-^ ^^^ 9. At $1 per yard, what is the cost of J yd. of silk?, off yd.? ofiyd.?, 7 REVIEW 166 10. If a man's expenses for 3 mo. amount to 8135, at the same rate, how much will his expenses amount to in lyr.? ?5'^'C> 11. If a horse is fed 1 bu. of oats in 5 da., how many- bushels will be necessary to feed it for 1 mo. (30 da.)? (> 12. If it costs $10 to pasture 6 horses for 1 month, how ; /) < much will it cost to pasture 9 horses for the same length \ of time ? 13. At 1 7.25 per ton, how much will 6. 75 T. of coal cost ? Vy . ^^ 14. How much more sugar is received for $ 1 by buying at 16 J lb. for a dollar rather than at 6|^ per pound ? T ^ 15. On the morning of March 7 a ship captain announced that he had on board enough provisions to last 80 da. Give the date on which the provisions would give out. 16. A boat that was due in port on Dec. 25 arrived on Jan. 6. How many days was she overdue ? ' 17. Change to decimals ; |, J, -J, |, -J, ^, |, -J, |, -^, /^ ., li, H, li, 2f, If. /^ 18. Change to common fractions : .125, .375, .25, .875, .6, .625, .8, .40. 19. Find .05 of $200; .06 of $18.75; 1.04of|80. ^ 20. $ 282 =.94 of ^-^; $375 =.75 of ; $60 =| of — . '^^;^.ixv 21. 160 = 1^ times ^-^; $60 = 1^ times = 1.20 of . 22. Change to lOOths : i, f , i, |, |, iV 23. If 4 of the cost of a city lot is $ 1200, how much is the cost of the lot ? ' -^ ^ 24. How many tons of hay at $12 a ton will equal in value the cost of laying a concrete sidewalk 40 ft. long and 6 ft. wide at 12^^ per square foot? z. / ' X PART III 7- PERCENTAGE 217. Hundredths as Per Cents. 1. Read each : ^% .60, .10, ^, .06, .85, \^, .05. 2. ^1^, or .05, may be written b^fo- It is then read 5 per cent. Per cent means hundredths. 5 per cent means 5 of the 100 equal parts. The sign (%) is called the per cent sign. 3. The unit 1 is equivalent to how many hundredths ? to how many per cent ? 4. Read the following: 4%, 8%, 25%, 40%, 75%» 100%, 150%, 200%, 6J%. 5. Express as per cents : y^, y^^, ^-f^, ^, -^^ \^^. 6. Express as per cents : .01, .03, .12, .18, .50, .90, .99, 1, 2, .125 (121%), .375, .625, .875. 7. Write as common fractions : 7%, 2%, 40%, 85%, 45%, 4%, 100%. 8. Write as decimal fractions : 1%, 5%, 7%, 30%, 3%, 75%, 80%, T%, 100%, 37J%, 33^ %, 14f %. 218. Finding some per cent of a number. 1. 4% of $500 is the same as 1500 multiplied by .04. Find 4% of $500; of $250; of $45.50; of $875. To find any per cent of a number^ multiply the number by the required per cent expressed as a decimal fraction. 2. Find 5% of $860; of $60; of $100; of $840.25. 106 HUNDREDTHS AS PER CENTS 167 3. Find 12% of 1400; of 1350; ofUlOO; of $247.25; of 11300. 4. Find 45% of 650 mi. ; 80% of 640 A. ; 62% of 400 bu.; 1% of $400. 5. Find 100% of $500. Compare 100% of $500 with $500. 6. 125% means -ifl, or 1.25. Find 125% of 300 mi. 7. Name a per cent of $600 that is the same as $600; that is less than $600 ; that is more than $600. 8. Is 80% of a number more or less than the number? What per cent of a number is equivalent to one half of the number? 9. A man owes 8% of $700. How much does he owe? 10. A man borrowed $800 and agreed to pay 8% of the amount borrowed for the use of it for one year. How much did he pay for the use of $800 for a year? 11. A man borrowed $700 and agreed to pay 8% of tlie amount borrowed for the use of the money each year. How much did he pay for the use of $700 for 1 year? for 2 years? for 3 years? 12. Money paid for the use of money is called interest. 13. A man borrowed $400 and agreed to pay 6% in- terest each year. How much interest did he pay in 1 year? in ^ year ? in 1^ years ? in 2 years? in 2^ years ? 14. Find the interest on $600 for 2 years at 6%. 15. A real estate agent sold a city lot for Mr. Thomas for $1500. He received for his services 5% of the selling price of the lot. How much did he receive for selling the lot? D 168 PERCENTAGE 16. A real estate agent sold a city lot for Mr. Brown for 12000. He received a commission of 6% of the selling price for his services. How much did he receive for sell- ing the lot? How much did Mr. Brown receive for the lot, after paying the commission? 17. A farmer shipped 25 tons of hay to a commission merchant in a city, who sold it for $8 per ton. The com- mission merchant received for his services 2% of the amount of the sale. Find the amount of his commission. 18. A commission merchant received a car of broom- corn containing 8 tons, which he sold at $120 per ton. He received a commission of 5% for selling it. Find the amount of his commission. 19. A farmer shipped 40 tons of hay to a commission merchant who sold it for f 10 per ton. He received a commission of 6%. Find the amount of his commission. How much did the farmer receive for the hay, after de- ducting the commission? 20. A farmer had 160 acres of land. He sold 40% of it. How many acres did he sell? What per cent of the land did he have left? If he received 1 85 per acre for the land sold, how much did he receive for it? 21. A farmer had 320 acres of land. He sold 60% of it for $80 per acre and the remainder for $75 per acre. How much did he receive for the land? 22. Mr. Evans borrowed $250 of Mr. White and paid 7% interest. At the end of the year, how much should Mr. White receive from Mr. Evans, if he received the interest and the money loaned ? 23. Find 50% of the number of children in your school- room. FRACTIONS AS PER CENTS 169 219. Fractions as Per Cents. 1. The unit 1 is equivalent to how many hundredths ? to how many per cent ? 2. What per cent of a number is equivalent to the number ? to J of the number ? to | of the number ? to ^^ of the number ? to 2 times the number ? to 5 times the number ? 3. State how a common fraction may be reduced to a decimal fraction. 4. Change f to per cent. go __ gQ of Change the fraction to a decimal, Model • t^q aq extending the reduction to two deci- mal places. Express hundredths as per cent. Another Method: Since 1 is 100%; f is f of 100%, or 60%. 20% „ Work: W%x| = 60%. 5. Change | to a decimal fraction. .375 is the same as .37 J, which is the same as Zl\ %. 6. State how a common fraction may be changed to hundredths, expressed as a decimal. What is the "mean- ing of per cent ? To change a common fraction to per cent^ divide the numera- tor hy the denominator^ and carry the reduction to two deci- mal places in the quotient. Express the quotient as per cent, 7. Change to per cents : |, |, |, |-, |, ^9_ 1, 8. Change to per cents : 11, if, -j^^ |f if, |f. 9. Change to per cent : 1|. 1.80 = 180 (fo Model: 1-| = |. 5)9.00 la Change to per cents : IJ, If, 1\, IJ, If, 1|. ro PERCENTAGE 11. Memorize the following : 1 = 100% 1 = 60 % T%= 30% i = 12J% 1 = 50% 1=80% T^= 70% f = S7^% i = 25% 2V= 5% A= 90% f=62J% 1 = 75% 2V= 4% iV= 8i% i = 87J % i = 20% 3V= 2% i = S3i % i = 16|% 1 = 40% tV = io% | = 66|% | = 14f% 220. Oral Exercises. 1. Findl6| % of $480. 16f % of $480 may be found by multiplying $480 by .16f, or it may be found by taking ^ of $ 480. Solve by both methods. Which method is the shorter ? 2. Certain per cents of quantities may be found more easily by the use of fractional equivalents. One of these is 33| %. Name others. Solve, using fractional equivalents : * 3. 16| % of 24 hr. 12. 62i % of 640 A. 4. 331 % of 1 15. 13. 87|- % of 320 rd. 5. 142 % of 1 35. 14. 16f % of $ 7.20. 6. 66f % of 36 mi. 15. 14f % of 184. 7. 37| % of 48 yd. 16. 66| % of 60 bu. 8. 25 % of 320 rd. 17. 33 J % of 12.10. 9. 50 % of 11.60. 18. 75 % cf 1400. 10. 8J % of 360 da. 19. 80 % of 1 25. 11. 12| % of 80)2^. 20. 16| % of 30 ft. * This exercise should be supplemented with oral drills until the pupils are able to find the above per cents readily by the use of their fractional equivalents. The fractional equivalents of the above per cents should be used in subsequent exercises. S. DISCOUNT 171 221. Oral Exercises. 1. At a sale the following discounts were advertised. (a) Find the amount of reduction and (5) the selling price : a. 16f % off on carpets marked 90^ per yard. h. 33^ % off on bric-a-brac marked f 6. c, 14f % off on ladies' hats marked $14. d, 66| % off on damaged cloth marked 30^ per yard. e, Ill % off on tables marked $16. /. 371 % off on cloaks marked $16. 2. At what price should the following be marked : a. Cloth that cost 80^ per yard, to make a profit of 2.5%? h. Suits that cost $15, to make a profit of 33^ % ? c. Hats that cost $2.40, to make a profit of 25% ? d. Shoes that cost $3 per pair, to make a profit of 331%? g. Silk that cost $1.50, to make a profit of 50 % ? /. Overcoats that cost $16, to make a profit of 37^%? g. Lace at 60 ^ per yard, to make a profit of 16| % ? 3. What per cent of a number remains after subtract- ing 25% of it? 20%? 40%? 75%? 5%? 66|%? 33i%? 50 % ? 60 % ? 10 % ? 2 % ? 100 % ? 90 % ? 4. What fractional part of a quantity remains after subtracting 50% of it? 20%? 25%? 30%? 40%? 75%? 33|-%?80%?66f%?10%? 16f%? 142%? 5^^? 121%? 100%? 90%? 37J%? 15%? 5. How much remains of $24 after deducting 50% of it? 25%? 75%? 16|%? 331%? 66|-%? 12i%? 37|%? 6. How much remains of $36 after deducting 25% of it? 50%? 75%? 33J%? 66f%? 16|%? 100%? 172 PERCENTAGE 7. Find 1 of 36; 50% of 80; ^ of 90; 83J% of 76; ^ of 200; 75% of 200; 20% of 15 • |- of 60; 40% of 120; ^ of 40 ; 12J% of 72 ; 66^% of 90 ; f of 64. 8. A merchant bought silk at $1.80 per yard and sold it at a profit of 33J % . How much did he make on each yard ? 9. A man bought hay at 1 8 per ton and sold it at a profit of 25 % . What was his profit on each ton ? "What was the selling price per ton ? 10. A grocer bought tea at 40^ per pound and sold it at a profit of 50 %. What was the selling price? 11. A suit of clothes marked $ 20 was sold at a reduc- tion of 20 %. Find the amount of the discount and the selling price of the suit. 12. A wagon that cost $72 was sold at a profit of 16| %. What was the selling price of the wagon ? ""^^^ 13- A merchant advertised a reduction of 25 % on all goods. Find the reduction on suits marked $ 30 ; on shoes marked $4 ; on hats marked $2; on cloth marked 80 ^ per yard ; on rugs marked $ 6. 14. A house owned by Mr. West was rented to Mr. James by a real estate firm for one year at $ 30 per month. If the firm received as commission 10 % of the first month's rent, what was the amount of the commission ? 15. A hardware merchant invested $ 5000 in his busi- ness. He cleared 15 % on the investment in one year. What was the amount cleared during the year ? 16. In the catalogue of a carriage manufacturer a cer- tain carriage was listed at $ 150. It was bought by a re- tail dealer at a discount of 20 % from the list price. How much did the carriage cost the retail dealer ? PERCENTAGE 178 17. A farmer shipped 50 boxes of apples to a commis- sion merchant, who sold them at 90 i per box. The com- mission merchant charged a commission of 5 % for his services. Find the amount of his commission. He paid freight charges amounting to $3.50. How much should he remit to the farmer after deducting for commission and freight ? 18. Mr. A bought a cow for 140 and sold it at a profit of 20 % . What was the selling price of the cow ? 19. Mr. A sold a cow for |- of the cost. He received % 48 for the cow. Find the cost of the cow. 20. Mr. A sold a cow at a profit of \ of the cost. His profit was % 8. Find the cost. 21. A real estate dealer bought a lot for % 600. After five years he sold it at a profit of 100 % of the cost. What per cent of the cost of the lot did he receive for it ? (22 J A merchant's stock of goods valued at % 4500 was damaged by fire. He was obliged to dispose of the goods for ^^\ % of their former value. What fractional part of their value did he receive for them ? How much did he receive for his stock ? 23. A dealer was asked the price of a certain carriage. He replied that he would sell the carriage for % 200 and allow the purchaser 60 days in which to make the pay- ment, or that he would allow a discount of 2 % for cash payment. Find the cash price of the carriage. 24. Mr. James pays \\% taxes on $4000. Find the amount of his tax. 25. A man bought a lot for f 1600. He sold it for % 1800. How much did he gain on the lot ? His gain was what part of the cost \ 174 PERCENTAGE ■f" 26. Frank Thomas borrowed $ 1200 for 1 yr. at 6% interest. How much did he pay for the use of the money ? 222. Finding the number of which a given number is a certain per cent. 1. If 4 % of a sum of money is $12, what is 1 % of it? If 1 % of a sum of money is $3, what is 100 % of it? 2. When 5% of a selling price is f 80, what is 1 % of the selling price ? What is the selling price ? 3. If 8 % of a number is 160, what is 1 % of the num- ber? What is 2 % of the number? What is the number? 4. When 6 % of a number is given, how may 1 % of it be found? How, then, may the number be found? Any number is equivalent to how many per cent of itself? 5. To find a number when a certain per cent of it is given, find what 1 % of it is, then find 100 % of it. By this method find the number of which 24 is 8 % . 6. When the multiplier and the product are given, how may the multiplicand be found? 7. Some number when multiplied by 4 is 216, what is the number? Some number when multiplied by .08 is 24, what is the number? 8. To say that 7 % of a number is 161, is the same as to say that some number when multiplied by .07 gives 161 as a product. The number may be found by divid- ing 161 by .07. To find the number of which a given number is a certain per cent, divide the given number by the given per cerU expressed as a decimal. PERCENTAGE 175 In each of the following, name the multiplier and the product, and state how the multiplicand may be found : 9. 15 % of a number is 60. 12. 8 % of some land is 25.6 A. 10. 9 % of a number is 135. 13. 6 % of some money is $ 108. 11. 18 % of a number is 81. 14. 45 % of a crop is 135 bu. 223. Written Exercises. 1. $11.49 is 12% of what amount? Model • S345 75 Some amount when multiplied by '. — .12 gives $41.49 as a product. The .l^^-lt>4i.4y amount is found by dividing the product (141.49) by the multiplier (.12). 2. 240 mi. is 12 % of how many miles ? 3. 128 tons is 8 % of how many tons? 4. A man paid $32 interest for the use of some money for one year at 8 % . What was the sum borrowed ? 5. A farmer received 40% of a crop as rent for his land. His share of the wheat amounted to 400 bu. in one year. What amount of wheat was raised on the farm in that year? 6. A man received |80 interest on some money which he loaned for a year at 10%. Find the amount of the loan. 7. On a certain day 4 of the pupils in a school were absent. This was 8% of the number enrolled. How many pupils were enrolled in the school? 8. During one season a baseball team lost 14 games, which was 40% of the number of games played. How many games did the team play? O X 176 PERCENTAGE 9. Two men entered into partnership in a retail hard- ware store. One agreed to furnish 40% of the capital and the other 60 % of the capital. The partner who con- tributed 40% of the capital invested $2400. Find the whole amount of the capital. 10. A farmer sold 120 acres of land, which was 30% of his entire farm. How many acres had he before mak- ing the sale? What per cent of his farm did he still own? 224. Oral Exercises. 1. A number is how many times 20% of itself? If 20 % of a number is 8, what is the number ? 2. A number is how many times 33 J % of itself? If 33 J % of a number is 25, what is the number? 3. What part of a number is each of the following per cents of the number : 25 %, 37^ %, 50 %, 62| %, 66|%, 75%, 871%, 16|%, 14f %, 8J%, 12|%, 20%, 33J%? 4. A number is how many times each of the following per cents of itself: 14f%, 75%, 25%, 16|%, 50%, 81%, 20%, 371%, 12|%, 331%, 62J%, 871%, 66|%? 5. If 50 % of the amount of money a boy has is f 12, how much money has he? How much money has he if 25% of his money is |5? if 10% of his money is f 3 ? if 33^% of his money is $8? if 16^ % of his money is 1 10 ? if 66| % of his money is $ 24 ? 6. $ 5 is 25 % of —. $8 is 33^ % of — . 4 mi. is 20 % of — . 6 gal. is 50 % of — . 12 yd. is 75% of — . 160 rd. is 10 % of — . 7. Find the number of which 16 is 25 % ; 30 is 20 % ; 18 is 66^ % ; 40 is 200 % ; 60 is 300 % ; 15 is 37J % ; 25 is 50 % ; 50 is 62^ % ; 70 is 33J % ; 75 is 100 %. 11 r PERCENTAGE 177 8. When 14| % of a number is given, how may the number be found? 14 1 % of a farm is 25 acres. How many acres are there in the farm ? 9. A man sold 45 head of cattle, which was 25% of the number he had. How many head of cattle had he? 10. A merchant sold goods at a discount of 16| % from the cost price and lost $60. What was the cost? 11. In 37 1 % of a farm there are 90 acres. How many acres are there in the farm? 12. A merchant made 12| % on the cost of some goods by selling them at a profit of $6. Find the cost of the goods. Find tlie selling price of the goods. 13. A number is how many times ^ of itself? |- of itself? I of itself? f of itself? 14. If 66|% of a number is 120, what is the number? 15. If 87 J % of a number is 70, what is the number? 16. In a spelling test a boy spelled correctly 30 words, which was 75 % of the number of words in the test. Find the number of words in the test. 17. Mr. A sold a cow at a profit of 25%. His profit amounted to f 10. Find the cost. 18. Mr. A sold a cow at a loss of 25 % of the cost. For what part of the cost did he sell the cow? He re- ceived $ 30 for the cow. Find the cost. 19. A fruit grower planted 120 apple trees. 20 of them died. What per cent of the trees died ? 20. If 12 trees are 25 % of the number planted by a fruit grower, how many trees did he plant ? 21. Eight pupils were absent from school on a certain day, which was 20 % of the pupils enrolled. How many pupils were enrolled in the school ? AK. — 12 D 178 PERCENTAGE 225. Written Exercises. 1. A f iirin was sold for f 6000, which was 25 % more than it cost. Find the cost of the farm. The fractional equivalents of per cents should be used whenever the work can be made easier or shorter by their use. Model A : f of the cost of the farm = $ 6000. $1200 I of the cost of the farm = i of ^jim, or $ 4800. Since the farm was sold for f (125%) of its cost, the cost of the farm is | of the seUing price. Since $6000 is 125% of the cost of f 48 00. the farm, the cost of the farm may be Model B : 1.25)^6000.00 found by dividing 1 6000 by 1.25. (See Sec. 223.) 2. A city lot was sold for $1200, which was 20 % more than it cost. Find the cost of the lot. 3. After increasing his stock 33J%, a merchant found that he had $12,000 invested. Find the amount of his investment before the increase. 4. A sum of money was borrowed for a year at 8 % interest. At the end of the year the money borrowed and the interest amounted to $432. What per cent was this of the sum borrowed ? Find the sum borrowed. 5. If the population of a certain city in 1905 was 81,250, and this was an increase of 25 % over the population in 1895, what was the population in 1895? 6. A dealer sold a carriage for $96, at a loss of 20 %. What per cent of the cost of the carriage did he receive for it ? How much did the carriage cost him ? 7. A firm sold a carriage to a retail dealer for $119, which was at a discount of 15% from the list price of the carriage. Find the list price of the carriage. -^•^ PERCENTAGE 179 226. Finding what per cent one number is of another. 1. Each of the following fractions is equivalent to what per cent ; J, -J, |, |, |, f , -|, -J, |, J, f J, f , |, |, ^j, -^q, -^, r5^ A' 2^' A ^ 2. Each of the following is equivalent to what per cent : ij, li, i{, H, If, If, H, li, H, i^v If. If' If If i| ? 3. 8 is what part of 16 ? What is the ratio of 8 to 16 ? 1^ of a number is what per cent of the number ? 8 is what per cent of 16 ? 4. 24 is what part of 36 ? 24 is what per cent of 36 ? 5. 12 is what per cent of 36 ? of 24 ? of 48 ? of 60 ? 6. Express as a common fraction the ratio of 6 to 8 ; of 20 to 25 ; of 25 to 20 ; of 30 to 35 ; of 40 to 60. 7. Express as hundredths in decimal form the ratio of 3 to 5 ; of 5 to 8 ; of 4 to 5 ; of 24 to 30. 