IN MEMORIAM FLOR1AN CAJORI K^^ -^ OW" 1 " w) HOPKINS AND UNDERWOOD'S NEW ARITHMETICS ADVANCED BOOK THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO ATLANTA SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, LTD. TORONTO HOPKINS AND UNDERWOOD'S NEW ARITHMETICS ADVANCED BOOK BY JOHN W. HOPKINS SUPERINTENDENT OF THE GALVESTON PUBLIC SCHOOLS AND P. H. UNDERWOOD TEACHER OF MATHEMATICS, BALL HIGH SCHOOL, GALVESTON, TEXAS gorfe THE MACMILLAN COMPANY 1908 All rights reserved K7-3 COPYRIGHT, 1903, 1907, BY THE MACMILLAN COMPANY. Set up and electrotyped. Published August, 1903. New edition revised. Published December, 1907. CAJOR1 NorfoooU J. 8. Gushing Co. Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE THIS book assumes a working knowledge of the four fundamental rules as applied to integers and United States Money. It contains the essentials of practical arithmetic arranged by topics in conformity with the courses of study in some of the best school systems, each chapter representing a year's work commencing with the fifth grade. It aims to teach principles rather than rules. As the unitary method is the one most natural to the young learner, the first two chapters give prominence to this style of reasoning. Chapters III and IV give a thorough review of arithmetic principles and practice. In these chapters the method of ratio is brought into prominence. Science to be of value must be more or less deductive in form. Characteristic features of the book are the early intro- duction of Decimals, the large number of problems based on the industrial resources of our country, the clearness and directness with which problems are illustrated, and the omission of complicated problems of doubtful utility. The aim of teaching arithmetic is culture, accuracy, and rapidity in the computation of problems arising in actual life, and the acquisition of correct methods of reasoning. This aim is always kept in view. However, as students possess the power of learning readily to work processes, and, furthermore, as the practice of arithmetic 918235 vi PREFACE is of more importance to the majority of people than the theory, attention is paid especially to the art of compu- tation. The book contains a short introduction to the method of obtaining approximate results correct to any required degree of accuracy. This matter is new, but it is believed that it is well worthy of consideration. Chapters III and IV contain all the arithmetic and men- suration that is most needful to be known, and, in fact, will be found comprehensive enough to suit the require- ments of pupils taking a survey of commercial arithmetic. JOHN W. HOPKINS, P. H. UNDERWOOD. GALVESTON, TEXAS, September 9, 1907. CONTENTS PAGE PREFACE v CHAPTER I NUMERATION AND NOTATION 1 DECIMALS 5 ADDITION AND SUBTRACTION 7 MULTIPLICATION AND DIVISION 11 BILLS 22 MEASURES AND MULTIPLES 24 TESTS OF DIVISIBILITY 27 GREATEST COMMON DIVISOR 30 LEAST COMMON MULTIPLE 31 FRACTIONS 33 REDUCTION OF FRACTIONS 35 ADDITION OF FRACTIONS 39 SUBTRACTION OF FRACTIONS 41 CANCELLATION 42 MULTIPLICATION OF FRACTIONS 44 DIVISION OF FRACTIONS. RATIO 48 DECIMALS 53 MULTIPLICATION AND DIVISION BY POWERS OF TEN . . 54 ADDITION 57 SUBTRACTION 57 MULTIPLICATION 59 DIVISION o 61 vii viii CONTENTS PAGE REDUCTION OF FRACTIONS TO DECIMALS AND REDUCTION OF DECIMALS TO FRACTIONS ....... 66 MISCELLANEOUS EXAMPLES 67 COMPLEX FRACTIONS 70 AREAS OF RECTANGULAR FIGURES 71 COMPUTATION ON THE BASIS OF 100, 1000, 2000 ... 73 PERCENTAGE 76 INTEREST AND PROPERTY INSURANCE . . ... 79 CHAPTER II COMPOUND QUANTITIES . 81 REDUCTION OF COMPOUND QUANTITIES 82 ADDITION . . . 93 SUBTRACTION ... . . . . . . .96 MULTIPLICATION . .99 DIVISION . . . . 100 EXPRESSION OF ONE QUANTITY AS A FRACTION OF ANOTHER QUANTITY . . . 102 MEASUREMENTS . ... . . . . . . 103 VOLUMES OF RECTANGULAR SOLIDS . . . . . . 105 AREAS OF PARALLELOGRAMS, TRIANGLES, AND TRAPEZOIDS . 107 BOARD MEASUREMENT ........ 110 MASONRY AND BRICKLAYING 112 CARPETING . . 113 MISCELLANEOUS EXERCISES 115 REVIEW OF FRACTIONS 120 PERCENTAGE . . . 122 To FIND A NUMBER WHEN A PER CENT OF IT is GIVEN . 124 TO FIND WHAT PER CENT ONE NUMBER IS OF ANOTHER NUMBER . . . . 127 COMMERCIAL DISCOUNTS 130 PROFIT AND Loss 133 CONTENTS ix PAGE COMMISSION AND BROKERAGE . . . . * . . . . 140 INTEREST . . 144 SPECIFIC GRAVITY . 148 RATIO . . ... . ... . . . 154 PROPORTION . . . . . 156 REVIEW . . . . < . 158 CHAPTER III GENERAL REVIEW BY TOPICS 163 ADDITION . . . 163 SUBTRACTION .... . . . 