Jffijln "Y. -c UNIVERSITY OF CALIFORNIA AT LOS ANGELES GIFT OF Dr. ERNEST C. MOORE ESSENTIALS OF ARITHMETIC ORAL AND WRITTEN BY J. W. McCLYMONDS CITY SUPERINTENDENT OF SCHOOLS, OAKLAND, CALIFORNIA AND D. R. JONES SUPERVISOR OF THE TEACHING OF ARITHMETIC STATE NORMAL SCHOOL, SAN FRANCISCO, CALIFORNIA NEW YORK : CINCINNATI : CHICAGO AMERICAN BOOK COMPANY COPYRIGHT, 1907, BY J. W. McCLYMONDS AND D. K. JONES. ENTERED AT STATIONERS' HALL, LONDON. ESSEN. OF A HIT II. W. P. 2 103 M 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 ^3 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 omitted from other courses have 2587 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 in 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 Numeration Addition Subtraction Multiplication Bills and Accounts Divi- sion by Measurement and Partition 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- estBank Discount Present Worth . . . 166-220 PART IV FORMS AND MEASUREMENTS Lines Angles Surfaces Solids Longitude and Time Ratio 221-243 5 6 CONTENTS PART V POWERS AND ROOTS PAGES Powers Square Root Right-angled Triangles Similar Sur- faces and Solids 244-255 PAKT VI 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 Metric System Tables of Denominate Measures Table of Compound Interest . . . 256-320 INDEX 321-324 ESSENTIALS OF ARITHMETIC PAET 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 $2. 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. 7 8 REVIEW OF INTEGERS AND DECIMALS 5. In 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 may be shown thus : one hundreds' unit one tens' unit one unit 10 1 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 order, or of units' order ; the units represented by 2 are called units of the second order, or of tens' 1 order ; and the units represented by 3 are called units of the third order, or of hundreds' 1 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/owr 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, hundredths, 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, as five, 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: CO DO CO OB O Q) 99 ORDERS : & oo-a cons < -5 1 8 3 III n g +3 a g *j S 5 3 S 5 S III 789, 605, 842, 031 PERIODS: Billions Millions Thousands Units NOTATION AND NUMERATION 11 3. Reading Integers. To read an integer of more than three figures, begin at the right of the number and point off periods of three places each. Mead 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 : a b c d 1. 3,625* 35,205 825,380 7,125,380 2. 8,017 20,007 308,016 6,000,150 3. 9,008 45,500 950,025 8,040,075 4. 6,303 12,012 404,040 5,505,050 5. What is the name of the first period? of the second period ? of the third period ? of the fourth period ? 6. How many periods are there in the numbers in column a? 6? 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. Read 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 first 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 highest 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 has 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, 33,630,592; buckwheat, 14,529,770. ROMAN NOTATION 13 ROMAN NOTATION 5. l. The letters used in Roman notation are: I V X L C D M 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 ? b. 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. o. 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 placed 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 MDCCLXXVL 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: $2.40. 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 $.001. 7. Reading United States Money. Read the following : 1. $425.15 5. $30.755 9. $8340.05 2. $301.08 6. $ 7.057 10. $9015.807 3. $220.20 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 15 9. Reading Decimals. j d xv w S H ii H ORDERS: "2 .2 -3 & a *? "& g j| g a 3 ffi H H W S 452876 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 Millionths (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. l. 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 & c d e / 9 h i j Jc 1 ra n 9 8 9 5 8 7 9 7 7 7 8 8 9 6 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. HCCL. & JONES'S ESSEN, or AH. 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. 4. 5. $ 345.67 $ 58.06 $ 9.045 $405.27 $ 68. 84.075 275.936 590. 73.435 125.87 650. 83.07 5.15 487.50 45.369 70.004 342.457 69.075 50.258 845.075 572.806 34.08 610.75 250.50 8.75 6.605 8.125 57.246 .375 100. 852.451 64. 540.375 62.50 58.268 6. Read aloud each of the above. 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 $68.05, $107.98, $730.04, $9.75, $894, $80, $740.40, $375.15, $486.75, $836.95, $.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. ADDITION 19 15* Oral Exercises. Add: a b 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. 70 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 66 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 Ib. and his little brother weighs 34 Ib. 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 $94 for a wagon and $36 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 f 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. l. 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, how 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 ? b. 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 12. _ r is read 5 and how many are 9? Or, Bfrom 9 leaves 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. l. Find the missing addend. (one addend) The other addend may be 2874 (one addend) found b ? addin to the S iven coo/3 f . ij j \ addend the number that will 5236 (sum of two addends) . ,, , n ' 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 Add to the subtrahend the number that will 2874 give the minuend, thus : 4 and 2 are 6 ; 7 and 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. Write 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. MODBL 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 10. 2,814,004-935,940 SUBTRACTION 23 20. Oral Exercises. Subtract : a 6 c d e / 9 h i 1. 40 140 150 100 120 150 120 90 110 20 30 20 40 30 60 90 40 50 2. 95 126 83 142 149 155 153 129 124 40 90 50 60 80 70 20 40 80 3. 124* 109 138 96 75 139 136 88 99 92 44 85 13 24 63 45 16 44 4. 75 f 142 34 57 83 74 42 36 52 Q Q Oo 96 19 29 68 18 27 19 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 between 92 and 124 is 30 and 2, or 32. t SUGGESTION. The difference between 38 and 75 is 30 (38 to 68) and 7 (68 to 75), or 37 ; or 40 less 3, or 37. 24 REVIEW OF INTEGERS AND DECIMALS 21. Before solving, represent each by a diagram. l. 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. 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 56 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 the park, Henry had gained 200 ft. on him. At the same rate of gain, how many times will Frank ride around the park before Henry overtakes him ? SUBTRACTION 26 22. United States Money. Write units of the same kind below one another. Do not supply unnecessary O's. 1. Subtract: a. $12.75 from $37.25; b. $12 from $37.25; c. $12.75 from $37. MODEL a: $37-25 MODEL b: $37.25 MODEL c: $37. 12.75 12. 12.75 $24.50 $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 IJ 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 Ib. and a cubic foot of petroleum weighs 54.875 Ib. How much heavier is a cubic foot of rain water than a cubic foot of petroleum (kerosene) ? 26 REVIEW OF INTEGERS AND DECIMALS 24. l. Show the effect, if any, upon the difference : (a) of adding the same number to both minuend and subtrahend; (J) 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, 76. Add the following exercises in a similar manner : a & c d e / 9 A i / Jc I m (5 7 8 4 1 5 7 4 5 8 7 8 9 J5 3 2 6 9 5 2 6 5 7 3 9 8 8 6 9 7 6 4 8 5 6 7 7 1 2 |5 3 8 4 1 3 4 7 6 3 8 9 7 15 7 2 6 9 7 6 3 4 8 2 4 6 6 8 7 9 8 8 6 6 7 6 6 6 4 |5 7 2 6 9 7 8 2 1 4 5 6 5 Is 3 8 4 1 3 2 8 9 8 5 5 9 7 5 5 8 6 9 9 7 7 2 3 2 1 f5 3 2 6 9 5 4 1 3 7 9 8 5 Is 7 8 4 1 5 6 9 8 9 1 7 9 6678999997783 98978652968 84 2. Write ten columns, in each of which some of the five combinations whose sums are 10 occur several times. Add these columns. SUBTRACTION 26. Written Exercises. 27 NEW EM, LAND STATES AKI:A IN SQ. MILES GREAT LAKES AREA IN S$12 2. Write a problem for each : 3)H8 4)20? |6)fl2 $7)$21 5)10 yd. 3. 4)12 may be either partition or measurement. State what is meant by 4)12 (a) when it is partition ; (5) when it is measurement. 4. Is the divisor ever concrete in partition? Is the quotient ever concrete in measurement ? Give reasons. 5. In division by measurement the quotient is always what kind of a number ? 6. When the, divisor is a concrete number, is the divi- sion partition or measurement ? 7. Make ten problems in division, and tell which are partition and which are measurement. 8. Make problems for each of the following. Tell which are partition and which are measurement. 82)$ 10 2)$10 4)16 yd. 8wk.)16wk. 3)$15 5)25^ 63. Oral Exercises. J^- indicates that 12 is to be divided by 3. Solve each : 1. - 1 / 6. 4 11. Y 16. Y- 2. ^- 7. Sf. 12. 9- 17. ^L 3. J^l 8 . jyL 13 . S 18. 4 y 3. 4. JUL 9. jyi 14. Jijt 19. Jj. 5 . s 10. AA is. 20 - DIVISION 53 64. Written Exercises.* 1. At $8 each, how many tables can be bought for $128? MODEL for measurement: 16, number bought for $ 128, cost of 1 table, $8)$128 2. A man spent $216 in 6 mo. What was the av- erage amount spent each month ? MODEL for partition : . , f 36, spent in 1 mo. 6) f 216, spent in 6 mo. Tell which of the following are partition and which are measurement, and solve: 3. If a boy saves $5 a month, in how many months will he save $120? 4. How many tons of coal at $7 a ton can a man buy for $161? 5. Five boys agreed to share equally the expenses of a camping trip. The trip cost them $21.70. What was each boy's share ? 6. A girl bought 8 yd. of cloth for $2.56. How much did the cloth cost her per yard ? 7. A hardware merchant bought some stoves at $9 each. His bill amounted to $198. How many stoves did he buy ? 8. If a boy worked 65 problems correctly in 1 school week (5 days), what was the average number worked correctly each day ? 9. A dealer bought 6 copies of a certain book. His bill amounted to $7.50. What was the price of the book ? * Give the oral analysis of Probs. 1-9. 54 REVIEW OF INTEGERS AND DECIMALS 65. Ratio or Comparison. 1. How many times must the measure 2 ft. be applied in measuring 6 ft. ? The number 3 expresses the ratio, or relation, of the quantity 6 ft. to the unit 2 ft. 2. In measuring 6 ft. by 2 ft. the quantity to be meas- ured is 6 ft., and the unit of measure is 2 ft. The ratio of 6 ft. to 2 ft. is found by dividing 6 by 2. 3. What is the ratio of 8 ft. to 4 ft.? of 12 ft. to 3 ft.? 4. What is the ratio of 6 da. to 3 da.? of 24 hr. to 6 hr.? of 25/to5^? of $1 to $.25? of 75^ to 25^? 5. What is the ratio of 100 to 50 ? to 25 ? to 10 ? to 20? to 5? 6. In measuring 2 ft. by 4 ft. the unit of measure is 4 ft., and the quantity to be measured is 2 ft. The meas- ure 4 ft. is applied ^ time ; that is, one half the measure is applied in measuring 2 ft. The ratio of 2 ft. to 4 ft. is ^. 66. Draw on the blackboard lines the length of the quantities to be measured. Make measures the length of the units of measure to be used in measuring each line. By applying the measure to the line to be measured, de- termine the ratio of the following: 1. Of 2 ft. to 1 ft. 7. Of 1* ft. to 3 ft. 2. Of 2 ft. to J ft. 8. Of J ft. to 1 ft. 3. Of 1 ft. to 1 ft. 9. Of f ft. to 2 ft. 4. Of 3 ft. to ft. 10. Of 11 ft. to 2 ft. 5. Of 1 ft. tO 1 ft. 11. Of f ft. to $ ft. 6. Of I ft. to 2 ft. 12. Of f ft. to ft. 13. What part of 14 da. are 7 da.? What is the ratio of 7 da. to 14 da.? of 14 da. to 7 da. ? DIVISION 55 14. 3 in. is what part of 6 in.? What is the ratio of 3 in. to 6 in..? The ratio tells the number of times the unit of measure must be applied to measure the given quantity. A 6-in. measure must be applied times to measure 3 in. A 3-in. measure must be applied times to measure 6 in. The ratio of 6 in. to 3 in. is . The ratio of 3 in. to 6 in. is . 15. What part of the measure 12 in. must be applied to measure 3 in.? What is the ratio of 3 in. to 12 in.? of 12 in. to 3 in.? 16. The ratio of 3 yd. to some quantity is ^. What is the quantity ? 17. The ratio of some quantity to $ 2 is 4. What is the quantity ? 18. If 4 is the ratio of some amount to f 20, what is the amount? 19. A -ft. measure was used 12 times in measuring the length of a line. How long was the line ? 20. Draw a line of such length that a 6-in. measure will be applied 1^ times in measuring it. Prove your work by applying the measure. 21. What is the ratio of 2 to 8 ? of 6 to 2 ? of 20 to 5 ? of 5 to 30 ? of 8 to 48 ? of 40 to 8 ? 22. Two tons of coal will cost what part of the cost of 8 tons ? of 6 tons ? of 12 tons ? 23. 3 yd. of cloth will cost what part of the cost of 9 yd.? of 15 yd.? of 6 yd.? of 12 yd.? 24. If the cost of 24 yd. of cloth is given, how may the cost of 8 yd. be found ? of 6 yd.? of 4 yd. ? of 12 yd.? 25. If the cost of 6 sheep is $24, what is the cost of 18 sheep? of 12 sheep? of 3 sheep? of 2 sheep? 56 REVIEW OF INTEGERS AND DECIMALS 67. Oral Exercises. l. If 5 desks cost $ 20, how much will 7 desks cost ? The quantities 5 desks and 7 desks are measured by the common unit 1 desk. In solving this problem, first find the cost of the unit 1 desk. Next find the cost of the required number of units. MODEL for oral recitation : If 5 desks cost <$ 20, 1 desk will cost of $ 20, or <$4. Since 1 desk costs $4, 7 desks will cost 7 times <$ 4, or 2. If 4 tons of hay cost $ 32, how much will 6 tons cost ? 3. If 7 tablets cost 35 ^, how much will 4 tablets cost ? 4. At the rate of 5 for 25^, how much will 8 spelling- blanks cost ? 5. A girl paid 30 ^ for 5 yards of ribbon. How much would 8 yards have cost at the same rate ? 6. Make ten additional problems similar to the above. 68. Written Exercises. 1. A farmer raised 220 bu. of oats on 4 acres of land. How much at the same rate would a 7-acre field have produced ? 2. If a train travels 138 mi. in 3 hr., at the same rate, how far will it travel in 8 hr. ? 3. From a farm of 160 acres 8 acres were sold for $500. At this rate, what was the value of the entire farm ? 4. 7 men picked 210 boxes of prunes in 1 day. At the same rate, how many boxes would 15 men have picked ? 5. The expenses of a family amounted to $328.75 for 5 mo. At the same rate, what would the expenses amount to in 1 yr. (12 mo.)? DIVISION 57 69. Oral Exercises. Finding a part of an amount, when the amount is given : 1. Find | of 12 ft. 12ft t |_ I v 3ft. 3ft. 3ft. j 3ft. 9ft To find \ of 12 ft., divide 12 ft. into 4 equal parts. Then f of 12 ft. will be 3 of these parts. 2. Show by a diagram that \ of 12 ft. is 4 ft. and that f of 12 ft. are 8 ft; that of 8 yd. are 6 yd. 3. Show with objects that -| of 10 objects are 6 ob- jects; that of 6 objects are 4 objects. 4. Show by a diagram that if a board is 8 ft. long, | of the length of the board is 6 ft. 5. What is | of $12? MODEL for oral recitation: Since \ of $12 is $3, f of $12 is 3 times $3, or $9. 6. What is J of $15? of $24? of $30? of $12? 7. What is | of 10 mi.? of 25 mi.? of 35 mi.? 8. What is | of 18 lb.? of 30 lb.? of 42 lb.? of 60 lb.? 9. At 20^ a pound, how much will f of a pound of cheese cost? 10. How many months are there in | yr.? in | yr.? 11. If Fred worked 15 problems and John worked % as many, how many did John work ? 12. A girl worked 16 problems, and | of them were correct. How many of them were correct ? 13. How many inches are there in f of a foot ? 14. At $8 a ton, how much will f of a ton of coal cost? 15. Make ten additional problems similar to the above. 58 REVIEW OF INTEGERS AND DECIMALS 70. Written Problems. 1. There are 2000 pounds in a ton. How many pounds are there in ^ of a ton of hay ? 2. There are 320 rods in 1 mile. How many rods are there in of a mile ? O 3. A girl read a book containing 210 pages. How many pages had she read when she had read | of the book ? 4. Two boys, Henry and Frank, bought out a news- paper route that cost them $4.50. Frank paid |- of the cost of the route and Henry paid ^ the cost. How much did each pay ? 5. A man had 320 acres. He rented f of his land. How many acres did he rent? 6. There are 5280 feet in 1 mile. How many feet are there in | of a mile ? 71. Dividing by 20, 30, 40, 200, 300, etc. 1. State a short method of dividing a number by 10 ; by 100. 2. Divide 476 by 40. MODEL : 11.9 First place a decimal point in the quotient above and 40^476 between 7 and 6. Then divide by 4. 3. State how you would divide a number by 50 ; by 400 ; by 4000. Solve. Before dividing, estimate each quotient: 4. 324-40 8. 1260-5-400 12. 4860 -5- 40 5. 1728-5-60 9. 5280-*- 600 13. 3600-900 6. 1720-80 10. 7854-500 14. 4240-800 7. 320-5-20 11. 6250-5-500 15. 1240-5- 30 DIVISION 59 72. Oral Exercises. Finding an amount, when part of the amount is given : 1. When f of the length of a board is 8 ft., what is of the length of the board ? 8ft. If | of the length of a board is 8 ft., ^ of the length of the board is what part of 8 ft. ? If ^ of the length of a board is 4 ft., what is f of the length of the board ? 2. Show by a diagram that if f of the length of a line is 6 ft., I of the length of the line is 2 ft. If \ of the length of a line is 2 ft., what is the length of the line ? 3. Show by a diagram that if f of a line is 6 ft. long, ^ of the line is 2 ft. long and the line is 8 ft. long. 4. Using 12 objects, show that since ^ of 12 objects is 4 objects, of 12 objects are 8 objects. 5. Show that since $ of 12 objects are 8 objects, ^ of 12 objects is of 8 objects. 6. Draw a diagram to show the length of a room, if | of the length of the room is 9 ft. 7. If $12 is | of the cost of a suit of clothes, what is ^ of the cost of the suit? What is the cost of the suit? MODEL for oral recitation : If $12 is f of the cost of a suit, | of the cost of the suit is \ of $12, or $4. Since $4 is of the cost of a suit, the cost of the suit is 4 times $4, or $16. 8. If $20 is of the cost of a cow, what is the cost of the cow? 9. If | of the cost of a book is 40/, what is the cost of the book? 60 REVIEW OF INTEGERS AND DECIMALS 10. Two boys together bought a baseball. One boy paid $.60, which was f of the cost of the ball. How much did the other boy pay ? What was the cost of the ball ? 11. A boy spent 90^, which was f of the whole amount of money he had. How much money had he ? How much money had he left ? 12. Fred weighs 100 Ib. This is ^ of George's weight. How much does George weigh ? 13. Make ten additional problems similar to the above. 73. Oral Exercises. Two addends whose sum is 10 or less may be taken as a single addend. Exercise a below may be added : 13, 23, 30, 47, 54, 62, 7-0. Add b-m in a similar manner : a b c d e / 9 h i J fc I m 8 8 8 7 2 3 8 7 3 2 3 2 4 f 4 6 1 6 5 6 5 4 6 7 3 1 7 {4 1 3 2 2 7 3 5 2 7 7 5 2 [2 3 2 2 5 6 8 8 9 2 3 6 4 i5 2 4 2 2 3 5 5 7 5 5 8 8 {0 4 5 5 5 3 3 4 6 2 4 3 3 9 2 2 4 2 6 5 7 2 7 3 4 1 {o 3 6 7 5 8 9 4 9 8 3 2 4 4 6 7 8 7 6 5 5 2 7 8 2 5 f 6 2 2 6 2 3 3 7 . 6 2 3 7 7 14 3 2 3 5 9 8 9 7 6 3 8 2 5 8 7 7 3 3 3 5 2 2 9 3 3 8 6 9 4 4 6 5 4 6 7 8 7 6 Write ten columns in which two addends whose sum is 10 or less may be taken as a single addend. Add your columns. Add the columns in Sec. 13. REVIEW . 61 REVIEW 74. Multiplication. Name the multiples of 2 to 24 ; of 3 to 36 ; of 4 to 48 ; of 5 to 60 ; of 6 to 72 ; of 7 to 84 ; of 8 to 96 ; of 9 to 108. Multiply the numbers in each column by the number at the head of the column, and add to each product the num- ber in parentheses : abed e f g h 2 (9) 3 (2) 4 (3) 5 (4) 6 (5) 7 (6) 8 (7) 9 (8) 654982 8 4 7 2 8' 2 7 3 3 8 . 9 9 3 3 6 4 7 5 8 7 6 8 2 8 4 9 4 6 9 6 9 7 6 2 5 3 7 5 3 5 5 6 7 8 9 7 6 9 7 8 9 6 6 9 8 8 6 6 6 9 8 6 7 6 8 7 8 7 7 8 9 7 9 75. Write the following in the forms of bills, and find the amounts in each. (See p. 42.) 1. Jan. 2, 1907 : 3 bars of soap at 6^ each ; 4 Ib. of prunes at 8/ per pound; 85 Ib. of potatoes at 2^ per pound. Jan. 5, 1907 : 2 Ib. of coffee at 33^ per pound; 2 Ib. of cheese at 18^ per pound ; 3 Ib. of tea at 55^ per pound. 2. Jan. 9, 1907 : 7 yd. dress cloth at $1.20 per yard ; 1 doz. handkerchiefs at $1.40 per dozen ; 3 shirts at $1.75 each. Jan. 10, 1907 : 5 pair socks at $.25 each ; 1 um- brella at $2.40 ; 1 pair scissors at $.75. 62 REVIEW OF INTEGERS AND DECIMALS 76. Division. l. Name the highest multiple of the number at the head of the column in each number in the column, thus for column a : 18, 9, 15, etc. a b c d e / 9 3 4 5 6 7 8 9 20 17 38 39 40 30 40 11 10 17 28 25 46 78 16 23 26 11 58 37 31 8 25 14 22 13 22 86 25 31 23 5.8 32 76 68 14 14 49 46 65 53 23 23 19 33 33 39 60 59 17 27 43 52 52 29 60 28 38 24 16 18 69 51 19 29 39 40 60 18 34 22 34 28 50 34 59 43 13 21 34 55 68 14 70 29 39 63 63 87 97 75 24 26 32 70 48 50 39 32 43 54 57 29 17 110 2. Name the highest multiple of the number at the head of the column in each number in the above columns, and give the difference between the multiple and the number in the column, thus for column a ; 18 and 2 ; 9 and 2 ; 15 and 1, etc. 3. Divide the numbers in each column by the number above the column, giving the quotient and remainder, thus for column a : 6 and 2 over; 3 and 2 over, etc. 4. Divide 21,487,249 by 6; 459,738,795 by 8; 59,482,395 by 7; 708,718,907 by 9. LONG DIVISION 63 LONG DIVISION 77. When all the steps in division are written, the process is called long division. Long division is generally used with divisors of two or more places. For the purpose of finding the quotient figure, all divi- sors are classified into two general cases, as follows : CASE I. All divisors in which the second figure* is the same as or less than the first figure, as 66, 65, 832. CASE II. All divisors in which the second figure is greater than the first figure, as 15, 68, 271, 795. Write in order all numbers from 13 to 100 that come under Case I, as 21, 22; all that come under Case II, as 13, 14, 15. CASE I 78. When the second figure of the divisor is the same as or less than the first figure, a trial quotient figure may be found, as in the following : Divide 2792 by 65. It will take three places at the left of 4^ 2792 to contain 65 at least once. MODEL : 65)2792 Step 1. 6 is contained in 27 four times, with 3 remainder. Is 5 contained in 39 as many as 4 times ? Yes. Try 4 as a quo- tient figure. Place 4 in quotient above 9. Step 2. Multiply the divisor 65 by 4, writing the product below 279. Step 3. Subtract 260 from 279. Step 4. Bring down the next figure in the dividend. The new dividend is 192. Repeat Step 1. 6 is contained in 19 three times, with 1 remainder. Is 5 contained in 12 as many as 3 times? No. Try 1 less than 3, or 2, as a quotient figure. Complete the work, writing the remainder as in short division. * In 65 regard 6 as the first figure and 6 as the second figure. 64 REVIEW OF INTEGERS AND DECIMALS With divisors of Case I, the trial quotient figure found as in the model is always the correct quotient figure when the divisor contains two places, as 42, 87, etc. It is also the correct quotient figure when the divisor contains three or more places, whenever the second figure of the divisor is not contained in its dividend as many times as the first figure is contained in its dividend. With divisors of Case I, when the first figure of the divisor is contained in its dividend ten times, the correct quotient figure is 9, thus : 9 532)52678 5 is contained in 52 ten times. Take 9 as the quotient figure. It is unnecessary to test the second figure of the divisor. 79. Written Exercises. Tell to which case each divisor in the following belongs : Solve only the exercises in which the divisors are of Case I. 1. 75,679 by 42 13. 15,672 by 29 25. 57,606 by 21 2. 73,496 by 24 14. 71,896 by 83 26. 40,000 by 20 3. 12,500 by 64 is. 83,678 by 95 27. 59,684 by 84 4. 62,847 by 66 16. 79,678 by 88 28. ' 85,678 by 96 5. 95,438 by 27 17. 53,678 by 61 29. 45,672 by 63 "~6. 18,245 by 29 18. 29,678 by 54 30. 456,783 by 15 7. 1'0,000 by 75 19. 38,006 by 47 31. 648,739 by 16 8. 60,000 by 85 20. 47,608 by 96 32. 457,820 by 14 9. 35,640 by 44 21. 64,896 by 88 33. 426,789 by 13 10. 11,045 by 19 22. 52,873 by 68 34. 480,068 by 96 11. 16,712 by 18 23. 49,678 by 79 35. 124,530 by 144 12. 27,672 by 28 24. 68,368 by 72 36. 231,672 by 772 LONG DIVISION 65 CASE II 80. For the purpose of finding a trial quotient figure that will seldom vary much from the correct quotient figure, divisors of Case II are classified into three groups. GROUP a. When the second figure of the divisor is 7, 8, or 9, as 17, 18, 19 ; 578, 588, 598, etc. GROUP b. When the first figure is more than 1 and the second figure is 3, 4, 5, or 6, as 23, 24, 35, 46, etc. GROUP c. When the first figure is 1 and the second figure is 3, 4, 5, or 6, i.e. 13, 14, 15, 16. Write the numbers from 13 to 100 that come under Group a, Case II ; that come under Group 6, Case II ; that come under Group c, Case II. 81. Group a. With divisors of Group a, the trial quotient figure may be found by using as a divisor 1 more than the first figure of the divisor, as in the following : 1. Divide 379,868 by 476. MODEL : 79 It will take four places to contain 476 at least d.7ftVV7Q$}8 once. 5 (1 more than 4) is contained in 37 seven times. Try 7 as a quotient figure. Complete the division. 4666 The trial quotient figure found as in the model will sometimes be 1 less than the correct quotient figure. 