8. 16 is what per cent of 20 ? ^6 = |. Reduce f to a decimal and .80 = 80 % carry the reduction to two decimal Model A : 5)4.00 places in the quotient. .80 is the same as 80%. Some per cent of 20 is 16. 20 is the multiplicand and 16 is the prod- ^- ^" '° uct. The multiplier may be found Model B : 20)16.00 ^^ ^.^.^^^ ^^^ p^^^^^^ ^^g^ ^^ ^^^ multiplicand (20). The multiplier is .80, which is the same as 80 %. To find what per cent one number is of another, express as a common fraction the ratio of the one to the other, and reduce the fraction to a decimal, carrying the reduction to two decimal places in the quotient. Express the result as per cent. 3. 50 is 20. 4. 25 is 50. 5. 18 is 15. 6. 48 is 60. 7. 54 is 27. 8. 240 mi. is 180 mi. 9. 360 bu. is 600 bu. 180 PERCENTAGE 227. Written Exercises. 1. What per cent of 120 mi. is 90 mi. ? 2. 145 is what per cent of $50 ? Find what per cent of : 10. 320 rd. is 80 rd. 11. 640 A. is 120 A. 12. 360 da. is 30 da. 13. 5280 ft. is 1760 ft. 14. 5000 ft. is 1000 ft. 15. 2000 mi. is 6000 mi. 16. 2000 lb. is 750 lb. 17. A man owned 320 A. of land. He sold 80 A. What per cent of his land did he sell ? What per cent of it did he have left? 18. A coal dealer bought 240 tons of coal. He sold 160 tons. What per cent of it did he sell? What per cent of it did he have left ? 228. Oral Exercises. 1. 1^ times a number is what per cent of the number ? If 1^ times a number is 36, what is the number? If 133 J 9^ of a number is 48, what is the number ? 2. What per cent of a number is 1 J times the number ? If 150 % of a number is 12, what is the number ? 3. If If times a number is 20, what is the number ? If 166^% of a number is 60, what is the number? 4. If 6 % of a certain amount is $30, what is 1 % of the amount ? GAIN AND LOSS 181 229. Per Cent of Gain or Loss. 1. Mr. A bought a cow for $40 and sold it at a gain of $S, $8, the gain, is what per cent of f 40, the cost ? 2. Mr. Clark bought a cow for $40 and sold it for $48. Find the gain. The gain is what per cent of the cost ? 3. Mr. Brown bought a horse for $120 and sold it for $100. Find the amount of his loss. His loss is what per cent of the cost of the horse ? 4. When the cost price and the selling price are given, how is the amount of the gain or loss found ? 5. The per cent which the amount of gain or loss is of the cost is called the gain or loss per cent. The gain or loss per cent is always some per cent of the cost. To find the gain or loss per cent, find what per cent the amount of gain or loss is of the cost. 6. A furniture dealer bought some rocking-chairs for $4 each and sold them for $6 each. How much did he make on each chair ? What was his gain per cent ? 7. A bicycle that cost $40 was sold for $30. What was the loss per cent ? 8. A fruit dealer bought berries at 6^ per box and sold them at 10 ^ per box. What was his gain per cent ? 9. A man bought a cow for $30 and sold it for $40. What was his gain per cent ? 10. A newsboy bought papers for Sfi each and sold them for 5 ^ each. What was his gain per cent ? 11. A newsboy bought papers for 1^ each and sold them for 2 ^ each. What was his gain per cent ? 182 PERCENTAGE 230. Oral Exercises. Find the gain or loss per cent Cost Selling Price Cost Gain Loss 1. $10 $15 6. $16 $4 2. $15 $10 7. $12 •4 3. $25 $30 8. $15 $3 4. $30 $25 9. $20 $2 5. $40 $45 10. $25 «5 231. Written Exercises. Find the value of x in each Cost Selling Price Gain Loss Gain% Loss% 1. $80 $100 X X 2. $75 X $25 X 3. X $120 $30 X 4. $50 X $5 X 5. X $60 $20 X 6. X $4.80 $1.20 X 7. $20 X X 8% 8. $36 X X hfo 9. X $80 X 20% 10. X ■ $24 X 20% 11. Ex. 1 above may be stated in the form of a prob- lem, thus : A man bought a horse for $ 80 and sold it for $100. Find the gain or loss per cent. State problems for Exs. 1-10 above. 12. A certain baseball team won 6 games out of 10. What per cent of the games did the team win ? GAIN AND LOSS 188 232. Written Exercises. 1. A real estate agent bought a city lot for 11200 and sold it for §1500. What was the gain per cent ? 2. A merchant disposed of a stock of goods valued at 18000 for $6000. What was the loss per cent? 3. An agent received $40 for selling hay at a com- mission of 5%. Find the selling price of the hay. 4. The interest on a sum of money for one year at 6 % was $ 72. On what amount was interest paid ? 5. A farmer lost 45 % of his wheat crop by fire. His loss amounted to 600 bushels. What was the amount of his entire crop? 6. After suffering a loss of 35% of the value of his stock of goods, a merchant found that the remainder of his stock was worth $13,000. What was the value of his stock before the loss ? 7. A stock of goods valued at $4500 was partly de- stroyed by fire. After the fire the stock was estimated to be worth $3000. What was the per cent of loss ? 8. Mr. Thomas bought a farm for $5250. He rented the farm for $420 a year. His rent amounted to what per cent of his investment ? 9. Mr. Bunker bought a lot for $1500 and built a house on it costing $3000. He rented his property for $300 a year. His rent amounted to what per cent of his investment ? 10. A business block in a city was advertised for sale for $75,000. This block rented for $500 per month. The income from the rent amounted to what per cent of the price asked for the property ? > 184 PERCENTAGE REVIEW 233. Oral Exercises. 1. By selling land at $25 per acre more than it cost him, a farmer gained 20 % of the cost of the land. Find the cost of the land. The gain, or $ 25 per acre, amounts to 20 % of the cost, or | of the cost. Since $ 25 per acre is ^ of the cost, the cost is 5 times ^25 per acre, or $ 125 per acre. 2. By selling a carriage for $15 more than it cost him, a dealer gained 12J % of the cost of the carriage. Find the cost of the carriage. 3. A city lot increased $200 in value, which amounted to an increase of 33 J % of its cost. Find the cost of the lot. 4. A gain of 66^% of the cost amounted to a gain of $120. Find the cost. 5. A horse was sold for $150, which was 120 % (|) of the cost. Find the cost of the horse. 6. By selling an overcoat for $35, a merchant made a profit of 16| % of the cost. What fraction expresses the ratio of the selling price to the cost ? Find the cost. 7. A boy sold a pony for $6 more than it cost him. His profit amounted to 16| % of the cost of the pony. Find the cost and the selling price. 8. After selling 80% of his land, a farmer had what per cent of it left ? After selling 80 % of his land, a farmer had left 40 acres. How many acres had he before making the sale ? 9. By selling a cow for $32, a farmer lost 20% of the cost price. What fraction expresses the ratio of the sell- ing price to the cost ? Find the cost of the cow. Y- REYIEW 186 10. A liveryman made 40 % on the cost of a horse by selling the horse for $140. What fraction expresses the ratio of the selling price to the cost ? Find the cost. 11. By selling a lot for 1640, a dealer lost 20% of the cost price. The selling price was what fraction of the cost of the lot ? Find the cost of the lot. 12. A field of wheat was damaged by floods to the ex- tent of 25 % of the expected yield. The yield amounted to 30 bushels of oats to the acre. This was what fractional part of the expected yield? What was the expected yield ? 13. A watch that cost $25 was sold for 200 % of the cost. Find the selling price of the watch. 14. A painting that cost f 60 was sold for 33^ % less than it cost. It was sold for what fractional part of its cost ? Find the selling price. V" 15. A merchant made a profit of 25 % of the cost of silk by selling it for | . 80 per yard. Find the cost of the silk per yard. 16. A sum of money loaned at 7 % yields f 42 interest each year. Find the sum loaned. 17. $ 20 is what part of % 100 ? A carriage that cost % 100 was sold for $120. It was sold for what per cent of the cost price ? 18. A stove that costs $40 is sold for 1 36. The loss is what part of the cost of the stove ? The loss is what per cent of the cost of the stove ? The selling price is what per cent of the cost price ? 19. A farm that costs % 60 per acre is sold for $ 70 per acre. The gain on each acre is what part of the cost per acre ? The gain is what per cent of the cost ? "b 18« PERCENTAGE 234. Written Exercises. 1. Hay that cost $40 for 5 tons was sold at $ 9 a ton. What was the profit on each ton? the gain per cent? 2. A clothing merchant advertised a reduction of 20 % on all goods. Find the amount of reduction and the sale price of suits marked $35, hats marked §2, suspenders marked 50^, shoes marked $3.50, neckties marked 25^, overcoats marked $20, cuffs marked 20^ per pair, collars marked 2 for 25^. 3. A jeweler sold a watch for $26, which was at a profit of 33^%. Find the cost of the watch. 4. Goods damaged by fire were sold for $2400, which was at a loss of 40 % . What was their original value ? 5. What per cent of his earnings does a man save who earns $ 80 a month and saves $300 each year ? 6. A farmer paid $4000 for a farm and sold it for $ 4200. Find the gain per cent. 7. A man's yearly income from a farm valued at $ 6000 is $ 1500. The income is what per cent of the value of the farm? 8. By selling a carriage for 12 J % more than it cost him, a dealer made a profit of $15. How much did the carriage cost him ? 9. Two men entered into partnership to purchase a boat that cost $300. Each contributed one half of the capital. One of the men sold his share of the boat for $120. Did he gain or lose, and what per cent? 10. A house that was valued at $ 2400 was rented so that the yearly rent amounted to 12 % of the value of the property. What was the monthly rent of the house ? ^ REVIEW 187 11. By selling a cow for $15 more than it cost him, a farmer gained 33^ % of the cost of the cow. Find the cost of the cow. Find the selling price. 12. Tea that was sold at 60 f^ per pound was sold at a profit of 33J%. Find the cost of the tea. 13. A piano dealer sold two pianos for $240 each. On one he made a profit of 20 % and on the other he lost 20 %, How much did each of the pianos cost him ? Did he gain or lose on the two pianos ? 14. How should goods that cost f 1.20 per yard be marked to sell at a profit of 20% ? 25% ? 33^% ? 50% ? 15. Three men bought some land for $ 3600. One fur- nished $1500, another $1200, and the third 1 900. They sold the land for $4200. What per cent of the capital did each furnish? What per cent of the profit should each receive ? What was each man's share of the profit ? 16. A man had $800 in a bank. He drew out first $200 and then $300. What per cent of his money did he draw out ? What per cent was left in the bank ? 17. The salary of a clerk was increased from $60 per month tp $75 per month. What per cent of increase was made in his salary ? The increase would amount to how many dollars in two years ? 18. The population of Los Angeles was 50,300 in 1890 and 102,479 in 1900. The population in 1900 was what per cent of the population in 1890 ? What was the in- crease in population from 1890 to 1900 ? What was the per cent of increase in the ten years ? 19. A newsboy sold 25 papers at 5^ each, which had cost him h^ each. What was the amount of his profit? What was his gain per cen<^ ^ 188 PERCENTAGE 235. Oral Exercises. 1. Find 25%, 50%, and 75% of each of the following: $100; ISO; |120; 40 A.; 36 in.; 2000 lb.; 16 oz.; 12 mo.; 24 hr.; 360 da.; 144 sq. in. 2. What is 33i % and 6G| % of each of the following: $120? $1200? 360 da.? 36 in. ? 180 mi.? 27 ft. ? $1500? 12 mo. ? 60 min. ? 24 yd. ? 3. Find 121%, 371%, 62| %, and 871% of each of the following: 24 hr.; $240; $1; 20 mi.; 2000 1b.; 144 sq. in.; 360 da.; $1200; 640 A.; 72 yd. ; 16 oz.; 216 cu. in.; 320 rd.; $4000; $.48. 4. Express each in per cent : ■^, J, |, |, f , J, f , ^^, |, h h h h h h h h h h ^^ H^ H' If' H^ 21, H, If, 2, 8, 10. 5. Express each as a common fraction in lowest terms: 80%, 50%, 33J%, 25%, 125%, 20%, 120 %, 70%, 40 %, 150%, 66|%, 14f %, 75%, 133J-%, 175%, 180%, 12J%, 140%, 160%, 112J%, 16|%, 90%, 37^%, 87J%, 62J%, 1371%, 187J%, 110%, 130%. 6. Find 125%, 150%, 175%, 112J%, 1371%, 162J%, and 1871% of each of the following: 24 hr.; 320 rd.; 640 A.; 360 da,; 16 oz.; 20001b.; $1200; $4000; $80. 7. Find 133J%, 120%, 166f %, 140%, and 160% of each of the following : $150; 30 da.; 360 da.; $6000; 120 rd.; $250; 60 ff; $1.80. 8. Find the number of which 30 is 33^%; 60 is 25 %; 20 is 40%; 36 is 66^%; 35 is 125%; 120 is 120%; 48 is 37J%; 90 is 150%; 180isl2l%; 180isll2}%; 42 is 175%; 50is200%; 24isl60%; 200is40%; 80ia66J%; 15 is 166f %; 24 is 4%; 30 is 5%; 18 is 6% ; 45 is 9%. 38 is 110%; 12 is 2%; 130 is 200%. PERCENTAGE 189 236. Written Exercises. 1. A sum of money borrowed, together with the in- terest on it for one year at 7%, amounted to $909.50. This was what per cent of the money borrowed? Find the sum borrowed. 2. A boy spelled correctly 45 words in a test of 50 words. What per cent should he receive as his standing in the test ? 3. A girl missed 4 problems in an arithmetic test con- taining 10 problems. What per cent of the problems did she miss ? What per cent did she have correct ? 4. 5 % of a certain amount is $20. Find the amount. 5. Find the amount when 8 % of the amount is $240 ; $80. 6. A farmer had 24 cows and sold 16 of them. What per cent of the cows did he sell ? What per cent did he have left ? 7. A house and lot was advertised for sale for $8000. This property was rented for $32.50 per month. The rent amounted to what per cent of the price asked for the property ? 8. If a man spent 60% of his savings in building a barn and had $400 left, how much had he saved ? 9. A liveryman made 40 % on the cost of a horse by selling it at a profit of $36. Find the cost of the horse. 10. An article that cost a retail merchant $ 14 was sold to a customer at a profit of 14| %. How much did the customer pay for the article ? 11. The total enrollment in a certain school was 180 pupils. On a certain day 150 pupils were present. The number present was what per cent of the enrollment ? 190 PERCENTAGE 237. Oral Exercises. 1. By selling a horse for 20 % more than it cost him a liveryman gained 130. How much did the horse cost him? For how much did he sell it ? 2. What is 2| o/o of $400 ? 31 % of $60 ? 5 % of $1400? 6% of $250? 10% of 2000 lb.? 5^% of $200? 7% of $150? 8% of $2500? 3. What is the difference between 1 % and .1 % ? be- tween ^ of a number and | % of a number ? 4. What is \ % of $200 ? .2 % of $400 ? ^ % of $8000 ? 5. 8 is what part of 24 ? 8 is what per cent of 24 ? 20 is what per cent of 25 ? $20 is what per cent of $30 ? $40 is what per cent of $30 ? 6. A boy missed 1 word in a spelling lesson of 20 words. At the same rate, how many would he have missed in a lesson of 100 words ? 7. After having his salary raised $10 a month, a clerk's yearly salary amounted to $1620. What was his monthly salary before receiving the increase ? 8. A carriage that cost $120 was sold for $80. The sale price was what per cent of the cost ? 9. A man's monthly salary was raised from $60 to $75. What per cent was his salary increased ? 10. 25^ is what per cent of 30^? The cost of 3 bars of soap when bought at 3 bars for 25^ is what per cent of the cost when bought at 10^ a bar ? 11. What per cent of profit is made when articles are bought at 40^ a dozen and sold at 5^ apiece? 12. What per cent of profit is made when articles are bought at 10^ a dozen and sold at 2 for 5^? PEROENTAQE 191 238. Oral Exercises. 1. If I of the value of a piece of property is $1500, what is the value of the property ? 2. If I of a man's yearly salary is f 1200, what is his yearly salary ? 3. A clerk saved $40 a month, which was | of his monthly salary. What was his monthly salary ? 4. What part of his income does a man save who saves $60 a month from an income of $1200 a year ? 5. Frank has a certain sum of money and James has I as much. They both together have 60^. How much money has each? The moDey of both together is how many thirds of Frank's money ? 6. Two boys took a piece of work to do for $6. One boy worked twice as many hours as the other boy. How much should each receive ? 7. A man gave Henry $3 as many times as he gave Walter $4. He gave $14 to the two boys. How much did each receive ? 8. Separate $45 into two amounts in the ratio of 5 to 4 ; $36 into three parts in the ratio of 2, 3, and 4. 9. In a school of 120 pupils there were 5 girls to every 3 boys. Find the number of boys and girls. 10. Rob, Fred, and Ada together received $2.40 from their father. For every 15^ that Rob received Fred re- ceived 10^, and Ada 5^. How much did each receive ? 11. A newsboy wished to make an estimate of his yearly earnings, so he kept account of his earnings for 3 weeks and found that he earned $6 the first week, $4 the second week, and $5 the third week. At the same rate, how much would he earn in a year ? 192 PERCENTAGE 2f39. Commission.* A person who transacts business for another frequently receives as his pay a certain rate per cent of the amount involved in the transaction. This is known as his com- mission. One who buys or sells for another on commis- sion is called a commission merchant, a broker, or an agent. 240. Written Exercises. 1. Find 2% of 12400. 2. A commission merchant sold $2400 worth of hay for a farmer and charged 2 % for his services. Find the amount of his commission. How much should he remit to the farmer, after deducting his commission and $300 for freight charges and $150 for cartage ? 3. An agent received $16 as his commission for sell- ing a bill of goods at a commission of 5%. Find the amount of his sales. 4. A farmer shipped 40 sacks of potatoes to a commis- sion merchant, who sold them at 95^ a sack. After de- ducting his commission of 5%, how much should he remit to the farmer ? 5. A merchant's profits averaged 15%. His total sales for January, 1906, amounted to $13,800. Find the cost of the goods sold. Find the profits for the month. 6. A farmer shipped 18 tons of hay to a commission merchant, who sold it at $9.50 per ton. How much did the merchant remit to the farmer, after deducting his com- mission of 5 % and freight and cartage charges amounting to $1.75 per ton? *For a more extended treatment of Commission, see Appendix, pp. 262-264. PERCENTAGE 193 7. Find the net proceeds of the sale of 860 lb. of butter at 18^ per pound, commission 6%. 8. A real estate agent received a commission of 5% for selling a city lot. Find the sale price of the lot, if the agent's commission amounted to $62.50. 9. If the salary of a traveling salesman is $20 a week and a commission of l^ % on the amount of his sales, how much does he earn in a week in which his sales amount to i 2254. 75? 10. A carriage dealer offered to sell a certain carriage for $250 on two months' time, or to allow a discount of 2 % for cash. Find the cash price of the carriage. 11. If a collector retains 10 % of the amount of a cer- tain bill for collecting it, what per cent of the amount of the bill does the creditor receive ? A collector remitted to a creditor $126 as the net proceeds of a collection, after retaining his commission of 10%. Find the amount of the bill collected. 12. After deducting his commission of 4%, an agent remitted $79.20 to a shipper. Find the amount of the sales. 13. The amount received by a shipper, after a commis- sion of 5 % has been deducted, is what per cent of the amount of the sales? A shipper received $60.80 as the net returns of a sale of some potatoes, after paying a com- mission of 5 % . Find the amount of the sale. 14. A dairyman shipped 1250 lb. of butter to a com- mission merchant, who sold it at 22^ per pound. If the cost of shipping was $2.40 and the cartage amounted to $1.75, how much did the shipment net the dairyman, after paying a commission of 4 % ? AB. — IS 194 PERCENTAGE 241. Oral Exercises. 