168 MULTIPLICATION 172 PARTICULAR SHORT METHODS OF MULTIPLICATION . . 173 DIVISION . . 177 LONGITUDE AND TIME . . ... . . . 181 STANDARD TIME 188 APPROXIMATIONS. CONTRACTED PROCESSES, GENERAL METHODS OF SOLUTION .... . . . 190 ' LANGUAGE OF MATHEMATICS 196 s\ 'RATIO 200 COMPOUND PROPORTION . . .... . . 205 PARTNERSHIP . .... . . . . . 208 PERCENTAGE . . 210 INTEREST . . . 217 EXACT INTEREST . . 218 INVERSE QUESTIONS IN INTEREST . 219 REVIEW .222 REVIEW QUESTIONS . 224 PROMISSORY NOTES . . . 225 BANKERS' INTEREST 228 COMMERCIAL DISCOUNT 231 PARTIAL PAYMENTS. UNITED STATES RULE . . . . 232 X CONTENTS PAGK MERCHANTS* RULE 234 EXCHANGE 236 VALUE OF FOREIGN COINS . . . . . . . . . 241 ENGLISH MONEY 242 FOREIGN EXCHANGE 244 STOCKS AND BONDS 246 CUSTOMS AND DUTIES 252 CHAPTER IV INVOLUTION 255 EVOLUTION 259 PROBLEMS INVOLVING SQUARE ROOT ..... 264 AREAS OF PLANE TRIANGLES 267 MENSURATION OF THE CIRCLE 269 SIMILAR FIGURES 275 SURFACES OF PRISM, PYRAMID, CYLINDER, CONE, AND SPHERE 277 VOLUMES OF SOLIDS 280 MEASURE OF TEMPERATURE 282 METRIC SYSTEM OF WEIGHTS AND MEASURES . . . 284 ANNUAL INTEREST 291 COMPOUND INTEREST 292 MISCELLANEOUS TOPICS 295 WORK AND TIME 295 MOTION IN THE SAME OR OPPOSITE DIRECTIONS . . . 298 CLOCKS 299 MISCELLANEOUS EXAMPLES BY TOPICS (A) .... 303 MISCELLANEOUS EXAMPLES (B) 318 TABLES . , 325 HOPKINS AND UNDERWOOD'S NEW ARITHMETICS ADVANCED BOOK ADVANCED BOOK OF AEITHMETIC CHAPTER I Arithmetic is the science of numbers and the art of com- putation. The fundamental operation in arithmetic is counting. The result of counting is a number. Any one of the natural numbers, one, two, three, etc., is called an integer, or -whole number. A unit is one thing, or a group of things regarded as a single thing. NUMERATION The number names are one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, hun- dred, thousand, million, billion, trillion, etc. By combin- ing these number names all numbers may be expressed in words. Ones, tens, hundreds, thousands, ten-thousands, are re- spectively called units of the first order, units of the second order, units of the third order, units of the fourth order, units of the fifth order. In our system of naming numbers ten units of any order are equal to one unit of the next higher order. On this account our system is known as the decimal system. Numeration is the naming of numbers. 2 ADVANCED BOOK OF ARITHMETIC NOTATION Every number can be expressed by one or more of the following figures, sometimes called Arabic numerals : 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine of these figures are called digits, or significant figures. Tens are written in the second place ; hundreds, in the third place ; thousands, in the fourth place ; millions, in the seventh place. For example, four thousand seven hundred eighty-nine is written 4,789. This number may be read four thousands seven hundreds eight tens nine ones. The nine ones occupy the first place ; the eight tens, the second place ; the seven hundreds, the third place ; the four thousands, the fourth place. To read a number expressed by more than three figures, begin at the right, that is, with units of the first order, and mark off with commas the figures in groups of three each. The three places, or orders, in which units of the first order occur constitute what is called the units' period ; the next three places, the thousands' period ; the next three places, the millions' period; the next three, the billions' period, etc. As an illustration take the number 1734902309 ; marking this number off into periods, it becomes 1,734,902,309; this is read one billion seven hundred thirty-four million nine hundred two thousand three hundred nine. Observe that each period has its hundreds, tens, and units. The periods most used are the units, thousands, and millions. The bil- lions period is rarely used. The billionth part of a great circle of the earth is less than 2 inches. The names of a few of the succeeding periods are trillions, quadrillions,, quintillions, and sextillions. Notation is the expression of numbers by means of symbols. NOTATION 3 EXERCISE 1 Express in words : 1. 45289. 10. 910003. 19. 8307308. 2. 90208. 11. 728000. 20. 8000401. 3. 75307. 12. 400098. 21. 7000014. 4. 392394. 13. 902023. 22. 6100079. 5. 738211. 14. 630006. 23. 3927173. 6. 328993. is. 1000000. 24. 5009020. 7. 401012. 16. 2227001. 25. 8000904. 8. 300287. 17. 3456000. 26. 6203003. 9. 200020. 18. 9287003. 27. 9000090. 