2. Solve all exercises in Sec. 79 in which the divisors are of Group a, Case II. 3. Write and solve five exercises in division, using divisors of Group a, Case II, and five exercises using divisors of Case I. MCCL. & JONES'S ESSEN. OF AR. 6 66 REVIEW OF INTEGERS AND DECIMALS 82. Group b. With divisors of Group 6, Case II, the trial quotient figure may be found by using as a divisor 1 more than the first figure of the divisor and adding 1 to the quo- tient, as in the following : ' 1. Divide 11,678 by 24. 4 , , 1 X-.-, nr , It will take three places to contain 24 JVlODEI * / &. ) 1 I r\i ?s at least once. 3 (1 more than 2) is con- "^ tained in 11 three times. Try 4 as a 207 quotient figure. Complete the division. The trial quotient figure found as in the model will sometimes vary 1 from the correct quotient figure. 2. Solve all exercises in Sec. 79 in which the divisors are of Group 5, Case II. 3. Write and solve ten exercises in division, using di- visors of Groups a and 6, Case II. 4. Write and solve five exercises in division, using divisors of Case I. 83. Group c. With divisors of Group c, Case II, the trial quotient figure may be found by using 2 as a divisor, and adding 2 to the quotient, as in the following : 1. Divide 12,678 by 142. It will take four places in the divi- dend to contain 142 at least once. 2 is MODEL: 142)12678 cont ained in 12 six times. Try 8 as a quotient figure. Complete the division. The trial quotient figure found as in the model will seldom vary more than 1 from the correct quotient figure. 2. Solve all exercises in Sec. 79 in which the divisors are of Group c, Case II. 3. Write and solve ten exercises in division, using divisors of Case IT. LONG DIVISION 67 84. With 1367 as a dividend, find the first trial quo- tient figure, using each of the following as divisors, and explain how each is found : 32, 16, 327, 48, 59, 24, 375, 698, 156, 426, 276, 149, 137, 161. 85. Written Exercises. 1. Write and solve ten exercises in division, using as divisors numbers of Case I between 100 and 1000. 2. Write and solve ten exercises in division, using as divisors numbers of Groups a, 6, and c, Case II, between 100 and 1000. 86. Written Exercises. 1. How many dozen eggs at 18^ a dozen must be sold to pay for 1 Ib. of tea at 60^, and 1 Ib. of coffee at 30^? 2. A man's yearly salary is $ 1860. Find his salary per month. What is the amount of his salary per week? 3. If a train travels at an average rate of 45 mi. an hour, in how many hours will it travel 2000 mi. ? 4. A ton is 2000 Ib. How many pupils of your own weight will it take to weigh IT.? 5. A bushel of wheat weighs 60 Ib. How many bushels will it take to weigh IT.? 6. The monthly rental of an apartment house amounted to $144. The average rental of an apartment was $24. How many apartments were there in the house? 7. The total annual expenditure of the United States government for the year ending June 30, 1905, amounted to $532,122,762.47. What was the average daily expenditure? 8. The number of school children in a certain city is 1640. If the average number of pupils to each room is 40, how many schoolrooms are there in the city? 68 REVIEW OF INTEGERS AND DECIMALS DIVISION OF DECIMALS 87. 1. How many $2 are there in $4? How many 2 tenths are there in 4 tenths? .4 -*- .2 = x? 2. How many 3 qt. are there in 9 qt. ? How many 3 hundredths are there in 9 hundredths? 9 qt. -i-3qt. =x? .09 -=- .03 = z? 3. How many times are 4 yd. contained in 8 yd. ? 4 tenths in 8 tenths? 4 hundredths in 8 hundredths? 4 thousandths in 8 thousandths? 4. What is the quotient in each of the following : 4)8; .4)78"; .04)708; .004)7008? Prove the correct- ness of your answer by multiplying the divisor by the quotient and comparing it with the dividend. 5. If the divisor contains tenths, tenths of the divi- dend may give a whole number in the quotient. _3 36 .4)1.2 .6)21.6 6. If the divisor contains hundredths, hundredths of the dividend may give a whole number in the quotient. __9 59 854 .05). 45 .04)2.36 .04)34.16 Place the decimal point in the quotient above and after the figure in the dividend occupying the same order as the lowest order in the divisor. Divide as in integers. 7. Divide 21.66 by 6. q -| MODEL : As the lowest order in the divisor is units, 6)21.66 place the decimal point in the quotient above and after the figure occupying units' order in the dividend. Divide as in integers. When the divisor is an integer, the lowest order in the divisor is units, and the decimal point in the quotient is directly above the decimal point in the dividend. LONG DIVISION 69 8. Divide 43.38 by .8. MODEL- 0x40 0*0 As the lowest order in the divisor is tenths, place the decimal point above and after the figure occupying tenths' order in the dividend. Divide. 9. Divide 2.4 by .006. MODEL : 006^2 400 ~^ s *^ e ^ owes ^ or der in the divisor ia thousandths, supply two ciphers in the dividend to make the lowest order thousandths. Place the decimal point above and after the figure in the dividend occupying thou- sandths' order. Divide. 10. With 45.06 as a divisor, the decimal point will be placed in .the quotient above and after the figure in the dividend occupying hundredths' order. State where the decimal point should be placed with each of the following as divisors: 5.05; 67; 3.15; 3.1416; .008; 26.1; .0045; 50; .05; 2150.42; 6. 88. Arrange as in the models and fix the decimal points in the quotients : a b c d 1. .05 + 2.5 .04-*-. 002 180 + .006 3 + 4 2. .6 + . 3 2.5 + 50 36 + 750 20 + 50 3. 1.44 + .12 .0048 -5- .6 .27 + 3 1.75 + .025 4. 2.7H-.9 3.6 + .12 .007+3.5 .075 + 2.5 5. .48-5-8 .16-1-20 14 + .007 120 + .004 6. .1 + .005 2.4 + 12 22.5 + .15 4.8 + .0012 7. .024 --.8 .012 -.03 2 + 10 .065 + 3.25 8. 3.6 + .006 .005 + .1 45 + 90 64 + .0008 9. 1 + .045 4 + 4.56 100 + .1 .75 + .6 10. 10 + .01 .01 + 10 .001 + .01 101 + 1.01 70 REVIEW OF INTEGERS AND DECIMALS 89. l. Divide 75.51 by 60.4, and carry the result to two decimal places. 1.25 + MODEL: 60.4)75.51 The or | ; of D to (7 is f, or |. 7. What is the ratio of D to E? ofEtoD? of 18 to C? 8. If A represents 10 acres, what does B represent ? C? D? El 9. If B represents 40 acres, what does A represent ? C? J)? E? 10. If the cost of the land represented by B is $100, what is the cost of the land represented by A? C? D? E? 11. If the area represented by E is 610 acres, what is the area represented by C? B? A? D? 12. Draw two lines such that the ratio of one to the other is |; \\ 2; 5. 13. Draw oblongs such that the ratio of one to the other is f ; 1; |; 3; f. 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 f 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 : |> , -|, f yd., -^ 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 \. 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 |^ mi.; ^ mi.; | mi. Show the part whose ratio to the whole line is ^, ^, , f, |. Show the part to which the ratio of the whole line is 2: 8: 4: 4: 4: *. 94 FRACTIONS 118. i. f wk., | yd., f gal., l| yr., f, ^ lb., f, f mi., ^, T V a. Read aloud each of the above fractions. b. 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. f. 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 f, ^, etc. Name ten proper fractions. 4. A fraction whose numerator is equal to or greater than the denominator is called an improper fraction, as I-, f , etc. Name ten improper fractions. 5. When a number is composed of an integer and a fraction, it is called a mixed number. 6 is a mixed num- ber. Its value is expressed in two different units. The 6 is expressed in units of ones; the % is expressed in units of one fifths. 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 5| may be expressed in the fractional unit ^. There are | in 1. In 5 there are 5 times |, or -^-. % and | are %-. What kind of a fraction is --? REDUCTION 95 REDUCTION 119. Changing Mixed Numbers to Improper Fractions. Change 4f 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 by the denominator of the fraction, add the numerator, and write the sum over the denominator of the fraction. 120. Oral Exercises. Change the following to improper fractions : * a b c d e f g h i j k 1. 61 7f 8f- 9| 5| 4f 3| 2| 1 8 8^ 3. 5|- 5J If 9 2f 8| 8} 7f 4} 6f 5f 4. 2f 9f 7f 4| 7| 5| 6| 3f 8| 2f 9| 5. 2f 9f 7| 6| 3| 7-| 6f 4^ 2| 7f 3| 6. 7. 8. Write ten mixed numbers and change them to im- proper fractions. 9. Express the value of the following integers in the fractional unit J : 3, 5, 7, 6, 9, 2, 8, 10, 12. 10. Write ten proper fractions. State what the frac- tional unit is in each. 11. 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 f ? in J^? in ? 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 by the denominator. 122. Oral Exercises. Change the following to whole or mixed numbers : * a b cdefghijk i- -V- 2 . ^ ^ ^ ^ ^ M. _4Jl M. JgL J ^ 3- V *' 5. e. tf W tt ft W ft 7 . i| f i w y '> ft tt ft '. ft ft 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 : f , f , ^ ? 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 2f ft., 5| ft., and 6 ft. MODEL : 2f ft. Add the fractions first : $ ft. and ft. and f ft. 5^ ft. are f ft., which are equal to 1J ft. Write $ ft. Q 1 f k below the column of fractions, and carry 1 ft. to iji P, the column of whole numbers. X i T'K It. 6. Add the following : abcdefgh i 6 i 3 3 i 4 f 4| 5 6f 7 8 7 9| 9 1% 6 7f 5 3f 8| 4 64 8i 8| 4 to 7 Z o y 6J 9f 8f 8f 61 2J 21 JONES'S ESSEN. OF AR. 7 98 FRACTIONS i. ^ da. = f da. w .~ rjp O =z 4| mi. 124. Oral Exercises. 1. | ft. \ ft. = | ft. I mi. | mi. = f mi. ^ yr. 7_ y r . = _*_ yr. A 2. 6f ft. - 4 ft. = ft. - 2 mi. = mi. 3. Subtract the fractions first and then the whole numbers : abed e 8f wk. $9f 5| yd. 5f yr. 6f Ib. If wk. $4* 3* yd. 21 yr. 44 Ib. I O / D */ 8f wk. 5 wk. h 24f yd. 17| yd. 29* yd. 19$ yd. 4. Find the sum of each of the above. 5. Subtract 3^ ft. from 6 ft. Since there is no fractional part in the minu- MODEL 6. Subtract : 8 hr. T da. 4| hr. 21 da. 7. Subtract : 7} yd. 9fyr. 2^ yd. 6 yr. end, the sum of the fraction of the subtrahend and the fraction of the difference is 1 ft. | ft. and | ft. are 1 ft. Carry 1 ft. to 3 ft. and sub- tract the integers. 4 ft. and 2 ft. are 6 ft. ft. ft. 8 A. 4JA. mi. 34 mi. o 5f wk. 3f wk. 8 Ib. 31 Ib. 6 yd. 3|yd. mi. mi. 8|hr. 4 hr. 8. If a boy attended school how many days was he absent ? da. in a certain week, ADDITION AND SUBTRACTION 99 9. A girl who was taking lessons on the piano prac- ticed as follows during one week : Monday, 1^ hr. ; Tuesday morning, 1 hr. ; Tuesday afternoon, | hr. ; Wednesday, 1| hr. ; Thursday, f hr. ; Friday, 1 lir. ; 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 3f mi. on foot and rode the remainder of the distance, how far did he ride ? 125. Oral Exercises. 1. Subtract 6 from 9|. MODEL : 91 The sum of f and the fraction of the difference . is Ig. Find what must be added to f to make 1 and 93 add it to | . | and j[ are 1. and | are f. Carry 1 to 6. 7 and 2 are 9. The numerator 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 : abode f g h i 5f 9f 7| 8J 9# 7A *# Jl' 'a M "M\ftt ft SI- i t 91 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 10^ yd. in making a dress. How much cloth was left ? 100 FRACTIONS REDUCTION 126. Changing to Higher and Lower Terms. B 1 _ 1 a 2 1. If A represents a unit divided into 2 equal parts, what does B represent ? 0? JO? El F? f 2. The fractional unit of B is ^; what is the fractional unit of O? D? JE? F? What part of the fractional unit of A is the fractional unit of^? O? D? El F? How many of the fractional units of B does it take to make one of the fractional units of A ? How many of (7? of D? of Jf? of I 7 ? 3. The fractional unit -| is what part of the fractional unit J ? \ = f . | is what part of f ? 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, -f^ = |- 5. The fractions i|, |, ^, ^ 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 -^V Compare their numerators. Sis times 4. Compare their de- nominators. 16 is times 8. How may -fy be derived from | ? How may | be derived from -^ ? REDUCTION 101 7. Compare in a similar way the terms of the fractions and ^; \ and T \; f and ^. 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 multiplying 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 ? | = f = -f% f^. 9. The fraction -^ is equal to the fraction f. Compare their numerators. 5 is what part of 10 ? Compare their denominators. 8 is what part of 16 ? Compare in a sim- ilar way -^2 with f ; $ with -|. What effect upon the value of a fraction has dividing both terms by the same number? ^ = | = -|=|. 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 by 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, , 1, ft, &. 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 f to 12ths. As the denominator 12 is 4 times the denominator of f , the numerator of the required fraction must be 4 times the numerator of f . MODEL: f =iV 3 is contained in 12 four times. 4 times 2 is 8. f = T 8 2 . Another method. 1 = Jf ; $ = of Jf , or T \ ; f = 2 times ^, or T \. 2. State how you would find the number that 3 must be multiplied by to change -| to 20ths. 3. Change to 12ths: J, f, f, f, J, f, f, , f . 4. Change to 18ths: J, |, f, |, f, |, J, , |. 5. Change to 24ths: |, f, |, f, f, T *-, f, ^. 6. Change to 20ths: }, f, f ^ f, ^, J, ^. 7. Change to 36ths: |, J, f, |, f, If, T ^, |, J. 8. Change to SOths: ^, ^, |, f, |, J, 1|, f, ^. 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, 3|, i|, J, 2, 2J, |, f, 7|, ^, ffr 1. 2. Find the sum of 3^, 2|, and 7. 3. Find the difference: f 7 1 1 _ _ 5 _ 5 4. Show that ^ of 2 yd. is equal to f of 1 yd. ; that of 8 ft. is the same as | of 1 ft. REDUCTION 103 129. Oral Exercises. 1. Show by a diagram that % of a line is equivalent to of the line ; that J ft. = f ft. ; that f ft. + J ft. = J ft., or 11 ft. 2. Show with objects that | of 12 objects = T ^ of 12 objects ; that ^ of 12 objects = -^ of 12 objects ; that f of 12 objects + I of 12 objects = T 9 2 of 12 objects + ^ of 12 objects, or ^ of 12 objects. 3. Why is it necessary to change | and i to 12ths be- fore rinding their sum ? 4. Show by a diagram that the difference between % ft. and \ ft. is -^ ft. 5. If | 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 \ 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 : |, $ , | , | ? 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. 131t Name the exact measures of : 1. 10 ft. 6. 30 mi. 11. 50 rd. 16. 2. 16 gal. 7. 36 yd. 12. 27 pt. 17. 3. 20 da. 8. 40 yr. 13. 28 da. 18. 4. 18 hr. 9. 48 Ib. 14. 64 ft. '19. 5. 24 in. 10. 45 qt. 15. 72 mi. 20. 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 ^ must be repeated how many times to equal the fractional unit ? The fractional unit T^ 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 -^ ? $ ? f ? f? J ? 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 , -f, f , f, if, i|, J$, |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 terms. Canceling the greatest common factor of both terms reduces the fraction to its low- est terms . 134. Oral Exercises. Reduce to lowest terms : i- A A* A' &' A* i* 9 - II' If If' If' It' II 2 * ST 2!"' 24' A' 2f 2? ^' T2' A' yf' yf' ?' fi" 3 - ftftfttttt'li " ft' f I' yf' Y!' yf' ft * A' ^T' A' A' A^ A 12 ' ft' f 2' W' II' f 9^ li 5 - 13 - - 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 numbei 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. The 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 2. 4 and 6 8. 6 and 9 14. 12 and 9 3. 3 and 4 9. 4 and 10 15. 3 and 7 4. 6 and 8 10. 5 and 10 16. 2, 3, and 6 5. 5 and 7 11. 10 and 4 17. 3, 4, and 8 6. 4 and 8 12. 5 and 8 18. 4, 5, and 15 137. 1. Find the least common multiple of 8, 10, 18, and 48. As 48 is a multiple of 8, cancel 8. As the MODEL $ factor 2 is common to 10 and to 48, cancel this -.n. r factor of 10, leaving the factor 5. As 6 is a factor common to 18 and 48, cancel this factor f-P & of 18, leaving the factor 3. 5 x 3 x 48, or 720, 48 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 ? Find by inspection the least common multiple of : 1. 4, 5, 8, 24 7. 10, 15, 25, 40 13. 7, 35, 45, 90, 70 2. 3, 12, 15, 30 8. 36, 48, 60, 72 14. 5, 14, 42, 60 3. 5, 8, 25, 40 9. 12, 18, 24, 36 15. 6, 7, 8, 9, 84 4. 2, 6, 15, 45 10. 3, 5, 30, 45 is. 20, 24, 30, 100 5. 7, 21, 49, 84 11. 4, 18, 27, 72 17. 4, 9, 20, 54 6. 9, 15, 36, 60 12. 8, 12, 15, 60 18. 6, 15, 24, 36 108 FRACTIONS ADDITION AND SUBTRACTION 139. Written Exercises. 1. Change to 12ths and add f, f, j. MODEL : | = -^ 2. 3. 4. 5. 6. 7. _9_ 1 5. 11 1 _9 4 ~ 12 66612612 .5 _ 10. 13211A 6 12 4 A ~f f f 6 I 2 "" * | 1 V 1 Y '2 O & O O O O 8. Add the fractions in Exs. 3-8, Sec. 127. 9. Change to 24ths and add 3|, 4|, fill, 7f . MODEL: ^f = 3J 41 _ 44 The sum of the fractions is f |, which reduces to 2f. Write 4 as the frac- b -^ b * 12 2 * tional part of the answer. Carry 2 to = *24 the column of integers. 22 10. Change to 12ths and add 4f, 3|, 4|, 5 11. Change to 18ths and add 8|, 7|, 9^, 6 12. Change to 24ths and add 7|, 6-^-, 9J, 8f, 4^. 13. Change to 36ths and add 3JJ, 7Jf , 8J, 6J, 9f . 14. Change to 48ths and add 7J, 9f, 3 T ^, 5i|, g^. 15. Change to 72ds and add 81, 9f, 7^, 18^, 3f. 16. Reduce to lowest terms : ||, ||, ^|, |^. 17. Change to improper fractions : 7|, 9|^, 8|, 7|. 18. Change to mixed numbers : -^, -^-, -^1-, -^-. 19. Add 4f , 6f 8f 8f 9f 7f 20. Change 4 to 12ths ; 3 to ISths ; 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 : |, f , and | ? 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 t. ? | ft. ? X ft. ? o D 1 _ These fractions may be expressed in the same unit of measure, T ^ ft. f ft. = T % ft. f ft. = T % ft - 5. Can the following be expressed in the same unit of measure : ^ ft., | da., and ^ gal.? Can the following : I ft., I ft., and iV ft.? 6. What is the least common multiple of the denomi- nators of the fractions , |> |> and ^ ? The least common multiple of the denominators of two or more fractions is called their least common denominator. 14>1. Reduce the fractions to fractions having the least common denominator, and add :' abed e f g Ti i 8* 91 4J 3^ 4l-t 31 5f 4^ 6f 6f S\ 6f 4f 7J 8| 8& m 8f 4} 91 6J 7| 110 FRACTIONS 142. Written Exercises.* l. From 5| subtract 3J. MODEL : 5 ^ = Oy 9 ^ Reduce the fractions to fractions having 31 = SA the least common denominator, and sub- tract. "12 Subtract : abode 9| 23} 43f 6 4| 4| 5| 51 4| 2f 4} 6| 7& 9f 18f 2f 41 7f 5. 21| 30| 20 26f 43| 16 18| 29f 10| 9 9f 4| 7 10| 81 9| 6. 87^- 79 T \ 90 T 6 F 68^2 74f 20 20 4f 64f 87^ 1A 16H 13if 143. l. Add each exercise in Sec. 142. 2. The lengths of the blackboards in a certain school- room are 12^ ft., 14| ft., 8| ft., and 6^ 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 221 ib. and the other 17| Ib. * See Sec. 126. ADDITION AND SUBTRACTION 111 144. Review Sec. 134. Reduce to lowest terms : 1- U> tf ' If' II 5 ' 2- 3 4 > 6. 3- _, _ 145. Review Sec. 120. Reduce to improper fractions : 1. 14f, 30, 161, 33^, 66f 4. 17} mi., 5 yd., 8f wk. 2. 16f, 37f, 111, 9J p 28f 5. 231 f t ., 8} lb., 9f wk. 3. 38^, 5f, 16f, 25|, 67| 6. 6& yd., 4^ mi., 811 A . 146. Review Sec. 122. Reduce to integers or mixed numbers : 1. 1^, 144, 104 JL4J1, IjJ! 5. Jf yd., ^ gal., ^ wk. 2. iJHs -MM 1 , -W, W, -W- 6. 7. -^ 5 - da., -^ mi., -^ g 147. Add: 43} 64| 18^ 77} 191 39} 56f 691 58| 791 24| 67^ 34f 76J 73f 74| 83^ 66^ 53| 65| 84} 56J 95| 73J 49| 47^ 59| 112 FRACTIONS 148. Subtract : a b c d e / 9 1. 190f 265| 398f 443^ 178 2961 467| 137f 124| 154| 217| 25| 180f 337f 2. Add each of the above exercises. 149. Review Sec. 137. Give the least common multiple of : 1. 2, 3, 4 8. 4, 5, 8 15. 7, 8, 9 2. 3, 4, 5 9. 3, 4, 7 16. 3, 4, 6, 8, 12 3. 4, 5, 6 10. 6, 7, 8 17. 5, 7, 8, 12, 4 4. 2, 3, 5 11. 3, 7, 9 18. 3, 5, 6, 8, 15 5. 3, 4, 7 12. 4, 7, 9 19. 7, 9, 12, 14, 21 6. 4, 6, 8 13. 6, 8, 12 20. 8, 10, 12, 15, 20 7. 5, 7, 8 14. 5, 8, 12 21. 4, 6, 10, 14, 20 150. When the least common multiple of the denomi- nators cannot be found readily by inspection, use the fol- lowing method : 1. Find the least common denominator : ^ ? , -g 3 ^, T ^, 1-*. MODEL: 2}24 50 72 80 Find the least common multi- 2) 25 36 40 P le of 24 ' 50 ' 72 > 80> Cancel 24 ^~ ^7 7^ ^T; as it is a factor of 72. Select a prime number that is a factor of c\ oc n -| n * *'" or more of the remaining 592 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 brought down have a common factor. The product of the several divisors and numbers remaining is the least common multiple of the denominators. I c m =2x2x2x5x Find the least common multi- 50 o or-nn P^ e f ^ ne same numbers by the X " X ^ = ooUU. ,, , , . , . o IOT method explained in Sec. 137. 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 | mi. Show the part that represents ^ mi. Compare the part f mi. with the part f mi. 3. What is the sum of |, f, and | ? 3 times | = f 4. Write 1 four times as an addend and find the sum. 7 State how the sum was found. 5. If | is written five times as an addend, what is the sum ? If | 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: f by 5; | by 3 ; | by 6 ; f by 7 ; f yd. by 4. 8. If | is multiplied by 3 by multiplying the numera- tor by 3, the result will not be in its lowest terms. Why? | may be multiplied by 3 by dividing the denominator 9 by 3. fx3 = f. 9. Dividing the denominator of a fraction by a whole number has what effect upon the value of the fraction ? 10. Multiply | 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 J . = V = 10J. MOCL. & JONES'S ESSEN. OF AR. 8 114 FRACTIONS 152. Oral Exercises. Solve each in the shortest way : abed e 1. f x5 11x5 fixlO ifx9 fx5 2. | x4 T ^x3 ||x7 fx8 i|x!2 3 v ^ 8v7 11 v 8 1& v 3 1J1 vfi ' 6 X O 2T X 1 T9 X IT X d 3 > 4. f x 5 if x 4 H x 6 f x 7 ff x 4 5. I x 20 I x!2 ^|x24 | x!6 f f x 48 6. -I x24 x7 _^ x30 | X 42 |4 X3 7. 11x18 f x5 ^ Y x8 ||x!6 Hx28 g 14 vx i A _4 x 5 ^ x 75 l^ x 24 i x 50 9. ^x36 11x28 ^|x7 ^x8 |i x 17 153. Written Exercises. 1. Multiply 43f by 8. MODEL : 43| 8 ^ First, multiply f 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 : abed e 2. 47f x 5 64f x 7 82f x 14 74911 x 5 708f x 30 3. 68f x 7 74| x 9 65f x 15 896i| x 7 580| x 15 4. 96f x 3 59f x 6 94f x 12 780f| x 3 496i x 48 5. 78f x 9 76| x 9 70| x 18 973f x 9 573| x 25 6. 56| x 9 381 x 8 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 | ? Name the multiplicand and the multiplier in each, and tell what each shows. 2. 4 ft. x | is the same as ^ of 4 ft. How may ^ of a number be found? How may ^ of a number be found? When you know what of a number is, how can you find $ of the number? 3. Show with objects what is meant by | of 9 things ; of 12 things ; of 6 things. 4. | 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 | of 24 yd. 5. Show by a diagram what is meant by | of 12 in. ; by | of 1 mi. ; by | of 6 mi. ; by % of 10 mi. 6. How many thirds of 18 are equivalent to 18? Are | 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 : 9 20 x -I ; 16 yr. x f ; 25 mi. x f ; 18 mo. x f ; 24 Ib. x f. B. Compare f of $20 with of 3 times $20. Compare of 25 mi. with of 4 times 25 mi. 9. f of 8 ft. is the same as $ of ft. f of $5 is the same as ^ of $ . 10. Show by a diagram that f of 1 yd. is the same as of 3 yd. 11. 5 divided by 7 may be indicated 7)7), or ^. Indi- cate ^ of 3 ; of 2 ft. ; of 5 ini. ; | of 5 mi. 12. The products of ^ x 18 and of 18 x f are the same. 116 FRACTIONS 155. Written Exercises. 1. Multiply 36 by JJ. " MODEL : Solve : a 2. 30 x f 3. 48xii 4. 36 X ^2 5. 21 X f 6. 4x1 -11 X JJ = ^ = 16 J. 12 is a factor common to 36 and 24 - Cancel the com- mon factor. 3 times 11 is 33 ; A = 16. 8xif 7.x f 8x^ 9xi 6xf 156. Written Exercises. 1. Multiply 845 by 4f 845 c ^ ~~ 5 factors and multiply. 5 1 Solve. Before multiplying, cancel common factors : 2- fxf fixf tfxtt J^XJ xtf 3- *X| JfxiJ fxf fxf ffxH & y * -1& x 4 -M x -^ 4 x -A-A- ' 7 x lf x 2 A 3 3; * TT 163. Written Exercises. Change the mixed numbers to improper fractions and solve : a b c d 1. 34 x 4 f x 24 2f x 44 If x 34 O 7 7o O7 oo 2. 41 x 5 | X 6| 6fx4f 4i x 2 ^. 3f x f f x 4| 7f x 4f 6f x 4| * ^1 x I A x H 9f x 6J 3| x 2 164. Written Exercises. Review Sees. 152, 153, and 156. l. Multiply 45 J by |. MODEL : 45f t of 45| is ?H; f of 45f = 5 times 7fti or 5 (fof45f) 122 FRACTIONS Solve without reducing the mixed numbers to improper fractions : a b c d e 2. 34| x f 84f x f 546f x f 654f x f 840f x f 3. 18|xf 55|xf 385f xf 235f x| 468f x f 4. 72fx| 401 x 3 4631 x 3. 900^ x f 479| x | 5. 48fxf 38xf 847 xf 783f xf 673| x f 6. 96fxf 941 x | 170JLxf 680^ x & 574| x f 7. 48ix| 63Jxf 4311 x 598| x| 6501 x J 165. Written Exercises, l. Multiply 349f by 3f . MODEL : 349| ' First multiply 349| by |. Next multiply 349| Q 3 by 3. Add the products. 5 (^ times 349$) 209f ( times 349f) 1049 (3 times 349f ) 1258| Solve without reducing the mixed numbers to improper fractions : abed 2. 645fx6| 584& x 4f 963| x 7J 642f x 7^ 3. 867fx5| 982f x 8f 333 x 6| 789^ x 7f 4. 694f x 7| 648| x 5^ 781^ x 9| 537^ x 61 5. 748x5f 457f x 7^ 450f x 7| 5211- x 2 | 6. 384fx4f 926| x 2| 467 x 91 830f x 4f 7. 412f x 6| 726| x 8| 940f x 3| 590-^ x 7| 8. 240|x4f 948f x 7 T 4 T 640f x 7f 810f x 3f 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 the multiplicand? than the multiplier? Why? Illustrate. 124 FRACTIONS 167. Dividing by a Fraction. 1. Draw a line 4 ft. long. Make a measure \ ft. long. Apply this measure to the line. How many times must the measure \ 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 \ ft. be applied to measure 6 ft. ? 10 ft. ? 12 ft. ? 5. To measure a line \ ft. long, the measure \ ft. must be applied \ times 2, or \ times. That is, one half of the measure must be applied. 