1. If a boy sells 1 newspaper for what 2 papers cost him, what per cent of profit does he make? 2. If a baker sells 2 pies for what 3 pies cost him, what per cent of profit does he make? 3. A merchant sold 5 yd. of cloth for what 6 yd. cost him. What per cent of profit did he make? 4. What per cent of profit does a grocer make who buys canned tomatoes at the rate of 3 cans for 25^ and sells them at the rate of 2 cans for 25/? 5. A dealer marked his goods so that he would make 30 % profit on them. In order to dispose of his goods, he was obliged to sell them at a discount of 10 % from the marked price. What per cent of profit did he make? 6. A collector was allowed a commission of 20% on a bill of $80. What amount did the creditor receive ? 7. A dealer marked his goods at 20 % above cost. In order to close out his stock, he was obliged to sell the goods at a discount of 25%. Did he gain or lose, and what per cent? 8. There are 4 boys and 8 girls in a class in arithmetic. What per cent of the pupils in the class are girls? 9. The enrollment of pupils in a class was 25 in 1905 and 30 in 1906. What was the per cent of increase ? 10. On a certain day a boy missed 3 words out of 15 in spelling. What was the per cent of words correctly spelled? 11. An agent received a commission of 5 % for selling a lot for $1500. Find the amount of his commission. 12. An agent's commission of 5% for selling a city lot amounted to $60. For what amount did he sell the lot? INSURANCE 195 242. Insurance.* i. Owners of buildings, merchandise, etc., generally protect themselves against loss by fire by having such property insured. Insurance of property against loss by fire is called fire insurance, against loss by sea marine insurance. What is life insurance? accident insurance? Name other forms of insurance. 2. The written agreement between the insurance com- pany and the person protected is called a policy. Examine a fire insurance policy. The amount paid for insurance is called the premium. The rates of insurance are expressed as a rate per cent on the face of the policy, or as a speci- fied sum for each |100, or for each 11000, of the face of the policy. 243. Written Exercises. 1. Mr. Wilson insured his store for $6000. The in- surance cost him 1J%. Find the premium. 2. Mrs. Hardy insured her house, valued at $8000, for I of its value. Find the amount of the face of the policy. The insurance cost her $1.40 on each $100 and extended for three years. How much did the insurance cost her? 3. If 90 % of a sum is $28.80, what is the sum ? 4. For what price was a city lot sold if the agent's commission of 5% amounted to $87.50? How much did the owner receive ? Find the premium on the following amounts of in- surance at the rates given : 5. $4000 at 1\%. 8. $5600 at $1.20 per $100. 6. $2400 at If %. 9. $1400 at $1.35 per $100. 7. $12,000 at 11%. 10. $4250 at $1.80 per $100. * For a more extended treatment of Insurance, see Appendix, pp. 278-283. 196 PERCENTAGE 11. Mr. Rogers built a house that cost him $4500. It cost him 11800 additional to furnish it. To protect him- self against the complete loss of his property by fire, he insured his house for $3000 and his household goods for $1200. The insurance for three years cost him !{% of the face of the policy. a. Find the cost of the insurance. h. If the house and contents were destroyed by fire, how much insurance would he receive ? c. What would be the amount of his loss, including the amount paid for insurance ? d. If the house were damaged to the extent of $400, how much would he receive? 12. Two men own a store in partnership. One has $16,000 invested in it, and the other has $10,000. What part of the store does each own? If the store were sold for $39,000, what part of this amount would each re- ceive? How much would each receive? If the store were damaged by fire to the extent of $13,000, how much would each lose? 13. A hotel valued at $80,000 was insured for $50,000 in one company and for $25,000 in a second company. How much would each company be liable for (a) if the hotel were totally destroyed ; (h) if it were damaged to the extent of $12,000? of $30,000? 14. What was the amount of commission received by an architect who charged a commission of 5 % for drawing the plans and supervising the construction of a house that cost $4500, exclusive of the architect's fees? 15. Write five insurance problems based on conditions in your community. PERCENTAGE 197 244. Oral Exercises. Express the part and the per cent the first quantity is of the second : 1. 130, 140. 6. 12.50, $3. 11. 11200, 11500. 2. $40, 150. 7. 80 A., 160 A. 12. 45 T., 60 T. 3. 20 mi., 25 mi. 8. 10 yd., 16 yd. 13. 80 A., 320 A. 4. 40 ft., 60 ft. 9. $4, 124. 14. 2000 ft., 2200 ft. 5. 1100,1120, 10. 60 lb., 100 lb. 15. |12, |200. 16. Express the ratio of the second quantity to the first in each of the above in the form of a fraction in lowest terms and in per cent. 245. Oral Exercises. 1. A collector's commission of. 5% amounted to $30. Find the amount of the bill collected. 2. After deducting his commission of 20%, a collector remitted f 24 to the creditor. Find the amount of the bill collected. 3. Mr. Wright has $4500 out on interest at 6 %. His annual taxes on the money are $20. What is his net annual income from the $4500? 4. The yield from a certain field was 30 bu. of oats to the acre in 1904 and 40 bu. to the acre in 1906. What was the per cent of increase in the yield in 1906 over the yield in 1904? 5. The enrollment in a certain school in 1906 was 36 pupils, which was an increase of 20 % over 1905. What was the number of pupils enrolled in 1905 ? 6. 40% of the pupils in a certain school are boys. There are 24 girls in the school. How many pupils are there in the school? 198 PERCENTAGE 246. Taxes. * l. What are some of the expenses of a city government? of a state government? of the national gov- ernment? The money necessary for the maintenance of state and local governments is derived mainly from taxes levied upon persons, property, and business. All movable property, such as household goods, money, cattle, ships, etc., is called personal property. Immovable property, such as lands, buildings, mines, etc., is called real estate, or real property. Both forms of property are subject to taxation. 2. For the purpose of taxation, the value of all taxable property is estimated by a public officer called an assessor. Property is not generally assessed at its full value. 3. The rate of taxation is expressed as a per cent on the assessed valuation, or as a specified sum on each $1, or on each ^100, of assessed valuation. Thus, a tax rate of 1^ % may be stated as a tax of 1|^ (on each |1), or of $1.50 (on each $100). 4. The national government is supported mainly by rev- enues derived from taxes levied upon goods imported from other countries, called duties, or customs, and from internal revenues, which consist chiefly of taxes levied upon the manufacture of liquors and tobacco products. Some imports are admitted without duty. These are said to be on the free list. Nearly all imports are subject either to an ad valorem or a specific duty^ or both. 5. An ad valorem duty is a tax of a certain rate per cent on the cost of the goods. 6. A specific duty is a tax of a specified amount per pound, yard, etc., without reference to the cost of the goods. 7. Customhouses have been established at all ports wliere vessels are authorized to land cargoes. The revenues are collected by federal officers stationed at ports of entry. * For a more extended discussion of Taxes, see Appendix, pp. 269-273. TAXES 199 247. Written Exercises. 1. A man had 16000. He invested 11500 in a city Ipt. What per cent of his money did he invest ? 2. A certain city had an assessed valuation of #8,000,000. The amount needed to defray the expenses of the city for a year was estimated at 1100,000. The amount needed for expenses was what per cent of the assessed valuation? 3. The assessed valuation of a certain city is $ 12,000,000 and the amount to be raised by taxation is 1180,000. What rate of taxation is necessary in order to raise this amount? 4. What is the amount of an agent's commission for selling real estate for $150,000 at a commission of 1^ % ? 5. What is the amount of a man's taxes on property assessed at 16000 if the tax rate is 11.20 on each 1 100? 6. A real estate agent received #84 for selling a piece of property at a commission of 2%. Find the selling price of the property. 7. The assessed valuation of a certain farm is -$3600. This is 40 % less than the actual value of the farm. Find the value of the farm. 8. What per cent on his investment did a boy make who bought a pony for $40 and sold him for $50? 9. The assessed valuation of the property in a county 'm $42,000,000, and $672,000 is to be raised by taxation. Express the rate of taxation in three ways. 10. Find the rate of taxation on: a. Valuation, $450,000; taxes, $6000. h. Valuation, $275,000; taxes, $2475. c. Valuation, $360,000; taxes, $6300. 200 PERCENTAGE 11. What rate of commission was charged by a col- lector who charged $15 for collecting a debt of $225 ? 12. The premium on an insurance of $4500 is $60. What is the rate of premium ? 13. The premium received for insuring a store at IJ % was $105. What was the amount of insurance? 14. At the rate of 1| %, how much is the tax on prop- erty assessed at $4500 ? 15. When the valuation and the rate of taxation are given, how may the tax be found ? Find the tax on : a, $12,000 at 1|%; at .8% ; at 1.4%; atlf%. h. $10,000 at $1.20 per $100; at $.80 per $100. c. $6000 at 8 mills on a dollar ; at 7.6 mills on a dollar. d. $3600 at $.007 on a dollar ; at $.014 on a dollar. 16. If a broker received a commission of IJ % for his services, find the amount of his brokerage for buying 2450 cwt. of wheat at $1.34 per cwt. If this wheat was bought for a milling company, what was the total cost of ♦^^he wheat to the company ? the cost per cwt. ? 17. If a traveling salesman sells on an average $400 worth of goods every week, which of the following offers should he accept from the wholesale firm : (a) a salary of $25 per week and expenses; (5) a salary of $15 a week and expenses, and a commission of 5 % on all sales over $300 per week; (c) or his expenses and a commis- sion of 8 % on all sales ? 18. When the tax and the rate of taxation are given, how may the valuation be found? Find the valuation: a. Tax, $60; rate of taxation, 1J%. h. Tax, $120 ; rate of taxation, 8 mills on a dollar. (?. Tax, $96 ; rate of taxation, $1.20 per $100. INSURANCE AND TAXES 201 19. What was the amount of insurance if the premium received for insuring a house at $1.40 per $100 was $49 ? 20. Furniture yalued at $600 was insured for $400. For what part of its value was the furniture insured? The premium paid for 3 years was $8. What was the rate of premium paid ? 21. The pupils of the advanced arithmetic class in a certain school were told that the school building was insured for | of its estimated value, and that the annual premium at 1 % was $75. They were asked to find the estimated value of the building. One pupil found the value to be $5625. Was his answer correct ? 22. A certain school district voted $12,000 to erect a new schoolhouse. The assessed valuation of the property in the district was $600,000. Find the rate of taxation. 23. If a tax collector in a certain city receives a com- mission of 2 % for collecting taxes, what per cent of the amount collected does the city receive ? Find the amount of taxes that must be levied in order that a city may receive $19,600, after allowing a collector a commission of 2% for collecting. 119,600 = 98% of the sum levied. 24. Property worth $9000 was assessed at $6000. The rate was $1.50 for each $100 of assessed valuation. Had this property been assessed at its full value, what rate of taxation would have yielded the same amount of taxes ? 25. Examine a tax receipt. Is a separate entry made for taxes on personal property and on real property? Is there an entry for school taxes ? 26. Make and solve five problems in taxes, using when possible the actual rates in your county or city. 202 PERCENTAGE 248. Customs and Duties. The following rates of cus- toms are from the schedule adopted by Congress in 1897, commonly known as the Dingley Tariff : Newspapers, periodicals, free. Hay, % 4 per ton. Coffee, free. Carpets (velvet), 60^ per sq. yd. Musical instruments, 45 % and 40% ad. val. ad. val. Table knives, 16^ each and 15% Potatoes, 25 f per bu. ad. val. Tea, free. Paintings, 20% ad. val. 249. Written Exercises. 1. Find the duty on 60 sq. yd. of velvet carpet worth $1.50 per square yard. 2. What is the duty on 45 tons of hay ? 3. What is the duty on a violin worth 180 ? 4. A painting valued at 12500 was purchased in Italy and brought to the United States. Find the amount of customs on it. 5. Find the amount of the duty on 6 doz. table knives worth $1.80 per dozen. 6. Why are tea and coffee on the free list, while a duty of 25^ per bushel is placed upon potatoes ? 7. What were the net proceeds of an auction sale, if the sales amounted to 11215.40, and the auctioneer received a commission of 10 % ? 8. After deducting his commission of 5% and $12.50 for freight and cartage, a commission merchant remitted $633.50 to the shipper. Find the amount of the sales. 9. A city lot that cost $1600 was sold for $1800. Find the gain per cent. PERCENTAGE 203 250, Oral Exercises. 1. What is the price of coal a ton when it is selling at i.25 a hundredweight ? 2. When hay is selling at $12 a ton, what is its price per hundredweight ? 3. If I of the length of a certain bridge is 240 ft., how long is the bridge ? 4. If the interest for one year at 5% is $80, what is the sum on which the interest is paid ? 5. A boy shot 10 times at a target and hit it 8 times. Express as per cent the ratio of the number of accurate shots to the number of shots taken. 6. On a certain day a girl missed 3 out of 12 words in a spelling lesson. What per cent of the words did she spell correctly ? 7. A baseball team played 8 games and lost 3 of them. What per cent of the games played did the team win ? 8. A girl was absent from school 4 days and present 16 days during a school month. What per cent of the time was she present ? 9. A man paid a tax of 1J% on property valued at $4000. Find the amount of his tax. 10. A commission merchant received $20 for selling $1000 worth of produce. What was his rate of com- mission ? 11. If a spelling lesson consists of 25 words, what per cent of the lesson is each word ? What per cent of the words does a boy spell correctly who misspells 4 words ? 12. A boy caught a ball 6 times and missed it 2 times. The number of times he caught the ball is what per cent of the number of chances he had to catch it ? 204 PERCENTAGE 251. Trade Discount.* i. Manufacturers and wholesale dealers issue catalogues describing articles sold by them and giving their list prices, A discount from the list price is made to retail dealers and sometimes to other customers, particularly when goods are purchased in large quantities. Such a discount is generally known as trade discount, or commercial discount. 2. Several successive discounts are sometimes allowed. Thus, an article may be sold subject to discounts of 25 %, 10 %, and 5 % ; that is, a discount of 25 % is made from the list price, and a second discount of 10 % is made from the price after making the discount of 25 %, and a third discount of 5 % is made on the price after the two dis- counts have been made. A separate cash discount is usually allowed when payment is made within a specified time after the purchase of the goods. 252. Written Exercises. 1. Find the net cash price to a retail hardware mer- chant of a stove listed at $45, trade discounts of 20 % and 10 (fo^ and a cash discount of 5 %. Model : $ 45, list price. ^^^ ^'^^ discount is 20% ^ n . 1. . of $45, or $9. The price _9, first discount. ^^^^^ ^^^^.^^ ^^^^ ^.^^^^^^ % 36, second price. is $ 45 - $ 9, or $ 36. The 3.6 0, second discount, second discount is 10% of $32.40, third price. ^36, or $3.G0. The price 1.62, cash discount. ^^^^ ^^^"^g *^« ««^^"^ discount is $ 36 - 1 3.60, $30.78, net price. " d^on a^k., futif /6, 1906. ^ioctAf dai^ after date, '^^, o-v &ttk&^ o-^ loQy, promise to pay to S^uin^ S^. /CenA^cyyv, or order, Q^Coc kicr^iecC dattaiO', with interest thereon at ^% per annum from date until paid. Value received. W^cult&v f. BityvkeA,, fo-kn, R. Jbav^^e^. 3. Each maker of a joint note is liable for its payment in full. 4. The following should appear in a note : a. The time and place where the note was executed. This is usually written at the top and toward the right PROMISSORY NOTES 215 h. The sum to be paid, including the rate of interest, if any is paid. The face of the note is usually written in figures at the top and toward the left and in words in the body of the note. c. The signature of the maker or makers. d. The time of payment. When no time of payment is specified, the note is payable on demand. e. Notes usually contain the words value received. Answer each concerning the two notes in Sec. 262. 5. When was the note executed ? Where ? 6. When is the note payable ? 7. What is its face ? 8. Who is the maker ? the payee ? 9. Is the note negotiable ? 10. When a note becomes due it is said to mature. In some states a note matures three days after the time specified in the note. The three additional days are called days of grace. Days of grace have been abolished in most states, and are not computed in the answers given in this book. 11. Each state has fixed its own legal rate of interest, which is the rate allowed on claims drawing interest when no rate of interest has been otherwise arranged. What is the legal rate in the state in which you live ? 12. Many states have fixed a maximum rate of interest that can be collected by agreement. A higher rate than that authorized by law is called usury. Is there a law against usury in the state in which you live ? 13. Write a demand note ; a time note ; a joint note. 216 PERCENTAGE 263. Partial Payments. — Mercantile Rule.* i. On Jan. 1, 1906, James Smith of Los Angeles, Cal., borrowed of Frank Adams $ 1000 for one year at 6 %, giving his note for this amount. Write the note. When the loan was made, it was agreed that if James Smith made any payments on the note before its maturity, he would be credited with each partial payment and would be credited with the same rate of interest as he was paying, 6 %, from the date of each payment until the time of final settlement. 2. On July 1, 1906, James Smith paid Frank Adams $400. Indorse this payment by writing " July 1, 1906, f 400 " on the back of the note. Final settlement was made Jan. 1, 1907. Under these conditions James Smith had the use of $1000 bor- rowed of Frank Adams for 1 yr., and Frank Adams had the use ox $400 paid by James Smith for 6 mo. (from July 1, 1906, to Jan. 1, 1907). At the time of final settlement James Smith owed Frank Adams $ 1000, with interest for 1 yr., at 6%, or $ 1060 ; and Frank Adams owed James Smith $ 400, with interest for 6 mo. at 6 %, or $ 412. In settling the note, James Smith paid Frank Adams $1060 - $412, or $648. Notes and accounts which do not run for more than one year, on which partial payments are made, are often settled by business men as above. Mercantile Rule. 1. Find the amount of the faee of the note at the time of settlement, 2. Find the amount of each payment from the date of payment to the date of settlement, 3. Subtract the sum of the amounts of the payments from the amount of the face of the note, 3. Write a note, naming some pupil as payee and your- self as payer. Make three partial payments and have them indorsed to your credit. Settle the note. • For the United States Rule of Partial Payments, see Appendix, p. 266. COMPOUND INTEREST 217 264. Compound Interest.* 1. When the unpaid interest is added to the principal, as it becomes due, to form a new principal on which interest is computed, the interest is called compound interest. Interest may be added to the principal annually, semiannually, quarterly, etc., according to agreement. The payment of compound interest cannot usually be enforced by law, but if the debtor is willing to pay compound interest, it may be collected without violating the law against usury. 