28. What is the largest whole number expressed by two figures? 29. What is the smallest whole number expressed by two figures? 30. What is the largest whole number expressed by three figures? 31. What is the smallest whole number expressed by three figures? 32. Write the largest number expressed by the figures 0, 4, 5. 33. Write the smallest number expressed by the figures 0, 4, 5. 34. Write three numbers expressed by the figures 2, 3, 4. 35. Write four numbers expressed by the figures 7, 6, 8. 36. What is the largest number expressed by the figures 7, 3, 2, 8? 37. What is the smallest number expressed by the figures 2, 5, 3, 4? 4 ADVANCED BOOK OF ARITHMETIC EXERCISE 2 Write in figures : 1. Four thousand, eight hundred twenty-seven. 2. Nine thousand, seven hundred one. 3. Sixty-eight thousand, four hundred fifty-two. 4. Forty-seven thousand, three hundred eight. 5. Ninety thousand, six hundred four. 6. Eighty-seven thousand, one hundred one. 7. Twenty-two thousand, three hundred eleven. 8. Twelve thousand, fifteen. 9. Eighteen thousand, eighteen. 10. Fourteen thousand, thirty-four. 11. Thirteen thousand, five. 12. Ninety thousand, nine. 13. Fifty-four thousand, eleven. 14. Seventy-three thousand, one. 15. Six hundred four thousand, two hundred one. 16. One hundred sixty-three thousand, ten. 17. One hundred one thousand, three hundred. 18. One hundred thousand, seven. 19. Four hundred ten thousand, one hundred twenty, seven. 20. Five hundred four thousand, three hundred eight. 21. Five hundred thousand, eleven. 22. Six hundred thousand, seventeen. 23. Nine hundred ninety thousand, fifteen. 24. Two hundred one thousand, one. 25. Seventy-two thousand, four. READING DECIMALS 5 DECIMALS NOTE. Pupils should draw and measure distances such as 9.3 centi- meters, 2.87 inches. The law pervading the decimal system of notation is the value of a digit in any place is always ten times the value of the same digit written in the next place to the right. A familiar illustration of this law is the notation of United States money. For example, $5.55. The 5 on the left is ten times the value of the second 5, and the second 5 is ten times the value of the 5 on the right. The period separating dollars and cents is called the decimal point. The first place to the right of the decimal point is called the tenths' place ; the second place, the hundredths' place ; the third place, the thousandths' place ; the fourth place, the ten-thousandths' place ; the fifth place, the hun- dred-thousandths' place; the sixth place, the million ths' place. READING DECIMALS Since .47 = 4 tenths 7 hundredths = Therefore, .47 is read forty-seven hundredths. Since .372 = 3 tenths 7 hundredths 2 thousandths = ^ + To -o + To 2 o o = iVoV Therefore, .372 is read 372 thou- sandths. In general a decimal is read by regarding it as a whole number and adding the name of the place the right-hand digit occupies. In reading decimals and should not be used except to connect the integral and decimal parts of the number. For example, 500.005 is read five hundred and five thousandths. .505 is read five hundred five thousandths. 8.0379 is read eight and three hundred seventy-nine ten-thousandths. .8379 is read eight thou- sand three hundred seventy-nine ten-thousandths. 6 ADVANCED BOOK OF ARITHMETIC EXERCISE 3 Read : 1. 6.2. 7. 6.201. is. 5.0067. 2. 7.9. 8. 4.027. 14. 7.0123. 3. 8.4. 9. 5.029. is. 9.1238. 4. 4.32. 10. 9.001. 16. .0003. 5. .12. 11. 6.034. 17. .0054. 6. 5.17. 12. 8.295. is. .4008. Write : 1. 33 hundredths. 3. 329 thousandths. 2. 2005 "thousandths. 4. 101 thousandths. 5. Two hundred three thousandths. 6. Seven hundred and four thousandths. 7. Nine hundred three thousandths. 8. Nine hundred and three thousandths. 9. Six thousand seven hundred ten-thousandths. 10. Six thousand seven hundred and one ten-thou- sandth. 11. Five hundred ninety ten -thousandths. 12. Five hundred and ninety ten-thousandths. 13. Six thousand one ten-thousandths. 14. Seven hundred ten-thousandths. 15. Seven hundred ten thousandths. 16. Five hundred thousandths. 17. Five hundred-thousandths. 18. Two hundred seven hundred-thousandths. 19. Two hundred and seven hundred -thousandths. 20. Five thousand two hundred two hundred-thou- sandths. ADDITION AND SUBTRACTION 7 21. Six thousand four hundred-thousandths. 22. Three thousand ten thousandths. 23. Three thousand one ten-thousandths. 24. Five hundred seventeen ten-thousandths. 25. One hundred eleven hundred-thousandths. 