6. The expression \ ft. -s- \ ft. indicates that a line \ ft. in length is to be measured by a measure \ ft. in length. What is meant by each of the expressions : 6ft.-lft.? lft. -1ft.? T ^ft,-lft.? 7. Draw a line \ ft. long. Determine how many times each of the following measures must be applied to measure it: \ ft., \ ft., -Jg- ft. 8. The measure ^ ft. must be applied how many times to measure a 1-ft. line? 11 =|. 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 | times f , or | times. 10. To measure a 1-ft. line, a 2-ft. measure must be ap- plied ^ 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. -4- ft. ; 3 ft. *- J f t. ; I ft. H- 2 ft. ; ft. -*- 3 ft. ; 4 ft. -* J ft. 12. To measure a 1-ft. line, the measure ^ ft. must be used 2 times. 2 is the reciprocal of ^. To measure a 1-ft. line, the measure 2 ft. must applied times. ^ is the reciprocal of 2. 13. The reciprocal of 4 is \ ; of 3 is ^ ; of ^ is 3 ; of | is | ; of -| is | . If | 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. Ifr. 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 f is contained in 1 times. Hence the following rule : To divide by a fraction, multiply the reciprocal of the di- visor by the dividend. 16. What is the reciprocal of each: f? f? f?'^? ? W f ? F ti? J? If? A? 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. 126 FRACTIONS 168. Written Exercises. 1. Divide 6| by |. 3 . 5 .27 6 81 M DEL: 6 ^6 = 7 X I = 10 Solve : ^ 2. 3f yd. -5- f yd. 10. 6f -4- 6f 18. If -s- f 3. 5J yd. -j- f yd. 11. 4| -4- 6| 19. 3J H- 21 4. 6f wk. -*- f wk. 12. 5| -H 2 T ^ 20. 8{i -4- 9f 5. 8|yd.^-|yd. 13. 7f + 8J 21. 7l|^4 T ^ 6. $7f-*-$f 14. f-| 22. 16f-=-14f 7. &fr in. -5- 1 in. 15. ^- T ^ 23. -6| + 12f 8. 8JH-4f 16. JL+1 24. 7^6|1 9. 7J-*-5f 17. 2\-f 25. -24f--8 26. Divide 100 by 331 . by 66| ; by 37| ; by 87J. 27. Divide if by 4 ; if by 5 ; 316| by 8 ; 435| by 27. 28. Multiply 635f by 8; 315| by 8; 80f by 9. 29. Add 8f , 4|, 31 6^, 8^. 30. Take 324 from 96*. P'rom 804 take 194. i V o O 31. Divide 8.125 by .04; 180.40 by .05; 725 by 1.25. 32. Multiply 3.1416 by 4f ; .7854 by 6; 2150.42 by 60f. 169. Oral Exercises. 1. Divide each by 100 : $43, 3.14,. 60.75, .9, 2000. 2. Add ^ and ^; 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. l. 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 ? 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 $ 5, at 4^ a pound? at 4|^ a pound? at 5^ a pound? 9. How many strips of matting 42 in. (1 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. l. 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 $ f a day, in how many days will he earn $ 15 ? 128 FRACTIONS 3. 1 yd. of cloth will cost how many times the cost of I yd-? 4. If a dealer charges $ 6 for | T. of coal, what is the price of the coal per ton ? 5. Find the cost of 1 yd. of lace if 2| yd. cost 75 ^. 6. At $ 1^ per yard, what will be the cost of 8^ yd. of silk? 7. Find the area of a rectangle 8^ in. by 6| in.; of a square whose side is 2| ft. 8. How many pounds of meat at 12|^ a pound can be bought for 75^? 9. If 3^ Ib. of coffee are sold for $ 1, how many pounds can be bought for $ 6 ? for f 3 ? for $ 9 ? for $ 12 ? 10. If 1 Ib. of tea costs $ f , how many pounds can be bought for f 2| ? for $ 6 ? for $ 12 ? 11. A tailor used 2| yd. of cloth for each pair of trou- sers. How many pairs can be made from 22 yd. ? 12. Change to feet : 6 in. ; 9 in. ; 10 in. ; 3 in. ; 4 in. 13. Change to inches : f ft. ; f f t. ; f ft. ; 1 ft. ; -^ ft. 14. Change to months : ^ yr. ; yr. ; | yr. ; | yr. ; f yr. 15. The atmosphere presses equally in all directions with a pressure of about 15 Ib. to the square inch. Find the pressure on the top of your desk. 16. If the circumference of the wheel of a bicycle is 6^ ft., how many times will the wheel turn in going 1 mi.? 17. The cost of laying a concrete sidewalk at 11| X P er square foot was $34.50. Find the area of the sidewalk. If the walk was 6 ft. wide, how long was it? MULTIPLICATION AND DIVISION 129 172. Review Exercises. 1. Find the perimeter of a rectangle 8 ft. 6 in. wide and 12 ft. 9 in. long. Find its area. 2. Find the number of square feet of blackboard sur- face in the schoolroom. 3. The diameter of a cylindrical tank is 6.5 ft. Find its circumference, (circum. = diam. x 3^.) 4. A farmer asked his two boys, George and Frank, to figure out the number of posts necessary to build a fence 28 rd. long, the posts to be placed rd. apart. George said it would take 56 posts, and Frank said it would take 57. Was either boy's answer correct? 5. The farmer (Prob. 4) asked the boys to find the number of posts necessary to build a fence around a gar- den 6 rd. by 8 rd., the posts to be placed | rd. apart. Both boys said it would take 57 posts. Was the answer correct ? 6. If a man had 60 sheep and sold | of them, how many did he sell ? How many did he have left ? 7. There are 640 acres in 1 square mile. How many are there in | of a square mile ? f 8. If -| of the distance between two cities is 15 mi., how far apart are the cities ? 9. Mary's age is 12 years. She is f as old as Ethel. How old is Ethel ? 10. The cost of 15 T. of hay was $112.50. What was the cost per ton? 11. At 75^ a yard, how many yards of cloth can be bought for $15? . MOCL. & JONES'S ESSEN. OF AR. 9 130 FRACTIONS 12. Find the value of the potato crop on 50 acres, if the average yield is 125 sacks to the acre and the potatoes are worth $1.20 per sack. 13. Draw three dials, as in Sec. 97, and fix the hands to read 8-4,500 cu. ft. ; 32,800 cu. ft. ; 47,000 cu. ft. 14. Is the height of any mountain given in your geog- raphy text ? If so, express the height of some mountains in miles. 15. Draw a line 20 in. long. Test it with a ruler. 16. If the height of your schoolroom is 12 ft. 6 in., find the distance from the highest point on your desk to the ceiling. 17. Find the area of your desk top. 18. Find the area of a blackboard in your schoolroom. 19. Draw a diagram to show the ratio of 2 in. to 4 in. ; of 4 in. to 6 in. j of 8 in. to 12 in. 20. What number expresses the ratio of 3 in. to 6 in.? of 6 in. to 3 in.? of 8 in. to 4 in.? of 4 in. to 12 in.? 21. Express the value of 5| in the fractional unit ^ ; of 4| in the fractional unit \ ; of 8| in the fractional unit |. 22. From a piece of cloth containing 22^ yd. a tailor used 17| yd. How many yards remained in the piece? 23. How much heavier is Mary than Ethel, if Mary weighs 101^ Ib. and Ethel weighs 92| lb.? Find the com- bined weight of the two girls. 24. What effect upon the value of a fraction has multi- plying or dividing both terms by the same number? Illustrate, using ^. 25. Show by a diagram that f^ of 3 ft. is the same as -| of 2 times 3 ft. MULTIPLICATION AND DIVISION 131 173. Finding what Fraction One Number is of Another. 1. What fraction of a dollar is 33}^? 33 1 MODEL : 33 \t is T-T-^ of a dollar. Performing the indicated divi- iOO sion : 33J -=- 100, or ifa x ^ = f To find what fraction one number is of another, take the number denoting the part for the numerator and the number denoting the whole for the denominator. Express the result in its simplest form. What fraction of : 2. 8 is 5? 9. 100 is 75? 16. 100 is 12}? 3. 15 is 10? 10. 100 is 60? 17. 100 is 37}? 4. 20 is 24? 11. 100 is 125? 18. 100 is 137}? 5. 16 is 10? 12. 100 is 175? 19. 100 is 133 J? 6. 36 is 30? 13. 100 is 120? 20. 100 is 66|? 7. 100 is 25? 14. 100 is 40? 21. 100 is 166|? 8. 100 is 150? is. 100 is 87}? 22. 100 is 112}? 23. What fraction of a dollar is 75^? 50^? 25^? 12}^? 20?? 60^? 37}^? 33}^? 66|^? 87^? 62^? 80^? 24. If a train travels at an average rate of 40 mi. per hour, in what part of an hour will the train travel 15 ini. ? 25 mi. ? 60 mi. ? 75 mi. ? 100 mi. ? 25. If a 20-acre field produced $600 worth of wheat in a certain year, at the same rate what part of this amount would a 10-acre field have produced ? a 30-acre field ? a 15- acre field? a 25-acre field? a 60-acre field? 132 FRACTIONS DRAWING TO A SCALE 174. l. Lines and surfaces are frequently represented by drawings. As most lines and surfaces are too large to be drawn full size, they have to be drawn on a reduced scale. 2. By letting ^ in. represent 1 ft., a line 8 ft. long may be represented thus: 8ft. Scale i in. = 1 ft. 3. Using the scale ^ in. = 1 ft., represent a line 12 ft. long ; 5 ft. long ; 20 ft. long ; 16 ft. long. 4. Using the scale ^ in. = 1 ft., represent a square whose side is 4 ft. ; 6 ft. ; 12 ft. 5. Using the scale \ in. = 1 yd., represent a line 12 yd. in length and a line 8 yd. in length. 6. Using a convenient scale, represent a rectangle 12 ft. by 10 ft. ; a garden 16 ft. by 12 ft. 7. Using the scale ^ in. = 1 ft., represent a school garden 24 ft. by 16 ft. Find the area of this garden. 8. Using a convenient scale, represent a sidewalk 6 ft. wide and 48 ft. long. Find the area of this walk. 9. Below is a diagram of a school yard, drawn to the scale \ in. = 20 ft. Find the dimensions of the yard. Find its area. SCALE DRAWING 133 175. l. Find the scale to which each of the following lines has been drawn : A 60ft. 90ft. /50ft. 2. Find the dimensions of the floor of your schoolroom. Using a convenient scale, draw a floor plan of your schoolroom. 3. Find the dimensions of the school grounds. Using a convenient scale, draw a plan of the school grounds. Show the ground plan of the schoolhouse properly located and in correct proportions. 4. The relative areas of the several oceans are repre- sented by the lines below, as follows : A, Arctic ; J?, Antarctic ; (7, Indian ; D, Atlantic ; E, Pacific. The num- bers above the lines indicate the number of millions of square miles in each. AJL R 7.5 C 23 7/ 5. Using a convenient scale, represent the relative population of Asia, Europe, Africa, North America, South America. 134 FRACTIONS 176. 1. This figure represents a section of land. Find the dimensions of each division ; the num- ber of acres in each ; the cost of the sec- tion; and of each division at $45 per acre. (Section = 1 sq. mi. = 640 A.) 2. A field containing 10 A. is 40 rd. long. How wide is it? Draw it to a scale. 6rd. 4-rd. 3. This figure represents a garden. What is the scale ? Draw the plan of this garden on a scale twice as large as the figure, viz. \ in. = l rd. Divide the garden into three rectangles. Find the area of each rec- tangle. Find the area of the garden ; the perime- ter. Find the cost of fencing this garden at $1.75 per rod. 4. Draw to a scale the side of your schoolroom and locate the openings. Mark the dimensions on your drawing. 5. Draw lines to represent the population of New York City, Chicago, Philadelphia, London, and Paris, using the same scale for each. 6rd. I6rd. 177. City Lots. l. 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 I I (J) Grove St. 100 ^ >> ^ ^ IV) Co Ch Co Co o -^ N ^ o is. I of 45 Ib. "1^27. 4 of $100 V V t> 180. Oral Exercises. Find the quotient of : 1. | H-4 5. f-r-6 9. l T \-5-4 13. 2J-f-5 2. if ^-6 6. |-5-10 10. 6f -j-3 14. 3f-!-8 3 . IA^T 7. f-f-2 11. 3{^20 15. 4f-*-4 4 . 1^5 8 . |^8 12. If -s-10 16. 181. Oral Exercises. Find the product of : 1. 2 of 1 7. f of 1 13. f X j 2. of -1 8. | of 1 14. | X 1 3. of f 9. 1 of 1 15. f X 1 4. I of i 10. A of tt 16. | X 1 5. -I of 11 11. f of I 17. | X t 6. 1 of 1 12. 1 of f 18. 1 X tt REVIEW 139 182. Oral Exercises. 1. If f of a ton of coal costs $6, what is the cost of a ton? 2. If | of the cost of a farm is $2400, what is the cost of the farm ? What is | of the cost of the farm ? o 3. If | 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 $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 ^ of the capital ? 6-- A man sold -| of his land for $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 $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. $20 is $ of | of $20 = 13. $35 is f of f of $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 % more than Edna. Ethel's weight is 105 Ib. 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. $200 is | of x. $60 is \\ of x. 20. What amount less \ of itself equals $100 ? 21. What amount less f of itself equals $60 ? 22. $1200 is 2| times x. ^ of some amount is $160. What is the amount ? 23. After gaining f 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. $80is|of . $120 is f of . $90isfof . 26. $60 is 11 times . $150 is 1 times . 27. $6 -5-$. 75. $9--$1.50. $50-*- $1.25. REVIEW 141 183. Oral Exercises. 1. If ^ of the cost of a pair of skates is 60 ^, the cost of the pair of skates is how many times 60 ^ ? 2. If ^ of the cost of a desk is $3, the cost of the desk is how many times $3? 3. Compare f with |. Show by a diagram that | is 1 times |, or | times f . 4. If | of the cost of a table is $9, the cost of the table is how many times $9 ? 5. |^ of the cost of a clock is f 8. In finding the cost of the clock we may find ^ of its cost, and then | of its cost. Show that multiplying $8 by |- is the same as find- ing first ^ of the cost, and then ^ of its cost. 6. If ^ 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. 8. A man sold | of his crop of apples for $120. At the same rate, what was the value of his entire 'crop ? 9. If f of a ton of coal costs $6, what is the cost per ton ? Are these two solutions identical in character ? A. 3)86, B. 82 $2, cost of \ ton. $P x i = 88 _4 88, cost of 1 ton. 1 10. If of the cost of a farm is 86000, 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 | of all he had. How many papers had he at first ? 142 FRACTIONS 13. After solving 8 problems, a girl had f 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 ^ of the cost. Find the cost of the article. 15. By selling a horse for $ 90, a man lost -^ 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 | of its cost and the other paying | of its cost. The boy who paid f of its cost paid 70 X. Find the cost of the sled. 18. After having his salary increased by ^, 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. l mi. is ^ 2. $20 is f 8. 320 rd. is % 14. 4 2 gal. is ^ 3. 75 mi. isf 9. $42.50 is f 15. 81 ft. is T 9 T 4. | is f 10. 36yd. is -| 16. $1 .20 is | 5. | T. is t 11. 90 ft. is f 17. 3} is T 2 T 6. $6400 isf 12. 5280 is | 18. 144 is f 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; ^ of the amount; ^ of the amount; f of the amount. 3. Find the whole amount when $120 is 1^ times the amount; 1^ times the amount; 1^ times the amount; 1 times the amount; 1|^ times the amount. 4. Find the whole amount when $240 is of the a \ amount; f of the amount; ^ of the amount ; of the amount; of the amount; % o f the amount. 5. Find the whole amount when $600 is -$fa of the amount; Iff of the amount; ^f |}of the amount; T 4 ^ of the amount; T 2 ^ 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 Ib. 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 | of . 90 mi. is | of . 144 FRACTIONS REVIEW 186. Written Exercises. 1. 47f-14f 9. 87-66f 17. 2. 93|-52| 10. 47f+62f is. 3. 48fx84| 11. 19fx38f 19. 4. 9f-4 12. 2fx6| 20. 5. 324|xf 13. 96|x7| 21. - = 3% 6. 453f-s-5 14. 30|x45f 22. 8| = f 7. 526f-=-f 15. 897f*6 23. fxfxf 8. 736fx5f 16. 7301 x | 2 4. 3ix4| rr o w 9 V O 187. Written Exercises. 1. Find the value of | of a farm of 160 A. at $85 per acre. 2. A man sold f 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| Ib. at 16^ per pound. 6. A turkey weighing 9^ Ib. was bought for $1.90. Find the price paid per pound. 7. Express in cents and find the sum of the following : 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 7 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 $3 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 $.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 Ib. to 8 Ib. ? of 5 Ib. to 20 Ib. ? 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 24 ft.? of 48 mi. to 36 mi.? of $25 to $50? 12. What fraction expresses the ratio of 5^ to 25^? of 10J* to 50^? of 20^ to 100^? of 25^ to 100**? of 25 X to 75 # of 10^ to 40^? of 5^ to 45^? of 20^ to JICCL. & JONES'S ESSEN. OF AB. 10 146 FRACTIONS 13. What fraction expresses the ratio of 3 qt. to 4 qt.? of 5 mi. to 8 mi.? of 4 Ib. to 61b.? of 8 bu. to 12 bu.? of 12 yd. to 9 yd. ? of 20 mi. to 15 mi.? of 25^ to 40^? of 18 yr. to 12 yr.? of 6 mo. to 9 mo.? of 9 mo. to 12 mo.? 14. What part of $1 is 25^? 50^? 75^? 40^? 70^? 80^? 90^? 5^? 10^? 20^? 15. What part of fl is 12^? 37|^? 62^? 87 J^? 16. What part of 1 ft. is 2 in.? 3 in. ? 4 in. ? 5 in. ? 6 in. ? Tin.? 8 in.? 9 in.? 10 in.? 11 in.? 17. What part of 1 yd. is 2 in.? 3 in.? 4 in.? 6 in.? 8 in.? 9 in. ? 12 in.? 18 in.? 20 in. ? 24 in. ? 30 in.? 13 in. ? 21 in. ? 18. Whatpartof lib. isioz.?4oz.? 8oz.? 12 oz.? 7 oz.? 19. What part of 1 yr. is 2 mo. ? 3 mo. ? 4 mo. ? 5 mo. ? 6 mo.? 7 mo.? 8 mo.? 9 mo.? 10 mo.? 11 mo.? 20. What part of 1 da. is 2 hr. ? 3 hr. ? 4 hr. ? 5 hr. ? 6hr.? 8hr.? 10 hr. ? 12 hr. ? 15 hr.? 16 hr.? 18 hr.? 20 hr.? 21. What part of 1 hr. is 5 min.? 10 min. ? 15 min. ? 20 min. ? 25 min. ? 30 min. ? 35 min. ? 40 min. ? 45 min. ? 50 min. ? 55 min. ? 17 min. ? 22. What part of 1 mi. is 10 rd. ? 20 rd. ? 40 rd. ? 80 rd.? 60 rd.? 90 rd.? 23. What part of 1 T. is 1000 Ib. ? 500 Ib. ? 250 Ib. ? 200 Ib. ? 100 Ib. ? 24. What part of 1 section of land (1 sq. mi., or 640 A.) is 320 A. ? 160 A. ? 80 A. ? 40 A. ? 20 A. ? ALIQUOT PARTS 147 189. Aliquot Parts. 1. Name several amounts that are exactly contained in 2. How many times is each of the following contained in$l: 50^? 25^? 12^? 10^? 20^? 5^? 4^?. 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^? 5. If 40 sheep can be bought for $100, how many sheep can be bought for $20 (J of $100)? for $25? for $12.50? for $10? 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 12X a yard? at 20^ a yard? at 16f ^ 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 1 each? at 12^ each ? at 33 each? 190. Memorize the following fractional parts of 1 : .50= i .20 = 4 .12^ = 4 .40 =| 31 '. O A o O OC 1 OO1 1 O^l Q /^/^ Q .ii5= ^ ^7 == i '^'2 = f '"v = f .10= T V .66} | .75= * .05 =A From the above table construct a table showipg the same fractional parts of $1 ; of 100 ; of $100 ; of 1000. 148 FRACTIONS 191. Oral Exercises. From the cost of 120 articles at $1 each find the cost: 1. At 50 j each. 7. At 20 ^ each. 2. At 25 ^ each. 8. At 37|^ each. 3. At 75 t each. 9. At 62|^ each. 4. At 12-i- ^ each. 10 . At 40 ^ each. 5. At 33^ each. 11. At 60 / each. 6. At 66f ^ each. 12. At 80 ^ each. 192. Written Exercises. Solve each 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 1 ^ 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 $25 each; at $250 each (J of $1000) ; at $125 each (J 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 66f ^ each? at 12^ each? at 20^ each? at 37^ each? at 87-|^ each? 193. Short Methods. Solve each, using the shortest method: 1. $40x25 5. $2040 x. 121 9 . 400 Ib. x.625 2. $120x25 6. 640 A. x 37J 10. $8.60x75 3. $80x250 7. 240 mi. x. 125 11. $5.60x750 4. 60 mi. x 33^ 8. 36ft. x 125 12. $4.64x12.5 ALIQUOT PARTS 149 194. l. Divide by 25 : 12 ; $ 16 ; 640 A. To divide, by 25, divide by 100 and multiply the quotient by 4. 2. State a short method of dividing a number by 250; by 50; by 33; by66|;by37; 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. Divide : 3. $400 by 25 10. 2240 Ib. by .25 4. $300 by 250 11. 2000 Ib. 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 33 15. $ 3200 by $ 625 9. 120 yr. by66f 16. $1500 by $2. 50 195. l. What is the cost of 24 yd. of cloth at 50 ^ per yard ? at 12| ^ per yard ? at 16| t 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 121 ^ a yar d ? at 6 j a yard ? at 37| ^ a yard? at 33^ a yard? at 66f bayard? at $1.50 a yard? at $1.33 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 -f- of 40 acres + of ^ 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. + l of 60 da.) ? for 120 da. ? for 70 da.? for 50 da. ? for 20 da. ? for 80 da. ? 150 FRACTIONS REVIEW 196. l. Draw a square and show the following parts of it : .50, .25, .75, .121, .371, .871, .33J, .66|. 2. Draw a square, and divide it into as many equal parts as are necessary to show either ^ or ^ 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 : %> i' i' i' Show on the square the parts f , |, f , f . 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 6" 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. i. Divide each by 10 : 47; $38.40; $.80; 3.1416. 2. Divide by 100: 2200 Ib. ; 5280 ft.; 1760yd.; $457.50. 3. State a short method for multiplying by 10 ; by 100; by 25; by 331; by 66|; by 75; by 121; by .25; by .87^ ; 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 REVIEW 151 5. Divide | ft. by ft ft.; 6.2 by .02; l* by 6; }f 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)1^4; 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, 5, 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 12|. Ask a question concerning a fractional part of 100 that is answered by each of the following : 20, 75, 25, 12|, 371, 5, 621, 66f, 87J, 33, 40, 10, 80, 16|. 10. The question, What is the ratio of 10^ to $1? is answered by the number J^. Ask a question concerning the ratio of some quantity to $1 that is answered by each : J, f , J, f , , , f , , , -|, |, ^ ^ J, f , , 2, f , 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. $16.40, 2. 30, 8. $.15, 3 14. $20, . 3. 15, .5 . 9. $.15, 5^ is. 96 ft., . 4. f, 2 10. $1.80, .06 16. 96 ft., | 5. f, | 11. $30, .05 17. $34.40, .08 6. .5, 10 12. $25.60, .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 i. $40 $50 $10 J 2(2; O tjp n_) 830 x X 3. 860 $50 X X 4. $80 x $20 X 5. X $100 $25 X 6. x $150 $50 X 7. $75 x $50 X 8. $75 x \ 9.8150 x X \ 10 X $110 x iV 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 cost of 12 articles be found? of 37^ articles? of 62 articles ? of 75 articles ? 202. i. 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. $45 is .09 of what amount? $75 is .15 of what amount? $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 ? \ = yf 7 . 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 ^ 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 $.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 Ib. 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 $ . 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 $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 $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 $6400. From the answer find .02 of $6400; .03 of $6400; .09 of $6400; .08 (J more than. 06) of $6400 ; .04 ( less than .06) of $6400 ; .05 (J less than .06) of $6400 ; .'07 of $6400 ; .12 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 $$$ 3. Change .875 to a common fraction. MODEL: .875 = rfflfr = 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, .65, .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, $6.40 = $6$: $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 |-? \1 |? ^? -|? f? 4V 1? 5 ? 3? 7? 7_9 _3? _9__? 8 8' 10' 17' IS' 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 DECIMAL 157 206. Changing Common Fractions to Decimal Fractions. 1. Change | to a decimal fraction. 3 -f- 4 is the same as . 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 | = . ' . The fraction f has been reduced to a decimal fraction. Tc^o.OO 2. Perform the indicated division. Continue the division until there is no longer a remainder. -|, |, , ^, t fr. fr fr i- To change a common fraction to a decimal fraction, divide the numerator by 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. 25. 26. 27. 28. $7f 29. 30. 1. 1 7. f 13. A 19. 2. f 8- 1 14. & 20. 3. i * 6 1 15. f 21. 4. \ 10. | 16. I 22. 5. f 11. f 17. i 23. 6. 12. | 18. A 24. 158 FRACTIONS 208. Changing Fractions to Hundredths. 1. Change -| to a fraction whose denominator is 100. 4 x MODEL : _ = . 5 j s contained in 100 twenty times. Multiply both terms of \ by 20. $ = -&V Since 1 is equivalent to |$, 5 is equivalent to y 2 ^, and f is equiva- lent to T <&. 2. Change to fractions whose denominators are 100 : 111133 6 _4 _3 12111357 1 A ' 2"' I' "&' 6' ?' TD~' 20"' 2?' ""' "3"' 6' T' IT' ' U" 8' T2"' 3. Express as decimal fractions each of the fractions in Ex. 2. 3 100 2 66| - = - a 3 100 l=^ 8 100 8=a 8 100 2 = ^ 8 100 Z = ?H 8 100 l = l^ 6 100 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 : 7f, 4|, 6|, 81, 9|, 4J, 3 T V 8^, 12ft, 9f . 7. Express as dollars and cents and add $8, f> 3|, ^, $12f, |15f, $14ft, f 7ft. 4. Memorize the following : 1 _100_ 1 1 _ 5 = .05 100 20 ~100 1 _ 50 _ .50 1 4 = .04 2 100 25 ~100 1 _ 25 _ .25 1 2 = .02 4 ~100~ 50 ~100 3 4 _ 75 _ 100 .75 1 12 -8L 100 = 08$ 1 _ 20 _ .20 1 _ 10 = .10 6 ~100~ 10 ~100 2 _ 40 _ .40 3 _ 30 = .30 5 ~100 10 ~100 3 _ 60 _ .60 7 _ 70 = .70 5 ~100~ 10 100 4 _ 80 _ .80 9 _ 90 = .90 5 100 10 100 REVIEW 159 209. Written Exercises. 1. There are 2000 Ib. in a ton. How many tons of hay are there in 5400 Ib. ? 2. At $8.50 per ton, how much will 6500 Ib. of hay cost? 3. At $10.50 per ton, how much will 950 Ib. of coal cost? 4. How many hundredweight (100 Ib.) are 575 Ib. ? At $5.60 per hundredweight, how much will a farmer receive for some hogs weighing 3750 Ib. ? 5. At $37.50 per ton, how much will a farmer receive for 12,400 Ib. of wheat? 6. At $1.25 each, how many hats can be bought for 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. V 9. Find the circumference of a cylindrical tank, the diameter of which is 4 ft. 9 in. (4.75). 10. At $38.50 each, how much will 14 cows cost? ..11 . The area of a rectangle is 42.625 sq. in. Its length is 7.75 in. How wide is the rectangle ? 12. If a train travels at an average rate of 46.75 mi. per hour, in how many hours will it travel 390.6 mi. ? 13. At 5|^ ($.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. l. 