2. Savings banks generally pay interest semiannually. When it is not collected by the depositor, it is added to his deposit and he is paid compound interest. 3. If interest is collected when due and reinvested at once at the same rate of interest, the result is the same as when compound interest is received. 4. Find the amount of |600 for 2 yr. 6 mo. at 8%, interest compounded annually. Find the difference be- tween the compound interest and the simple interest. Model : f 600 = principal for first year. 48 = interest for first year. $648 = amount, or principal, for second year. 51.84 — interest for second year. $699.84 = amount, or principal, for third year. 27.99 = interest for 6 mo. $727.83 = amount for 2 yr. 6 mo. at 8%. Compound interest = $727.83 - $600 = $127.83. Siniple interest = 120. Difference = $ 7.83. 5. If a man invests $1000 at compound interest at 6% when he is 30 years of age and keeps it earning at the same rate until he is 50 years of age, what will be the amount of the f 1000 at that time ? (Use table, p. 320.) * For table of compound interest, see Appendix, p. 320. 218 PERCENTAGE 265. Bank Discount and Proceeds. 1. Banks usually collect interest in advance on sums loaned. Thus, if George White borrows $100 at a bank for 60 da. at 6%, his note will be made out for flOO, and the bank will deduct from this amount the in- terest on $100 for 60 da. at 6%, or $1. Mr. White wiU receive $99. At the end of 60 da. he will pay the bank the face of the note, or $100. 2. If interest is collected in advance, how much money will a person receive at a bank on a note for $2000 for 60 da., if the bank charges 8 % interest ? 3. On April 8 J. J. Dow bought $ 600 worth of goods of D. C. Brown, on 90 da. time, giving his note for the amount without interest. On the same day D. C. Brown sold the note to a bank, the bank deducting 6 % interest for the term of the note (90 da.). Find the amount received for the note by D. C. Brown. 4. Interest paid in advance upon the amount due on a note at its maturity is called bank discount. Bank dis- count is computed from the date of the purchase of the note by the bank to the legal date of maturity. Some banks include both the day of purchase and the day of maturity in the discount period. "When days of grace are allowed, these are included in the discount period. 5. The sum paid for a note when sold is called the proceeds of the note. The proceeds on a note is the amount due at maturity, less the bank discount. 6. C. W. Smith held a note against R. E. Orr for $4000 for 60 da. without interest. After 20 da., he sold it to a bank at a discount of 6 % ; that is, the bank deducted 6 % int. on the note for the 40 da. between its purchase and expiration. Find the bank discount and the proceeds. BANK DISCOUNT 219 7. On April 24, 1906, James J. Hall sold a horse to G. M. Bruce for f 150, taking in payment his note for 1 year with interest at 6 % . Find the amount of the note at maturity. 8. Mr. Hall (Ex. 7) needed money, so he sold the note to a bank on the same day, the bank discounting it at 6 % . How much did Mr. Hall receive for the note ? Model : Face of note = $ 150 Interest for 1 yr. at 6 % = 9 Amount at maturity = $ 159. Discount 1 yr. at 6 % = 9.54 (computed on $ 159) Proceeds = $ 149.46 9. If the note (Probs. T and 8) had been discounted at 8 % instead of 6 % , what amount would Mr. Hall have received ? 10. If the note (Probs. 7 and 8) had been discounted three months after date of issue, or on July 24, 1906, the bank would have deducted interest on the amount due at maturity (1 159) for the exact number of days from July 24, 1906 to April 24, 1907 (7 da. 4- 31 da. + 30 da. + 31 da. 4- 30 da. + 31 da. + 31 da. + 28 da. + 31 da. + 24 da.), or for 274 da. Find the amount which Mr. Hall would have received. 11. A 90-da. note for $ 500, without grace, dated Aug. 5, 1905, with interest at 5%, was discounted at a bank on Aug. 25 at 6%. Find the day of maturity, the amount at maturity, the bank discount, and the proceeds. 12. A man borrowed flOOO of a bank for 1 yr. at 6%, paying interest in advance. 6% interest in advance on $1000 is equivalent to what rate paid at the end of the year ? MO PERCENTAGE 266. Review. 1. During a certain school month a boy worked 209 problems, of which 194 were correct. Find the per cent of correct work. 2. A baseball team won 43 games and lost 15 games one season. Find the per cent of games won. 3. The rent of a house was raised from $ 30 a month to I 35 ; this was an increase of what per cent ? 4. A person bought a house for $6000. The taxes, insurance, repairs, and other expenses connected with the property amounted to $120 a year. For how much a month must the property be rented to net 6% on the investment ? 5. A man built two flats costing him $ 5000 on a lot which cost $2000. He rented one of the flats for $40 a month and the other for $35. The expenses connected with the property amounted to $200 a year. The net income amounted to what per cent on the investment ? 6. An electric light meter registered 80,000 watt hours on Oct. 9, and 106,000 watt hours on Nov. 9. Find the amount of the bill for the month at 9^ for each 1000 watt hours. 7. A gas meter registered 29,800 cu. ft. on April 24, and 31,800 cu. ft. oh May 24. Find the amount of the bill for the month at $ .90 per 1000 cu. ft. 8. The population of a certain city was 47,235 in 1900, and 60,624 in 1910. Find the increase per cent. 9. At 66 ff a sack (100 lb.), what is the price of coal per ton ? 10. Find the interest on $2800 from June 8 to Jan. 16 at 6 % per annum. PART IV FORMS AND MEASUREMENTS* 267. 1. Lines are vertical j , horizontal — , and oblique \ /. 2. These are right angles. | | [ | 3. A rectangle has four right angles. QJ | [ 4. These are right triangles. \ yx\ 5. These are acute angles. /\ v 6. These are acute-angled triangles. ^ \J 7. These are obtuse angles. ^-^"-\ ^^..^ 8. These are obtuse-angled triangles. ^-^^ ^^^Z^ 9. Perpendicular (p) means at right . |^^ angles to. -1^ l^ 10. These figures have a base (5) ^i i^ y^a A and an altitude (a). ^-t "^""^ b" 11. These lines are parallel. 12. These are quadrilaterals. V~\ /~~J / \ ^ | 13. These quadrilaterals are parallelograms. I I I 7 13 ^Pl^] 14. These are rectangular prisms. ♦ With complete reviews. 221 222 FORMS AND MEASUREMENTS 15. These are triangular prisms. 16. Circumference, diameter, and radius belong to the circle. 17. These are cylinders. B ^ circle. 268. Relation of Forms. Study the relation of these forms L Right Angle. Rectanqle. Rectangular Prism. L [X Right Angle. Right Tbl^u^gle. Right Triangular Prism. A A Acute Angle. Acute Triangle. Acute Triangular Prism. Obtuse Angle. Obtuse Triangle. Obtuse Triangular Prism. O Cl&CLiB. m Sphere. Ctlindbb. LINES AND ANGLES 223 269. Lines. 1. Lines that extend in the same direction and are the same distance apart are called parallel lines. — 2. Suspend a weight by a string. p^^allel Lines. When the weight is at rest, the line represented by the string is called a vertical line. 3. The surface of the water in a tank or a pond is said to be level, or horizontal. A slanting line is called an oblique line. A vertical line is represented on a page by a line parallel to the sides, and a horizontal line by a line parallel to the top and bottom. 4. Hold your pencil in a vertical position ; in a hori- zontal position ; in an oblique position, 5. Point to surfaces in the schoolroom that are hori- zontal, vertical, oblique, parallel. 6. Draw two vertical parallel lines on the blackboard; two horizontal parallel lines; two oblique parallel lines. 270. Right Angles. 1. Two lines that meet form an angle, Z. When two lines form a square corner, the angle between them is called a right angle. 2. Draw four right angles. . . . — — . 3. Point to surfaces in your school- — - — ' ' ^ room that meet at right angles. 4. Two lines that form a right angle are said to be per- pendicular to each other. Draw perpen- i \_p_ p/ dicular lines. — ■— ' ^ Perpendiculab 5. Point to lines or surfaces in the Lines. schoolroom that are perpendicular to each other. 224 FORMS AND MEASUREMENTS n n Square. Ilurizontal. Vertical. Bectanolbs. 271. Rectangles. 1. A figure whose angles are all right angles is called a rectangle. Reel means right. Rectangle means having right angles. 2. A rectangle whose sides are all the same length is called a square. A rectangle having two opposite sides longer than the other two opposite sides is sometimes called an oblong. 3. Draw a square ; a vertical rectangle ; a horizontal rectangle. Point to surfaces in your schoolroom that are rectangles. Are any of these squares? 4. How many sides has a rectangle ? How many angles has a rectangle? Are the sides of a rectangle parallel? Name surfaces not in your schoolroom that are rectangles. 272. 1. Draw a square whose side is 1 foot. This is called a square foot. Draw and name a square whose side is 1 inch. 2. Draw a square yard. Divide it into square feet. How many square feet are there in a square yard? 3. Divide a square foot into square inches. How many square inches are there in a square foot ? 4. A square 16 J feet each way is called a square rod. Mark out a square rod on the school grounds. 5. Draw a square whose side is 2 inches. Divide it into square inches. How many are there ? 6. Draw 3 inch squares and a 3-inch square. Compare the size of a 3-inch square and 3 square inches. 7. The number of square units in a surface is called its area. RECTANGLES 225 273. Areas of Rectangles. 1. Repeat the Table of Linear Measure (§ 99). Re- view p. 81. Repeat the Table of Square Measure (§ 101). 2. Using the scale J in. = 1 rd., make a drawing to represent a rectangle 16 rd. long and 10 rd. wide. Ex- press the area of the rectangle in acres. 3. Find the area of a flower bed that is 12 ft. 9 in. long and 8 ft. 4 in. wide. 4. How many acres are there in a tract of land 80 rd. by 80 rd. ? 5. A farm that contains 80 A. is J mi. wide. How long is the farm ? 6. Find the number of square yards of surface in the walls and ceiling of your schoolroom, deducting for the doors and windows. 7. Find the value of a field 40 rd. long and 20 rd. wide at f 85 an acre. 274. Written Exercises. 1. Reduce 2 yd. 2 ft. 7 in. to inches. 2. Find the sum of 8 ft. 6 in., 7 ft. 4 in., 9 ft. 11 in., and 6 ft. 5 in. 3. Find the perimeter of a rectangle whose length is 24 ft. 8 in. and whose width is 15 ft. 10 in. 4. How many rods of fence are required to inclose a rectangular 20-acre field whose length is 80 rd. ? 5. How many bundles of shingles are necessary to shingle a surface 50 ft. by 16 ft., if the shingles are laid 4 in. to the weather ? 6. How many yards of carpet are necessary to cover a floor 16 ft. by 12 ft., if the carpet is 27 in. in width ? AB. — 15 226 FORMS AND MEASUREMENTS 275. Right Triangles. 1. Draw a right angle. The point at which the lines meet is called the vertex of the angle. 2. Draw a rectangle. Draw a straight line joining the vertices of the opposite angles of the rec- tangle. This line is called the diagonal of the rectangle. The diagonal divides the rectangle into two equal triangles. 3. A figure having three angles is called a triangle. Tri means three. Triangle means having three angles. 4. A triangle having one right angle is called a right triangle. 276. Area of Right Triangles. 1. The base of a figure is the side on which it is as- 6 sumed to rest, and the altitude is the perpendicular distance between the top a and the base, or the base produced. Consider the length of a rectangle as its and the width as its altitude. 6 RiaHT Triangles. 2. The area of a right triangle is what part of the area of a rectangle having the same base and altitude? The area of a right triangle is equal to one half the prod- uct of its base and altitude. The work is sometimes shorter if the altitude is multiplied by one half the base, or if the base is multiplied by one half the altitude. Dimensions must be expressed in like units. By the product of the lines is meant the product of the numbers denoting them. 3. Find the area of right triangles of the following dimensions: base, 12 in., altitude, 8 in. ; base, 6J ft., alti- tude, 9 J ft. ; base, 40 rd., altitude, 80 rd. Parallelogram. PARALLELOGRAMS 227 277. Parallelograms. 1. A plane (flat) figure bounded by four straight lines is called a quadrilateral. Quadri means four and lateral means sides. Quadrilateral means having four sides. 2. A quadrilateral whose opposite sides are parallel is called a parallelogram. Is a rectangle a parallelogram? Draw a quadrilateral that is not a parallelogram. 3. This figure represents a city lot. Is the form of the lot a parallelogram ? The form of the lot may be regarded as composed of a rectangle, a^ and two right triangles, b and o. Since the figure is a par- allelogram, triangle h is the same size as triangle c. If triangle h were cut off and placed alongside triangle c, with the side fd on the side ge^ what change, if any, would this make in the size of the lot ? What change, if any, would it make in the form of the lot? What would be the dimensions of the resulting lot? How should you find its area? 4. The sum of the areas of triangles h and c is equiva- lent to the area of a rectangle having the same base and altitude as the triangles. Therefore the number of square feet in the lot may be found by multiplying 120 by 40. The area of a parallelogram is equal to the product of its base and altitude. 5. Cut several parallelograms out of paper, and show by the method of Prob. 3 that each parallelogram has the same area as a rectangle having the same base and altitude. 6. Draw several parallelograms. Assign the dimen- sions, and find the area of each. "5- " K 26rd. Fig. 1.— Tkapkzoid. 228 FORMS AND MEASUREMENTS 278. Trapezoids. 1. Fig. 1 represents a field whose area is to be found. It may be regarded as composed of a rectangle, a, and a right 2Q , triangle, h. Its area is the sum of the areas of these two parts. The number of square rods in the rectangle is 10 x 20, and in the tri- angle is 10 X 3. The number in the entire field is 10 x 23. Explain. 2. What is the average length of the two sides of the field (Fig. 1) ? The field has the same area as a field of the same width whose length is one half of the sum of 20 rd. and 26 rd., or 23 rd. Explain. 3. Draw five figures similar to Fig. 1. Assign the dimensions, and find the area represented by each. 4. Figure 2 represents a cross section of a foundation wall. It may be regarded as composed of a rectangle, a, and two right triangles, h and c. What is the combined length of the bases of the two triangles ? If triangle h were cut off and placed in an inverted position alongside triangle c, what change would it make in the form of the figure ? What would be the dimensions of the resulting figure ? How would you find its area ? 5. The area of triangle h (Fig. 2) is equivalent to the d 4ft e area of a rectangle of the same altitude whose base is one half of the base of the triangle. Is the same true of the area of triangle c 2 The area of the sur- face represented by the entire figure is equivalent to the area of a rectangle of Fio. 2. the same altitude but whose base is one half the sum of the two bases de and fg. TRIANGLES 6. A quadrilateral that has only two sides parallel is called a trapezoid. See Figs. 1 and 2, p. 228. The area of a trapezoid is equal to the product of its alti- tude and one half the sum of its bases. 7. Draw five figures similar to Fig. 2, p. 228. Assign the dimensions, and find the area of each. The area of any quadrilateral may be found by resolving it into triangles or into rectangles and triangles. 279. Area of Triangles. 1. An angle that is less than a right angle is called an acute angle. Acute means sharp or pointed. 2. An angle that is greater than a right angle is called an obtuse angle. Obtuse means dull or blunt. 3. A triangle whose angles are all acute is called an acute-angled triangle. 4. A triangle one of whose an- gles is obtuse is called an obtuse- angled triangle. Obtuse-anqled triangles. 5. Can you draw a triangle having more than one ob- tuse angle ? Draw an acute-angled triangle; an obtuse- angled triangle. 6. Any triangle may be divided into two right triangles. Acute Angles. Obtuse Angles. AV Acute-angled TblAuNGles. 7. Triangle Imn in each of the above figures is one half of the rectangle opln. Explain. ^-80 FORMS AND MEASUREMENTS 8. Any triangle may be considered as one half of a parallelo- a c gram of the same base Fig- 1- and altitude. Triangle ahe in Fig. 1 is one half of the parallelogram ahdc. 9. Draw five triangles. Show by the methods given in Probs. T, 8 that the triangles are each one half of a parallelogram having the same base and altitude. The area of a triangle is equal to one half the product of its base and altitude. 10. Draw five trian- gles. Assign their di- mensions and find the area of each. The area of any figure may be found by resolving it into triangles. 11. Figure 2 repre- sents a farm 80 rd. by 160 rd. Find the area of each field. 12. The two sides of a field, one 132 rods, the other 152 rods in length, are parallel. The perpendicular dis- tance between the two sides is 80 rods. FiQ. 2. Diagram and find area. r-'^ £Ord 40rd. ^"^ t eOrd. "P ^ ^ c ^ E CU J 40 rd. t / ^ f 05 ^ ? // t 5 25 rd. / ^ D *ti V. / ^ / 15 rd / I'/ 40 rd. O) '^ / . ^ ^ / ^' ^ § J B ? 40rd / ^ ^ ^ / 80 rd <\j CIRCLES 231 280. Circles. . 1. Draw a circle. Mark the center ((y) (^ "^ of the circle. The line bounding the circle is called the circumference. Draw a line from the center of the circle to the cir- cumference. This is called a radius of the circle. Draw a straight line through the center of the circle from circumference to circumference. This is called the diameter of the circle. Compare the length of the diameter with the length of the radius. 2. Measure the diameter of a circle. Measure the cir- cumference of the same circle. Divide the circumference by the diameter. The answer should be nearly 3.1416 (3|). This is the ratio of the circumference to the diameter. This ratio is commonly denoted by the symbol TT, which is a Greek letter named pi. 3. The circumference of a circle is 3.1416 times the di- ameter. How can you find the circumference when the radius is given ? When the circumference is given, how can you find the diameter ? the radius ? 4. If a wagon wheel is 3| ft. in diameter, what is its circumference ? How many times will it turn in going 1 mile ? 5. Find the circumference of a circle whose diameter is 24 in. ; 7J in. ; 2 ft. 6 in. ; 40 mi. ; 80 rd. 6. Find the circumference of a circle whose radius is 2 in. : 2 ft. 4 in. ; 6 ft. ; 40 rd. ; 3J yd. 7. Find the diameter and radius of a circle whose cir- cumference is 24 ft. ; 12 J in. ; 1 mi. ; 36 rd. 8. Find the equatorial diameter of the earth if its equatorial circumference is 24,900 mi. 232 FORMS AND MEASUREMENTS 281. Area of a Circle. 1. A circle may be regarded as com- posed of an infiuite number of triangles, the sum of whose bases is the circumfer- ence of the circle and whose altitude is the radius of the circle. Therefore the area of a circle is the area of the triangles composing it. The area of a circle is equal to one half the product of its circumference and radius. 2. The circumference of a circle is 3.1416 times the diameter, or tt times 2 times the radius = 2 7rr. Area of circle (Prob. 1) = ^lli2ij:. Substituting 2 irr for Cir., Area of circle = - — = tt x r x r, or tt r^, read pi r square. The area of a circle is equal to irr^ (3.1416 X r x r). 3. Find the area of a circle whose radius is 2 in. ; 12 in. ; 24 ft. ; 4.5 rd. ; 64 yd. ; 18 ft. ; 3 ft. 7 in.; 8| in. 4. Find the area of a circle whose circumference is 24 in. ; 2 ft. 4 in. ; 40 rd. ; 28 yd. ; J mi. ; 1 mi. 5. The diameter of a circular flower bed is 6 ft. What is its area ? 