26. Seventy-eight ten-thousandths. ADDITION AND SUBTRACTION The result of combining two or more numbers into a single number is called the sum. Addition is the process of finding the sum of two or more numbers. Only numbers of the same kind can be added. The symbol for addition is +, and it is read plus. The symbol = is read equal, or equals ; thus, 5 + 8 = 13 is read five plus eight equal thirteen. The difference between two numbers is the excess of one over the other. Subtraction is the process of finding the difference be- tween two numbers. The number subtracted is the subtrahend. The num- ber from which the subtrahend is taken is the minuend. The result of subtraction is called the remainder, or difference. The sign of subtraction is , and is read minus. Thus, 8 5 = 3 is read eight minus five equals three. ILLUSTRATIVE EXAMPLE. From 913 take 537. 913 800 + 100 + 13 = 913 537 500+ 30+ 7 = 537 376 300 + 70 + 6 = 376 7 from 13 leaves 6 ; 3 from 10 leaves 7 ; 5 from 8 leaves 3. This is the usual explanation given by teachers. 8 ADVANCED BOOK OF ARITHMETIC When a figure in the subtrahend cannot be taken from the corresponding figure in the minuend, a unit of the next higher order in the minuend is changed to ten units and then added to the figure in the minuend. A better way of subtracting is : 7 and 6 are 13. Write 6, carry 1. 1 and 3 are 4; 4 and 7 are 11. Write 7, carry 1. 1 and 5 are 6 ; 6 and 3 are 9. Write 3. EXERCISE 4 The following table gives the number of children of school age, number enrolled, average daily attendance, and total expenditures for the public schools by states and territories for the school year 1904 : STATE OR TERRITORY NUMBER OP CHILDREN NUMBER ENROLLED AVERAGE DAILY ATTENDANCE TOTAL EXPEND- ITURES North Atlantic Division. Maine 163,931 131,176 98,257 $ 2,080,109 New Hampshire 91,847 65,673 48,673 1,376,899 Vermont 81,358 66,535 48,845 1,176,784 Massachusetts 673,690 494,042 391,771 16,436,668 Rhode Island 108,471 70,843 51,692 1,804,762 Connecticut 223,174 163,141 123,317 3,795,260 New York 1,859,824 1,300,065 963,780 43,750,277 New Jersey 514,585 352,203 239,505 8,838,515 Pennsylvania 1,782,740 1,200,230 900,234 26,073,565 South Atlantic Division. Delaware 50,695 36,895 25,300 453,670 Maryland 347,594 209,978 130,065 2,755,288 District of Columbia 64,766 49,789 39,300 1,576,354 Virginia 611,555 375,601 224,769 2,137,365 West Virginia 319,874 244,040 158,264 2,531,655 North Carolina 666,782 491,838 318,055 2,075.566 South Carolina 490,214 292,115 214,133 1,191,963 Georgia 789,939 502,014 310,400 2,240,247 Florida 180,501 122,636 83,631 945,848 ADDITION South Central Division. Kentucky 700,272 501,482 309,836 $2,662,863 Tennessee 678,782 502,330 344,882 2,602,141 Alabama 652,518 365,171 240,000 1,252,247 Mississippi 563,019 403,647 233,175 1,868,544 Louisiana 483,967 208,737 155,794 1,551,232 Texas 1,128,934 722,904 461,938 6,200,587 Arkansas 467,821 339,542 212,131 1,729,879 Oklahoma 164,882 152,886 93,495 1,359,624 Indian Territory 162,641 38,422 23,053 643,616 North Central Division. Ohio 1,151,007 835,607 618,495 15,802,002 Indiana 732,172 550,732 416,047 9,363,450 Illinois 1,428,613 978,554 783,563 21,792,751 Michigan 684,369 497,299 388,092 9,158,014 Wisconsin 658,474 461,214 288,300 7,885,050 Minnesota 566,397 423,663 272,500 8,073,323 Iowa 672,271 545,940 373,023 10,696,693 Missouri 965,598 731,410 464,706 9,878,198 North Dakota 110,938 95,224 58,442 2,316,346 South Dakota 130,844 106,822 73,700 2,239,135 Nebraska 321,822 278,930 180,771 4,774,146 Kansas 455,943 390,236 270,878 5,684,579 Western Division. Montana 63,106 44,881 31,471 1,236,253 Wyoming 24,960 14,512 9,650 253,551 Colorado 145,799 134,260 95,117 3,984,967 New Mexico 64,094 39,704 29,582 353,012 Arizona 35,365 21,088 13,022 438,828 Utah 98,762 75,662 56,183 1,657,234 Nevada 9,013 7,319 5,182 257,501 Idaho 54,700 54,480 39,817 1,001,394 Washington 147,302 161,651 110,774 4,053,468 Oregon 118,977 103,877 72,464 1,803,339 California 363,846 299,038 222,182 9,401,465 Find the totals for each of the above divisions. Also find the total in each instance for the entire country. 