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 numeral "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. (l 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. (| ft.) wide, and 1 in. thick ? 1 ft. long, 8 in. ( ft.) wide, and 1 in. thick ? 1 ft. long, 4 in. (^ 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 board 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: Take either 3 " or 4 " as the width. Taking 4" as the 16 x n X 3 X 4 board feet = 7 idth ' * h mb * r f b ard ffl feet in 1 ft. of the lengbh is 192 board feet. 3 x & board feet ' and in l piece 12 ft. long, 12 x 3x^j board feet ; and in 16 pieces, 16 x 12 x 3 x T \ 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. 1" 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. I" X 16", 15 ft. long 212. Written Exercises. 1. Find the number of feet in 120 pieces of lumber, each 2" by 4" by 16'. 2. Measure various pieces of lumber. 3. Find the cost of lumber for a bridge 10 ft. Jong, if planks 3" x 12" x 14' are laid over four timbers 8" x 8" X 14'. Boards costing $18 per M; timbers $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 $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 $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, -J in. thick, and 12 ft. long is used. What is the cost of the flooring at $40 per M? . & JONES'S ESSEN. OF AR. 11 162 FRACTIONS 3. What part of 5|- in. is J in. ? of 2 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 ^ square foot. It will thus take about 900 shingles to cover a square. Allowing for waste, 1000 shingles are estimated for a square. A lunch 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 $2.25 per thousand shingles. MODEL: 24 x 40 x 2 x .01 x $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. 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 ^*" a ( ' evident that some number of days added to 26 da. equal 1 mo. and 4 da. Subtract 1893 5 26 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. to 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 $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 $1.50 a day, how much did he owe for the use of the team? 164 FRACTIONS /6O/-of. 40rd. 4OrJ. SOrcf. I 40rcf. 40raf. I I Scale* "=20rc/. 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 80 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 figure at $1.25 per rod. 3. A field containing 20 A. is 40 rd. wide. How long is it? 4. Mr. James bought Lot 2 (p. 135) for $40 per front foot. After paying for a 6-ft. cement sidewalk costing 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 | as much land left. How many acres had he before making the sale ? 6. If 80 A. of land cost $4000, how much at the same price per acre will 320 A. cost ? 7. If hay is worth $12 a ton, how much is 500 Ib. of hay worth ? How much is 400 Ib. 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 yd. of silk? of | yd.? of f yd.? REVIEW 165 10. If a man's expenses for 3 mo. amount to $135, at the same rate, how much will his expenses amount to in lyr.? 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 $7.25 per ton, how much will 6.75T. of coal cost? 14. How much more sugar is received for $ 1 by buying at 16^ Ib. for a dollar rather than at 6^ per pound ? 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 : f , J, |, f , J, J, f , |, f , ^, IJ, H, 1J, 2f , If 18. Change to common fractions : .125, .375, .25, .875, .6, .625, .8, .40. 19. Find .05 of $200; .06 of $18.75; 1.04 of $80. 20. $282 =.94 of ; $375 = .75 of ; $60 =| of . 21. $60=1 times ; $60 = 1 times ; $60 = 1.20 of . 22. Change to lOOths : |, f , J, f, f, ^. 23. If | 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? PART III PERCENTAGE 217. Hundredths as Per Cents. 1. Read each : Jfo, .60, .10, T ^, .06, .85, iff, .05. 2. y|o> or .05, may be written 5%. 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%, 6|%. 5. Express as per cents : T fo, T jfo, ^, T 2 ^, T \V \U- 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%, , 75%, 80%, T ^ 7 , 100%, 371%, 331 %, 14f %. 218. Finding some per cent of a number. 1. 4% of $500 is the same as $500 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 $360; of $60; of $100; of $840.25. 166 HUNDREDTHS AS PER CENTS 167 3. Find l'2% of $400; of$350; of $100; of $247.25; of $1300. 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 -iff, 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 the 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? 168 PERCENTAGE 16. A real estate agent sold a city lot for Mr. Brown for $2000. He received a commission of 5% 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 $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 $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 $250 of Mr. White and paid 1% interest. At the end of the year, how much should Mr. White receive from Mr. Evans, if he received the interest and the money loaned? 23. Find 50% of the number of children in your school- room. FRACTIONS AS PER CENTS 169 219. Fractions as Per Cents. 1. The unit 1 is equivalent to how many hundredths ? to how many per cent ? 2. What per cent of a number is equivalent to the number ? to of the number ? to ^ of the number ? to fa of the number ? to 2 times the number ? to 5 times the number ? 3. State how a common fraction may be reduced to a decimal fraction. 4. Change f to per cent. Aft _ gQ of Change the fraction to a decimal, rTcTTiTv extending the reduction to two deci- MODEL: 5)3.00 p , , ,,, mal places. Express hundredths as per cent. Another Method: Since 1 is 100%; f is f of 100%, or 60%. 20% Work: W%xf=60%. w 5. Change f to a decimal fraction. .375 is the same as .37J, which is the same as 37- %. 6. State how a common fraction may be changed to hundredths, expressed as a decimal. What is the mean- ing of per cent ? To change a common fraction to per cent, divide the numera- tor by the denominator, and carry the reduction to two deci- mal places in the quotient. Express the quotient as per cent. 7. Change to per cents : f , }, f , f , f , -fa, J. 8. Change to per cents : ^, |^, ^, ff J$, f f . 9. Change to per cent : 1. 1.80= 180% MODEL: 1 = f. 5)9.00 10. Change to per cents : 1, 1-|, 1, 1^, If, If. 170 PERCENTAGE 11. Memorize the following : 1 = 100 % I = 60 % T 3 o = - 30 % | = 121 % 50 % f = 80 % A = 70 % f = 371 % fa = 5 % tfr = 90 % f = 621 efo Wt = 4 % T V = 81 % I = 871 oi A \J J. A O ' A i" ^ = 2 % = 331 % J = 16| % Jj = 10 % | = 66| % j = 14f % 220. Oral Exercises. 1. Find 16| % of $ 480. 16f % of $ 480 may be found by multiplying $ 480 by .16f , or it may be found by taking ^ of f 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. 13f % of 24 hr. 12. 62| % O f 640 A. 4. 331 % of $ 15. 13, 871 % of 320 rd. 5. 14f % of $ 35. 14. 16| % of $ 7.20. 6. 66| % of 36 mi. 15. 14f % of $ 84. 7. 371 % of 48 yd. 16. 66| % of 60 bu. 8. 25 % of 320 rd. 17. 331 % O f 1 2.10. 9. 50% of $1.60. 18. 75% of $400. 10. 81 % of 360 da. 19. 80 % of $ 25. 11. 121 % of 80^. 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. DISCOUNT 171 21. Oral Exercises. 1. At a sale the following discounts were advertised. (a) Find the amount of reduction and (7>) the selling price: a. 16f % off on carpets marked 90^ per yard. b. 33| % off on bric-a-brac marked $6. c. 14| % off on ladies' hats marked $14. d. 0(5 1 % off on damaged cloth marked 30^ per yard. e. 12| % off on tables marked $16. /. 37| % 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 25%? b. 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 e. 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|%? 33%? 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%? 14f%? 5%? 12|%? 100%? 90%? 37^%? 15%?- 5. How much remains of $24 after deducting 50% of it? 25%? 75%? 16|%? 33%? 66f%? 12|%? 37*%? 6. How much remains of $36 after deducting 25% of it? 50%? 75%? 33^%? 66f%? 16f%? 100%? 172 PERCENTAGE 7. Find l- of 36 ; 50 % of 80 ; l of 90 ; 33 % of 75 ; i- of 200 ; 75 % of 200 ; 20 % of 15 ; f of 60 ; 40 % of 120; ^ of 40 ; 12| % o f 72 ; 66f % of 90 ; f of 64. 8. A merchant bought silk at $1.80 per yard and sold it at a profit of 33|- % . How much did he make on each yard? 9. A man bought hay at $ 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 173 17. A farmer shipped 50 boxes of apples to a commis- sion merchant, who sold them at 90 ^ 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 $40 and sold it at a profit of 20 % What was the selling price of the cow ? 19. Mr. A sold a cow for f 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 fonit ? 22. A merchant's stock of goods valued >at $ 4500 was damaged by fire. He was obliged to dispose of the goods for 66f % 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 1 % taxes on $ 4000. Find the amount of his tax. 25. A man bought a lot for $1600. He sold it for $1800. How much did he gain on the lot? His gain was what part of the cost ? 174 PERCENTAGE 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 $80, what is 1 % of the selling price ? What is the selling price ? 3. If 8 /o 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 12, how much money has he? How much money has he if 25% of his money is $5? if 10% of his money is $3 ? if 331% of his money is $8? if 16| % of his money is $ 10 ? if 66f % of his money is $ 24 ? 6. f 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 37 % ; 25 is 50 % ; 50 is 62^ % ; 70 is 33 % ; 75 is 100 %. PERCENTAGE 177 8. When 14f % of a number is given, how may the number be found? 14 f% 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 \ % 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 the selling price of the goods. 13. A number is how many times f of itself? of itself? f of itself? of itself? 14. If 66f % of a number is 120, what is the number? 15. If 87|% 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 $ 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 ? MOCL. & JONES'S ESSEN. OF AR. 12 178 PERCENTAGE 225. Written Exercises. 1. A farm was sold for .$ 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 : of the cost of the farm = $ 6000. $1200 | of the cost of the farm = 1 of $0000, or $4800. f Since the farm was sold for f (125%) of its cost, the cost of the farm is f of the selling price. Since $6000 is 125% of the cost of $48 00. the farm, the cost of the farm may be MODEL B : 1.25) $6000.00 found by dividing $ 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 33^%, 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 $119, which was at a discount of 15% from the list price of the carriage. Find the list price of the carriage. PERCENTAGE 179 226. Finding what per cent one number is of another. 1. Each of the following fractions is equivalent to what per cent : f f f , f f , f , f f f, f f f |, , f ^, ^, ^, lV' sV' 2*S' ^0 ' 2. Each of the following is equivalent to what per cent : 11 if ij, if i|, if, if 11, if I T V, if, if, if if, IF 3. 8 is what part of 16 ? What is the ratio of 8 to 16 ? ^ of a number is what per cent of the number ? 8 is what per cent of 16 ? 4. 24 is what part of 36 ? 24 is what per cent of 36 ? 5. 12 is what per cent of 36 ? of 24 ? of 48 ? of 60 ? 6. Express as a common fraction the ratio of 6 to 8 ; of 20 to 25 ; of 25 to 20 ; of 30 to 35 ; of 40 to 60. 7. Express as hundredths in decimal form the ratio of 3 to 5 ; of 5 to 8 ; of 4 to 5 ; of 24 to 30. 8. 16 is what per cent of 20 ? 1$ = f . 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- on an o/ ' u h uct. The multiplier may be found by dividing the product (16) by the 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. 180 PERCENTAGE 227. Written Exercises. 1. What per cent of 120 mi. is 90 mi. ? 2. $ 45 is what per cent of $ 50 ? Find what per cent of: 3. 50 is 20. 10. 320 rd. is 80 rd. 4. 25 is 50. 11. 640 A. is 120 A. 5. 18 is 15. 12. 360 da. is 30 da. 6. 48 is 60. 13. 5280 ft. is 1760 ft. 7. 54 is 27. 14. 5000 ft. is 1000 ft. 8. 240 mi. is 180 mi. 15. 2000 mi. is 6000 mi. 9. 360 bu. is 600 bu. 16. 2000 Ib. is 750 Ib. 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^ % of a number is 48, what is the number ? 2. What per cent of a number is 1| times the number ? If 150 % of a number is 12, what is the number ? 3. If 1| times a number is 20, what is the number ? If 166f % 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 $8. $ 8, the gain, is what per cent of $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 G^ 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 3^ 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 1. $10 $15 6. $16 2. 115 $10 7. $12 3. $25 $30 8. $15 4. $30 $25 9. $20 5. $40 $45 10. $25 GAIN $2 231. Written Exercises. Find the value of x in each : Loss $3 COST SELLING PRICE GAIN Loss GAIN % Loss % i. $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 s% 8. $36 X x 5% 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 183 232. Written Exercises. 1. A real estate agent bought a city lot for $1200 and sold it for $1500. What was the gain per cent ? 2. A merchant disposed of a stock of goods valued at $8000 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 12|% of the cost of the carriage. Find the cost of the carriage. 3. A city lot increased 1 200 in value, which amounted to an increase of 33^ % of its cost. Find the cost of the lot. 4. A gain of 66f % 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. REVIEW 185 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 $640, 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 $ 60 was sold for 33^ % less than it cost. It was sold for what fractional part of its cost ? Find the selling price. 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 $ 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 $ 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 ? 186 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 t> I' I' I' Y> -t 2 i> ^'-H, If, If, 2J, 1J, If, 2, 8, 10. 5. Express each as a common fraction in lowest terms: 80%, 50%. 331%, 25%, 125%, 20%, 120%, 70%, 40%, 150%, 66f %, 14f %, 75%, 1331%, 175%, 180%, 12-|%, 140%, 160%, 1121%, 16|%, 90%, 371%, 871%, 6 2| %, 1371%, 1871%, 110%, 130%. 6. Find 125%, 150%, 175%, 1121%, 1371%, 162|-%, and 1871% of each of the following: 24 hr.; 320 rd.; 640 A.; 360 da.; 16 oz.; 2000 lb.; $1200; $4000; $80. 7. Find 1331%, 120%, 166f %, 140%, and 160% of each of the following : $150; 30 da.; 360 da.;- $6000; 120 rd.; $250; 60^; $1.80. 8. Find the number of which 30 is 331 % ; 60 is 25 % ; 20 is 40%; 36 is 66|%; 35 is 125%; 120 is 120%; 48 is 371%; 90 is 150%; 180 is 121%; 180 is 1121%; 42 is 175 % ; 50 is 200 % ; 24 is 160 % ; 200 is 40 % ; 80 is 66f % ; 15 is 166f % ; 24 is 4 % ; 30 is 5 % ; 18 is 6 % ; 45 is 9 %. 33 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 ; 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 $30. How much did the horse cost him? For how much did he sell it ? 2. What is 2|% of $400? 3%of$60? 5%of$1400? 6% of $250? 10% of 2000 Ib. '{ 5| % of $200? 1% of $150? 8% of $2500? 3. What is the difference between \% 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 wprd 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^? PERCENTAGE 191 238. Oral Exercises. 1. If | of the value of a piece of property is $1500, what is the value of the property ? 2. If f of a man's yearly salary is $1200, what is his yearly salary ? 3. A clerk saved $40 a month, which was f 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 | as much. They both together have 60^. How much money has each ? The money 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 239. 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 $2400. 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 Ib. of butter at 18^ per pound, commission 6%. a. 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 1| % on the amount of his sales, how much does he earn in a week in which his sales amount to $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 Ib. 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 % ? MCCL. & JONES'S ESSKX. OF AR. 13 194 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 $ 1000, of the face of tne policy. 243. Written Exercises. 1. Mr. Wilson insured his store for 86000. The in- surance cost him 1| r%. Find the premium. 2. Mrs. Hardy insured her house, valued at 88000, for | 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 H%. 8. $5600 at $1.20 per $100. 6. $2400 at If %. 9. $1400 at $1.35 per $100. 7. $12,000 at 1| %. 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 $1800 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 \\/o of the face of the policy. a. Find the cost of the insurance. b. 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 ; (J) 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. $30, $40. 6. $2.50, $3. 11. $1200, $1500. 2. $40, $50. 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, $24. 14. 2000 ft., 2200 ft. 5. 1100, $120. ^M>. 601b.,1001b. is. $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 $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. * 1. 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 f> 1), or of $1.50 (on each f 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 where 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 $6000. He invested 11500 in a city lot. 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 $100,000. 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 $180,000. 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 \\ %? 5. What is the amount of a man's taxes on property assessed at $6000 if the tax rate is $1.20 on each $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 is $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. b. Valuation, $275,000; taxes, $2475. c. Valuation, $360,000; taxes, $125,400. 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 1| % 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 If % ; at .S% ; at 1.4%; at If %. 6. $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 1| % 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 the 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; (>) a salary of $15 a week and expenses, and a commission of 5 % on all sales over $300 per week; ( each and 15 % Potatoes, 25 ^ 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 $80 ? 4. A painting valued at $2500 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 $1215.40, 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 $.25 a hundredweight ? 2. When hay is selling at $12 a ton, what is its price per hundredweight ? 3. If |^ 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 1^% 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.* l. 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. l. Find the net cash price to a retail hardware mer- chant of a stove listed at $45, trade discounts of 20 % and 10 %, and a cash discount of 5 %. MODEL : $ 45, list price. f Jhe first discount is 20% of 945, or 9 9. The price _9, first discount. after making this discount second price. is $45 - $9, or $36. The 3.60, second discount, second discount is 10% of 132.40, third price. $36, or $3.60. The price 1.62, cash discount. after makin S the * econd discount is 9 36 - 9 3.60, ).78, net price. OT |32>4a From this a cash discount of 5 % is deducted, leaving the net price $30.78. Instead of deducting each discount separately, the sum of the several discounts may be stated as a single discount. This may be *For a more extended discussion of Trade Discount, see Appendix, p, 265. TRADE DISCOUNT 205 found thus : A discount of 20 % from the list price reduces the cost to 80% of the list price; and the second discount reduces it to 90% of this, or to 90 % of 80% of the list price, which is 72 % of the list price, and the cash discount reduces it to 95% of 72% of the list price, or to 68.4 % (68f %) of the list price.* Compare 68| % of $45 with the answer found. In figuring discounts, use the shortest method in every part of the problem. 2. Which is the greater discount, a single discount of 25 % or a discount of 20 % and a second discount of 5 / ? 3. If the hardware merchant (Prob. 1) sold the stove for $45, how much was his profit if he paid out $ 2.50 for freight and cartage ? What per cent profit did he make ? 4. What was the net cash price to a jeweler of a watch listed at $35, discounts 30 %, 15 %, 5 %, and a cash dis- count of 2 % ? If the jeweler sold the watch for $ 35, what per cent profit did he make on it ? 5. A piano firm bought a piano listed at $350 and received discounts of 40 %, 20 %, and 10 %. How much did the firm make on the piano by selling it at $350? What per cent profit was made by the firm ? 6. How much does a dealer make on a carriage listed at $120, if he buys it at a discount of 20 %, 5 %, and takes advantage of a cash discount of 2 %, and sells it at the list price ? 7. How much less does a dealer pay for a wagon listed at $ 150, if he is allowed a single discount of 35 % than if he is allowed successive discounts of 15 %, 15 %, and 5 1200, a merchant had $ 4200 invested in his business. What amount had he invested before increasing his capital ? By what per cent of itself was the original capital increased ? 9. By selling a city lot for $ 1500, a man gained 25 % Find the cost of the lot. 10. Property worth $ 6000 was assessed for purposes of taxation at $ 4000. For what per cent of its value was the property assessed ? 11. What monthly rental must a man get from property valued at $ 3000 to yield a net income of 6 %, if it costs him $ 60 a year to maintain his property ? INTEREST 207 INTEREST 254. 1. On July 1, 1907, Charles H. Thomas borrowed of Joseph R. White $ 300, which he promised to return in one year, with interest at 6 per cent. As an acknowledg- ment of his indebtedness and as a promise to pay, Mr. Thomas gave Mr. White his note, of which the following is a copy: Berkeley, &d., fuly /, 1907. y&av _____ after date, for value received,. __c/__ _ promise to pay to *j.o-v,&jaA, R. Whites, or order, __________ with interest thereon at 6k per annum from date until paid. 2. The sum specified in a note is called its face, or the principal. 3. What is the principal and the rate of interest speci- fied in the above note ? When and where was the note executed ? On what date did the above note fall due ? 4. The person to whom, or to whose order, the amount named in a note is to be paid is called the payee, and the person by whom the note is to be paid is called the payer. In the above note who is the payer and who the payee ? What is the meaning of the words or order? per annum? s. Has any provision been made in the above note for the payment of interest beyond the period of one year ? 208 PERCENTAGE 6. Where in the note did Mr. Thomas acknowledge that he had received of Mr. White something of the value of $ 300 ? 7. As the note was made payable to Mr. White, or order, it is said to be negotiable ; that is, it may be passed from one person to another, and it becomes payable to the person to whom it is ordered paid. Six months after the note was executed, Mr. White bought a city lot of J. C. Anderson, and as part payment for the lot, he transferred the note to Mr. Anderson. In making the transfer, Mr. White wrote on the back of the note, over his own signature, " Pay to J. C. Anderson, or order." By this indorsement, the note was made payable to Mr. Anderson. 8. Find the interest on the note to July 1, 1908. 9. The sum of the principal and interest is called the amount. 10. What was the amount of the note on July 1, 1908 ? 255. Method of Aliquot Parts. 1. The unit of time for which interest is computed is usually one year. The interest on a given note for three years is how many times the interest for one year? The interest for six months is what part of the interest for one year? 2. What part of the interest for one year is the interest for 3 months ? for 2 months ? for 4 months ? for 1 month ? 3. The interest for 1 month is what part of the in- terest for 6 months ? for 4 months ? for 3 months? for 2 months ? 4. The interest for 5 months is the interest for 4 months plus the interest for what part of 4 months ? INTEREST 209 5. Find the interest on $400 for 1 yr. and 5 mo.at 1%. MODEL: $400 .07 $28.00 = interest for 1 yr. 4 mo. = ^ yr. 9.33 = interest for 4 mo. 1 mo. = \ of 4 mo. 2.33 = interest for 1 mo. $39.66 = interest for 1 yr. and 5 mo. EXPLANATION. First find the interest for 1 yr. by multiplying the principal by .07. Next find the interest for 4 mo. by dividing the interest for 1 yr. by 3 ; then find the interest for 1 mo. by dividing the interest for 4 mo. by 4. The sum of these three amounts is the interest for 1 yr. and 5 mo. In dividing drop all fractions of cents. 6. From the interest for one year, the interest for any number of months may be found by taking the following parts: 1 mo. = 3^ yr. 3 mo. = J yr. 2 mo. = I yr. 4 mo. = yr. 6 mo. = yr. 5 mo. (4 mo. and 1 mo.) = % yr. plus \ of \ yr. 7 mo. (6 mo. and 1 mo.) = \ yr. plus \ of \ yr. 8 mo. (4 mo. and 4 mo.) = \ yr. plus \ yr. 9 mo. (6 mo. and 3 mo.) = \ yr. plus \ of \ yr. 10 mo. (6 mo. and 4 mo.) = \ yr. plus \ yr. 11 mo. (6 mo. and 5 mo.) = \ yr. plus \ yr. plus \ of J yr. Into what other suitable parts for finding interest may each be divided : 9 mo. ? 8 mo. ? 10 mo. ? 256. Written Exercises. Find the interest and the amount of : 1. $250, lyr. 6 mo., 8%. 6. $260, 1 yr. 10 mo., 8%. 2. $560, 2 yr. 4 mo., 6%. 7. $720, 1 yr. 3 mo., 5%. 3. $875, 3 yr. 5 mo., 1%. 8. $1200, 10 mo., 6%. 4. $100, 1 yr. 9 mo., 7%. 9. $2400, 2 yr. 3 mo., 6%. 5. $620, 7 mo., 4j%. 