6. The atmospheric pressure is about 15 lb. to the square inch. Find the pressure on a surface of a watch crystal \\ in. in diameter. 7. Over how many square feet of surface can a horse graze when tied with a rope 20 ft. long ? 8. Which has the greater surface, a rectangular table top that is 3' 6" by 3' 6'' or a circular table top that is 3' 9" in diameter? PRISMS AND CYLINDERS 233 282. Volume of Prisms and Cylinders. 1. Draw a rectangle. A rectangle has two dimensions; namely, length and width. If it is given a third dimension, thickness, it becomes a rectangular solid or a rectangular prism. 2. Anything that has length, breadth, and thickness is called a solid. /^ hi Rectangles. Rectangular Prisms. 3. How many rectangular faces has a rectangular prism ? Name some rectangular prisms that jo\i have seen. 4. A rectangular prism whose faces are all equal squares is called a cube. 5. Construct rectangular prisms out of cardboard or paper. 283. 1. Draw a triangle. How many dimensions has a triangle ? If it is given a third dimension, it becomes a triangular prism. The ends of a triangular prism are tri- angles and the sides are rectangles. Triangles. Triangular Prisms. 2. How many faces has a triangular prism ? How many of the faces are triangles ? How many are rectangles ? 3. Construct a triangular prism out of cardboard or paper. 234 FORMS AND MEASUREMENTS 284. 1. Draw a circle. If a circle is given three di- mensions, it becomes a cylinder. Mention some cylindrical objects. Cylinders. 2. Construct a cylinder out of paper. 3. The number of cubic units that a solid contains is called its volume, or capacity. s ^ 9 square units. Fig. 1. 9 cubic units. Fig. 2. 8 times 9 cubic unlta. Fig. 3. 285. 1. The area of the end, or base, of a prism or a cylinder tells how many square units that surface contains (Fig. 1). There are as many cubic units in one unit of length as there are square units in the surface of the end, or base (Fig. 2). Explain. There are as many cubic units in the prism or cylinder as the product of the number of units in the area of the end, or base, and the number of units in the length or altitude of the prism or cylinder (Fig. 8). Explain. 2. The volume of a prism or a cylinder is equal to the product of the area of the end, or base, and the length, or altitude. PRISMS AND CYLINDERS 236 3. Find the capacity of a cylindrical tank whose diameter is 18 in. and whose height is 4 ft. 6 in. Number of square feet in area of base = 3.1 416 x | x f(| X | = r'^). Number of cubic feet in capacity = 3.1416 x | x | x 4|. 4. There are 231 cubic inches in a gallon. How many gallons will the tank (Prob. 3) hold ? 5. Find the number of cubic feet in a bin 8 feet long, 4 feet wide, and 6 feet deep. 6. There are 2150.42 cubic inches in a measured bushel. How many bushels will the bin (Prob. 5) hold ? 7. Find the number of cubic feet of air in a room 16 feet long, 10 feet wide, and 9 feet high. 8. Find the number of cubic yards of earth that must be removed in excavating a basement 8 feet deep, 36 feet long, and 24 feet wide. 286. Surfaces of Prisms and Cylinders. 1. How many surfaces has a cube? a rectangular prism ? a triangular prism ? Construct each out of card- board. State how the area of the combined surfaces of a prism may be found. 2. Find the area of the surfaces (excluding the ends) of a timber 12" by 12" and 16' in length. What name is given to such a solid ? 3. Bring together the ends of a sheet of paper so that the sides form circles. What name is given to the fig- ure formed by the sheet? The length of the sheet be- comes the circumference of the base of the cylinder and the width of the sheet becomes its altitude. The area of the cylinder (excepting the bases) is therefore the area of the rectangle forming its convex surface. 4. Find the convex surface of a cylindrical tank whose diameter is 6 feet and whose altitude is 8 feet. 286 FORMS AND MEASUREMENTS 287. Written Problems. 1. If it takes .98 cu. yd. of crushed stone, .47 cu. yd. of sand, and 1.56^hbl^of cement to make 1 cu. yd. of concrete, how much di each will make 100 cu. yd. of concrete ? \y 2. How much crushed stone (Prob. 1), sand, and cement will be required to build a concrete wall 6 ft. high, 18 in. thick, and 60 ft. long, if 3j cu. ft. of cement is a barrel? (Express each answer as a whole number, since a_fractional part of these units cannot be purchasedL)_^- 3. If the sand costs 80^ per cubic yard, the cement $2 per barrel of 3| cu. ft. each, the stone $1.50 per cubic yard, and the labor for building the wall 80^ per cubic yard, find the cost of the wall (Prob. 2). 4. Mr. Adams owns a 50-ft. lot on which he has re- cently built a house. He wishes to have a 6-ft. cement sidewalk laid along the front of the lot 2 ft. in from the D ( ^ curb, and a 4-ft. cement sidewalk laid from the curb to the front steps of the house, 20 ft. in from the curb. Find the cost of the walks at $.12 J per square foot. First make a diagram of the walks. 95' 73 5. The figure represents a lot owned by Mr. Morse. Find the area of the lot. Find the cost of excavating a basement 8 ft. deep on the property at $1.25 per cubic yard. 6. How much will it cost at $ .32 a cubic yard to re- move 1 ft. of dirt from a lot 100 ft. by 150 ft. ? 7. At $5.50 a cord, how much will a pile of wood 24 ft. long, 4 ft. wide, and 8 ft. high cost? CIRCULAR MEASUREMENTS 237 MEASUREMENT OF CIRCLES AND CIRCUMFERENCES 288. 1. For the purpose of measurement circumfer- ences of circles are considered to be divided into 360 equal parts, called degrees (°). 2. A portion of a circumference is called an arc. 3. What portion of the circumfer- ence of the larger circle is arc a5? What portion of" the circumference of the smaller circle is arc dV ? 4. How many degrees are there in arc ah ? in arc ac ? in arc aJW ? in arc a! c^ ? 5. If the circumference of the larger circle is 24,90^0. mi., how long is each degree on the circumference ? If the circumference of the smaller circle is 6000 mi., how long is each degree on the circumference ? 6. As the angle at 0^ the common center of the two circles, increases or diminishes as fast as the arc suspended by its sides increases or diminishes, the angle is also measured in degrees. Thus, when the arc between two radii is 90°, the angle formed at the center of the circle by the radii is an angle of 90°, or a right angle. 7. How many arcs of 90° are there in a circumference ? How many right angles can be formed at the center of a circle ? 289. Arc and Angle Measure. 60 seconds ('') = 1 minute (') 60' = 1 degree (°) 860° = a circumference 360° = 4 right angles (rt. -4) 238 FORMS AND MEASUREMENTS 29t^. Latitude and Longitude. 1. Using a map, point to the equator ; to a meridian. The equator is midway between what two points? Do the meridians extend around the earth or only from pole to pole ? What is a meridian ? How must two or more places be loc^,ted to have the same meridian ? How must two or more places be located so as not to have the same meridians? Are the meridians of all places shown on the map ? 2. Places on the earth's surface may be located by two measures taken from two lines intersecting at right angles. 3. The lines taken for locating places on a map are the equa- tor and some selected meridian, called the prime meridian. The meridian of the Royal Observatory at Green- wich (near London), England, is taken by most nations as the prime meridian. 4. The distance in degrees north and south from the equator is called latitude ; and the distance in degrees east and west from the prime meridian is called longitude. Places north of the equator are in north latitude. What places are in south latitude ? in east longitude ? in west longitude ? 5. In the figure, point to a place located 60° west longitude and 45° north latitude ; 90° west longitude and LATITUDE AND LONGITUDE 239 45° north latitude ; 30° east longitude and 45° south lati- tude ; 45° west longitude and 45° south latitude. Give the latitude and longitude of each point at which lines intersect in the figure. 6. What is the difference in degrees between two places, one 60° west longitude and the other 15° east longitude? one 105° west longitude and the other 30° west longitude ? one 45° north latitude and the other 60° south latitude ? 7. How many degrees is it from the equator to the North Pole ? from the equator to the South Pole ? from the North Pole to the South Pole ? How many degrees is it from any point on the equator halfway around the earth ? one fourth way around the earth ? 8. The equatorial circumference of the earth is 24900 mi. What is the length in miles of a degree on the equator ? 9. What is the latitude of your home ? If the polar circumference of the earth is 24800 mi., how far do you live from the equator ? from the North Pole ? 10. Which is longer, a degree on the Arctic Circle, on the Tropic of Capricorn, or on the equator ? Explain. 11. What is the greatest latitude that a place can have ? 12. What is the greatest longitude that a place can have either east or west from the prime meridian ? Why ? 13. Where must a place be located to have a latitude of 0° ? Where must a place be located to have a latitude of 0° and a longitude of 0° ? 14. If the prime meridian in the figure on p. 238 is the meridian of Greenwich, locate on the figure your own home ; the city of New York ; Chicago ; San Francisco ; Rio Janeiro : Berlin. 7 -A 240 FORMS AND MEASUREMENTS 291. Longitude and Time — Local Time. 1. What part of the earth's surface receives the light of the sun at any one time ? Why does the sun appear to move from east to west? 2. How many hours does it take the earth to rotate once on its axis ? Through how many degrees does any meridian pass during each rotation ? 3. How many degrees of longitude pass under the sun's rays during 24 hours ? during 1 hour ? 4. Which passes under the sun's rays first, the meridian of your home or that of a place 15° east of you ? west of you? 5. When the vertical rays of the sun fall on any part of the meridian of your home, it is noon by sun time at all places on the meridian. Is it then before noon or after noon by sun time at places east of your home ? west ? 6. Since the earth rotates through 360° in every 24 hours, it must rotate through 15° each hour. Therefore the difference in sun time between places 15° apart, in an east and west line, is 1 hour. 7. When it is noon by sun time on the prime meridian, what is the time at a place 15° E.? 15° W.? 30° E.? 30° W.? 45° W.? 60° E.? 75° W.? 90° W.? 105° W.? ^ tr 6 7 8. When it is noon by sun time on the principal meridian, what is the longitude of a place at which it is 11 A.M.? 1 P.M.? 10 A.M.? 2 P.M.? 9 A.M.? 4 P.M.? 9. What is the difference in sun time between places 30° W. and 30° E. ? 45° W. and 60° E. ? 30° W. and 60° W.? 60° W. and 105° W.? 10. When it is noon on the prime meridian, where is it midnight? 9 a.m.? 9 p.m.? 6 a.m.? 6 p.m.? LONGITUDE AND TIME 941 292. Standard Time. To avoid the confusion that would arise if every place used its own local time, in 1883 the railroads of the United States and Canada agreed upon a system known as standard time. Under this system the United States is divided into four time belts, each approximately 15° in width, and each having the local time of its central me- O'idian, which is some multiple of 15°. These divisions are named after the sections of the country embraced by them as follows : Eastern, having the time of the meridian of 75° W. ; Central, having the time of the meridian of 90° W. ; Mountain, having the time of the meridian 105° W. ; and Pacific, having the time of the meridian 120° W. 5 A.M. 6 A.M. 7 A.M. 8 A.M. STANDARD TIME BELTS 105 While the time belts are theoretically 15° in width, they are actually wider or narrower than 15°. The irregularities of the divisions are due to the fact that the railways find it convenient to make the changes in time at the division termini that are nearest to 7|* east or west of the central meridians. ~" * ^ 242 FORMS AND MEASUREMENTS 293. Map Questions. 1. When it is noon in Philadelphia, what time is it in Chicago? in Denver? in San Francisco? in New York? 2. When it is 9 A.M. in Chicago, what time is it in San Francisco ? in Washington ? in New Orleans ? in Denver? in Seattle? in New York? 3. A telegram was sent from Washington at 2 p.m. and was received in San Francisco at 11 A.M. of the same day. Explain. 4. At 10 P.M. the people of Los Angeles, Cal., were reading the election returns of New York, which had f been compiled at 11 p.m. of the same day. Explain. 5. The passengers on a west-bound train arrived in Sparks, Nev., at 6.05 A.M. and after a stop of 10 min. started on their journey at 5.15 a.m. Explain. 6. On leaving North Platte, Neb., the passengers on an east-bound train found that their watches were all 1 hr. behind time. Explain. 7. How many times must a person reset his watch in traveling from Boston to San Francisco, if he wishes to have correct time on the journey? 8. If the telegraph office in Chicago, 111., closes at 6 p.m., what is the latest time a message can be sent from San Francisco in time to reach this office before it closes, allowing 30 min. for delays in transmission ? 9. When it is noon by standard time in western Iowa, is it earlier or later than noon by local time? Name some place where it is 6 P.M. by standard time before it is 6 p.m. by local time. RATIO 243 RATIO ^ 294. 1. What is the ratio of 4 ft. to G ft. ? Compare the ratio of 4 ft. to 6 ft. with the ratio of 2 ft. to 3 ft., and with the ratio of 8 ft. to 12 ft. 2. What effect upon the ratio of two quantities has (a) multiplying both terms by the same number ? (6) dividing both terms by the same number ? 3. Name two quantities whose ratio is -|. Name two other quantities having the same ratio. ( 4. Name numbers whose ratio is expressed by the fraction f, f, |, |, f , |, J. 5. Name whole numbers whose ratio is as | to 2 ; as 3 to J ; as | to 6 ; as 6 to |. 6. Write ratios equivalent to the following, but with one or both terms a fraction : 1 to 2 ; 1 to 4 ; 2 to 3 ; 2 to 1 ; 3 to 7. 4 X 7. Supply the number in place of x: -=-— . The fractions are stated as equivalent fractions. Compare their terms to find the number in the place of x. 8. The ratio between two numbers may be stated in the form of a fraction. In | = | we have an equality of ratios. The equality between ratios is called a proportion. Qi 2_4 6_x 5_10 ^_12 ^_10 ^ ''^- 5~x' i0~30' 7~ X ' 8~16' 4""l2' 10. If 20 bbl. of flour cost $ 80, how many barrels of flour can be bought for $120 ? ($120 is one lialf again as much as $80.) 11. In a certain city the ratio of the number of school- census children to the total population is 1 to 4J. If the school census is 20,000, what is the population of the city ? . / ; PART V POWERS AND ROOTS 295- 3x3 = 9. 3 is used twice as a factor to give 9. 9 is called the second power of 3. What number is the second power of 2 ? of 4 ? of 5? of 10 ? of 1 ? of 12 ? 2. The second power of a number is called its square, as the number of units in the area of a square surface is found by taking the second power of the number denoting the length of a side of the square. 3. The square of 3 may be indicated thus : 3^. Indi- cate the square of 4 ; of 5 ; of 1 ; of 10 ; of 12. Give the value of each : 1\ 82, 2\ 62. The small figure written at the right and above indicates how many times the number is to be taken as a factor and is called the exponent of the number. 4. 3 X 3.x 3 = 27. 27 is the third power, or cube, of 3. What number is the cube of 1 ? of 2 ? of 4 ? of 5 ? of 6 ? of 10? of 12? 5. The cube of 3 may be indicated by an exponent, thus : 33. Indicate the cube of 7 ; of 8 ; of 9. Give the value of each : 1^, 2^, lO^, 123. 5* is read the fourth power of 5, or 5 to the fourth power ; it means 5x5x5x5. Read and tell meaning of : 6*, 3^ 2^. 6. Find the volume of a cube whose edge is 5 in. Find the cube of 5. 7. Give the square of each of the numbers from 1 to 12. 8. Square J, |, j, .5, 1.5, .04, ^, 2J. 9. Find and memorize the cubes of 1, 2, 3, 4, 5, 6, 10, 12. 244 POWERS AND ROOTS 246 The process of finding a power of a number is some- times called involution. 10. A number that is the square of some integer or fraction is called a perfect square. Thus, 25 (5 x 5) and II (5 X I) are perfect squares. Is 24 a perfect square ? 11. Square each : 20, 30, 40, 50, 60, 70, 100. 12. Is the square of 2 plus the square of 3 the same as the square of 5 ? 296. 1. Which is the more and how much, 20^ + 5^ or 252? 2. The square of any number composed of tens and units may be found thus : 20 + 5 The square of 25 20 + 5 is seen to be the 100 + 25 (20 + 5)x5 square of the tens, ^ ' y plus twice the 400 + 100 (20 + 5) X 20 ^..^uct of the 400 + 2(100) + 25 = 202+ 2(20 X 5) + 52 tens and the units, plus the square of the units. 3. Square as above : 23, 47, 105 (100 + 5). 4. The figure represents a square whose side is 25 units. The square whose side is 20 units contains 400 square units. The two rec- tangles 20 by 5 contain 100 square units "^ each. The square is completed by the addition of the small square 5 by 5, o containing 25 square units. The area of the square is (400 + 2(20 X 5) + 25), ^^ or 625 square units. 20 5 too 25 •^ 400 100 5. Construct a square whose side is 10 + 5 units. >( 246 POWERS AND ROOTS 297. Roots. 1. Since 9 is the square of 3, 3 is the square root of 9 ; that is, it is one of the two equal factors of 9. What number is the square root of 4 ? of 25 ? of 64 ? of 36 ? of 49 ? of 16 ? of 144 ? of 100 ? of 81 ? of 121 ? of 1 ? 2. Since 27 is the cube of 3, 3 is the cube root of 27. What is meant by the cube root of a number? What number is the cube root of 1 ? of 125 ? of 8 ? of 1000 ? of 1728? 3. The sign ( V ') is called the radical, or root sign, and is placed over a number to show that its root is to be taken. The root to be taken is indicated by a small figure, called an index, written in the radical thus, VST, which is read the cube root of 27. The index 2 for square root is usually omitted. 4. Read and give the roots : V64, "v^, V49, VTOO, a/125, V81, V36, V144, -v^. The process of finding the root of a number is some- times called evolution. 298. Finding Roots by Factoring. Roots of perfect squares may be found by factoring. 1. Find the square root of 324. By factoring, 324 = 2 x 2 x 3 x 3 x 3 x 3. Arranging the factors into two like groups, 324 = (2 X 3 X 3) X (2 X 3 X 3). V324 = 2 X 3 X 3, or 18. 2. Find the cube root of 2744. Factor 2774. Group the factors into three like groups. The product of one of these groups is the cube root. 3. The square root of a fraction is the square root of its numerator over the square root of its denominator, thus : V|"= |. SQUARE ROOT 247 299. Find the roots indicated : t^ 5 4. ^3875 7. ^/^OOO 10. V1296 Wo 5. Vl29,600 8. Vff 11. V15,625 3. Vll,664 6. ^/'5T2 9. V^ 12. V6| 300. 1. Compare Vl = 1, Vlp = 10, and Vl|00|00 = 100. Notice that there is one figure in the square root for each period of two figures each into which the square can be separated, beginning at units. The period at the left may contain only one figure. By separating any number into such periods, the number of figures in the square root may be told. 2. How many figures are there in the square root of each: 11,664? 129,600? 11,025? 3. 1.22 = 1.44 ; 9.92 = 98.01; 1.222 = 1.4884. Notice that there are two decimal places in the square for each decimal place in the root. 4. How many decimal places are there in the square root of each : 4.1616 ? 1190.25 ? 2550.25 ? 