10 ADVANCED BOOK OF ARITHMETIC EXERCISE 5 The following table gives the gross and net earnings of the principal railroads of Texas for the eleven months ending May 31, 1907 : RAILROAD GROSS EARNINGS NET EARNINGS 1. C. R. I. and G. $3,082,314.24 $ 895,096.73 2. F. W. and D. C. 4,094,588.28 1,345,055.55 3. G. H. and S. A. 11,130,030.96 2,536,237.69 4. F. W. and R. G. 1,075,725.13 342,223.14 5. H. and T. C. 6,572,660.00 2,242,022.12 6. G. C. and S. F. 12,510,936.83 3,365,234.82 7. H. E. and W. T. 1,284,929.31 511,161.15 8. I. and G. N. 8,204,579.38 2,079,764.06 9. M. K. and T. 9,989,708.55 2,239,523.41 10. St. L. S. W. of T. 3,936,613.38 506,646.38 11. S. A. and A. P. 3,518,565.80 1,517,563.03 12. T. and N. O. 4,103,849.13 968,031.21 13. T. and P. 15,456,714.54 5,427,188.11 14. Texas Central 1,149,069.36 489,109.71 Find the operating expenses (difference between gross and net earnings) of each of the above roads. 15. The increase in net earnings of the G. C. and S. F. road for the eleven months ending May 31, 1907, over that of the corresponding months of the preceding year was $1,361,052.87. Find the net earnings for the eleven months ending May 31, 1906. 16. A like increase for the I. and G. N. road was $1,235,847.06. Find its net earnings for the eleven months ending May, 1906. MULTIPLICATION AND DIVISION 11 MULTIPLICATION AND DIVISION 97 + 97 + 97 + 97 + 97 = ? If these numbers are added, the sum will be 485. In examples of this character the usual process is as follows : 97 Five 7's are 35. Write 5, carry 3. Five 9's are _5 45, and 3 are 48. The result is 485. 485 This short method of adding is called multiplication. Multiplication is a short method of addition when the numbers to be added are all the same. The number to be repeatedly added is the multiplicand. The number in- dicating how often the multiplicand is to be taken as an addend is the multiplier. The result of a multiplication exercise is the product. The multiplicand and multiplier are factors of the product. Since the multiplier denotes number of times, it must always be a pure number, or an abstract number. The multiplicand may be either a con- crete or an abstract number. The product is concrete or ab- stract according as the multiplicand is concrete or abstract. The sign of multiplication is x , and is read multiplied by, or times. Thus, $ 34 x 7 means $ 34 is to be multi- plied by 7. 7 x $ 34 means 7 times $ 34. Is 7x8 = 8x7? Is 9x4 = 4x9? Is 7x3x8 = 3x8x7? Is 9x6x5 = 6x9x5? The order in -which numbers are multiplied is immaterial. (a) How many times is $4 contained in $24 ? (5) What is the sixth part of $24 ? These two examples illustrate the two kinds of division; the first is to determine the number of times one number or quantity is contained in another number or quantity. This is often called measuring. The second is to deter- mine a part of a number or quantity, and is often called parting, or dividing. 12 ADVANCED BOOK OF ARITHMETIC Division is the process of determining how often one num- ber is contained in another, and also of determining any given part of a number. The first number is the divisor; the second is the dividend ; the result is the quotient. When divisor and dividend are concrete numbers, the quotient is abstract. When the dividend is a concrete number, and the divisor an abstract number, the quotient is a concrete number. The signs of division are -*-, and a horizontal stroke, the dividend written above, the divisor below. Thus, 27 -5- 3, and -2JL indicate that 27 is dividend and 3, divisor. ILLUSTRATIVE EXAMPLES (a) The area of Massachusetts is 8,040 square miles, and the average number of inhabitants per square mile, for the year 1900, was 348.9. Find its population. 348.9 8040 One square mile averages 348.9 inhab- 139560 itants ; therefore, 8,040 square miles will 27912 have 8,040 times 348.9 inhabitants. 2805156.0 (5) The area of the United States, exclusive of posses- sions, is 3,026,000 square miles, and the estimated popu- lation for the year ending June 30, 1905, was 83,143,000. Find the average number of inhabitants per square mile. 27.4 3026)83143. Observe the area is 3,026 thousands 6052 of square miles and the population is 22623 83,143 thousands of inhabitants. Hence, 21182 the required quotient will be obtained 14410 by dividing 83,143 by 3,026. MULTIPLICATION 13 EXERCISE 6 1. An office desk costs $25. How much will 3 such desks cost? 8 desks? 36 desks? 49 desks? 2. Eggs sell for 26 / per dozen. Find the cost of 8 dozen ; 18 dozen ; 94 dozen. 3. There are 5,280 feet in a mile. How many feet are in 19 miles? in 76 miles? 4. How many days are in 39 weeks? 5. A contractor pays in wages $78 a day. How much will he pay in 78 days? 6. How many hours are in 89 days? 7. A train travels at the rate of 34 miles an hour. How far will it run in 47 hours? 