10. $500, 1 yr. 8 mo., 4%. . & JONES'S ESSEN. OF AR. 14 210 PERCENTAGE 257. Interest for Years, Months, and Days. 1. It is sometimes necessary to find the interest for years, months, and days, in which case thirty days are usually regarded as one month. 2. When the interest for one month is known, how may the interest be found for 15 da. ? for 10 da. ? for 6 da. ? for 5 da. ? for 3 da. ? for 1 da. ? When the interest for 6 da. is known, how may the interest be found for 1 da. ? 3. Find the interest on $ 150 for 1 yr. 7 mo. 14 da. at 8 % MODEL: $150 .08 $12.00 = interest for 1 yr. 6 mo. = \ yr. 6.00 = interest for 6 mo. 1 mo. = \ of \ yr. 1.00 = interest for 1 mo. 10 da. = \ mo. .33 = interest for 10 da. 3 da. = fa mo. .10 = interest for 3 da. 1 da. = \ of 3 da. .03 = interest for 1 da. = interest for 1 yr. 7 mo. 14 da. 4. From the interest for one month, the interest for any number of days may be found as in the following : 22 da. (10 da. and 10 da. and 2 da.) = \ mo. plus \ mo. plus \ of \ mo. 18 da. (6 da. and 6 da. and 6 da.) = \ mo. taken 3 times. Is this easier than to separate 18 da. into the parts 15 da. and 3 da., or into the parts 10 da. and 6 da. and 2 da. ? Explain and illustrate. 5. Determine for each number of days, from 1 to 29, how the interest can be found most readily, when the in- terest for one month is known. Compare your results with those determined by other pupils, to see who has the best method. Test each method by taking some amount as the interest for one month. 6. Write a note, naming some pupil as payee and your- self as the maker, and find the amount of the note for 1 yr. 5 mo. 12 da. INTEREST 211 258. Written Exercises. Find the interest on: 1. $250 for 1 yr. 9 mo. 15 da. at 1%. 2. $700 for 2 yr. 8 mo. 21 da. at 5%. 3. $684.50 for 7 mo. 25 da. at 6%. 4. $1200 for 3 yr. 4 mo. 14 da. at 5| %. 5. $300 for 1 yr. 10 da. at 8%. 6. $45.75 for 2 yr. 8 mo. 12 da.- at 4%. 7. $2500 for 9 mo. 13 da. at 8%. 8. $560 for 2 yr. 3 mo. 23 da. at 7 %. 9. $645.40 for 2 yr. 7 mo. 26 da. at 9%. 10. $820.15 for 1 yr. 5 mo. 20 da. at 4%. 11. $125 for 11 mo. 17 da. at 8 %. 12. $214.45 for 4 yr. 2 mo. 19 da. at 6 %. 13. $750 for 1 yr. 6 mo. 28 da. at 7 %. Find the interest on each of the following at 6%; at 5%; at 1%; at 4|%. 14. $800 from Oct. 1, 1904 to May 10, 1906. 15. $475 from June 11, 1903 to Nov. 18, 1904. 16. $240.60 from April 8, 1904 to Feb. 21, 1905. 17. $350 from Jan. 1, 1904 to Nov. 20, 1904. 18. $1340 from June 8, 1903 to Dec. 29, 1904. 19. $26.48 from Sept. 12, 1905 to Aug. 10, 1906. 20. $1700 from March 24, 1905 to Aug. 15, 1906. 21. $48.62 from Nov. 18, 1902 to July 20, 1904. 22. $5000 from Sept. 7, 1903 to Dec. 23, 1903. 23. $467.89 from April 4, 1904 to July 26, 1905. 212 ' PERCENTAGE 259. Sixty Days Method. 1. Money loaned for less than one year is usually loaned for 90 da., 60 da., or less. The best unit of time to use in finding the interest is 60 da., and the best rate is 6%, as the interest at 6 % for 60 da. is 1 % (.01) of the principal, found by. moving the decimal point. 2. Find the interest on $2700 for 60 da. at 7 %. MODEL : $ 27 = interest at 6 % for 60 da. 4.50 = interest at 1 % for 60 da. $ 31.50 = interest at 7 % for 60 da. 3. What part of 60 da. is 30 da.? 10 da.? 20 da.? 5 da.? 15 da.? 12 da.? 6 da.? 3 da. ? 2 da. ? 4. From the interest for 60 da., how may the interest be found for 30 da. ? for 15 da. ? for 6 da. ? for 20 da. ? for 10 da.? for 5 da. ? for 90 da. ? for 120 da. ? 5. Find the interest on $4000 at 6 % for 60 da. ; for 90 da. ; for 30 da. ; for 20 da. ; for 120 da. 6. From the interest for 30 da., how may the interest be found for 15 da.? for 45 da. ? for 10 da. ? for 5 da. ? 7. From the interest at 6%, how may the interest be found at 7 % ? at 8 % ? at 9 % ? at 5 % ? at 5 J % ? 8. Find the interest on $500 for 90 da. at 7 %. MODEL : $ 5 = interest at 6 % for 60 da. 2.50 = interest at 6 % for 30 da. $ 7.50 = interest at 6 % for 90 da. 1.25 = interest at 1 % for 90 da. $ 8.75 = interest at 7 % for 90 da. Find the interest on : 9. $600 at 6 % for 30 da. 11. $ 1200 at 5 % for 60 da. 10. $1000 at 7 % for 90 da. 12. $10,000 at 7 % for 45 da. INTEREST 213 260. Cancellation Method. (For problems, see Sec. 258.) 1. What part of 360 da. are 90 da. ? The interest for 90 da. is what part of the interest for 1 yr. (360 da.)? 2. Find the interest on $600 for 90 da. at 7 %. MODEL : $000 x .07 x ^- = $ 10.50. pfift i The interest for 1 yr. is $ 600 x .07, and the interest for 90 da. is s%, or \ of $ 600 x .07. 3. What part of 1 year's interest is the interest for 6 mo. ? for 2 mo. ? for 8 mo. ? for 9 mo. ? for 10 mo. ? for 15 da. ? for 45 da. ? for 1 mo. 15 da. ? for 3 mo. 20 da. ? for 1 yr. 3 mo. ? for 1 yr. 6 mo. ? Find the interest on $600 at 5% for each of these periods. 261. Six Per Cent Method. (For problems, see Sec. 268.) 1. Interest is sometimes calculated by a method commonly known as the Six Per Cent Method. By this method, the interest at 6% for the given time is found, and from this the interest at the required per cent. 2. As 1 mo. is y 1 ^ of 1 yr., the rate of interest for 1 mo. is -fa of 6 %, or ^% (.005); and as 1 da. is -fa of 1 mo., the rate of interest for 1 da. is -fa of .005, or .000^. The interest at 6% for 1 yr. = .06 of the principal. The interest at 6% for 1 mo. = .005 of the principal. The interest at 6% for 1 da. = .000^ of the principal. 3. Find interest on f 350 for 2 yr. 7 mo. 21 da. at 1 %. MODEL : Rate for 2 yr. at 6 % ....... 12 Rate for 7 mo. at 6 % ....... 035 Rate for 21 da. at 6 % ...... .0035 Rate for 2 yr. 7 mo. 21 da. . . . . .1585 $ 350 x .1585 = 9 55.475 = int. on $350 for 2 yr. 7 mo. 21 da. at 6 %. 9.245+= int. on $350 for 2 yr. 7 mo. 21 da. at 1 %. 9 64.72 = int. on $350 for 2 yr. 7 mo. 21 da. at 7 %. 214 PERCENTAGE 262. Promissory Notes. 1. A written promise to pay a definite sum of money at a specified time is called a promissory note. A promissory note is usually called a note. 2. Compare the promissory note on p. 207 with the fol- lowing. The note on p. 207 is a time note, as the time of payment is specified in it. A DEMAND NOTE $300 focukUncl, &a,L, fam,. , 1907- far value received <$ promise to pay to R. 10JiLt&, or order, t/iis&& JiuncU&d ctoltai^, with interest thereon at 6% per annum, from date until paid. /if. A JOINT NOTE tf&atttz, lOaali., f-ufy /6, 190$. after date, W-&, ov zitheA, o^ IM,, promise to pay to S^icvnfc &. fCewyon,, or order, 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. on 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 65^ 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. PAKT IV FORMS AND MEASUREMENTS* 267. i. Lines are vertical J , horizontal , and. oblique \ /. 2. These are right angles. |_ 3. A rectangle has four right angles. j [ { 4. These are right triangles. |xx^ 5. These are acute angles. f\ \f 6. These are acute-angled triangles f\ 7. These are obtuse angles. ^ 8. These are obtuse-angled triangles, ^ 9. Perpendicular (jt?) means at right angles to. b 10. These figures have a base (0) a and an altitude (a). , p l~ 11. These lines are parallel. 12. These are quadrilaterals. /~~J / \ 13. These quadrilaterals are parallelograms. I I / 7 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. fif j^ *^J 268. Relation of Forms. Study the relation of these forms : Circle. L RIGHT ANGLE. RECTANGLE. RECTANGULAR PRISM. RIGHT ANGLE. RIGHT TRIANGLE. RIGHT TRIANGULAR PRISM. A A OBTUSE ANGLE. OBTUSE TRIANGLE. OBTUSE TRIANGULAR PRISM. o CIRCLE. SPHERE. CYLINDER. 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. PARALLEL LINKS. 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, /.. 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- l RIGHT ANGLES. 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- dicular lines. PERPENDICULAR 5. Point to lines or surfaces in the LINES. schoolroom that are perpendicular to each other. 224 FORMS AND MEASUREMENTS Square. Horizontal. Vertical. RECTANGLES. 271. Rectangles. 1. A figure whose angles are all right angles is called a rectangle. Meet 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 ftie sides of a rectangle parallel? Name surfaces not in your schoolroom that are rectangles. 272. l. 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 16J 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 ^ 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 ^ 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 ajid windows. 7. Find the value of a field 40 rd. long and 20 rd. wide at $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 ? MCCL. & JONES'S ESSEN. OF AR. 16 226 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- 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 base " and the width as its altitude. RIGHT 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 i>he 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, 6|- ft., alti- tude, 9| ft. ; base, 40 rd., altitude, 80 rd. b 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 c. - PARALLELOGRAM. Since the figure is a par- allelogram, triangle b is the same size as triangle e. If triangle b were cut off and placed alongside triangle f ec t 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 2 -f- 5 2 or 25 2 ? 2. The square of any number composed of tens and units may be found thus : 20 + 5 The square of 25 20 -f- 5 is s 6611 t be the 100 + 25 (20 + 5)X5 square of the tens, plus twice the 400 + 100 (20 + 5)x20 product of the 400 + 2(100; + 25 = 20 2 + 2(20 X 5) -f 5 2 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- 20 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. 5. Construct a square whose side is 10 -f 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 o 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, v27, which is read the cube root of 27. The index 2 for square root is usually omitted. 4. Read and give the roots : V64, -\/64, V49, VIOO, ^125, V81, V36, V144, ^81. 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 = 2x2x3x3x3x3. 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: VJ= *. V O SQUARE ROOT 247 299. Find the roots indicated: 4. A/3375 7. A/8000 10. V1296 2. A/9025 5. V129,600 8. Vfj- 11. A/15,625 300. l. Compare VI = 1, VTJOO = 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 apy 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. 1.2 2 = 1.44; 9.9 2 = 98.01; 1.22 2 = 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 fourtd 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 numbr of periods of two places each, beginning with units into which the number can be separated. MODEL: 5' 29123 20 2 = 4 2 x 20 = 40 129 (40+3) x3 = 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 thfe 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 D3 <\J b. 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 apart 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. 60 9 A 400 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 7. 1600 2. 104,976 5. .6724 8. 10.24 3. 3844 6. 160 9. .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 304r. l. 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 O be found ? 6. If the number of squares in B and O are given, how may the number in A be found? 7. If the number of squares in A and O are given, how may the number in B be found ? RIGHT TRIANGLES 251 305. Written Exercises. 1. Find the length of the third side of each : 15ft 20ft 12ft QOrcL. 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 c, a corner of the rectangle, and extended one cord in the direction of cd and the other in the direction of ce. To make the angle at c a right angle and thus to get the positions of the two sides of the rectangle, they measured off from c 15 ft. on one cord and 20 ft. on the other cord. With these as the legs of a right triangle, they adjusted the position of the cords so as to make the distance hi 25 ft. Having determined the direction of the sides cs and ee, they measured 24 ft. on cs and 16 ft. on ce and marked the corners s and i. From s they extended a line in the direction so, and from i a line in the direction io. They measured 16 ft. on the line so and 20 ft. on the line io. They marked the point o where the two measured lines met. They tested their work, finding that the diagonals co and si were equal. 3. Using cords and a measure, lay out on the school grounds rectangles 12 ft. by 16 ft. ; 20 ft. by 30 ft. 4. Lay off a baseball " diamond " whose side is 60 ft. SIMILAR FIGURES 253 SIMILAR SURFACES AND SOLIDS 307. l. Draw squares whose sides are 1 inch; 2 inches; 3 inches. Find their areas. The areas of the three squares are to each other as I 2 , 2 2 , and 3 2 . 2. Express the ratio of the areas of a 4-inch square and a 6-inch square. 3. Draw circles whose diameters are 4 inches ; 6 inches. As the area of a circle is Trr 2 , the ratio of the areas of these two circles is as 2 2 is to 3 2 . Explain. 4. Is the ratio of the squares of the diameters of two circles the same as the ratio of the squares of their radii ? 5. Is the ratio of the squares of the diagonals of two squares the same as the ratio of the squares of their sides ? 6. Figures that are of exactly the same shape are called similar figures. Draw two similar figures. Are similar figures necessarily 'the same in size ? The areas of similar plane figures are proportional to the squares of their corresponding lines. 7. Find the volume of a 2-inch cube ; of a 3-inch cube. Their volumes are in the ratio of 2 3 to 3 3 . The volumes of similar solids are proportional to the cubes of their corresponding lines. 8. Compare the volumes of two spheres, one 5 inches in diameter and the other 10 inches in diameter. As the diameter of the larger sphere is twice the diameter of the smaller, the volume of the larger is 2 8 times the volume of the smaller. Explain. 9. Compare the weights of two solid iron spheres of the same density, if one is 2 inches in diameter and the other is 4 inches in diameter. 254 MISCELLANEOUS EXERCISES MISCELLANEOUS EXERCISES 308. l. A merchant gained $28 by selling some goods at a profit of 20 %. Find the cost of the goods. 2. An agent who canvassed for a book received 40 / of the amount of the sales for selling and delivering the books. Find the amount of his commission for selling and delivering 60 copies at $1.25 per copy. 3. If the amount received for goods sold averages 20 % more than the cost of the goods, find the net profit for a month when the sales amounted to $24,000 and the expenses for the month amounted to $3000. 4. 20 % of the pupils enrolled in a certain school were absent one stormy day. Twenty-four pupils were present. Find the number enrolled. 5. A commission merchant sold $6000 worth of prod- uce at a commission of \\/o. Find the amount of his commission. 6. If a collector received 20% commission for collect- ing a bill of $17.75, what was the amount of his commis- sion? What per cent of the amount of the bill did the creditor receive? 7. A man paid $42 taxes when the tax rate was 2%. What was the assessed valuation of his property ? 8. If a person pays $50 tax on property when the rate is 1^ %, what is the assessed valuation of his property ? 9. A miller bought a ton of wheat through a broker, who charged a commission of 2 %. What was the amount paid for the wheat if the cost of the wheat and the broker- age amounted to $25.50 ? The cost of the wheat plus 2% of the cost of the wheat, or 102% of the cost of the wheat, was (25.50. Prove your answer. MISCELLANEOUS EXERCISES 255 10. A fruit grower shipped 25 boxes of apples to a commission merchant, who sold them at 85^ per box, charging 4% commission. He was directed to invest the proceeds in groceries, after deducting a commission of 2 % for making the purchase. Find the amount expended for the groceries. The not proceeds of the sale of the apples was $20.40, which was 102% of the amount invested in groceries. Prove your answer. 11. After, selling 25% of his interest in a flour mill, a man considered his remaining interest worth $ 9000. At this rate, what was the value of his interest before making the sale ? 12. A real estate broker bought two 60-ft. lots, adjoin- ing, for $ 1500 apiece, and divided them into three lots of equal frontage which he sold for $ 1200 apiece. What per cent of profit did he make ? 13. At $ 9 a ton, find the cost of two sacks of coal, each weighing 100 Ib. 14. A man bought a lot for 1200 and sold it for $ 1400. He bought it back for $ 1500 and resold it for $ 1600. How much did he make on the lot ? 15. A man bought four 50-ft. lots and divided the land into 40-ft. lots, which he sold at the same price per lot as he had paid. Find his gain per cent. 16. At 22 ^ a square foot, find the cost per front foot of paving a street 60 ft. wide. Find the cost per front foot to a property owner who pays for half the width of the street. Find the cost to the property owner of paving the street in front of a 45-ft. lot. 17. If 27 tons of coal cost $ 243, how many tons can be bought for $ 189 ? PAET YI APPENDIX CORPORATIONS, STOCKS, AND BONDS 309. 1. Corporations. A large business enterprise frequently requires more capital than one person may care to invest in it. Provision is made in the laws of the various states whereby a number of persons may organize a company, called a corporation, to engage in business as one body. Sometimes all the necessary capital is subscribed by the persons who organ- ize the corporation, but often the organizers of a company sub- scribe only a part of the capital. The laws regulating the incorporation of companies differ considerably in the several states. Frequently a corporation intending to transact busi- ness in one state will incorporate in another state, because of certain ad- vantages to be derived thereby. 2. Railway companies, mining companies, express com- panies, oil companies, and banking institutions are among the largest business corporations. 310. Shares of Stock. 1. Each corporation is capitalized for a special amount, as f 25,000, $ 50,000, $ 1,000,000, etc. The capital is divided into shares, usually of $ 100 each or of $ 1 each. Thus, a corporation that is capitalized for $ 100,000 may issue 1000 shares of the face value of $ 100 each, or 100,000 shares of the face value of $ 1 each, etc. These shares are bought by persons who invest in the enterprise. Each person who owns one or more shares of stock is called a stockholder. The several stockholders constitute the corporation. 256 CORPORATIONS, STOCKS, AND BONDS 257 2. Every stockholder receives a certificate of stock, showing the number of shares he owns and the face value, or par value, of each. These certificates are negotiable, and a record of their transfer is usually made on the books of the corporation. The affairs of a corporation are managed through a board of directors, elected by the stockholders, each stockholder having as many votes as the number of shares of stock he owns. Value of Stock, l. The price at which stocks are bought and sold in the stock market is called their market value. When the market value of stock is the same as its face value, the stock is said to be at par. Stock is said to be at a premium, or above par, when its market value is more than its face value, and at a discount, or below par, when its market value is less than its face value. 2. Examine the stock quotations in a newspaper. Can you tell from the quotations what the face value of the stock is? Tell which stock is at par, above par, below par. 312. Dividends. The net earnings of a corporation, after a surplus sufficient to cover the probable needs has been re- served, are divided among the stockholders according to the number of shares owned by each. These divided profits are called dividends. Dividends are computed on the par value of the stock, and are declared annually, semiaunually, quarterly, etc. ILLUSTRATION. If the amount of capital stock is $ 500,000 and the amount to be divided among the stockholders is $25,000, the rate of dividend is $25,000 -5- $500,000, or 5%. A stockholder who owns stock of the face value of $10,000 will receive 5% of $10,000, or $500. Stock that regularly pays a large dividend is usually regarded as a good investment, and is likely to be above par. When the dividends are not equivalent to a fair rate of interest on the investment, the stock is likely to be below par. MPCL. & JONES'S ESSEN. OF AR. 17 258 APPENDIX 313. Corporations sometimes issue two kinds of stock, called preferred and common. When both kinds of stock are issued, the holders of common stock are not entitled to participate in the profits until a fixed rate of dividend has been paid to holders of preferred stock. 314. Stock Brokers. 1. A person who is engaged in buying and selling stocks for others is called a stock broker. Stocks are usually bought and sold through brokers, generally at a regu- lar meeting place for transacting such business, called a stock exchange. 2. The commission of a broker is called brokerage. The rate of brokerage varies in different parts of the country from \/o of the par value for buying and also for selling, to \/ or more. A minimum amount is fixed for making small sales and purchases. The standard amount of stock bought and sold is 100 shares, although a smaller amount may be negotiated. As a rule, fractions of a share cannot be bought. In stock ^quotations fractions are always expressed in halves, fourths, and eighths. See quotations, p. 259. 315. Assessments. When the funds of a corporation are not sufficient to carry on its business, an assessment is sometimes levied upon the stock. The assessment is usually some number of cents per share, and the failure to pay it is generally pun- ishable by the forfeiture of the stock. 316. Examine a newspaper for stock quotations. What is meant by par value ? premium ? dividend ? market value ? brokerage ? Name some of the corporations engaged in busi- ness in the state in which you live. How many shares of stock of the par value of $ 100 each are issued by a corporation that is capitalized for $ 500,000 ? for $ 2,000,000 ? What is a stockholder ? CORPORATIONS, STOCKS, AND BONDS 259 STOCK QUOTATIONS 317. NEW YORK STOCK SAN FRANCISCO STOCK AND EXCHANGE EXCHANGE BOARD Mar. 9, Jan. 22, Mar. 9, Jan. 22, 190T 1907 190T 1907 MINING Adams Express 295 300 Caledonia .43 .67 American Express 223 242 Confidence 1.05 1.30 C. & N. W. 154i 190J Combin. Frac. 4.46 5.50 Denver & Rio G. 81| 39J Jumbo Exten. 1.85 1.90 Denver & Rio G. , pfd .72 81 Mustang .27 .21 Illinois Central 147* 166 Ophir 2.65 3.10 Northern Pacific 137f 155fc Red Top Exten. .50 .37 Southern Pacific 84* 95f Silver Pick 1.35 1.46 Pullman Car Co. 166 173 St. Ives 1.82 .93 Union Pacific 156 176f Utah .06 .08 Do. preferred 88 92^ Vernal .20 .26 Wells Fargo Ex. 280 300 West End 1.40 1.90 Western Union 804 83| Yellow Jacket 1.05 1.15 The par value of the stock quoted in the left-hand column is $100 per share, and in the other column is ft per share. On Jan. 22 the mar- ket was high, and on March 9 low. Which of the stocks quoted are above par ? below par ? At the above prices, find the cost without brokerage, on March 9, of 100 shares of Wells Fargo Express stock; of 100 shares of Chicago and Northwestern Railway stock ; of 100 shares of Western Union stock ; of 100 shares of stock in the Ophir mine ; of 100 shares of stock in the Silver Pick mine. Find the same for the prices given for Jan. 22. 318. Written Exercises. 1. Find the cost of 100 shares of Western Union stock at 80f , including \ cj brokerage. Market price of each share, $804,. Brokerage on each share, f \. Cost of each share (including brokerage), $80f, or $80.76. Cost of 100 shares (including brokerage), $8075. 260 APPENDIX 2. Find the cost of 100 shares of Northern Pacific stock as quoted for March 9, including i- % brokerage. 3. Find the net proceeds of the sale of 100 shares of Southern Pacific stock as quoted for Jan. 22, allowing ^% brokerage. Market price of each share, $ 95f . Brokerage on each share, $ \. Net proceeds on each share, Net proceeds on 100 shares, 4. Find the net proceeds from the sales of 400 shares of Illinois Central stock as quoted for March 9, allowing \/ brokerage. 5. How much would a man clear by buying 100 shares of stock at 142^ and selling them at 150f , allowing \ % broker- age both for buying and for selling ? 6. If the Pullman Car Company declares a dividend of 10 % > how much will a person receive who owns 100 shares of the Stock ? Par value of 100 shares, $10,000. His dividend will amount to 10% of $10,000. 7. If the Union Pacific Eailway declared a dividend of 8 % , how much did a person receive who owned 500 shares of the stock ? 8. Find the cost of 100 shares of stock at $ 1.20 a share, in- cluding | % brokerage. If a 10 % dividend is declared on this stock, how much does a person receive who owns 100 shares of the stock ? 9. What per cent does a person receive on the amount in- vested who buys 100 shares of stock at $ 1.20, paying ^ % brokerage, and receives a dividend of 10 % ? The dividend amounts to 10% of $100. This amount is what per cent of $120.50, the cost of the stock ? CORPORATIONS, STOCKS, AND BONDS 261 10. A man bought 100 shares of Red Top Extension stock at 37 cents, paying $ 1 brokerage. Find the cost of the stock. If he received a dividend of 5 %, what per cent did he receive on his investment ? $5, the dividend, is what per cent of $38, the amount invested ? 11. What per cent did a man make on his investment who bought 100 shares of stock at 45 ^ and sold them at 55^, pay- ing $ 1 brokerage both for buying and for selling ? Cost of the shares, $46; received for the shares, $54 ; net profit, $9. $9 is what per cent of $46 ? 