301. Square Root. a. Find the square root of 529. b. Find the side of a square whose area is 529 square units. As the square root of some numbers cannot be found by factoring, another method of finding the square root of numbers is necessary. From Sec. 296 we see that the square of a number is the square of the tens, plus twice the product of the tens and units, plus the square of the units; and from Sec. 300 we see that the number of fig- ures in the square root of any number is the same as the 248 POWERS AND ROOTS number of periods of two places each, beginning with units into which the number can be separated. Model: 5'29|23 202 = 4 2x20 = 40 129 (40+3) X 3=129 a. As 529 can be separated into two periods, its square root consists of tens and units. Since the square of tens is hundreds, 5 hundreds must include the square of the tens of the root. The largest perfect square in 5 hundreds is 4 hundreds. The square root of 4 hundreds is 2 tens. Write this in the answer at the right. The square of 2 tens is 4 hundreds. Subtract 4 hundreds from 529. The remainder is 129. This remainder must be twice the product of the tens and the units, plus the square of the units. Twice 2 tens is 40. The units' figure of the root is found by taking 40 as a partial divisor. 40 is contained in 120 (omitting the 9, as it is evidently the square of the figure in units' place, or a part of its square) three times. Write 3 as the units' figure of the root. Use 43 as the complete divisor. 3 x 43 = 129, which exhausts the remainder. 20 B 60 A AGO 20 D3 h. As the largest perfect square in 5 hundred square units contains 4 hundred square units, its side is 20 units {A). 129 square units remain to be added in such form as to keep the figure a square. It is evident that these units must be added along two adjacent sides, as B and C, and at the corner, as D. The combined length of the two rectangles, B and C, is 40 units. Their width may be deter- mined from the fact that their com- bined areas, plus the area of the small square D, is 129 square units. Omitting the 9, as it evidently is the number (or a part of the number) of square units in the small square, 120 square units -^ 40 square units = 3, the number of units in the width of the rectangles, and also in the side of the small square. SQUARE ROOT 249 302. To extract the square root of a number : 1. Separate the number into periods of two figures each, beginning at the decimal point. 2. Find the greatest square in the left-hand period, and write its root for the left-hand figure of the required root. 3. Subtract the square from the left-hand period, and bring down the next period to form the complete dividend. 4. Double the part of the root already found, and place it at the left of the dividend for a partial divisor. Disregarding the right-hand figure of the dividend, divide by the partial divisor. The quotient (or quotient di- minished) will be the next figure of the root. 5. Annex the root figure last found to the partial divi- sor for a complete divisor. Multiply the complete divisor by the root figure last found. Subtract the product from the dividend, bring down the next period to form the complete dividend, and continue as before. 303. Written Exercises. Square roots of numbers that are not perfect squares may be ap- proximated by annexing periods of two decimal ciphers and continu- ing the process to several decimal places in the roots. Extract the square root of each : 1. 841 / 4 56.25 ^771600 ;2. 104,976 5. .6724 8. 10.24 3844 6. 160 U^. 007225 10. Find the side of a square whose area is 256 sq. rd. 11. Find the side of a square field whose area is 10 A. 12. Find the perimeter of a square 40-acre field. \ 250 POWERS AND ROOTS D 304. 1. Draw a right angle. Draw a rectangle. Draw a diagonal through the rectangle. Into how many equal triangles does the diagonal divide the rectangle? What kind of triangles are they ? 2. The longest side of a right triangle is called its hypotenuse, and the other two sides are called its legs. 3. Draw a right triangle whose legs are 6 in. and 8 in. Measure the length of the hypotenuse. Construct squares upon each of the three sides, and divide them into square inches. Compare the number of square inches in the square on the hypotenuse with the number in the other two squares together. 4. The figure represents a right triangle whose legs are 3 units and 4 units and whose hypotenuse is found to be 5 units. Compare the number of units in the square upon the hypotenuse with the number of units in the sum of the squares upon the other two sides. The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Answer the following from the figure: 5. If the number of squares in A and B are given, how may the number in C be found ? 6. If the number of squares in B and C are given, how may the number in A be found ? 7. If the number of squares in A and C are given, how may the number in B be found ? ^/<>' xyi 3 ri " B A -<- RIGHT TRIANGLES 261 305. Written Exercises. 1. Find the length of the third side of each 15ft 12ft 20ft eOrd. 9ft 2. Find the length of the diagonal of the floor of a rectangular room 14 ft. by 16 ft., to the nearest thousandth of a foot. 3. A boy stood on the ground 45 ft. from the foot of a tree 60 ft. in height. How far was it in a straight line from the boy's feet to the top of the tree ? 4. How much less is the distance along a diagonal path across a rectangular field 40 rd. by 80 rd. than the dis- tance around two sides of the field ? 5. How long must a rope be to reach from the top of a 60-ft. pole to a point on the ground 30 ft. from the foot of the pole ? ^ 6. Find the diagonal of a square field whose side is 40 rd. 7. Find the side of a square whose diagonal is 60 ft. 8. Find the diagonal of a rectangular room 20 ft. by 26 ft. If the ceiling of the room is 10 ft. from the floor, what is the distance from one of the lower corners of the room at one end of the diagonal on the floor to the upper corner at the other end ? 9. If A is 85 mi. south of B, and C is 75 mi. west of B, how far is it from A to C ? 10. One side of a rectangular field is 40 rd. The diag- onal is 50 rd. Find the other side. 11. Find the diagonal of a 10-acre square field. 252 POWERS AND ROOTS 306. Laying off a Rectangle. 1. When the two sides of a rectangle are in the ratio of 3 units to 4 units, the diagonal is one fourth more than the longer side, thus : If the sides of a rectangle are 18 ft. and 24 ft., the diagonal is 24 ft. plus 6 ft., or 30 ft. Prove that this is correct and that it holds with various rectangles when the ratio of the sides is as 3 to 4. 2. A farmer asked two schoolboys to lay off a rec- tangle 16 ft. by 24 ft. to mark the foundation of a car- riage house. The boys used two pieces of cord and a measure. They tied the cords to two stakes so that they crossed at ^, a corner of the rectangle, and extended one cord in the direction of cd and tlie other in the direction of ce. To make the angle at ^'-v/-%^DoLLAKS 2. Checks are indorsed in the same manner as promissory- notes. 3. To cash this check, James E. Thomas will indorse it and present it at the bank, or will deposit it to his credit at the bank where his accounts are kept. The check will finally be returned to R. E. Davies after it has been paid. It will then serve as a receipt from James E. Thomas, since it bears his indorsement. 4. A depositor may draw money for himself from his bank by making his check payable to " Cash," in which case no in- dorsement is necessary. 349. Certificate of Deposit. When a person who does not intend to become a regular customer of a bank makes a deposit, he is given a certificate of deposit, showing the amount of his deposit. Such a deposit is not subject to check. The amount may be withdrawn upon the return of the certificate with the proper indorsement. Certificates of deposit are usually issued to persons who deposit money with banks when interest is paid upon the deposit. 278 APPENDIX 350. Drafts. Banks usually keep money on deposit in some bank, called a correspondence bank, in Boston, New York, Chicago, San Francisco, or other financial center, against which to draw checks. When a person wishes to make a payment in a distant place, he may purchase the check of his local bank on a bank of correspondence, which will honor this check when presented for payment. Such a check is called a bank draft. Thus, Mr. A, in Seattle, Wash., may wish to pay Mr. B, in Peoria, 111., $45.60. Mr. A may purchase a draft of a bank in Seattle on a bank in Chicago, which will honor the draft when presented either by Mr. B, or by some bank which has purchased it from Mr. B. The bank in Seattle may charge Mr. A a small amount for making this exchange of money. 351. Clearing House. Daily settlements of accounts between banks are made through an association called a clearing house. At a fixed hour each day representatives from each bank that is a member of the clearing house visit the clearing house and settle the accounts of their bank with other banks. All large cities have clearing houses and nearly all banks in these cities are members. By means of the clearing house the American Exchange National Bank one day transacted a business of $ 18,000,000 in checks, with a bal- ance of only 12 cents to pay. The clearing house sheets showed that $9,049,255.40 in checks, drawn by the depositors of the bank, had been turned in by other institutions. Against these the bank had .$ 9,049,255.28 in checks of other banks belonging to the clearing house, which had been deposited with the American Exchange National Bank. The clearance was made by a payment by the bank of 12 cents to the clearing house. LIFE INSURANCE 352. Personal Insurance. There are various kinds of per- sonal insurance. Of these the most common are: Accide^it insurance, which is an indemnity for injuries sustained by accident ; health insurance, which is an indemnity for loss of time caused by illness ; life insurance, the principal forms of which are discussed on p. 279. LIFE INSURANCE 279 353. 1. Persons who insure their lives usually do so to pro- vide for those who are dependent upon them. The cost of life insurance depends (a) upon the age of the person insured; and (b) upon the kind of policy taken out. Life insurance premiums are always stated at so much on each $1000 of insurance. The most common kinds of policies are : 2. Ordinary Life Policies, called also straight life policies and life policies. The insured pays a premium, usually annually, at the beginning of each year from the time he insures his life until his death. At his death, the company pays the face of the policy to the person (or persons) named in the policy as his beneficiary. 3. Limited Payment Life Policies. The insured pays a pre- mium for a limited number of years, as 20 years, at the expira- tion of which the policy is said to be paid tip. The face of the policy is paid to the beneficiary at the death of the insured. 4. Endowment Policies. Premiums are paid for a period of years, as 10, 15, or 20 years, and the face of the policy is paid at the end of the period specified, or at death if the insured should die before the expiration of the period. 5. Term Policies. The insurance extends for a specified period, as for 10, 15, 20 years, etc., at the expiration of which the insurance ceases. The face of the policy is paid if the insured dies within the period specified. The amount of the annual premium on $1000 of insurance for a life policy, a limited life policy, and for an endowment policy, ages 20 years to 40 years, is given in the table on j). 280. 6. Which is the more likely to live twenty years longer, a person twenty years of age or a person forty years of age? Statistics have been carefully compiled showing the ages at which persons die. From these statistics insurance companies are able to determine the average number of years a healthy person of a given age may be expected to live. The rates of annual premiums are based upon these statistics. 280 APPENDIX Tablb or Annual Premiums for $ 1000 (Ages 20 years to 40 years) Policies Non-fobfbitable and Participatinq Premiums may also be paid half-yearly or quarterly ; and if desired, may be paid iu 10, 15, or 20 years instead of during the whole term. Ordinary Life 20 Endowments J Payment Life Age Yearly 10 Year 15 Year 20 Year 20 $17.30 $24.16 $99.27 $62.34 $44.10 21 17.80 24.00 99.40 62.40 44.25 22 18.30 25.10 99.50 62.45 44.40 23 18.70 25.70 99.00 62.50 44.55 24 19.30 26.20 99.75 62.G0 44.70 25 19.80 26.75 99.90 62.70 44.82 26 20.30 27.30 100.00 62.80 44.95 27 20.90 27.90 100.05 62.90 45.10 28 21.50 28.50 100.10 63.05 45.25 29 22.10 29.10 100.20 63.20 45.45 30 22.70 29.70 100.30 63.34 45.63 31 23.40 30.35 100.40 63.i30 45.85 32 24.10 31.00 100.50 63.70 46.05 33 24.80 31.72 100.60 63.90 46.25 34 25.60 32.50 100.75 64.05 46.45 35 26.50 33.28 100.90 64.20 46.70 36 27.40 34.10 101.15 64.40 46.85 37 28.30 34.96 101.45 64.65 47.05 38 29.30 35.88 101.75 64.95 47.25 39 30.40 36.84 101.95 65.30 47.45 40 31.50 j 37.84 102.14 65.67 48.64 364. Dividends. The premium charged represents the esti- mated cost of insurance and is based upon conservative assump- tions as to future death rate, the rate of interest which the company may expect to receive for loans, etc. The actual cost of insurance is determined by experience from year to year. The difference between the estimated cost and the actual cost J 8 called the profit. Policy holders are usually allowed to LIFE INSURANCE 281 participate in the profits, either by having them applied to reduce the yearly premiums or by having them accumulate in the possession of the companies until the expiration of the term of insurance. An insurance policy in which it is stipu- lated that no dividend shall be paid until the close of the term of insurance is called a tontine policy. Examine a life insurance policy. Kead all its provisions. 355. Use the table in answering the following : 1. How much will it cost annually to carry an ordinary life policy for $ 1000, if it is taken out at the age of 20 ? at the age of 25 ? at the age of 35 ? 2. How much will it cost annually to carry a 20-payment life policy for $2000, if it is taken out at the age of 20? at the age of 27 ? at the age of 40 ? 3. How much will it cost annually to carry a 10-year en- dowment policy for $ 5000, if it is taken out at the age of 20 ? at the age of 30 ? at the age of 40 ? 4. Suppose that a young man 20 years old takes out a 20- payment life policy for f 1000 and dies after paying 8 annual premiums. Find the net cost of the insurance, if dividends amounting to $40 were applied to reduce the premiums. How much would the beneficiary named in the policy receive at his death ? 5. How much will it cost to carry a 20-year endowment policy for $ 1000 for the term of the policy, if it is taken out at the age of 30 ? How much would the insured receive from the insurance company at the end of the term, not including the dividends ? 6. If the insured (Prob. 5) died after paying 15 premiums, how much more than the amount paid as premiums would the beneficiary receive ? 7. What is meant by a non-forf citable and participating policy ? by a tontine policy ? 282 APPENDIX Table of Loak and Surrender Values 356. The following table shows the loan and surrender values on a 20-payment life policy for $ 1000 taken out when the insured was 25 years of age, the annual premium being $ 26.95 : At End of Loan Cash Value Pald-up Insurance Extended Insurance Years Days 3d «150 4 342 4th 200 6 291 5th $54 $60 250 8 232 6th 68 76 300 10 317 10th 130 145 500 19 17 15th 224 249 750 26 134 19th 315 351 950 31 111 20th 342 380 Policy full-paid Answer the following from the above table : 1. If the insured wished to borrow money, how much would the company loan him at the end of the 10th year, if he assigned to the company his policy as security ? how much at the end of the 15th year ? 2. If the insured surrendered his policy at the end of the 5th year, how much would the insurance company pay him for his policy ? How long would they continue his insurance without the payment of premiums ? 3. Find the amount of the annual premiums for 20 years. What is the cash value of the policy at the end of 20 years ? If the dividends average $ 6.50 a year, how much will they amount to in 20 years ? What is the sum of the cash value and dividends at the end of the insurance term ? How does this sum compare with the total cost of the insurance for the term ? THE EQUATION 283 4. Using the compound interest table on p. 320, find the amount of $ 26.95 (the premium) for 20 years. If money is worth" 6%, find the total amount of the premiums paid at the end of 20 years, the premium being paid at the beginning of each year. 5. If the insured should die at the age of 40, how much would the beneficiary receive ? THE EQUATION 357. 1. The relation of the quantities involved in some prob- lems can be stated in a simpler and clearer way by the use of the equation. In an equation, the vahie of the unknown quantity is usually represented by the letter x. Thus, in 6 -f 4 = 0^, X is called the unknown quantity, and the expres- sion 6 -h 4 = ^' is called an equation. 2. An equation may be compared to a balance scale. In an equation the quantities on the two sides are equivalent — they balance one another. 3. If a package weigh- ing 4 lb. is placed in one pan of a balance scale, what weight must be placed in the other pan to make the scale balance ? 4. A package weighing 5 lb. was placed in the pan on the right of a balance scale and a 2-lb. weight was placed in the pan on the left. What additional weight must be placed in the pan on the left to make the scales balance ? 5. Would the scales as represented in the figure still bal- a!ice if a 5-lb. weight were added to the weights in each pan ? Would they balance if a 5-lb. weight were added to the weight in one pan ? 6. Would the scales as represented in the figure still bal- ance if 2 lb. were removed from both pans ? Would they balance if 2 lb. were removed from only one pan ? 284 APPENDIX 7. Would the scales as represented in the figure balance if the weight on both sides were doubled ? Would they balance if the weight on only one side were doubled ? 8. Would the scales as represented in the figure balance if one half of the present weight were removed from each pan ? Would they balance if one half of the weight were taken out of only one pan ? 9. What is the value of the unknown quantity in the equations lb. =5 lb. + a;? in 6 lb. + a; = 10 lb. ? inZlb. — « = 2 1b.? 10. What is the value of x in 9 + a? = 12 ? Finding the value of the unknown quantity in an equation is called solving the equation. 11. If 3 is added to both sides of the equation a; + 4 = 0, the result is a; + 7 = 12. How does the value of a; in a; + 4 = 9 compare with the value ofa;ina;-l-7 = 12? 12. Write 5 equations. Add some number to both sides of each of the equations. Compare the value of x in the result- ing equation with the value of x in the original equation. 13. State what effect adding the same number to both sides of the equation has upon the value of x in the equation. Prove the truth of your statement. 14. If 2 is subtracted from both sides of the equation a; -I- 4 = 9, the result is a; + 2 = 7. How does the value of x in a; -f- 2 = 7 compare with the value of a; in a; + 4 = 9 ? 15. Write 5 equations. Subtract some number from both sides of each equation. Compare the value of x in each of the resulting equations with the value of x in the original equation. 16. State what effect upon the value of x in any equation subtracting the same number from both sides of the equation has upon the equation. Prove the truth of your statement. 17. If 4 is subtracted from both sides of the equation jc -f 4 = 9, what is the result ? THE EQUATION . 285 18. If 3 is subtracted from both sides of the equation x -\- S — 8, the result is a? = 5. If 2 is subtracted from both sides of the equation 9 = 2 + a;, the result is 7 =x. What number must be subtracted from both sides of each of the following equations to leave x bj itself on one side: a? 4- 5 = 8? a; + 7 = 15? x + 6 = 9? S = 6-\-x? 12 = x + o? 9-f-ic=15? 74-aJ = 10? 19. If 4 is added to both sides of the equation a; — 4 = 5, the result is a; = 9. 