8. How many acres are in a ranch containing 98 sections of land? (1 section = 640 acres.) 9. A degree on a meridian of the earth is about 69 miles. How many miles are in 17 degrees ? 10. A cubic foot of rock weighs 148 pounds. How many pounds do 3,297 cubic feet weigh? 11. The rent of a dwelling is $28 per month. Find the rent for 3 years. 12. A gallon of water contains 231 cubic inches. How many cubic inches are in 368 gallons? 13. A book has 360 pages, each page has 32 lines, and each line averages 9 words. How many words are in the book? 14. A carpenter earns $3.20 a day. At this rate, how much wages will he receive in 299 days? 15. A brick mason earns $ 5.60 a day. How much will he earn in 310 days ? 14 ADVANCED BOOK OF ARITHMETIC EXERCISE 7 1. A city block is 100 yards long and 90 yards wide. Find its area. 2. Find the area of a square whose side is 84 yards. 3. Find the area of a square having 320 rods for a side. 4. Find the area of a rectangle, the length being 140 yards and the width 84 yards. 5. Find the area of a rectangle 238 yards long and 96 yards wide. 6. A farm, rectangular in shape, 440 yards long and 380 yards wide, contains how many square yards ? 7. Find the area of a rectangle 75 rods long and 63 rods wide. 8. Find the area of a grass plot 240 feet by 84 feet. 9. A sheet of paper 18 inches long and 14 inches wide contains how many square inches ? 10. A township is 6 miles long and 6 miles wide. How many square miles does it contain ? 11. A county, having the shape of a rectangle, is 24 miles long and 18 miles wide. How many square miles are in its area ? 12. A street is 1760 yards long and 23 yards wide. How many square yards does it contain ? 13. A garden is 50 yards long and 44 yards wide. How many square yards are in its area ? 14. A city lot is 43 feet wide and 124 feet deep. How many square feet are in its area ? 15. A window is 60 inches by 48 inches. How many square inches are in its area ? 16. A yard is 36 inches. How many square inches are in a square yard ? MULTIPLICATION 15 EXERCISE 8 Find the product : 1. $ 79.94 x 8. 12. $ 79.29 x 7. 2. $ 32.20 x 7. 13. | 29.97 x 8. 3. $ 79.49 x 6. 14. $179.38 x 6. 4. $128.29 x 7. is. 1373.39 x 5. 5. $399.39 x 9. 16. $799.94 x 8. 6. $454.59 x 12. 17. $822.50 x 9. 7. $729.38 x 11. 18. $998.78 x 11. 8. $237.38 x 9. 19. $778.75 x 12. 9. $720.99 x 7. 20. $732.75 x 9. 10. $285.68 x 8. 21. $928.34 x 7. 11. $ 51.33 x 7. 22. $653.82 x 5. 23. When shoes sell for $3.90 a pair, how much will 24 pairs of shoes cost ? 24. If overcoats sell for $7.98, find the price of 20 overcoats. 25. Mackintoshes sell for $6.95 apiece. How much will 27 mackintoshes cost? 26. When wheat is 84^ per bushel, how much will 384 bushels bring ? 27. Find the price of 349 bushels of corn at 56^ a bushel. 28. Cheese costs 13^ per pound. Find the price of 54 pounds. 29. Find the cost of 325 pounds of sugar at 6? per pound. 30. An acre of land is worth $60.75. Find the value of 100 acres. 16 ADVANCED BOOK OF ARITHMETIC EXERCISE 9 On the map of the United States published by the General Land Office, Department of the Interior, 1 inch represents 37 miles. On this map the distances in inches between the cities named are given below : 1. New Orleans to Chicago 22.75. 2. Savannah to Indianapolis 16.6. 3. Mobile to Toledo 21.89. 4. Richmond to St. Louis 19.2. 5. Washington to San Antonio 37.88. 6. Boston to Jackson 34.6. 7. Atlanta to Des Moines 20.4. 8. Newport to St. Louis 27.9. 9. New York to Lincoln 32.2. 10. Chicago to San Francisco 50.8. 11. St. Louis to Portland, Oregon, 47.1. 12. Memphis to Seattle 51.2. Find the distance in miles between each of the above- named cities. Find the number of inhabitants in the states named: STATE AREAS IN SQ. Mi. NUMBER OF INHABIT- ANTS PER SQ. Mi. 13. Georgia 58,980 37.6 14. Iowa 55,475 40.2 15. Illinois 56,000 86.1 16. Louisiana 45,420 30.4 17. Michigan 57,430 42.2 18. New Jersey 7,525 250.0 19. Ohio 40,760 102.0 20. Pennsylvania 44,985 140.0 DIVISION 17 EXERCISE 10 Divide and prove your answers correct : 1. 77,354 -;- 16. 9. 99,392 - 36. 17. 828,374 + 56. 2. 79,358 -*- 18. 10. 59,738 + 35. is. 528,739 ^ 64. 3 . 97,854 H- 20. 11. 49,399 - 40. 19. 629,394 -*- 72. 4. 92,738 ^ 21. 12. 99,988 -f- 42. 20. 273,579 -5- 81. 5. 100,000 + 24. is. 68,698 -*- 48. 21. 179,246 -*- 84. 6. 73,948 -5- 25. 14. 123,456 -^ 49. 22. 739,264 -r- 90. 7. 69,593-^23. is. 876,543-^-50. 23. 543,293 -f- 56. 8. 85,376 -v- 32. 16. 789,295 -r- 54. 24. 