319. Corporation Bonds. The promissory note of a corpora- tion, issued under seal, is called a bond. The bond of a cor- poration is secured by a mortgage. 320. A city or an incorporated village is called a municipal corporation. 321. 1. Governments, states, cities, counties, etc., are some- times obliged to issue bonds to meet urgent demands for money and to provide needed improvements. Such bonds are not secured by mortgages. The integrity of the government issuing a bond is accepted as sufficient security for its pay- ment. The bond of a government is a certificate of in- debtedness, with a promise to pay a certain sum to the holder of the bond, with a fixed rate of interest at a specified time, as at the expiration of five years, twenty year's, fifty years, etc. 2. Under what conditions is it sometimes necessary for the United States government to borrow money ? For what pur- poses are cities frequently bonded ? What is a bond election ? Which is the better security, the note of an individual or the bond of the national government ? 322. Bonds that are registered by number and in the name of the holder are called registered bonds. Bonds having interest certificates attached in the form of coupons are called coupon bonds. 262 APPENDIX BOND QUOTATIONS 323. Examine a newspaper for bond quotations. The fol- lowing quotations are from a newspaper report of prices on the New York Stock Exchange. Japan 6's 99| Southern Pacific 4's . . 84 Mexican Central 4's . 83 U. S. New 4's reg. . . . 129J Northern Pacific 4's . 100| Do. coupon .... 129| The par value of the bonds quoted is $100 each. Bonds are bought and sold in the market in the same manner as stocks. Government bonds are exempt from taxation. Find the cost of Mexican Central bonds of the face value of $1000, including % brokerage. COMMISSION AND BROKERAGE 324. Producer and Consumer. A person who grows agri- cultural products or manufactures useful articles out of crude materials is called a producer. A person who uses up products is called a consumer. Name some producers of foods, of cloth- ing, of fuel, of building materials. Name some products that are consumed by nearly every one. Name some classes of persons who are consumers but are not producers. In early times there was comparatively little buying and selling, and the exchange of products was very limited. Each family produced nearly everything that it consumed. At that time, most of the trade was directly between the producer and the consumer. With the invention of machin- ery and with improved means of transportation, cities increased rapidly in number as centers of manufacture and trade. People began to devote themselves more particularly to special lines of work. As trade con- ditions grew more complex, it became more difficult for the producer to trade directly with the consumer. Wholesale and retail establishments developed as agencies for marketing products. Middleman. A person who deals between the producer and the con- sumer is known in trade as a middleman. Products are often handled by several middlemen before they reach the consumers. The middlemen are generally persons who make buying and selling products their special occupation. Is the retail dealer a middleman ? Which of the following COMMISSION AND BROKERAGE 263 are middlemen : farmers ? stock buyers ? hardware merchants ? carpen- ters ? shoemakers ? hay and grain dealers ? What are some of the con- ditions that make it inconvenient for persons living in large cities to buy agricultural products directly from farmers ? What are some of the conditions that make it inconvenient for farmers to buy their clothing and tools directly from the manufacturers ? Which contributes to the wealth of a country : the producer, the consumer, or the middleman ? 825. Commission. A person who transacts business for an- other frequently receives as his pay a certain rate per cent on the amount involved in the transaction. This is known as his com- mission. One who buys or sells produce for another, receiving as his pay a certain rate per cent on the cost of the products bought or on the selling price of the products sold, is called a com- mission merchant. A commission house is an establishment con- ducted by a commission merchant, where products are received and sold to retail dealers or consumers. Why are the commission houses located in the cities ? If your home is in a city in which there are commission houses, tell in what section of the city they are located. If your home is in the country, tell what products are shipped from your community to commission merchants. Farmers frequently dispose of their products by shipping them to commission merchants in the cities, who sell the products at the market prices and retain as their pay a certain per cent of the selling price. Commission merchants do not usually buy the products shipped to them, but they act merely as the agents of the shippers in receiving and selling the goods. The freight charges for shipping and the cartage charges for hauling the goods are generally paid by the Commission merchants from the proceeds of the sale of the products. After deducting their commis- sion and the charges for freight and cartage, the commission merchants remit to the shippers the balance of the sum received for the products sold. The entire amount received from the sale of goods, before any deduc- tion is made for expenses, etc., is called the gross receipts of the sale. The amount remaining from the sale of goods after all expenses have been deducted is called the net receipts of the sale. Where do city retail dealers buy their supplies of fruits, vegetables, etc. ? Farmers frequently sell or exchange small quantities of agricul- tural products at the general merchandise stores in small cities and villages. How do these merchants dispose of the products ? 264 APPENDIX 326. Market Reports. Newspapers publish market reports, giving the prices at which grain, live stock, dairy products, fruits, etc., were sold on the previous day. Of what use are such reports ? Read a recent market report. The price of produce is affected by the supply offered for sale and the demand for it. What effect upon prices has an increase in supply and a decrease in demand ? How is the price affected by an increase in demand and a decrease in supply ? 327. Brokerage. 1. One who acts as an agent for others to contract for the purchase or sale of goods, receiving as his pay a certain rate of commission, is called a broker. The com- mission of a broker is called brokerage. Commission mer- chants usually take possession of the goods bought and sold by them, while brokers merely contract for the sale or purchase of goods in the name of the person buying or selling, without taking possession of the goods. Brokers deal in stocks, bonds, grain, etc. 2. What is real estate ? One who buys and sells lands, ex- changes and leases property, etc., is called a real estate agent, or a real estate broker. He usually receives as his commission a certain rate per cent on the selling price of property, or on a month's rent when property is leased by the month. The rates of commission charged for selling and renting property vary in different communities. Find what rate of commission is charged by agents for selling and renting property in your community. What are some of the conditions that make it inconvenient for each person to sell or rent his own property ? When a person wishes to rent a house in a city, how does he find out what houses are for rent ? 3. Traveling salesmen, store clerks, auctioneers, insurance agents, etc., frequently receive as pay for their services a com- mission on the amounts of their sales. Can you name any other business in which those employed receive a commission for their services ? TRADE DISCOUNT 265 TRADE DISCOUNT 328. 1. Manufacturers and wholesale dealers issue cata- logues and price lists in which the articles manufactured by them are described and their prices given. Most manufacturers sell their goods to wholesale dealers, who supply, in turn, the retail dealers. The prices quoted in catalogues and price lists are commonly known as the list prices of goads. A discount from the list prices is generally made to retail dealers, and sometimes to others when goods are bought in large quanti- ties. This discount is reckoned as a certain rate per cent of the list price. Such a discount is generally known as trade discount. 2. Often several discounts are allowed. Thus, an article may be sold subject to discounts of 25 %, 10 %, and 5 % from the list price. In such a case, the first discount is from the list price, and the second discount is from the price after deducting the first discount, and the third discount is from the price after deducting the second discount. Usually the amount of discount allowed is varied as the market prices change. 3. A special discount is usually made when cash is paid for goods. The amount of cash discount varies considerably in different lines of trade, ranging generally from i % to 6 % of the amount of the bill after deducting the trade discount, and averaging about 2 %. Generally the retail dealer is given until about the tenth day of the" following month in which to make cash remittances. Instead of allowing a special discount for cash payments, sometimes an arrangement is made by which the retail purchaser is given 30 days, 60 days, 90 days, 6 months, or even a longer time in which to make his payment. The bill then becomes due at the end of the specified time, and in case it is not paid when the time has expired, the purchaser of the goods agrees to pay a specified rate of interest upon the amount of the bill from the time the bill is due until it is paid. 266 APPENDIX PARTIAL PAYMENTS 329. Instead of paying the whole amount of a note at one time, the maker sometimes pays it in two or more parts. Such payments are called partial payments. A record of each pay- ment is indorsed on the back of the note. 330. United J5tates Rule. The method of computing in- debtedness when partial payments have been made, illustrated in the following problem, was adopted by the Supreme Court of the United States, and is commonly known as the United States Rule. This method is the legal one in most states. In states where other methods are legal, teachers should follow them. 1. A note for $ 2000 dated May 1, 1906, at 6%, was indorsed as follows: July 25, 1906, $150; Dec. 16, 1906, $40; Feb. 12, 1907, $ 100. Find the amount due May 1, 1907. Principal, May 1, 1906 $ 2000 Interest on $ 2000 to July 25 (2 mo. 24 da.) . . . . 28 Amount, July 26, 1906 2028 First payment (July 25, 1906) 150 New Principal, July 25, 1906 $ 1878 Interest on $ 1878 to Dec. 16, 1906 (4 mo. 21 da.) . 44.13 Second payment, which is less than the interest due, Interest on $ 1878 from Dec. 16, 1906, to Feb. 12, 1907 (1 mo. 26 da.) 17.53 Amount, Feb. 12, 1907 $ 1939.66 Third payment, $100, which is to be added to the second, $ 40 140 New Principal, Feb. 12, 1907 $ 1799.63 Interest on $1799.66 to May 1, 1907 (2 mo. 19 da.) . 23.69 Amount due May 1, 1907 $ 1823.35 If the second payment, $ 40, which was less than the interest due, had been deducted from the amount due at the time the payment was made, and if the remainder had been regarded as a new principal, the effect would have been to increase the amount on which interest was paid. Hence, the interest must be reckoned to the time of the third payment. INTEREST 267 RULE. Find the amount of the principal to a time when a payment, or the sum of two or more payments, equals or exceeds the interest due. Subtract the payment or payments from the amount. Treat the remainder as a new principal and proceed as before. 2. Write a note for $ 800, naming the teacher as payee and yourself as maker. Make three partial payments and have them indorsed on the note. Find the amount due at the time of settlement. 3. Write a note for $ 1000, naming some pupil as payee and yourself as maker. Make two payments, such that the first is less than the interest to date, but the sum of both exceeds the interest to time of the second payment. Find the amount due at time of settlement. INTEREST 331. BANKERS' TABLE OF DAYS BETWEEN DATES JAN. FEB. MAR. APR. MAY JUNE JULY AUG. SEPT. OCT. Nov. DEC. Jan. 365 31 59 90 120 151 181 212 243 273 304 334 Feb. 334 366 28 69 89 120 160 181 212 242 273 303 Mar. 306 337 365 31 61 92 122 153 184 214 245 275 Apr. 275 306 334 365 30 61 91 122 163 183 214 244 May 245 276 304 335 365 31 61 92 123 153 184 214 June 214 245 273 304 334 365 30 61 92 122 153 183 July 184 215 243 274 304 335 365 31 62 92 123 153 Aug. 153 184 212 243 273 304 334 365 31 61 92 122 Sept. 122 153 181 212 242 273 303 334 365 30 61 91 Oct. 92 123 161 182 212 243 273 304 335 365 31 61 Nov. 61 92 120 151 181 212 242 273 304 334 365 30 Dec. 31 62 90 121 151 182 212 243 274 304 335 365 The exact number of days from any day of one month to the same day of another month within a year is found thus : Find the number of days from April 12 to Aug. 12. Starting from April, in the left-hand col- umn, pass the pencil across to the column headed August. The number 122 in the column headed August denotes the number of days from any day in April to the corresponding day in August. Hence it is 122 days from April 12 to Aug. 12. 268 . APPENDIX 1. How many days must be added to 122 days if you are required to find the number of days from April 12 to Aug. 14 ? to Aug. 20 ? How many days must be subtracted from 122 days if you are required to find the number of days from April 12 to Aug. 8 ? to Aug. 3? 2. What change would be made in the number of days given in the table if Feb. 29 of a leap year should intervene between dates ? 332. Written Exercises. Using the table in Sec. 331, find by the 60-day method (p. 212) the interest on : 1. $ 2000 from April 1 to July 4 at 6 %. 2. $ 4500 from Dec. 20 to Feb. 15 at 7 %. 3. $ 20,000 from May 12 to Aug. 5 at 5 %. 4. $ 80,000 from Apfil 16 to June 4 at 6 %. 5. $ 25,000 from July 4 to Sept. 16 at 8 %. 6. $ 4370 from Aug. 4 to Dec. 7 at 6 %. EXACT INTEREST 333. 1. In the methods of computing interest in Sees. 255 261, 360 days were taken as one year in reckoning the interest for periods less than a year. Interest computed on the basis of 365 days to the year is called exact interest. 2. Exact interest is found by taking such a fractional part of a year's interest as the exact number of days is of 365 days. 3. Find the exact interest on $ 400 from March 2 to June 15 at 8%. Find the exact number of days from March 2 to June 15. Do not count March 2, but count June 15. From March 2 to June 15 there are 29 da. + 30 da. -f 31 da. + 15 da. or 105 da. MODEL : 21 $400 x .08 x;0ff=,$9.20-f. m 78 TAXES 269 4. Compare the interest on $ 400 for 105 da. at 8 % com- puted in the ordinary way with the exact interest. 5. Exact interest for whole years is the same as interest computed by the ordinary method. For periods of less than a year, the exact interest is ff of the interest computed by the ordinary method, or -^^ (^) less than the ordinary interest. Compared with interest computed by the ordinary method, is exact interest favorable to the person loaning money or to the person borrowing it ? 6. Exact interest is computed by the United States govern- ment and sometimes by others. 7. How much interest does the United States government save by paying exact interest rather than ordinary interest on $ 5,000,000 from Jan. 1 to April 1, 1907, at 3 % ? 8. Find exact interest on each of the amounts in Sec. 332 for the time specified. TAXES 334. 1. Every person in the United States derives some benefits from one or more of the following divisions of govern- ment : national (also called federal), state, county, toivnship, school district, city, and village. Explain the purpose of each of these divisions of government and name some of the benefits you derive from each. The maintenance of these several forms of government necessarily involves the expenditure of money. This money is derived from taxes levied upon the persons, property, incomes, or business of individuals. Mention some purpose for which money is expended by each division of government named above. 2. Direct taxes are sums of money levied upon persons, prop- erty, incomes, or business of individuals for the support of state or local governments. They are called direct taxes be- cause they are levied directly upon the persons, lands, build- ings, etc., of individuals and are payable directly to public officials authorized to collect taxes. 270 APPENDIX 3. Indirect taxes are sums levied upon goods imported from other countries, upon certain products manufactured in our own country, and .upon the privilege of engaging in certain pursuits, as selling liquors, etc. They are called indirect taxes because they are paid indirectly by the consumers. 4. Ascertain how the money is raised for building school- houses, for repairing roads or streets, etc., in the community in which you live. STATE AND LOCAL TAXES 335. Poll Tax. Each male citizen of voting age (generally with certain exceptions) is usually taxed a fixed sum annually, regardless of the ownership of taxable property. This is called a poll tax. Poll means head. Is a poll tax a direct or an indirect tax ?t It is the only form of personal tax levied in the United States. What are the provisions relating to poll tax in your state ? 336. 1. Property Tax. Property is classified either as per- sonal property or as real property. All movable property, as household goods, money, cattle, ships, etc., is called personal property. Immovable property, as lands, buildings, mines, etc., is called real property. Both forms of property are subject to taxation. 2. For the purpose of taxation, the value of all taxable prop- erty is estimated by a public officer called an assessor. This officer prepares a list of all taxable property in his district, showing the names of the owners, the location of the property, and its assessed valuation. Such a list is commonly called an assessment roll. The assessed valuation of property is gen- erally less than its actual value. 3. Taxes are usually collected by a public officer called a tax collector, who deposits the amounts collected with another public officer called a treasurer. 4. What is public property ? What properties, other than public property, are exempt from taxation in your state ? CUSTOMS 271 5. The total assessed valuation of the property in any state is the sum of the assessed valuation of the property in the several counties of the state. How is the assessed valua- tion of the county in which you live determined ? The state and local taxes are usually levied and collected together. As- certain how state, county, and local taxes are levied and col- lected in your state. 6. Examine a receipt^ for the payment of taxes on real property. How is the location of the property described? 337. Written Exercises. 1. The assessed valuation of a certain county is $8,000,000. Find the rate of taxation that will yield revenue sufficient to pay for a new courthouse costing $ 100,000. How much must Mr. Thomas pay toward the building of the courthouse if his taxable property is assessed at $4000? 2. Find the value of Mr. White's property if his taxes amount to $67.50 when the tax rate is $.01 and his property is assessed at f of its value. 3. The rate of taxation in a certain county is $.01| when the assessed valuation is f of the actual value of the property. What would the rate of taxation have been to yield the same amount had the property been assessed at its full value ? 4. Make and solve five problems in taxes, using if possible the actual tax rates of the community in which you live. CUSTOMS AND INTERNAL REVENUE 338. 1. Funds for the maintenance of the national gov- ernment are derived chiefly from customs and internal revenue. 2. Customs, or duties, are taxes levied by the national govern- ment upon goods imported from other countries. Internal revenue is a tax levied by the national government upon the manufacture or sale of certain articles in the United States. 272 APPENDIX 339. The following tables show the receipts and expendi- tures of the national government during the fiscal year ended June 30, 1906: REVENUES, FISCAL YEAR 1906 Customs $300,251,877.77 Internal Revenue 249,150,212.91 Lands . . . ... . . ' . . 4,879,833.65 Miscellaneous 40,172,197.34 Total, exclusive of Postal .... $594,454,121.67 EXPENDITURES, FISCAL YEAR 1906 Civil and Miscellaneous $159,823,904.50 War Department . . . ... . . 119,704,113.09 Navy Department 111,166,784.35 Indians 12,746,859.08 Pensions 141,034,561.77 Interest 24,308,576.27 Total, exclusive of Postal .... $568,784,799.06 The following table shows the receipts from the principal objects of internal taxation during the same fiscal year: INTERNAL REVENUE RECEIPTS, FISCAL YEAR 1906 Distilled Spirits $143,394,055.00 Tobacco 48,422,997.88 Fermented Liquors 55,641,858.56 Oleomargarine 570,037.93 Mixed Flour 2,667.23 Adulterated Butter . . . . . . . 9,258.43 Renovated Butter 138,078.09 Playing Cards 489,347.26 Penalties . . . . . . . . 283,991.62 Collections 160,494.88 340. 1. Customs. A list or schedule of goods with the rates of import duties adopted by Congress is called a tariff. Under our tariff laws some imported articles are admitted without the payment of duties. These articles are said to be on the free list. Articles not on the free list are subject to an ad valorem duty, a specific duty, or to both. CUSTOMS 273 2. An ad valorem duty is a duty reckoned according to the value or cost of the goods in the country from which they are imported. Thus, the duty on jewelry is 60 % of the value. 3. A specific duty is a tax of a specified amount on each pound, yard, gallon, bushel, etc., regardless of the cost of the goods. Thus, the duty on onions is 40^ a bushel. 4. Certain ports are designated as ports of entry, where duties on cargoes are payable. A customhouse has been established at each port of entry for the collection of customs. The col- lection of customs at each port is under the direction of a government officer called the collector of the port. 5. The following rates of custom are from the schedule adopted by Congress in 1897, and commonly known as the Dingley Tariff. Cheese, 6^ per Ib. Paintings, 20 % ad val. Hay, $4 per T. Penholders, 25 % ad val. Coffee, free. Carpets (Axminster), 60^ per Apples, Dried, 2^ per Ib. sq. yd. and 40 % ad val. Tea, free. Carpets (Brussels), 44^ per sq. Bacon and Hams, 5^ per Ib. yd. and 40% ad val. Honey, 20 ^ per gal. Oats, 15^ per bu. Soap (Castile), 1^ per Ib. Wheat, 25^ per bu. Musical instr'ts, 45% ad val. Hops, 12^ per Ib. 341. Written Exercises. 1. How much is the duty on a violin worth $ 80 ? 2. A painting costing $2500 was purchased in Italy and brought into the United States. Find the amount of the duty charged for its admission. 3. What is the duty on 200 sq. yd. of Brussels carpet valued at $1.20 a square yard ? 4. What per cent of the expenditures of the national gov- ernment were defrayed from customs receipts in 1906 ? from internal revenue receipts ? MCCL. & JONES'S ESSEN. OF AR. 18 274 APPENDIX DEPOSIT SLIP. BANKING 342. Give the names of some banks that you know of. Of what use to a community are banks ? Why is a bank a safer place in which to keep money than a house or an office ? If a person wishes to make a payment to another, he may do so by giving him the money. Do you know of any other method commonly used in making payments ? 343. Savings Banks. Savings banks are banks organized under the laws of the different states for the purpose of re- ceiving and investing the savings of people. Their capital consists of the money put into the bank by the depositors, and their profits are divided among the depositors in pro- portion to the amount that each has on deposit. The profits are paid to the depos- itors in the form of interest, usually ranging from 3% to 4% annually, which is paid monthly, quarterly, or semi- annually. If the interest is not collected by the depositor when it becomes due, it is entered to his credit on the books of the bank and it thereafter draws interest in the same manner as the ordinary deposits. Do sav- ings banks pay compound interest ? 344. Savings Bank Ac- counts. Any reliable person who wishes to deposit money for safekeeping and invest- ment may open an account SAVINGS DEPOSIT No. /y-88 BAL. f 3/6.4- 6 DEPOSITED WITH UNION SAVINGS BANK FOR ACCOUNT OF Los ANGELES, CAL., March 22, 1907. DOLLARS GOLD .... SILVER . . . /o CURRENCY . . CHECKS . . . < BANKING 275 with a savings bank. On opening an account, the depositor is usually required to answer certain questions and to leave his signature with the bank, to protect the bank against fraudulent demands upon his accounts. The depositor then hands the "receiving teller" or the "cashier" the money which he wishes to deposit, together with a deposit slip, showing the amount of his deposit. He is then given a savings bank book, usually bearing a number, with the amount of his deposit credited to his account. This bank book must be presented whenever the depositor wishes to make a deposit or to draw against his credit in the bank. The smallest amount received for deposit is usually one dollar. When a depositor withdraws money from a savings bank, he is required to give the bank a receipt for the amount he has withdrawn. KECEIPT SAVINGS ACCOUNT Los ANGELES, CAL., afauJ, t6, ttfOY No. /V-88 RECEIVED FROM THE UNION SAVINGS BANK o&ffo&H, (Mwi ^^~xx-N~^^N^-^^-^DOLLARS f 15.00 BALANCE, $285.15. As the usage of banks differs considerably in the various sections of the country, no attempt is made here to give details. Pupils should familiarize themselves with local customs. Deposit slips, receipt blanks, or check blanks, sam- ple bank books, note forms, etc., should be examined by the pupils and their use explained to them. Where possible, a visit to a neighboring bank should be made during banking hours. 