20. What number must be added to both sides of each of the following equations to leave x by itself on one side : iB-5 = 8? a;-7 = 10? a;-4 = ll? x-0 = 2? 12 = a;-15? 10 = a;-5? 4 = a;-G? 2 = a;-3? 21. The value of a; in a; + 4 = 9 may be found by subtract- ing 4 from the left side of the equation and indicating the sub- traction of 4 from the other side, thus : a; = 9 — 4. Find the value of X in each of the following equations : a; -f- 6 = 15 ; fl; + 8 = 15; a; + 9 = 16; a;-f-20 = 45; a;-f345 = 670. 22. The value of a; in a; — 4 = 10 may be found by adding 4 to the left side of the equation and indicating the addition of 4 to the other side, thus : a; = 10 + 4. Find the value of x in each of the following equations : a; — 7 = 13 ; a; — 5 =18 ; a;-12 = 20; a;-14 = 17; a;-46 = 35; a;-80 = 120. 23. Write each of the following equations with x by itself on the left side of the equation : a? + 3 lb. = 10 lb. ; 5 f t. + a' = 13 ft. ; 24 yd. +a; = 45yd.; $ 7.50 + aj= $12.75; a; + $15 = $80; a; -12 ft. =20 ft.; a; - $3.45 = $1.20. 24. Compare 2 + 3 = 5 with 5 = 2 + 3. Compare a; + 4 = 9 with 9 = a; + 4. State what effect, if any, writing the equation with the sides changed has upon the equation. 25. Write each of the following equations so the side con- taining X is on the left : 45 ft. + 33 f t. = a; ; $ 2.45 = a; - $ 1.20 ; 14 yr. = 9 yr. + a; ; 10 yr. = a; — 7 yr. 286 APPENDIX 358. Solve each of the following without using x. Then write the equation for each, using x^ and find the value of x : 1. If 45 is added to a certain number, the sum is 73. What is the number ? Model : Let x = the unknown number a; + 45 = 73 a = 73 - 46 a; = 28 2. If 27 is subtracted from a certain number, the remain- der is 56. What is the number ? 3. If a certain number is increased by 347, the result is 591. What is the number ? 4. If a certain number is diminished by 274, the result is 483. What is the number ? 5. A boy deposited ^ 17 in a savings bank. He then had % 61 in the bank. How much money had he in the bank before depositing the $17? 6. After drawing out $35 from a savings bank a boy had left $ 7.45 in the bank. How much money had he in the bank before drawing out the $35 ? 7. After gaining 7 lb. a girl weighed 103 lb. How much did she weigh before gaining the 7 lb. ? 8. George and Frank together have as much money as Walter. George has $2.15 and Walter has $4.10. How much money has Frank ? 9. A man owns three farms amounting together to 240 acres. Two of the farms contain 80 acres and 120 acres re- spectively. How many acres are there in the third farm ? 10. A house and lot together cost $4500. The lot cost $ 1500. Find the cost of the house. 11. The sum of two numbers is 238. One of the numbers is 79. What is the other number ? THE EQUATION ^87 12. The sum of the three sides of a triangle is 24 in. One of the sides is 8 in. and another is 9 in. What is the length of the third side ? 13. After selling 40 sheep a farmer had 236 sheep. How many sheep had he before selling the 40 sheep ? 14. Write 3 problems similar to each of Probs. 1-13 and write the equation for each. 359. 1. If ^ is added to both sides of the equation 7 — aj'= 2, the result will be 7 = 2 + «• What will be the result if x is added to both sides of the equation 10 — a; = 7 ? 2. If X is added to both sides of the equation 14 = 25 — a;, the result will be 14 + a; = 25. The value of x is found by- adding to both sides of the equation some number that will leave x by itself on the left side. 3. Write each of the following equations so that x will be by itself on the left side of the equation. First, add x to both sides of the equation, then write the equation so that the side containing x will be on the left. 18 = 43 — iK ; 21 = 72 — a; ; 60-0^ = 37; 33-aj = 19; 54 = 62-a;; 68-ic = 28. Write each of the following statements so that x will be by itself on the left side of the equation, and solve : 4. 167-a; = 100. 8. 74-a; = 18. 5. a;-$36 = |75. 9. $ 45.75- a? = $30.50. 6. 15 lb. = 25 lb. -a.-. 10. 65.4 -a; = 18.45. 7. 78 ft. = 135 ft. -X. 11. 125 da. - a; = ^h da. 12. Write 10 equations and find the value of x in each. 360. 1. The sum of x and x and x, or 3 times x, is written Zx. Write the sum of x and x. Write the product of 4 times x\ of 5 times a. 2. If X is 4, what is the value of 2 a; ? Compare a; = 4 and 2aj = 8. What must both terms of a; = 4 be multiplied by to give 2 a; = 8? Multiply both terms of a; = 3 by 5. Has this changed the value of x in the equation ? 288 APPENDIX 3. State what effect multiplying both sides of the equation by the same number has upon the vahie of x in an equation. Prove the truth of your statement. 4. Divide both sides of the equation Go; = 18 by 2; by 3; by 6. Has this changed the value of x in the several equations ? 5. State what effect dividing both sides of tlie equation by the same number has upon the value of x in an equation. Prove the truth of your statement. 6. If 2a; + 4 = 21, what is the value of 2a; ? of a; ? of 3a; ? 7. If x = 6, what is the value of 7a; ? of 3a; ? of 5a; ? Find the value of x in each of the following equations. Where the equation shows an unknown quantity to be sub- tracted from one side of the equation, add this unknown quantity to both sides of the equation ; then write the equation with the unknown quantity on the left side and solve : 8. 2a; -45 = 69. 13. 146 -3a; = 83. 9. 24 = 78 -2a;. 14. $35.40 - 3 a; = f 10.95. 10. 45 ft. -4a; = 13 ft. 15. f.85 = $.40-f-3a;. 11. 345 -4 a; =135. 16. $90-6a;=$48. 12. 240 A. = 880 A. -4a;. 17. 24yr. -3a;=6yr. 361. Solve each of the following without using x. Then solve each, using x. 1. If 3 times a certain number, plus 25, is 55, what is the number ? 2. If 4 times a certain number, less 20, is 40, what is the number ? 3. Mary is 20 years old. This is 2 years more than twice Edna's age. What is Edna's age ? 4. Walter has $45. This is $13 more than 4 times the amount of money James has. How much money has James? 5. If 4 times a certain number, plus 3 times that number, is 28, what is the number ? (4a; -|- 3a; = 7a;.) THE EQUATION 289 6. If 6 times a certain number, plus 4 times that number, is 160, what is the number ? 7. If 4 times a certain number is the same as 6 times 18, what is the number ? 8. A man bought three railroad tickets, each costing the same amount, and paid $1.50 for bus rides. He paid out $ 6.90 in all. Find the price paid for each ticket. 9. The sum of two numbers is 48, and one number is 5 times the other. What are the numbers? (Let x and 5x represent the numbers.) 10. A man bought two carriages. For one he paid twice what he paid for the other. Both carriages cost him $210. Find the cost of each. 11. Write problems similar to each of the above, and state the equation for each. 12. Draw an oblong whose length is twice its width. Let x represent its width. What will represent its length ? its perimeter? If the perimeter of the oblong is 30 in., how wide is it ? How long is it ? 13. Draw two lines, one of which is 3 times the length of the other. If the sum of their lengths is 24 ft., how long is each line ? 14. Two men together own 540 acres of land. One owns twice as much as the other. How many does each own ? 15. A man offered to divide $10 between two boys in pro- portion to their ages, provided the boys could tell how much each should receive. The boys were 12 years and 13 years respectively. After solving the problem the boys stated that the younger should receive $4.50 and the older $5.50. Did they solve it correctly ? If not, what is the correct answer ? AK. — 19 290 APPENDIX 16. A man offered some boys $ 1.50 for weeding his garden. The boys found that they could not all work at the same time, so the man agreed to pay each boy the same wages per hour for the work done. One boy worked 7 hours, another worked 5 hours, and the third worked 3 liours. How much of the money should each boy receive ? 17. A man wished to leave $ 3500 to his three sons so that the second son would receive twice what the youngest received and the eldest would receive 4 times what the youngest re- ceived. How much should each son receive ? 362. 1. The expression f is used to denote ^ of x. Write the expression that denotes J of a; ; -J- of x; -J of a; ; ^ of a; ; -f of X. 2. By what number must f be multiplied to make x? By multiplying both sides of the equation f = 3 by 4, the equation is changed to a; = 12. 3. Multiply f by the number that will give x as the re- sult. Multiply ^ by the smallest whole number that will give a whole number of a;'s as the result. 4. What is the smallest number that both sides of the equation f = 12 can be multiplied by to leave only whole numbers in the equation? Multiplying both sides of an equation by some number that will leave the equation without fractional quantities is called clearing the equation of fractions. 5. Clear the following equations of fractions : f = 8 ; 1=14; ¥ = 12; |t = 36; 45=^; 60 = VV ; -¥--8 = 1; 72-^^ = 56. 6. Clear the following of fractions and find the value of x : Solve each of the following without using x. Then solve each, using x : 7. Divide 60 into two numbers such that the first is J of the second. THE EQUATION 291 8. Separate 36 into two parts whose ratio is -|. 9, Divide $2.10 into two amounts whose ratio is the same as the ratio of 15^ to 20 /. 10. If ^ of a certain number, plus | of it, is 39, what is the number ? 11. Solve : 6 times 8 = 12 times x ; 4 times 9 = 6 times x. 1 9 Snl VP • 6 a;, a — 12. « — 42 J.4S. OOlVe . 9- — -3 , -g- — T6 J T — 4 9"- 13. The equation J = ^ may be cleared of fractions by mul- tiplying both sides of the equation by the least common mul- tiple of the denominators. This is 2 x. The equation is thus changed to the form 12 = a;. 14. Solve: i = ii; ,\ = i; V"=l«; I = Il- ls. Solve: w = J^; M = ¥; A=^; ^=^- 16. Write ten exercises similar to exercises 8-12 and find the value of x in each. 363. Proportion. Solve each without using x. Solve each, using x : 1. The shadow of a post 5 ft. high is 3 ft. 6 in. long. How high is a telephone pole whose shadow is 28 ft. long ? a. The height of the post is -^ times the length of its shadow. The height of the telephone post is -^ times 28 ft. Explain. b. The length of the shadow of the post is in the same ratio to the height of the post as the length of the shadow of the telephone pole is to the height of the pole. The equality of these ratios may be expressed thus : 3^^28 5 X Solve to find the value of x, the number of feet in the height of the telephone pole. c. The equality of the two ratios may be expressed thus : 3.5 ft. : 5 f t. : : 28 ft. : x, which is read, 3.5 ft. is to 5 ft. as 28 ft. is to X. The first and last terms (as 3.5 ft. and x) of a 292 APPENDIX proportion are called the extremes, and the two middle terms the means. Tlie product of the extremes in a proportion is always equal to the product of the means. Hence, 3.5 times x = 5 times 2S, or 3.5 a; = 140. Solve to find the value of x, the number of feet in the height of the telephone pole. This method of solving a proportion differs only inform of expres- sion from the method (b) given on p. 201. 2. How high is a tree whose shadow is 34 ft. 6 in., if the shadow of a boy whose height is 4 ft. 9 in. is 3 ft. 3 in. ? 3. If the distance traveled by a trail* in 1 hr. 45 min. is 80 mi., how long, at the same rate of speed, xWU it take the train to travel 475 mi. ? 4. Find by the method used in solving Probs. 1 and 2 the height of objects near the schoolhouse. MEASUREMENT OF SURFACES AND SOLIDS 364. Areas of Surfaces. 1. Draw a vertical line; a horizontal line; an oblique line. 2. Draw a line perpendicular to another line ; parallel to another line. 3. Draw a right angle ; an acute angle ; an obtuse angle. 4. Draw a rectangle. Is a rectangle a parallelogram? Draw a parallelogram that is not a rectangle. 5. How many dimensions has a rectangle ? Is a rectangle a quadrilateral ? Draw a quadrilateral that is not a parallelo- gram. 6. State how the area of a parallelogram is found. Find the area of a parallelogram whose base is 20 ft. and whose altitude is 18 ft. 7. A quadrilateral that has only two parallel sides is called a trapezoid. 8. State how the area of a trapezoid is found. Draw a trapezoid. Assign its dimensions and find its area. MEASUREMENT OF SURFACES AND SOLIDS 293 9. TVTiat is a triangle ? Draw a right triangle ; an acute- angled triangle ; an obtuse-angled triangle. 10. State how the area of a triangle is found. Draw a tri- angle. Assign its dimensions and find its area. 11. Make a drawing to show the relation of the area of a triangle to the area of a parallelogram having the same base and altitude. 12. Draw a parallelogram. Draw its diagonals. Do they cross at the middle of the parallelogram ? 13. What is meant by the perimeter of a figure? Find the perimeter of your schoolroom. 14. Draw a circle. Draw its radius ; its diameter. Point to its circumference. 15. State how the circumference of a circle is found when the length of its radius is known. State how the diameter of a circle is found when the length of its circumference is known. 16. State how the area of a circle is found. Assign the necessary dimensions and find the area of a circle. 17. State how the area of the convex surface of a cylinder is found. Find the area (including the ends) of a cylinder whose diameter is 6 ft. and whose length is 8 ft. 365. Regular Polygons. 1. Mention a surface that is a plane surface. A plane figure bounded by straight sides is called a polygon. A polygon whose sides are all equal and whose angles are all equal is called a regular polygon. o o Triangle . Square Penta^'ou Hexagon Rbgdlak Pulygons 294 APPENDIX 2. A regular polygon of three sides is called an equilateral triangle; of four sides, a square; of five sides, a pentagon; of six sides, a hexagon ; of seven sides, a heptagon ; ^'^N. of eight sides, an octagon. Draw an octagon. ^ C-'^ ^* ^ straight line from the center of a \ f ' / regular polygon to any vertex is called its \ ; / radius (r). 4. The perpendicular from the center of a regular polygon to any side is called its apothem (a). 5. The area of a regular polygon is the sum of the areas of the triangles formed by its radii and sides. The apothem is the altitude of each of the triangles, and the perimeter is the sum of the bases of the triangles. Hence, The area of a regular polygon is equal to one half the product of its perimeter and apothem, 6. Draw a pentagon. Assign its dimensions and find its area. 7. Draw a hexagon. Assign its dimensions and find its area. 8. Draw an octagon. Assign its dimensions and find its area. 9. The area of a circle is one half the product of its radius and circumference. Compare the method of finding the area of a regular polygon with this method of finding the area of a circle. 366. Solids. 1. How many dimensions has a plane surface? Name them. 2. How many dimensions has a solid ? Name them. 3. What name is given to a solid whose faces are all rec- tangles ? to a solid whose faces are equal squares? MEASUREMENT OF SURFACES AND SOLIDS 295 4. What name is given to a solid whose ends are triangles and whose sides are rectangles ? 5. State how the volume of a prism is found. Draw a prism. Assign its dimensions and find its volume. 6. Name solids that are rectangular prisms. 7. State how the volume of a cylinder is found. Draw a cylinder. Assign its dimensions and find its volume. Find its area, including the ends. 367. Pyramids and Cones. 1. A solid whose base is a polygon and whose faces are triangles meeting at a point (vertex) is called a pyramid. 2. The area of the surface of a pyramid is the sum of the areas of the triangular faces. 3. The perpendicular distance from the base to the vertex of a pyramid is called its altitude (vb). 4. The altitude of one of the triangular faces of a pyramid is called its slant height (vs). 5. Construct a pyramid of cardboard. Which is the greater, the altitude of a pyramid or its slant height ? The apothem of a polygon forming the base of a pyramid may be regarded as the base of a right triangle (bs), the altitude as the other leg (vb), and the slant height as the hypotenuse (vs). How may the altitude be found when the apothem of the base and the slant height are given ? 6. Draw a regular polygon. Draw its radius and the apothem of an adjacent side. The figure formed by the radius, apothem, and one half of the adjacent side is what kind of a triangle ? if the radius and side of a regular polygon are given, how tnay the apothem be found ? 296 APPENDIX 7. If the altitude of a pyramid, the radius of its base, and the adjacent side are given, how may the slant height be found ? 8. A solid whose base is a circle and which tapers to a point called the vertex or apex, is called a cone. 9. A cone may be regarded as a pyramid whose surface is an infinite number of narrow triangles. Its altitude and slant height correspond to the altitude and slant height of a pyramid. The area of the surface of a pyramid or a cone is equal to one half the product of its slant height and the perimeter of its base. Prism Pyramid Cylinder Cone 10. The volume of a pyramid is equal to one third the volume of a prism of the same base and altitude, and the volume of a cone is equal to one third the volume of a cylinder of the same base and altitude. Hence, TJie volume of a pyramid or a cone is equal to one third the product of its altitude and the area of its base. 11. A cj^indrical granite stone 3 ft. in diameter and 4 ft. in height was cut down into a cone of the same base and altitude. What part of the stone was cut away ? 368. Spheres. TJie area of the surface of a sphere is four times the area of a great circle (irr') of the sphere. 1. As (2?-)', or 4 r^ is equal to d^, 4 irr'^ is equal' to tk?. PUBLIC LANDS 297 77i« area of tlie surface of a sphere ia equal to the square of the dmineter x vy or trd^, 2. Whicli is the greater and lio"w much, the area of a cube whose side is 1 ft. or the area of a sphere whose diameter is 1 ft. ? 3. A sphere may be divided into an infinite number of figures that are essentially pyramids. The combined volume of these pyramids is the volume of the sphere. The convex sur- face of the sphere may be re- garded as the sum of the bases of the pyramids and the radius of the sphere as the altitude of the pyramids. Hence, Tlie volume of a sphere is equal to one third the product of its radius and its convex surface, or | ttt^ (| of r x 4 trr^). 4. As d^, or (2iry is equal to 8r^, ^irr^ is equal to ^ird^. Hence, To find the volume of a sphere, multiply the cube of its diameter by. 5236 a of ^). 5. The earth is how many times the size of the moon, if the diameter of the earth is 8000 mi. and the diameter of the moon is 2000 mi.? Volumes of spheres are to each other as the cubes of their like dimensions. The ratio of the earth and moon is 8^ (8000^) to 2^ (2000^), or 4:^ to 1^. MEASUREMENT OF PUBLIC LANDS 369. 1. At the time the colonial settlements were made, no uniform system of measuring lands was used. Generally, each settler was permitted to occupy whatever lands he wished, and the boundary lines were often designated by such convenient natural objects as rocks, streams, trees, hilltops, etc. Later these boundaries were recorded as the legal " metes and bounds '' of their several possessions. These tracts of land 298 APPENDIX were often so irregular in shape as to make it difficult to fix their exact boundaries and to determine their exact areas. 2. Shortly after the close of the Kevolutionary War, the Continental Congress appointed a committee, of which Thomas Jefferson was chairman, to draw up some plan for the survey of public lands. This committee reported a plan which, after being slightly amended, was adopted by Congress in 1785, and thus became the government system of measuring public lands. 