665,670 + 81. EXERCISE 11 1. When sugar sells for 6 cents per pound, how many pounds can be bought for 84 cents ? 2. If a boy walks at the rate of 3 miles per hour, how long will it take him to walk 87 miles ? 3. In a peck there are 8 quarts. How many pecks are there in 3000 quarts ? 4. If a bicyclist rides 9 miles an hour, how many hours will it take him to go from St. Louis to Indianapolis, a distance of 265 miles? After riding 19 hours, how far from Indianapolis will he be ? 5. When coal costs 9 dollars a ton, how many tons can be bought for 3456 dollars ? 6. A teacher receives a salary at the rate of 4 dollars a day for every day he teaches. His yearly salary is 716 dollars. How many days are in the school year ? 7. A brick mason receives 4 dollars a day for every day he works. How many days must he work to earn 900 dollars ? 18 ADVANCED BOOK OF ARITHMETIC 8. How many feet are there in 2,500 inches ? 9. How many weeks are there in 364 days ? 10. If a dozen penknives cost 9 dollars, how many dozen penknives can be bought for 126 dollars ? 11. Plows cost 12 dollars a piece. How many can be bought for 192 dollars ? 12. A box of oranges is worth 3 dollars. How many such boxes can be bought for 111 dollars ? 13. When a barrel of pork sells for 12 dollars, how many barrels must be sold to realize 5004 dollars ? 14. A section foreman rides on a velocipede at the rate of 11 miles an hour. How long will it take him to go from Cincinnati to Cleveland, a distance of 264 miles ? 15. Divide 1000 dollars among 8 persons, giving to each the same sum of money. 16. A flock of sheep sells for 966 dollars. How many sheep are in the flock, if each sheep sells for 6 dollars? 17. A man has 795 dollars in 5-dollar gold pieces. How many coins has he ? 18. Hogs sell for 8 dollars apiece. At this price how many can be purchased for 360 dollars ? 19. A box of soap is listed at 4 dollars. How many such boxes can be purchased for 980 dollars ? 20. How many barrels of flour can be bought for 1002 dollars, when flour sells for 6 dollars a barrel ? 21. Oyster crackers cost 5 cents a pound. How many pounds can be bought for 95 cents ? 22. By buying horses at 75 dollars each and selling them at 84 dollars each, a jobber makes a profit of 324 dollars. How many does he sell ? DIVISION 19 EXERCISE 12 1. How many bags of Rio coffee can be bought for 882 dollars, if one bag costs 21 dollars ? 2. Currants sell for 14 dollars a barrel ; at this price how many barrels can be bought for 546 dollars ? 3. Granulated sugar is worth 16 dollars a barrel. How many barrels must be sold to realize 1264 dollars ? 4. There are 36 inches in one yard. How many yards are in 100,000 inches ? 5. There are 32 quarts in one bushel. How many bushels are in 7712 quarts ? 6. How many days are in 3000 hours ? 7. A degree on a meridian of the earth's surface is 69 miles long. Two places on the same meridian are 2484 miles apart. How many degrees apart are they ? 8. How many square yards are in 3276 square feet ? 9. A gallon contains 231 cubic inches. How many gallons are in a barrel containing 8316 cubic inches ? 10. Oolong tea costs 15 dollars a chest. How many chests can be purchased for 495 dollars ? 11. A barrel of sugar weighs 325 pounds. How many barrels are in 105,625 pounds ? 12. How long will it take a train, rate 30 miles an hour, to go from New York to San Francisco, a distance of 3270 miles, if 5 hours are allowed for stops ? 13. There are 10 square chains in an acre. How many acres are in 10,000 square chains ? 14. Rhode Island contains in round numbers 800,000 acres. Find its area in square miles. (640 acres = 1 square mile.) 20 ADVANCED BOOK OF ARITHMETIC EXERCISE 13 1. If land is worth $68 an acre, how many acres can be bought for $4,624? 2. A tract of land is sold for $4,795.50 at the rate of $69.50 an acre. How many acres are in the tract ? 3. When horses sell for $85.40 apiece, how many can be bought for $36,465.80 ? 4. If the price of wheat is 76^ per bushel, how many bushels can be bought for $4,043.20 ? 5. When cans of asparagus sell for 35^ each, how many can be bought for $85.75 ? 6. If a pair of patent leather shoes sells for $3.85, how many pairs must be sold to bring $1,482.25 ? 7. A clothier invests in men's trousers $392. 04. How many does he buy, supposing each pair to cost $1.98 ? 8. If a keg of pickles cost $1.70, how many kegs can be purchased for $28.90 ? 9. Chipped beef is bought at 17^ a pound. At this rate, how many pounds can be bought for $361.25? 