276 APPENDIX 345. Illustrative page from a savings bank book, showing entries of deposits, withdrawals, and interest : STATEMENT DATE WITHDRAWN DEPOSITED BALANCE Dec. 6, 1907 .... 45 50 128 75 Dec. 21, 1907 . . . 15 143 75 Dec. 24, 1907 . . . 35 108 75 Int. to Jan. 1, 1908 . 4 74 113 49 Jan. 3, 1908 .... 50 163 49 Feb. 2, 1908 .... 74 50 237 99 March 7, 1908 . . . 100 137 99 March 25, 1908 . . . 80 217 99 346. Banks of Deposit. Banks other than savings banks are sometimes called banks of deposit. They are variously known as national banks, state banks, commercial banks, private banks, etc. These banks are organized for the purpose of receiving deposits, making loans, etc. National banks are organized under the national law and are under the direct supervision of the Comptroller of Currency, who is appointed by the President of the United States. In addition to carrying on a general banking business, national banks have authority to issue paper money, called bank notes. The payment of these notes is secured by government bonds deposited with the Secretary of the Treasury. All other banks are organized under the laws of the states in which they are located. 347. Deposits. Accounts are opened with banks of deposit hr much the same manner as with savings banks. Business men and others who wish to keep money on hand for the pay- ment of bills, etc., usually have an account with a bank against which they may draw checks whenever they wish. Such ac- counts usually do not draw interest. Each depositor receives a bank book in which all deposits must be entered. BANKING 277 348. 1. Checks. When the account is opened, the depositor receives a check book which he uses in making demands against his accounts. STUB CHECK SAN FRANCISCO, CAL., 7n 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 ft. : : 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. The product of the extremes in a proportion is always equal to the product of the means. Hence, 3.5 times x = 5 times 28, or 3.5 x = 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 in form of expres- sion from the method (6) given on p. 291. 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 train in 1 hr. 45 min. is 80 mi., how long, at the same rate of speed, will 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. What 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 Pentagon Hexagon REGULAR, POLYGONS 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 ; of eight sides, an octagon. Draw an octagon. 3. A straight line from the center of a 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 (vfy. 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 (6s), the altitude as the other leg (v6), 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 may 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. Tlie 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, The 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 cylindrical 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. The area of the surface of a sphere is four times the area of a great circle (Trr 2 ) of the sphere. 1. As (2r) 2 , or 4 r 2 is equal to d 2 , 4 Trr 2 is equal to ird 2 . PUBLIC LANDS 297 TJie area of the surface of a sphere is equal to the square of the diameter x IT, or nd 2 . 2. Which is the greater and how 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, The volume of a sphere is equal to one third the product of its radius and its convex surface, or % Ti-r 3 (^ of r X 4 Trr 2 ). 4. As d s , or (2r) 8 is equal to 8r 3 , %irr s is equal to %-n-d s . Hence, To find the volume of a sphere, multiply the cube of its diameter fy. 5236(10/70. 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 3 (8000 3 ) to 2 s (2000 3 ), or 4 3 to I 3 . 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 Eevolutionarj 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, ui 6 <*: ui o (*J 1 5 T. 5N. Parallel 4 A B T. 4 N. ^ 3 C T. 3 N. .a s V, 2 D T. 2 N. 3 2 ,* 1 1 2 3 4 5 T. IN. G s 1 E T 1 5. .-. (J c 2 T 2 S. I 3 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 (R. 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 G. Write the description of each, using abbreviations. 12. Locate in the figure each of the following described town- ships : T. 2 N., R. 3 E.; T. 4 N., R. 5 E. ; T. 1 K, R. 1 W. ; T.2S.,R.4B.; T.1S.,R.3W.; T.4K,R.2W.; T.2S.,R.2E. 300 APPENDIX 13. 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., R. 1 E. ; T. 6 N., 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 should have a conspicuous location, visible from distant points on Hues ; 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 convergency 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. 6 5 4. 3 2 I 7 8 9 10 II 12 18 17 m IS 14 13 19 20 21 22 23 24 30 29 28 27 26 25 31 32 33 34 35 n A TOWNSHIP DIVIDED INTO SECTIONS. 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 302 APPENDIX sections of 640 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 Bis 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. J of the N.E. i of the section. How many acres does it contain ? 3. Describe the part marked G and tell how 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. . 8. S.W. of the N.W. . A \ \ D C j !- S\ -- B A SECTION SUBDIVIDED. PUBLIC LANDS 303 9. E. % of the S.W. . 10. W. of the N.W. . 11. S.E. of the S.W. of the KE. . 12. S. | of the S.E. of the N.E. . 372. Using the scale 1 in. = 1 mi., draw a plot to represent a township, say T. 6 N., E. 4 E. ; locate and find the area of each of the following : 1. E. | of the S.E. \ of Sec. 9, T. 6 K, E. 4 E. 2. N. W. of the S.E. of Sec. 22, T. 6 K, E. 4 E. 3. S.E. \ of the S. W. \ of the S.E. of Sec. 32, T. 6 N., E. 4 E. 4. E. \ of the S.W. \ of the N.E.. of Sec. 24, T. 6 N., E. 4 E., which is a farm owned by Mr. Thomas. 5. S.E. \ of the N.W. \ of Sec. 18, T. 6 N., E. 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 ? a Which are the school sections ? Why are they so called ? 304 APPENDIX . 9. Can 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 equator to the North Pole, measured on the meridian of Paris, was METRIC SYSTEM 305 selected as the primary unit of linear measure. This unit is called the meter. Meter is the French word for measure. The meter is a little longer than the yard. As it is based upon a measurement of the earth's polar circumference, the meter is a fixed natural unit.* 4. An International Bureau of Weights and Measures has been established in Paris, and is now supported by the contri- butions of more than twenty nations. A standard meter, made from an alloy of platinum and iridium, is carefully pre- served by this bureau. All the nations of the world have been furnished with duplicates of this standard meter. These duplicates are made of the most durable and least expansible metals known. The United States Bureau of Standards has fixed the legal equivalent of the meter as 39.37 inches. 5. The metric system is a decimal system. Units larger than the primary units are 10 times the primary units, 100 times the primary units, and 1000 times the primary units; units smaller than the primary units are -fa the primary units, Y^ the primary units, and T ^ny the primary units. Units of any given denomination are therefore reduced to units of a larger denomination by dividing by 10, by 100, and by 1000 ; and units are reduced to units of a smaller denomination by multiplying by 10, by 100, and by 1000. Quantities are not generally expressed in terms of two or more units, but in some single unit, parts of the unit being expressed as a decimal of the unit, as 6.35 meters. 6. Names of units larger than the primary units are formed by prefixing to the names of the primary units prefixes de- rived from the Greek words meaning ten, one hundred, and one thousand, etc.; and names of units smaller than the primary units are formed by prefixing to the names of the primary units prefixes derived from the Latin words meaning ten, one hundred, and one thousand, as follows : * Subsequent calculations have shown that the meter is not exactly a ten-millionth part of the distance from the equator to the North Pole. MCCL. & JONES'S ESSEN. OF AR. 20 806 APPENDIX GREEK PREFIXES deka, meaning 10; dekameter, meaning 10 meters. hekto, meaning 100 ; hektometer, meaning 100 meters. kilo, meaning 1000 ; kilometer, meaning 1000 meters. myria, meaning 10,000 ; myriameter, meaning 10,000 meters. The prefixes deka and hekto are sometimes written deca and hecto. LATIN PREFIXES deti, meaning 10 ; decimeter, meaning .1 meter. centi, meaning 100 ; centimeter, meaning .01 meter. milli, meaning 1000 ; millimeter, meaning .001 meter. Very small linear measurements are expressed in mikrons. Mikron is a Greek word meaning small. 375. Measures of Length. In the following exercises use a meter stick on which the centimeters and millimeters are marked off. Practice drawing these units until you can estimate their lengths quite accu- rately. Test all estimates by actual measurements. 1. Draw on the blackboard a line 1 meter long ; 2 meters long ; 3 meters long. 2. Fix two points on the floor 1 meter apart ; 2 meters apart ; 3 meters apart ; 4 meters apart. 3. Estimate the length, width, and height of your school- room in meters. 4. Measure the length of a blackboard in meters. Express fractional parts as a decimal of a meter, thus : if the black- board is 4 meters land 12 centimeters long, its length may be stated as 4.12 meters. 5. Estimate the length and width of the school yard in meters. 6. Draw a line 1 decimeter in length. Name some object in the schoolroom that is one decimeter in length, width, or thickness. METRIC SYSTEM 307 7. Draw a line 1 centimeter in length, 2 centimeters in length, 3 centimeters in length. 8. Measure the length and width of this book in centi- meters. Express fractional parts as a decimal of a centimeter. 9. Measure the thickness of this book in millimeters. How many millimeters make a centimeter ? a decimeter ? a meter ? 10. A kilometer is 1000 meters. It is equivalent to about | of a mile. Select some place that is about 1 kilometer from the schoolhouse. 11. Using rulers on which the units are marked off, com- pare the millimeter with -^ of an inch. 12. Which is the longer, a centimeter or an inch ? 376. Reduction of Linear Units. 1. A meter is how many decimeters ? how many centi- meters ? how many millimeters ? 2. 67 centimeters may be expressed as a decimal of a meter, thus : .67 meter. Express as meters : 34 centimeters, 15 centimeters, 76 centimeters. 3. A decimeter is what part of a meter ? 4 decimeters may be expressed as a decimal of a meter, thus : .4 meter. Ex- press as meters : 3 meters and 4 decimeters ; 7 meters and 32 centimeters ; 9 meters, 2 decimeters, and 4 centimeters. 4. Write a millimeter as a decimal of a meter. Write 8 millimeters as a decimal of a meter. Write 3 centimeters and 8 millimeters as a decimal of a meter. 5. Write 2 kilometers as meters. Write 2 kilometers and 430 meters as meters. Write as meters : 24.5 kilometers ; 4.25 kilometers. 6. Reduce to meters: 304 centimeters; 2.467 kilometers; 245.376 kilometers ; 30 centimeters. 308 APPENDIX 377. Table of Measures of Length. The following is the complete table of linear measure. The units most commonly used are the millimeter, centimeter, meter, and kilometer. 1000 mikrons (jt) = 1 millimeter (mm.) 10 mm. = 1 centimeter (cm.) 10 cm. = 1 decimeter (dm.) 10 dm. = 1 meter (m.) 10 m. =1 dekameter (Dm.) 10 Dm. = 1 hektometer (Hm.) 10 Htn. = 1 kilometer (Km.) 10 Km. = 1 myriameter Abbreviations of the names of the units that are multiples of the pri- mary unit are written with Capital letters to distinguish them from the abbreviations of the names of the units that are parts of the primary unit. 378. Measures of Surface. 1. Draw on the blackboard a square whose side is 1 meter in length. This is called a square meter. 2. Divide a square meter into square decimeters. How many square decimeters are there in a square meter ? 3. Divide a square decimeter into square centimeters. How many square centimeters are there in a square decimeter? 4. How many square millimeters are there in a square centimeter ? 5. How many square centimeters are there in a square meter ? 6. In what square unit should you express the area of the surface of the cover of this book? of the floor of your school- room ? 7. Draw on the school grounds a square whose side is 10 meters. This is called an are. It is the primary unit of land measure. The are is equivalent to 119.6 square yards. METRIC SYSTEM 309 8. A square whose side is 100 meters is called a hektare. 9. A square whose side is 1 kilometer is called a square kilometer. The area of gardens, etc. is usually given in ares ; of fields, etc. in hektares ; and of countries, etc. in square kilometers. 10. Estimate the number of square meters in the surface of the floor of your schoolroom. Test your estimate. 11. Estimate the number of ares in the school yard. Test your estimate. 12. How long is the side of a hektare ? of a square kilo- meter ? The hektare is nearly 2 acres. 379. Table of Measures of Surface. 100 square millimeters (qmm.) = 1 square centimeter (qcm.) 100 qcm. = 1 square decimeter (qdm.) 100 qdm. = 1 square meter (qm.) 100 qm. = 1 square dekameter (qDm. ) 100 qDm. = 1 square hektometer (qHm.) 100 qHm. = 1 square kilometer (qKm. ) 380. Table of Land Measure. 100 ceutares (ca.) = 1 are (a.) 100 a. =1 hektare (Ha.) 381. Measures of Volume. 1. From a piece of cardboard construct a cube whose edges are each 1 decimeter. This is called a cubic decimeter. 2. How many cubic decimeters are there in 1 cubic meter ? 3. From a piece of cardboard construct a cubic centimeter. Estimate the capacity of a crayon box in cubic centimeters. 4. Estimate the number of cubic meters of air in your schoolroom. Using a meter stick, make an approximate test of your estimate. 310 APPENDIX 5. The primary unit of volume is the cubic meter. The cubic meter is equivalent to 1.308 cubic yards. 6. The primary unit of wood measure is the stere, which is a cubic meter. 382. Table of Measures of Volume. 1000 cubic millimeters (cu. mm.) = 1 cubic centimeter (cu. cm.) 1000 cu. cm. = 1 cubic decimeter (cu. dm.) 1000 cu. dm. = 1 cubic meter (cu. m.) Units higher than the cubic meter are seldom used. 383. Measures of Capacity. 1. The primary unit of capacity for both liquid and dry measure is the liter, which contains 1 cubic decimeter. Using the measures, compare the capacity of a liter and a quart. The liter is equivalent to 1.0567 liquid quarts or .908 dry quart. 2. How many cubic centimeters are equivalent to 1 liter ? 3. 100 liters are 1 hektoliter. The liter is used to measure comparatively small quantities ; the hektoliter is used to meas- ure grain, produce, etc., in large quantities. The hektoliter is equivalent to 2.8377 bushels. 4. Mention some things that are bought or sold by the quart, dry measure ; by the quart or gallon, liquid measure. Where the metric system is used, these are bought and sold by the liter, or by the hektoliter if the quantities are large. 5. How many liters of water will a tank hold whose inside dimensions are 3.45 m. by 80 cm. by 60 cm.? 345 x 80 x 60 1000 number of liters in the tank. Explain. 6. Find the capacity in liters of a cylindrical tank whose diameter is 2.85 m. and whose altitude is 3.68 m. METRIC SYSTEM 311 384. Table of Measures of Capacity. 10 milliliters = 1 centiliter (el.) 10 cl. = 1 deciliter (dl.) 10 dl. = 1 liter (1.) 10 1. =1 dekaliter (Dl.) 10 Dl. = 1 hektoliter (HI.) 385. Measures of Weight. 1. The primary unit of weight is the gram, which is the freight of 1 cu. cm. of pure water at its greatest density. 2. Heft a gram weight. How 'many grams does a liter of pure water at its greatest density weigh ? 3. The weight of 1000 cubic centimeters of water (a liter) is called a kilogram, or a kilo. Heft a kilogram weight. A kilogram is equivalent to 2.2046 pounds avoirdupois. How many grams are equivalent to an ounce avoirdupois ? 4. The gram is used in weighing precious metals, medicines, etc. ; the kilogram in weighing meat, groceries, etc. Express your weight in kilograms, calling 2.2 pounds 1 kilogram. 5. 100 kilograms are 1 metric quintal, and 1000 kilograms 1 metric ton. A metric ton is equivalent to 2206 pounds or 1.1023 tons. 6. Express as grams : 2.125 Kg. ; 3.4 Kg. Express as kilo- grams : 245 g. ; 28 g. ; 362 M. T. ; 4.25 M. T. ; 4 Kg. 72 g. 386. Table of Measures of Weight. 10 milligrams (mg.) = 1 centigram (eg.) 10 eg. = 1 decigram (dg.) 10 dg. = 1 gram (g.) 10 g. =1 dekagram (Dg.) 10 Dg. = 1 hektogram (Hg.) 10 Hg. =1 kilogram (Kg.) 10 Kg. = 1 myriagram (Mg.) 100 Kg. =1 metric quintal (Q.) 1000 Kg. = 1 metric ton (M. T.) 812 APPENDIX 387. Equivalents of Metric Units. The following equivalents are given for comparison and for reference : METRIC TO COMMON COMMON TO METRIC 1m. =39.37 in., or 1.0936 yd. lyd. = .9144 m. IKm. =.62137 mi, 1 mi. =1.60935 Km. 1 sq. m. = 1.196 sq. yd. Isq.yd. = .836 sq. m. 1 Ha. =2.471 A. 1 A. . = .4047 Ha. 1 cu.m. = 1.308 cu. yd 1 cu. yd. = .766 cu. m. 11. = .908 qt. (dry) 1 qt. (dry) = 1.1012 1. 11. = 1.0567 qt. (liquid) 1 qt. (liquid) = .94636 1. 1 HI. = 2.8377 bu. 1 bu. = .35239 HI. 1 g. = 15.43 gr. (troy) 1 oz. (troy) = 31.10348 g. 1 Kg. = 32.1507 oz. (troy) 1 lb. (av.) = .45359 Kg. 1 Kg. = 2.2046 lb. (av.) 1 M. T. = 1.1023 T. 1 T. = .90718 M. T. TABLES OF DENOMINATE MEASURES (For Reference) 388. Measures of Time. 60 seconds = 1 minute 365 days = 1 year 60 minutes = 1 hour 366 days = 1 leap year 24 hours = 1 day 10 years = 1 decade 7 days = 1 week 100 years = 1 century 1. The day is the primary unit of time measure. It is the time taken by the earth to make one rotation on its axis. Is it a natural or an artificial unit ? The earth revolves around the sun in 365 days 5 hours 48 minutes 46 seconds (nearly 365^ days). This period is the solar (sun) year. 2. As the exact period taken for the earth to make a revolu- tion around the sun is a little less than 365^ days, an extra day (Feb. 29) is added to the common year once in four years (leap year), except in centennial years not exactly divisible by 400. 3. Centennial years divisible by 400 and other years divisible by 4 we leap years. Was 1700 a leap year ? Will 2000 be a leap year ? TABLES OF DENOMINATE MEASURES 313 4. More than four thousand years ago the Chaldeans, a people living in the valley of the Euphrates, calculated the length of the year to be 360 days. They believed that the sun traveled around the earth in .a circle in this period. They therefore divided the circular path of the sun into 360 equal parts, called degrees one for the part traversed each day. Hence there are 360 degrees in a circle. They observed twelve clusters of stars (constellations) in the zone in the heavens (zodiac) in which the paths of the sun and planets lie, and the occurrence of twelve full moons in successive parts of the zodiac each year. They therefore divided the course of the sun into twelve equal parts, one for each constellation. Hence there are twelve months in a year. The exact length of the lunar month is 29.53059 days. The Chaldeans divided the day into twelve " double hours." The number 60 was used by them as a unit, and they therefore divided the hour and the degree into 60 minutes ; and the minute into 60 seconds. 5. Seven days were made to constitute a unit of time meas- ure (week), either in accordance with the Mosaic law or from the fact that seven planets were known to the ancients. The days of the week were originally named after seven heavenly bodies. The English names of the days of the week are derived from the Saxons, a Germanic people who invaded and con- quered England in the fifth and sixth centuries. The Saxons borrowed the week from some eastern nation and substituted the names of their own divinities for those of the Grecian deities. NAMES OF THE DATS OF THE WEEK LATIN SAXON ENGLISH Dies Solis (Sun) Sun's day Sunday Dies Lunae (Moon) Moon's day Monday Dies Martis (Mars) Tiw's day Tuesday Dies Mercurii (Mercury) Woden's day Wednesday Dies Jovis (Jupiter) Thor's day Thursday Dies Veneris (Venus) Friga's day Friday Dies Saturni (Saturn) Seterne's day Saturday 314 APPENDIX 6. Until the time of Julius Caesar (46 B.C.) the calendar was in almost constant state of confusion, owing to the fact that the number of days allowed for a year was more or less than the actual number of days taken for one revolution of the earth in its orbit. As a result of this error, in the time of Julius Caesar the winter months had been carried back into autumn, and the autumn months into summer. To correct the error, Caesar decreed that 90 days should be added to the year to restore the time of the vernal equinox, and that the year should consist of 365^ days. He ordered that the common year should thereafter consist of 365 days and that every fourth year should consist of 366 days. The extra day was added to February, which at that time had 29 days. This arrangement is known as the Julian Calendar, or Old Style. The month of July was named after Julius Caesar. 7. Augustus Caesar ordered that the month following that which bore the name of Julius (July) should be named after himself ; and in order that the month bearing his name should have as many days as the month bearing the name of Julius, he ordered that one day be taken from February and added to the month which should bear his name. Hence the eighth month is named August and consists of as many days as July. 8. The year established by the Julian Calendar (365| days) was .00778 of a day longer than the actual time taken for one revolution of the earth in its orbit. This error had amounted to 10 days by 1582, when Pope Gregory XIII undertook the correction of the calendar. To adjust the time of the vernal equinox, Pope Gregory ordered that ten days be skipped, from October 5th to the 15th, and that only centennial years that are exactly divisible by 400 and other years that are exactly divis- ible by 4 be made leap years. This arrangement is known as the Gregorian Calendar, or New Style, and is the one in common use. Russia still follows the Julian or Old Style. The error in the Gregorian Calendar will amount to one day in about 5000 years. TABLES OF DENOMINATE MEASURES 315 389. Measures of Length. 12 inches = 1 foot 3 feet = 1 yard 16^ feet (5^ yd.) - 1 rod 320 rods = 1 mile 1 mile = 1760 yards = 5280 feet 1. The yard is the primary unit of length. All the other units of length are derived from it. 2. A furlong is % mile. It is little used at the present time. 3. A hand, used in measuring the height of horses at the shoulder, is 4 inches. 4. A fathom, used in measuring the depth of the sea, is 6 feet. 5. A knot, or nautical mile, used in measuring distances at sea, is 6080.27 feet, or approximately 1.15 (about 1|) miles. The speed of vessels is expressed in knots. A vessel that travels 18 knots an hour travels about 21 miles an hour (18 mi. plus of 18 mi.). 6. For the supposed origin of the inch, foot, fathom, etc., consult a dictionary or an encyclopedia. 390. Measures of Surface. 144 square inches = 1 square foot 9 square feet = 1 square yard 30J square yards = 1 square rod 160 square rods = 1 acre 640 acres = 1 square mile 1. A square acre is 208.71 + feet on a side. 2. A tract of land 1 mile square is called a section. A town- ship is a tract of land 6 miles square and consists of 36 sections. 3. 100 square feet of flooring, roofing, or slating is called a square. 391. Measures of Volume. 1728 cubic inches = 1 cubic foot 27 cubic feet = 1 cubic yard 316 APPENDIX 1. A pile of wood 8 feet long, 4 feet wide, and 4 feet high, or 128 cubic feet of wood, is called a cord. For the origin of the name, consult the dictionary. Stonework is sometimes meas- ured by the cord. 2. In measuring stonework, a pile of stone 16 feet long, 1 feet wide, and 1 foot high, or 24f cubic feet of stone, is called a perch. 392. Surveyors' Measures of Length. 100 links (!.) = ! chain (ch.) 80 chains = 1 mile The chain in common use is called Gunter's chain. It is 4 rods, or 66 feet long. A link is .66 foot. Links are written as hundredths of a chain, thus : 30 chains 45 links is written 30.45 chains. 393. Surveyors' Measures of Surface. 10 square chains = 1 acre 640 acres = 1 square mile Square chains are reduced to acres by moving the decimal point one place toward the left. Explain. 394. Avoirdupois Weight. 16 ounces = 1 pound 100 pounds = 1 hundredweight 2000 pounds = 1 ton 1. The English ton, known in the United States as the long ton, is 2240 pounds. It is used in United States custom- houses and in weighing coal and mineral products at the mines and sometimes in retailing coal. 2. The smallest unit of weight is the grain. A pound avoir- dupois is 7000 grains. Consult a dictionary for an explanation of the origin of the name. TABLES OF DENOMINATE MEASURES 317 3. Grains, vegetables, etc., are commonly sold by weight or measure. The weight of 1 bushel of the most common of these articles is as follows : wheat = 60 Ib. oats = 32 Ib. beans = 60 Ib. barley = 48 Ib. peas = 60 Ib. sweet potatoes = 65 Ib. clover seed = 60 Ib. rye = 66 Ib. Irish potatoes = 60 Ib. shelled corn = 56 Ib. 395. Troy Weight. Troy weight is used in weighing precious metals. 