3. In accordance with this system, all public lands, except "waste and useless lands," have been laid out in tracts 6 miles square called townships. The exact location of each township is determined by north and south lines called principal meri- dians, and by east and west lines called base lines. Study the following diagram : Standard. Base. Uj 6^ B T. 5n. _ Parol lef T. 4 N. T. 3 N. T. 2 N. T. I N. — Line T I S. T 2 S. 4. In surveying a tract of land, a prominent point that is easily identified and is visible for some distance is established astronomically, and is known as the initial (beginning) point. In the figure, the initial point is at 0. PUBLIC LANDS 299 5. A line extending north, or south, or both north and south, from the initial point is taken as a principal meridian. The principal meridian is the true meridian at the initial point. Locate the principal meridian in the figure. 6. A line extending either east or west, or both east and west, through the initial point, or a line perpendicular to the principal meridian, is taken as a base line. The base line is always a true parallel of latitude. Locate the base line in the figure. 7. East and west lines 6 miles apart, called town lines, are run parallel to the base line, and north and south meridian lines 6 miles apart, called range lines. These lines divide the tract into townships 6 miles square. Point to the township lines in the figure. How far apart are these lines ? Point to the range lines. How far apart are these lines ? 8. Point to a township in the first tier of townships north of the base line. Point to a township in the second tier of townships north of the base line. Point to a township in the first tier of townships south of the base line. 9. A township in the third tier of townships north of a base line is said to be in township 3, north (T. 3 N.). A township in the second tier of townships south of a base line is said to be in township 2, south (T. 2 S.). 10. Point to the first north and south row of townships, east of the principal meridian. These townships are said to be in range 1, east (K. 1 E.). Point to a township in range 3, east-, in range 2, west. 11. The township marked A is numbered township 4 north, range 2 east (T. 4 N., R. 2 E.). Describe the location of town- ships B, C, D, E, and O. Write the description of each, using abbreviations. 12. Locate in the figure each of the following described town- ships : T. 2N., R. 3E.; T. 4N., R. 5E.; T. IK, R. IW.; T.2S.,R.4E.; T.1S.,R.3W.; T.4K,R.2W.; T.2S.,R.2E. 300 APPENDIX la Draw a diagram showing a principal meridian, a base line, and townships and ranges as in the figure on p. 298. In your diagram, locate the following : T. 4 S., K. 1 E. ; T. 6 K., R. 5 W.; T. 6K, R. 6 E. 14. Locate on a map a principal meridian and a base line from which ranges and townships in your state are numbered, if the land has been measured by this system.* Give the number of the township in which you live. Can you tell the width of the state in which you live from the number of town- ships along the base line? Is there any similarity between the method of locating townships by means of principal me- ridians and base lines and the method of locating places on the earth's surface by means of degrees of longitude and latitude ? 15. The lands of Florida, Alabama, Mississippi, of the states west of Pennsylvania and north of the Ohio River, and of all states west of the Mississippi River, except Texas, have been surveyed in the manner described. Can you tell from your study of United States History why the lands of the other states were not surveyed in this manner ? 16. The initial points are located somewhat arbitrarily. Sometimes they are located on the east or west boundaries of states, at other times they are located at the junction of rivers, or on the summits of elevations. They are at irregular intervals apart. Consequently, the land in a single state may be meas- ured from more than one principal meridian, or a single me- ridian may be used for measuring the land in several states. Much care is taken to preserve the exact location of all initial points. "An initial point sliould have a conspicuous location, visible from distant points on lines ; it should be perpetuated by an indestructible monument, preferably a copper bolt firmly set in a rock ledge ; and it should be witnessed by rock bearings, without relying on anything perishable like wood." Manual of Surveying Instructions, 1902. • Unmounted land maps of the various states may be purchased from the Department of Interior, Washington, for a few cents. PUBLIC LANDS 301 17. As the lines that bound the ranges on the east and west are true meridians, they converge as they extend north from a base line. As a result, townships are not true squares. To correct the effect of the convergeucy of the meridians, standard parallels (formerly called correction lines) are established at regular intervals (now 24 miles apart) from the base line, and new meridians are established 6 miles apart on the stand- ard parallels. Guide meridians are also established at inter- vals (now 24 miles apart), east and west of the principal meridi- ans, to correct inaccuracies in measurement. 370. Townships. 1. A township is a tract of land 6 miles square. It contains 36 square miles of land. A square mile of land is called a section. The sections of a township are numbered as shown in the dia- gram. The sections of a town- ship are numbered, respectively, beginning with number 1 in the northeast section and numbering west and east alter- nately. Draw a township and number the sections. 2. Section 16 of each township in the state was granted by Congress to the states for educational purposes. This section is therefore commonly known as the school section, and all moneys derived from the rent or sale of these sections is placed in the public school fund of the state. States that have been organized since 1852 have been granted two sections in each township for the support of public schools, sections 16 and 36. Owing to the convergency of the meridians that bound the townships on the east and west, a township is never exactly 6 miles from east to west, and does not therefore contain 36 full 6 5 4 3 2 1 7 8 9 10 n 12 18 17 B 15 14 13 19 20 2/ 22 23 24 30 29 28 27 26 25 31 32 33 34 35 ■ A Township divided into Sections. 302 APPENDIX sections of G40 acres each. The survey of the sections in each township is begun in the southeast corner of the township, and all sections except those along the western and northern bound- aries of the township are 1 mile square, and contain 640 acres each. All excess or deficiency is added to or deducted from the sections along the western and northern boundaries of the township. These sections generally contain less than 640 acres. The sections along the western boundary of a township often contain less than 630 acres. Section 6 is frequently re- duced to about 620 acres. 371. Sections. 1. A section is subdivided into quarter sections, and these are again subdivided into quarters, etc., as shown in the diagram. 2. The part of the section marked A is described as the west one half (W. ^) of the section, and contains 320 acres. The part marked 5 is described as the south- east quarter of the section, and contains 160 acres. C is the west one half of the northeast one fourth of the section (W. | of N.E. J). How many acres does it contain ? The part marked F is described as the S.E. \ of the S.E. \ of the N.E. J of the section. How many acres does it contain ? 3. Describe the part marked O and tell bow many acres it contains. 4. Describe the part marked E and tell how many acres it contains. 5. Describe the part marked D and tell how many acres it contains. 6. Draw a section and subdivide it to show the following and give the number of acres in each : 7. N.W. \ of the N.E. J. 8. S.W. J of the N.W. J. D .\G \F A Sbction Subdivided. PUBLIC LANDS 303 9. E. I of the S.W. J. 10. W. I of the N.W. f 11. S.E. -1- of the S.W. i of the KE. \. 12. S. i of the S.E. i of the N.E. i. 372. Using the scale 1 in, = 1 mi., draw a plot to represent a township, say T. 6 'N., K. 4 E. ; locate and find the area of each of the following : 1. E. i of the S.E. ^ of Sec. 9, T. 6 N., E. 4 E. 2. N. W. 1 of the S.E. J of Sec. 22, T. 6 N., K. 4 E. 3. S.E. 1 of the S.W. i of the S.E. Jof Sec. 32, T. 6 N., R.4 E. 4. E. i of the S.W. i of the N.E. i of Sec. 24, T. 6 N., R. 4 E., which is a farm owned by Mr. Thomas. 5. S.E. 1 of the K.W. ^ of Sec. 18, T. 6 N., R. 4 E., which is the description of a piece of property on which Mr. White pays taxes. 373. Review. 1. The unit of land measure is the township, which is theo- retically 6 miles square. The word town is commonly used for township. 2. What are initial points ? principal meridians ? base lines ? 3. What is a range ? How many sections are there in a township ? How are they numbered ? 4. How many acres are there in a section ? in a quarter section ? 5. Public lands are generally sold in sections, half sections, quarter sections, and in half quarter sections. What part of a section is 80 acres ? 40 acres ? 20 acres ? 6. How many acres are there in a full township ? in a full section ? 7. What are standard parallels, or correction lines ? 8. Which are the school sections ? Why are they so called ? 804 APPENDIX 9. Cau you tell from your study of United States history why some uniform system of surveying public lands was necessary soon after the close of the Revolutionary War ? 10. "What sections generally contain less than 640 acres ? Why? 11. Locate the principal meridian and the base line used in measuring the land in which your schoolhouse is located. 12. A new standard parallel is located at intervals of 24 miles north or south of the base line, and a new guide meridian is located at intervals of 24 miles east and west of a principal meridian. Make a diagram showing these lines. THE METRIC SYSTEM OF WEIGHTS AND MEASURES 374. 1. The system of denominate units of measure in com- mon use in the United States is practically the same as that in use in Great Britain, with the exception of the units used in measuring value. Nearly all the other civilized nations use a decimal system of denominate numbers, called the metric system. The metric system has been legalized by the United States and Great Britain, and has been adopted as the sys- tem for use in the Philippines and Porto Rico. It is exten- sively used in scientific work. 2. A little more than a century ago the French government invited the nations of the world to a conference to consider an international system of weights and measures. Later, the French government appointed a committee to devise a conven- ient system of denominate units. The committee originated what is known as the Metric System of Weights and Meas- ures. The metric system includes measures of length, surface, capacity, volume, and weight. The primary unit of linear measure is the meter The primary unit of each of the other measures is based upon the meter. 3. One ten-millionth part of the distance from the e(]na<^or to the North Pole, measured on the meridian of Paris, \\6 1.3(50489 5 1.216653 1.246182 1.276282 1.338226 1.402562 1.469328 6 1.265319 1.302260 1.340096 1.418519 1.500730 1.586874 7 1.315932 1.360862 1.407100 1.503630 1.606782 1.713824 8 1.368569 1.422101 1.477456 1.693848 1.718186 1.850930 9 1.423312 1.486095 1.651328 1.689479 1.838459 1.991HXI5 10 1.480244 1.652969 1.628895 1.790848 1.967151 2.168925 11 1.539454 1.622863 1.710339 1.898299 2.104852 2.331639 12 1.601032 1.095881 1.796856 2.012197 2.252192 2.618170 13 1.665074 1.772196 1.886649 2.132928 2.409845 2.719(524 14 1.731676 1.851946 1.979932 2.260904 2.578634 2.937194 15 1.800944 1.935282 2.078928 2.396668 2.75iK)32 3.172169 16 1.872981 ?.022370 2.182875 2.640352 2.952164 S. 426943 17 1.947901 2.113377 2.292018 2.692773 3.168816 3.700018 18 2.026817 2.208479 2.406619 2.864339 3.379932 3.9J)6020 19- 2.106849 2.307860 2.526950 3.026600 3.616528 4.316701 20 2.191123 2.411714 2.653298 3.207136 3.869684 4.660967 INDEX Abstract number, 16. Accident insurance, 278. Accounts, 42, 43, 274. Acute angle, 79, 221, 229. Acute-angled triangle, 221, 229. Ad valorem duty, 198, 273. • Addend, IG. Addition, of denominate numbers 77. effractions, 97-99, 108-112. of integers and decimals, 16-19. Additive method of subtraction, 22. Aliquot parts, 147, 208. Altitude, 221, 226, 295. Amount, in addition, 16. in interest, 208. Angle measure, 237, 318. Angles, 79, 221, 229, 318. Apothecaries' measures, 317. Apothem, 294. Appendix, 256-320. Approximate ansvrers, 28. Arabic numerals, 10. Arc, 237, 318. Are, 309. Areas, 78, 80, 225-232, 235, 292. Assessed valuation, 198, 270. Assessors, 198, 270. Austrian method of subtraction, 22. Avoirdupois weight, 316, 317. Bank, of deposit, 276. savings, 274. Bank accounts, 274. Bank discount, 218. Bank notes, 276. Banking, 274-278. Base, 221, 226. Base line, 299. BiUs, 42, 43. and receipts, 44. Board foot, 160. Bonds, 261. Broker, 192, 258, 264. Brokerage, 258, 262-264. Calendar, 812-314. Cancellation, 105. Cancellation method, 213. Capacity, measures of, 233, 234, 310, 811. Capital, 256. Carat, 317. Cash discount, 204. Certificate of deposit, 277. Check, 277. Cipher, 10. Circle, 231, 232, 237. area of, 232. Circular measure, 237, 318. Circumference, 222, 231, 237. City lot, 134. Clearing house, 278. Coins, value of, 319. Collector, of the port, 273. of taxes, 270. Commercial discount, 204. Commission, 192, 193, 262-264. Common divisor, factor, or measure, 104. Common multiple, 106. Common stock, 258. Composite number, 87. Compound denominate numbers, 72, 77. Compound interest, 217. table of, 320. Concrete number, 16. Cone, 295, 296. Consumer, 262. Corporation, 256. Corporation bond, 261. Correspondence bank, 278. Counting measure, 318. Coupon bond, 261. Credit, creditor, 42. Cube (rectangular prism), 233. Cube of numbers, 244. Cube root, 246. Cubic measure, 84, 85, 233, 309, 310, 815. Customhouse, 198, 273. Customs and duties, 198, 202, 203, 271-278. Cylinder, 222, 234, 235. — 21 321 322 INDEX Dates, difference between, 168, 267. Days of grace, 215. Debit, debtor, 42. Decimal point, 9. Decimal system, 7. Decimals, addition of, 18. division of, C8-70. multiplication of, 89-41. notation and numeration of, 15. reduction of, 156-158. subtraction of, 25. Degree, 237. Denominate numbers, 72, 70-86. tables of, 312-319. Denominator, 93, 109. Deposit, bank of, 276. Deposit slip, 274. Diagonal, 226. Diameter, 222, 231. Difference, 20. Direct taxes, 269. Discount, 171. bank, 218. cash, 204. trade or commercial, 204, 265. true, 220. Dividend, in division, 45. in insurance, 280. in stocks, 257. Divisibility tests, 87. Division, effractions, 117, 118, 124-131. of integers and decimals, 45-70. Divisor, 45. Draft, bank, 278. Dry measure, 318. Duties, 198, 202, 208, 271-278. Endowment policy, 279. Equation, 283-292. Equator, 238. Equivalents, 312. Even number, 87. Evolution, 246. Exact interest, 268. Exponent, 244. Extremes, 292. Face of note, 207. Factor, 45, 87, 104. Factoring, 89, 104. roots found by, 246, 247. Farm problems, 86. Fire insurance, 195. Flooring, 161. Foreign money, 819. Forms, 221-248. Fraction deflued, 98. Fractional unit, 93. Fractions, 90-165, addition and subtraction of, 97-99, 108-112. multiplication and division of, 113-131. reduction of, 96, 96, 100-105, 166-168. Gain and loss, 181-183. Government expenses, 272. Gram, 311. Greatest common divisor, factor, ormeasure, 104. Gregorian calendar, 314. Health insurance, 278. Heptagon, 294. Hexagon, 294. Horizontal lines, 221, 228. Hypotenuse, 250. Improper fraction, 94. Index of roots, 246. Indirect taxes, 270. Insurance, 195, 196, 278-283. Integers, defined, 9. Integers and decimals, 7-89. Interest, by aliquot parts, 208. cancellation method, 218. compound, 217, 320. defined, 167. exact, 268, 269. simple, 207-213. six per cent method, 218. sixtv dav method, 212. tables, 267, 320. Internal revenue, 198. Joint note, 214. Land measure, 297, 809. Latitude, 288. Law of commutation, 32. Least common denominator, 109. Least common multiple, 106. Life insurance, 278-283. Life policies, 279. Like quantities, 16. Linear measure, 76, 808, 816. Liquid measure, 318. List prices, 204. Liter, 310. Local taxes, 198, 270. Long division, 63-67. Long measure, 76, 808, 816. Longitude and time, 240-242. Loss and profit, 181-188. Lowest terms, 105. Lumber measure, 160-162. INDEX 323 Maker of note, 214. Marine insurance, 195. Market reports, 264. Maturity of note, 215. Means, 292. Measurement, division by, 48. Measurements, 76-86, 160-165, 221-243, 292-297. Measures, 312. Merchants' rule for partial payments, 216. Meridian, 238, 299. Meter, 804. Meter reading, 75. Metric system, 304-312. Minuend, 20. Mixed number, 94. Model bill, 42. Multiple, 45, 106. Multiplicand, 31. Multiplication, 31. effractions, 113-116, 119-123. of integers and decimals, 81—44. Multiplier, 31. Municipal corporation, 261. Negotiable note, 208. New style calendar, 314. Notation and numeration, 10-15. Note, 207, 208, 214. Number relations, 145, 146. Numerals, 10. Numeration, 10-15. Numerator, 98. Oblique angle, 79. Oblique line, 221, 223. Obtuse angle, 79, 221, 229. Obtuse-angled triangle, 221, 229. Octagon, 294. Odd number, 87. Old style calendar, 814. Par of stock, 257. Parallel lines, 80, 221, 223. Parallelogram, 221, 227. Partial payments, 216, 266. Partition, 48, 51. Payee, 207. Payer, 207. Pentagon, 294. Percentage, 166-220. Perfect square, 245. Periods, 10. Perpendicular lines, 79, 221, 223. Personal insurance, 278. Personal property, 198, 270, Pi {T), 231, Policy, 195, 279. Poll tax, 270. Polygons, regular, 293-294. Powers and roots, 244-251. Preferred stock, 258. Premium, on policy, 195. stock at, 257. Price lists, 265. Prime meridian, 238. Prime number, 87. Principal, 207. Prism, 221, 222, 283-235. Proceeds, 218. Producers, 262. Product, 81. Profit and loss, 181-188. Promissory note, 207, 214. Proper fraction, 94. Property, 198, 269. Property tax, 270. Proportion, 243, 291. Public lands, 297-299. Pyramid, 295, 296. Quadrilateral, 221, 227. Quotient, 45. Radical, 246. Radius, 222, 231. Railway time table, 74. Range lines, 299. Rate, of dividend, 257. of interest, 215, 274. of taxation, 198. Ratio, 54, 91, 145, 243. Real estate or real property, 198, 264, 270. Receipts, 44. Reciprocals, 125. Rectangle, 80, 81, 221, 224, 252. Rectangular solid, 84, 221, 224, 225, 233, Reduction of fractions, 100. Registered bond, 261. Reviews, 71-75, 137-144, 150-155, 184-187, 254, 255. Right angle, 79, 221, 223, 237. Right-angled triangle, 221, 226, 250, 251. Roman notation, 13. Roots, 244-251. Savings banks, 274. Scale drawing, 132-136. Section, 301, 802. Shares of stock, 256. Shingling, 162. Short methods, 88, 148. Similar figures, 253. 324 INDEX Similar fractions, 97. Similar surfaces and solids, 253. Six per cent method, 213. Sixty day method, 212. Slant height, 295. Solar year, 812. Solids, 84, 294-297. Specific duty, 198,273. Sphere, 29C, 297. Square, rectangle, 80, 224, 294, second power, 244. Square measure, 78, 315. Square root, 247-250. Standard time, 241. State taxes, 198, 270. Stere, 310. Stock quotations, 259, Stocks and bonds, 250-202, Subtraction, of denominate numbers, 77. effractions, 97-99, 108-112. of integers and decimals, 20-30. Subtrahend, 20. Sum, 10. Surface measure, 78, 308, 809, 815. Surveyors' measure, 81G. Tariff, 272. Tax collector, 270. Tax rate, 198. Taxes, 198-201, 269-271. Term policy, 279. Terms effraction, 94. Time measure, 812, 318. Tontine policy, 281. Town lines, 299. Townships, 298, 801. Trade discount, 204, 205, 265. Trapezoid, 228, 229, 292. Triangle, 221, 226, 229, 230, 294. Triangular prisms, 222, 233. Troy weight, 817. Unit, 7. Unit, fractional, 93. Unit of measure, 7. United States money, 14, 25, 819. United States rule, 2CG. Unknown quantity, 283. Usury, 215. Value, table of, 319. Vertex, 226. Vertical line, 221. Volume, 233-235, 309, 810, 315, 816. Weight, measures of, 811, 816, 317. re 35874 ivi55974 ^^ Mp: THE UNIVERSITY OF CALIFORNIA LIBRARY