24 grains = 1 pennyweight 20 pennyweights = 1 ounce 12 ounces = 1 pound A pound troy is 5760 grains. It is f$[f pound avoirdupois. Precious stones and pearls are weighed by the carat. A carat equals 3 grains troy. The term carat is used also to express the proportion of gold in an alloy. It then signifies a twenty-fourth part. Thus, gold that is 18 carats fine is ^|, or f pure gold. 396. Apothecaries' Weight. Consult a dictionary for the meaning of the word apothecary. This system of weights is used to .some extent in filling pre- scriptions. The pound, ounce, and grain are the same as in troy weight, but the ounce is subdivided differently. 20 grains (gr.) = 1 scruple . . . sc. or 3 3 scruples = 1 dram . . . dr. or 3 8 drains = 1 ounce . . . oz. or 5 12 ounces = 1 pound . . . Ib. or Ib 397. Apothecaries' Liquid Measures. 60 drops (gtt.) or minims (TT\J = 1 fluid dram . . . /3 8 fluid drams = 1 fluid ounce . . . / 5 16 fluid ounces = 1 pint O. 8 pints = 1 gallon Cong. 318 APPENDIX 398. Liquid Measures. 4 gills = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon Quart means one fourth. A quart is one fourth of a gallon. A gallon is 231 cubic inches. A gallon of water weighs about 8^ pounds. A cubic foot of water (about 7^ gal.) weighs about 62 pounds. In measuring the capacity of cisterns, etc., 31^ gallons are called a barrel. 399. Dry Measures. This system is but little used in some parts of the United States. Where it is not used, articles are usually sold by weight. 2 pints = 1 quart 8 quarts = 1 peck 4 pecks = 1 bushel The dry quart contains 67.20 cubic inches, the fluid quart 57.75 cubic inches. A bushel contains 2150.42 cubic inches. The standard bushel in the United States is the Winchester bushel. It is the volume of a cylinder 18^ inches in internal diameter and 8 inches in depth. 400. Measures of Angles and Arcs. 60 seconds (") = 1 minute (') 60 minutes = 1 degree () 360 degrees = 4 right angles, or 1 circumference 90 of angle = 1 right angle ; 90 of arc = 1 quadrant For an explanation of the origin of 360 degrees in a circum- ference, etc., see Measures of Time, p. 313. 401. Counting Table. 2 units = 1 pair 20 units = 1 score 12 units = 1 dozen 12 dozen = 1 12 gross = 1 great gross TABLES OF DENOMINATE MEASURES 319 402. Measures of Value United States Money. 10 mills = 1 cent 10 dimes = 1 dollar 10 cents = 1 dime 10 dollars = 1 eagle The standard unit of value is the gold dollar. A gold dol- lar (no longer coined) contains 23.22 grains of pure gold and 2.58 grains of alloy. A silver dollar contains 371.25 grains of pure silver and 41.25 grains of alloy. The symbol for dollar is $, which is taken from U.S. The coins of the United States are bronze, 1^ ; nickel, 5 ^ ; silver, 10 25 X, 50^, $1; and gold $2^, $5, $ 10, and $20. The mill is not coined. These are coined at mints located in Philadelphia, New Orleans, Denver, and San Francisco. The paper currency is issued in the denominations of $1, f 2, f 5, $ 10, $ 20, $ 50, $ 100, $ 500, and $ 1000. Paper cur- rency consists of bank notes, silver certificates, and gold cer- tificates. Examine some paper currency. The provision made for the redemption of each piece of paper currency is printed on each bill. Paper currency issued by national banks is commonly called bank notes. Their payment is guaranteed by deposits of government bonds with the national government. 403. Values of Common Coins. VAUTK IN TERMS R OTOH COUNTRY MONETARY UNIT OP ^ ^^ >LI> EQUIVALENT Austria-Hungary Crown $ .203 $.20 British Possessions, N A. (except New- foundland) Dollar ($) $1.000 $1.00 France Franc (F.) $ .193 $ .20 German Empire Mark (M.) $ .238 $ .25 Great Britain Pound Sterling () $4.866^ $5.00 Italy . Lira (L.) $ .193 $ .20 Japan Yen (Y.) $ .498 $ .50 Mexico Peso $ .498 $ .50 Philippine Islands Peso $ .500 $ .50 Russia Ruble $ .515 f .50 320 APPENDIX 404. Table of Compound Interest. Amount of $ 1, at various rates, interest compounded annually. YEARS 1% iy% 3% 2V a % 3% 3V 2 % 1 1.010000 1.015000 1.020000 1.025000 1.030000 1.035000 2 1.020100 1.030225 1.040400 1.050625 1.060900 1.071225 3 1.030301 1.045678 1.061208 1.076891 1.092727 1.108718 4 1.040604 1.061364 1.082432 1.103813 1.125509 1.147523 5 1.051010 1.077284 1.104081 1.131408 1.159274 1.187686 6 1.061520 1.093443 1.126162 1.159693 1.194052 1.229255 7 1.072135 1.109845 1.148686 1.188686 1.229874 1.272279 8 1.082857 1.126493 1.171659 1.218403 1.266770 1.316809 9 1.093685 1.143390 1.195093 1.248863 1.304773 1.362897 10 1.104622 1.160541 1.218994 1.280085 1.343916 1.4.10599 11 1.115668 1.177949 1.243374 1.312087 1.384234 1.459970 12 1.126825 1.195618 1.268242 1.344889 1.425761 1.511069 13 1.13-8093 1.213552 1.293607 1.378511 1.468534 1.563956 14 1.149474 1.231756 1.319479 1.412974 1.512590 1.618695 15 1.160969 1.250232 1.345868 1.448298 1.557967 1.675349 16 1.172579 1.268986 1.372786 1.484506 1.604706 1 733986 17 1.184304 1.288020 1.400241 1.521618 1.652848 1.794670 18 1.196148 1.307341 1.428246 1.559659 1.702433 1.857489 19 1.208109 1.326951 1.456811 1.598650 1.753506 1.9-22501 20 1.220190 1.346855 1.485947 1.638616 1.806111 1.989789 YEAKS 4% 4V 2 % 5% 6% 7% 8% 1 1.040000 1.045000 1.050000 1.060000 1.070000 1.080000 2 1.081600 1.092025 1.102500 1.123600 1.144900 1.166400 3 1.124864 1.141166 1.157625 1.191016 1.225043 1.259712 4 1.169859 1.192519 1.215506 1.262477 1.310796 1.360489 5 1.216653 1.246182 1.276282 1.338226 1.402552 1.469328 6 1.265319 1.302260 1.340096 1.418519 1.500730 1.586874 7 1.315932 1.360802 1.407100 1.503630 1.605782 1.713824 8 1.368569 1.422101 1.477455 1.593848 1.718186 1.850930 9 1.423312 1.486095 1.551328 1.689479 1.838459 1.999005 10 1.480244 1.552969 1.628895 1.790848 1.967151 2.158925 11 1.539454 1.622853 1.710339 1.898299 2.104852 2.331639 12 1.601032 1.695881 1.795856 2.012197 2.252192 2.518170 13 1.665074 1.772196 1.885649 2.132928 2.409845 2.719624 14 1.731676 1.851945 1.979932 2.260904 2.578534 2.937194 15 1.800944 1.935282 2.078928 2.396558 2.759032 3.172169 16 1.872981 2.022370 2.182875 2.540352 2.952164 3.425943 17 1.947901 2.113377 2.292018 2.692773 3.158815 3.700018 18 2.025817 2.208479 2.406619 2.854339 3.379932 3.906020 19 2.10(5849 2.307860 2.526950 3.025600 3.616528 4.315701 20 2.191123 2.411714 2.653298 3.207136 3.869684 4.660957 INDEX Abstract number, 16. Accident Insurance, 278. Accounts, 42, 48, 274. Acute angle, 79, 221, 229. Acute-angled triangle, 221, 229. Ad valorem duty, 198, 278. Addend, 16. Addition, of denominate numbers 77. of fractions, 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, 818. Angles, 79, 221, 229, 818. Apothecaries' measures, 817. Apothem, 294. Appendix, 256-320. Approximate answers, 28. Arabic numerals, 10. Arc, 237, 818. 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. Bills, 42, 43. and receipts, 44. Board foot, 160. Bonds, 261. Broker, 192, 258, 264. Brokerage, 258, 262-264. MOCL. & JONES'S ESSEN. OF AR. Calendar, 312-314. Cancellation, 105. Cancellation method, 218. Capacity, measures of, 288, 284, 810, 811. Capital, 256. Carat, 317. Cash discount, 204. Certificate of deposit, 277. Check, 277. Cipher, 10. Circle, 231, 232, 287. area of, 232. Circular measure, 237, 818. 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, 820. Concrete number, 16. Cone, 295, 296. Consumer, 262. Corporation, 256. Corporation bond, 261. Correspondence bank, 278. Counting measure, 818. Coupon bond, 261. Credit, creditor, 42. Cube (rectangular prism), 288. Cube of numbers, 244. Cube root, 246. Cubic measure, 84, 85, 233, 809, 810, 815. Customhouse, 198, 278. 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, 68-70. multiplication of, 89-41. notation and numeration of, 15. reduction of, 156-158. subtraction of, 25. Degree, 237. Denominate numbers, 72, 76-86. tables of, 312-319. Denominator, 98, 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, of fractions, 117, 118, 124-131. of integers and decimals, 45-70. Divisor, 45. Draft, bank, 278. Drv measure, 818. Duties, 198, 202, 208, 271-278. Endowment policy, 279. Equation, 283-292. Equator, 288. 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, 36. Fire insurance, 195. Flooring, 161. Foreign money, 319. Forms, 221-243. Fraction defined, 98. Fractional unit, 98. Fractions, 90-165. addition and subtraction of, 97-99, 108-112. multiplication and division of, 118-131. reduction of, 95, 96, 100-105, 156-158. Gain and loss, 181-183. Government expenses, 272. Gram, 311. Greatest common divisor, factor, or measure, 104. Gregorian calendar, 814. 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, 278-288. Integers, defined, 9. Integers and decimals, 7-89. Interest, by aliquot parts, 208. cancellation method, 213. compound, 217, 820. defined, 167. exact, 268, 269. simple, 207-218. six per cent method, 218. sixty day method, 212. tables, 267, 320. Internal revenue, 198. Joint note, 214. Land measure, 297, 809. Latitude, 238. Law of commutation, 82. Least common denominator, 109. Least common multiple, 106. Life insurance, 273-283. Life policies, 279. Like quantities, 16. Linear measure, 76, 808, 815. Liquid measure, 318. List prices, 204. Liter, 310. Local taxes, 198, 270. Long division, 63-67. Long measure, 76, 808, 815. Longitude and time, 240-242. Loss and profit, 181-183. 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, 221-2' 292-297. Measures, 312. Merchants' rule for partial payments, 216. Meridian, 288, 299. Meter, 804. Meter reading, 75. Metric system, 804-812. Minuend, 20. Mixed number, 94. Model biU, 42. Multiple, 45, 106. Multiplicand, 81. Multiplication, 81. of fractions, 118-116, 119-123. of integers and decimals, 31-44. Multiplier, 31. Municipal corporation, 261. Negotiable note, 208. New style calendar, 814. Notation and numeration, 10-15. Note, 207, 20S, 214. Number relations, 145, 146. Numerals, 10. Numeration, 10-15. Numerator, 93. Oblique angle, 79. Oblique line, 221, 228. 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, 228. 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, 228. Personal insurance, 278. Personal property, 198, 270. Pi (T), 281. 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, 238-235. Proceeds, 218. Producers, 262. Product, 31. Profit and loss, 181-188. Promissory note, 207, 214. Proper fraction, 94. Property, 198, 269. Property tax, 270. Proportion, 248, 291. Public lands, 297-299. Pyramid, 295, 296. Quadrilateral, 221, 227. Quotient, 45. Radical, 246. Radius, 222, 281. 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, 238. Reduction of fractions, 100. Registered bond, 261. Reviews, 71-75, 187-144, 150-155, 184-187, 254, 255. Right angle, 79, 221, 228, 237. Right-angled triangle, 221, 226, 250, 251. Roman- notation, 18. Roots, 244-251. Savings banks, 274. Scale drawing, 132-186. Section, 301, 802. Shares of stock, 256. Shinning, 162. Short methods, 88, 148. Similar figures, 253. 324 INDEX Similar fractions, 97. Similar surfaces and solids, 258. Six per cent method, 218. Sixty day method, 212. Slant height, 295. Solar year, 312. Solids, 84, 294-297. Specific duty, 198, 273. Sphere, 296, 297. Square, rectangle, 80, 224, 294. second power, 244. Square measure, 78, 815. Square root, 247-250. Standard time, 241. State taxes, 198, 270. Stere, 310. Stock quotations, 259. Stocks and bonds, 256-262. Subtraction, of denominate numbers, 77. of fractions, 97-99, 108-112. of integers and decimals, 20-30. Subtrahend, 20. Sum, 16. Surface measure, 78, 808, 309, 815. Surveyors' measure, 316. Tariff, 272. Tax collector, 270. Tax rate, 198. Taxes, 198-201, 269-271. Term policy, 279. Terms of fraction, 94. Time measure, 312, 318. Tontine policy, 281. Town lines, 299. Townships, 298, 301. Trade discount, 204, 205, 265. Trapezoid, 228, 229, 292. Triangle, 221, 226, 229, 230, 294. Triangular prisms, 222, 238. Troy weight, 317. Unit, 7. Unit, fractional, 93. Unit of measure, 7. United States money, 14, 25, 819. United States rule, 266. Unknown quantity, 283. Usury, 215. Value, table of, 319. Vertex, 226. Vertical line, 221. Volume, 233-235, 309, 310, 315, 316. "Weight, measures of, 311, 316, 317. UNITED STATES HISTORIES By JOHN BACH McMASTER, Professor of American History, University of Pennsylvania Primary History, $0.60 School History, $1.00 Brief History, $1.00 THESE standard histories are remarkable for their freshness and vigor, their authoritative statements, and their impartial treatment. They give a well- proportioned and interesting narrative of the chief events in our history, and are n'ot loaded down with extended and unnecessary bibliographies. The illustrations are his- torically authentic, and show, besides well-known scenes and incidents, the implements and dress characteristic of the various periods. The maps are clear and full, and well executed. ^J The PRIMARY HISTORY is simply and interestingly written, with no long or involved sentences. 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At the same time, there is included a graphic description of the curious animals, rare birds, wonder- ful physical features, natural resources, and great industries of each country. The illustrations for the most part are repro- ductions of photographs taken by the author. The maps show the route taken over each continent. ^| The Readers on Commerce and Industry give a personal and living knowledge of the great world of commerce and industry. The children visit the great food centers and see for themselves how the chief food staples are produced and prepared for use, and they travel in the same way over the globe investigating the sources of their clothing. The journeys are along geographical lines, and while studying the industries the children are learning about localities, trade routes, and the other features of transportation and commerce. AMERICAN BOOK COMPANY SUPPLEMENTARY READING By EDWARD EGGLESTON STORIES OF GREAT AMERICANS FOR LITTLE AMERICANS $0.40 THIS book is eminently suited to second year pupils. Not only does it make learning to read an easy task, but it provides matter which is stimulating and enjoy- able. By means of interesting personal anecdotes, the child is made familiar with the history of our country and some of its leading figures. Famous warriors and patriots, states- men, discoverers, inventors, men of science and letters, find a place in these tales. Some of the stories should be known to every American, because they have become a kind of national folk-lore. The words are not too difficult, while the sentences and paragraphs are short. STORIES OF AMERICAN LIFE AND ADVENTURE $0.50 HERE are presented for third year pupils exciting stories which tell of the adventurous pioneer life of this country, and which show why the national character is distinguished by traits of quick-wittedness, humor, self- reliance, love of liberty, and democratic feeling. These historical anecdotes include stories of Indian life, of frontier peril and escape, of adventures with the pirates of Colonial times, of daring Revolutionary feats, of dangerous whaling voyages, of scientific explorations, and of personal encounters with sa'/ages and wild beasts. With them are intermingled sketches of the homes, the food and drink, the birds and animals, the schools, and the children's plays of other times. AMERICAN BOOK COMPANY (7) HISTORICAL READERS By H. A. GUERBER Story of the Thirteen Colonies $0.65 Story of the Great Republic 65 Story of the English 65 Story of the Chosen People 60 Story of the Greeks 60 Story of the Romans 60 A LTHOUGH these popular books are intended primarily J~\, for supplementary reading, they will be found quite as valuable in adding life and interest to the formal study of history. Beginning with the fifth school year, they can be used with profit in any of the upper grammar grades, ^j In these volumes the history of some of the world's peoples has taken the form of stories in which the principal events are centered about the lives of great men of all times. Through- out the attempt has been made to give in simple, forceful lan- guage an authentic account of famous deeds, and to present a stirring and lifelike picture of life and customs. Strictly mili- tary and political history have never been emphasized. ^[ No pains has been spared to interest boys and girls, to impart useful information, and to provide valuable lessons of patriotism, truthfulness, courage, patience, honesty, and in- dustry, which will make them good men and women. Many incidents and anecdoies, not included in larger works, are interspersed among the stories, because they are so frequently used in art and literature that familiarity with them is in- dispensable. The illustrations are unusually good. . ^| The author's Myths of Greece and Rome, Myths of Northern Lands, and Legends of the Middle Ages, each, price $ 1. 50, present a fascinating account of those wonderful legends and tales of mythology which should be known to everyone. Seventh and eighth year pupils will delight in them. AMERICAN BOOK COMPANY (18) SPENCERS' PRACTICAL WRITING By PLATT R. SPENCER'S SONS Books i, 2, 3, 4, 5, 6, 7, and g . . . . Per dozen, $0.60 SPENCERS' PRACTICAL WRITING has been devised because of the distinct and wide-spread reaction from the use of vertical writing in schools. It is thoroughly up- to-date, embodying all the advantages of the old and of the new. Each word can be written by one continuous move- ment of the pen. ^[ The books teach a plain, practical hand, moderate in slant, and free from ornamental curves, shades, and meaningless lines. The stem letters are long enough to be clear and un- mistakable. The capitals are about two spaces in height. ^[ The copies begin with words and gradually develop into sentences. The letters, both large and small, are taught systematically. In the first two books the writing is some- what larger than is customary because it is more easily learned by young children. These books also contain many illustra- tions in outline. The ruling is very simple. ^[ Instruction is afforded showing how the pupil should sit at the desk, and hold the pen and paper. A series of drill move- ment exercises, thirty-three in number, with directions for their use, accompanies each book. SPENCERIAN PRACTICAL WRITING SPELLER Per dozen, $0.48 THIS simple, inexpensive device provides abundant drill in writing words. At the same time it trains pupils to form their copies in accordance with the most modern and popular system of penmanship, and saves much valuable time for both teacher and pupil. AMERICAN BOOK COMPANY (39) HICKS'S C HAM PI ON SPELLING BOOK By WARREN E. HICKS, Assistant Superintendent of Schools, Cleveland, Ohio Complete, $0.25 - Part One, $o. 18 - Part Two, $0.18 THIS book embodies the method that enabled the pupils in the Cleveland schools after two years to win the Na- tional Education Association Spelling Contest of 1908. ^[ By this method a spelling lesson often words is given each day from the spoken vocabulary of the pupil. Of these ten words two are selected for intensive study, and in the spelling book are made prominent in both position and type at the head of each day's lessons, these two words being followed by the remaining eight words in smaller type. Systematic review is provided throughout the book. Each of the ten prominent words taught intensively in a week is listed as a subordinate word in the next two weeks ; included in a written spelling contest at the end of eight weeks ; again in the annual contest at the end of the year ; and again as a subordinate word in the following year's work; used five times in all within two years. ^J The Champion Spelling Book consists of a series of lessons arranged as above for six school years, from the third to the eighth, inclusive. It presents about 1,200 words each year, and teaches 3 I 2 of them with especial clearness and intensity. It also includes occasional supplementary exercises which serve as aids in teaching sounds, vowels, homonyms, rules of spell- ing, abbreviated forms, suffixes, prefixes, the use of hyphens, plurals, dictation work, and word building. The words have been selected from lists, supplied by grade teachers of Cleve- land schools, of words ordinarily misspelled by the pupils of their respective grades. AMERICAN BOOK COMPANY DAVISON'S HUMAN BODY AND HEALTH By ALVIN DAVISON, M.S., A.M., Ph.D., Professor or Biology in Lafayette College. Intermediate Book . $0.50 Advanced Book . $0.80 THE object of these books is to promote health and pre- vent disease and at the same time to do it in such a way as will appeal to the interest of boys and girls, and fix in their minds the essentials of right living. They are books of real service, which teach mainly the lessons of health- ful, sanitary living, and the prevention of disease, which do not waste time on the names of bones and organs, which furnish information that everyone ought to know, and which are both practical in their application and interesting in their presentation. ^j These books make clear: ^[ That the teaching of physiology in our schools can be made more vital and serviceable to humanity. ^[ That anatomy and physiology are of little value to young people, unless they help them to practice in their daily lives the teachings of hygiene and sanitation. ^| That both personal and public health can be improved by teaching certain basal truths, thus decreasing the death rate, now so large from a general ignorance of common diseases. ^[ That such instruction should show how these diseases, colds, pneumonia, tuberculosis, typhoid fever, diphtheria, and malaria are contracted and how they can be prevented. ^[ That the foundation for much of the illness in later life is laid by the boy and girl during school years, and that in- struction which helps the pupils to understand the care of the body, and the true value of fresh air, proper food, exercise, and cleanliness, will add much to the wealth of a nation and the happiness of its people. AMERICAN BOOK COMPANY THE ELEANOR SMITH MUSIC COURSE By ELEANOR SMITH, Head of the Department of Music, School of Education, University of Chicago, Director of Hull House Music School. First Book . . . . 80.25 Second Boole 30 Third Book .... $0.40 Fourth Book 50 THIS music series, consisting of four books, covers the work of the primary and grammar grades. It contains nearly a thousand songs of exceptional charm and interest, which are distinguished by their thoroughly artistic quality and cosmopolitan character. The folk songs of many nations, selections from the works of the most celebrated masters, numerous contributions from many eminent Ameri- can composers, now presented for the first time, are included. |J The Eleanor Smith Music Course is graded in sympathy with the best pedagogical ideas according to which every song becomes a study, and every study becomes a song. Technical points are worked out by means of real music, instead of manufactured exercises; complete melodies, instead of musical particles. Each technical point is illustrated by a wealth of song material. A great effort has been made to reduce to the minimum the number of songs having a very low alto. |J The course as a whole meets the demands of modern education. Modern life and modern thought require the richest and best of the past, combined with the richest and best of the present, so organized and arranged as to satisfy existing conditions in the school and home. The series is world wide in its sources, universal in its adaptation, and modern in the broadest and truest sense of the word. AMERICAN BOOK COMPANY A SYSTEM OF PEDAGOGY By EMERSON E. WHITE, A.M., LL.D. Elements of Pedagogy $ I . oo School Management and Moral Training I.oo Art of Teaching I.oo BY the safe path of experience and in the light of modern psychology the ELEMENTS OF PEDAGOGY points out the limitations of the ordinary systems of school education and shows how their methods may be har- monized and coordinated. The fundamental principles of teaching are expounded in a manner which is both logical and convincing, and such a variety and wealth of pedagogical principles are presented as are seldom to be found in a single text-book. ^[ SCHOOL MANAGEMENT discusses school govern- ment and moral training from the standpoint of experience, observation, and study. Avoiding dogmatism, the author carefully states the grounds of his views and suggestions, and freely uses the fundamental facts of mental and moral science. So practical are the applications of principles, and so apt are the concrete illustrations that the book can not fail to be of interest and profit to all teachers, whether experienced or inexperienced. ^[ In the ART OF TEACHING the fundamental princi- ples are presented in a clear and helpful manner, and after- wards applied in methods of teaching that are generic and comprehensive. Great pains has been taken to show the true functions of special methods and to point out their limita- tions, with a view to prevent teachers from accepting them as general methods and making them hobbies. The book throws a clear light, not only on fundamental methods and processes, but also on oral illustrations, book study, class instruction and management, written examinations and pro- motions of pupils, and other problems of great importance. AMERICAN BOOK COMPANY This book is DUE on the last date stamped below NOV 1 1934 *OV4 1938 MUL 8 193; 16 MAY 1 3 1954 MRY 1 4 RUCO JLTN7 1954 NOV 6 1958 Form L-9-15m-7,'32 UC SOUTHERN REGIONAL LIBRARY FACILITY A 000933212 3 UNIVERSITY of CALIFORNIA ..us JLNGELJbitt LIBRARY