10. A 12-pound sack of flour retails at 45^. How many sacks can be bought for $322.65? 11. When a can of sardines retails for 27^, how many cans will $218. 70 buy? 12. A farmer gets for his apples $816.35 at the rate of $1.45 per barrel. How many barrels does he sell ? 13. Corn is worth 45^ per bushel. How many bushels can be bought for $7,876.35 ? 14. Oats are worth 36 ^ per bushel. How many bushels can be bought for $142.56 ? 15. A share of railway stock is quoted at $78.50. How many shares must be sold to realize $3,061.50 ? DIVISION 21 EXERCISE 14 The following table gives the area and population of some of the principal countries : COUNTRY 1. Austria-Hungary 2. Belgium 3. Denmark 4. France 5. German Empire 6. Italy 7. Japan 8. Netherlands 9. Russia 10. Spain 11. United Kingdom AREA IN SQ. Mi. 241,300 11,370 15,360 207,050 208,800 110,600 147,700 12,560 8,660,000 194,800 121,370 POPULATION 47,355,000 7,161,000 2,574,000 39,300,000 60,478,000 33,604,000 47,975,000 5,592,000 141,000,000 18,618,000 43,221,000 Find the population per square mile of each of the above countries. 12. Reduce 1 to a decimal. 17.00000 2.83333 .40476 Reduce to decimals : 42 can be resolved into two factors, 6 and 7. The simplest way of dividing by 42 is to divide by the factors 6 and 7. ; H; rife; " M; H; i 15 - ft; W; A; IJ; iflh Reduce to decimals, correct to four figures : 16 - i 5 T ; i ; T\ ; iV > yV 5 i 1 ! 5 iV 5 iV- 17 * 2 5 Y ; M ; A 5 FT 5 II 5 yV' 22 ADVANCED BOOK OF ARITHMETIC SPECIMEN BILL GALVESTON, TEXAS, Jan. 31, 1907. MR. A. B. C. In account with K. M. & CO., DEALERS IN FURNITURE, CARPETS, RUGS, &c. Jan. 2 3 Chairs @ $2.25 $6 75 10 1 Library Table 25 00 15 3 Rugs @ $6.75 20 25 20 40 yd. Matting @ 45 $ 18 00 24 2 Wardrobes @ $17.50 35 00 $105 00 Paid Feb. 1, 1903. K. M. & CO. Per M. \ MR. P. Q. R. DALLAS, TEXAS, Feb. 5, 1907. Bought of M. R. S., RETAIL GROCERS. Jan. 5 3 Ib. Tea @50$ $1 50 a 28 Ib. Sugar @5\$ 1 4^ it 3pks. Potatoes @ 40$ 1 20 (( 7 Ib. Bacon @ 15$ 1 05 10 8 Ib. Butter @ 35$ 2 80 12 3 cans Salmon @ 17 $ 51 13 6 Ib. Sausage @ 12$ 72 o $>y 25 Paid Feb. 8. M. E. S. Per X. DECIMALS 23 EXERCISE 15 Make out the following bills and receipt them : 1. Mr. John Rye bought of William Merchant, 12 yd. Calico ...... @ 9 1 15 yd. Sheeting ..... @ 7 ^ 11 yd. Flannel ..... @ 35 f 2 Hats ....... @ 13.75 18 yd. Carpet ..... @ 75 1 3 Smyrna Rugs ..... @ 110.50 2. Mr. V J. Hill bought of F. Warner & Co., 5 Stoves ....... @ $6.50 3 doz. Knives ..... @ 14.80 2 Saws ....... @ $1.50 5 Iron Beds ...... @ 115.75 6 Wrenches ...... @ $1.25 3. H. Van Oppen bought of Hegel & Co., 2 bu. Potatoes ..... @ $1.50 5 Ib. Tea ....... @ 75 ^ 2 boxes Herring . . . . @ $1.95 25 Ib. Ham ...... @ 15 f 45 Ib. Sugar ...... @ 4. Mr. James Kay bought of Simpson, Perdue & Co., 50 Ib. Sugar ...... @ 4%f 15 cans Tomatoes . @ 13 ^ 27 cans Corn ...... @ 16 ^ 10 packages Breakfast Food . @ 12J ^ 8 cans Salmon ..... @ 18 ^ 5 gal. Maple Sirup . . . @ $1.30 25 Ib. Butter ...... @ 37^ 61b. Y. H. Tea. . . . . 75^ 24 ADVANCED BOOK OF ARITHMETIC MEASURES AND MULTIPLES NOTE. Measures and Multiples in this book have reference only to numbers which are both integral and abstract. A number is prime, or is said to be a prime number, when it is exactly divisible by only itself and unity. Thus, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc., are prime numbers, or prime factors. A number which is exactly divisible by other numbers, as well as by itself and unity, is called & composite number. Thus, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc., are compos- ite numbers. An even number is one which is exactly divisible by 2. Thus, 4, 6, 8, 10, 12, etc., are even numbers. An odd number is one which is not exactly divisible by 2. Thus, 3, 5, 7, 9, 11, etc., are odd numbers. One number is said to be a measure of another number when it is contained an exact number of times in that other number. Instead of the word " measure," factor, divisor, and sub- multiple are often used. Example: 5 is a measure, factor, divisor, or submultiple of 10, 15, 20, 25, etc. By Greatest Common Measure of two or more numbers is understood the greatest measure which these numbers have in common. Thus, 6 is the greatest common measure of 12, 18, 30. Greatest Common Measure is usually designated by the letters G. C. M. It is called also Greatest Common Divisor (G. C.D.). If a number measures each of two or more numbers, it is said to be a common measure of those numbers. SURES AND MULTIPLES 25 Thus, 2, 3, and 6 are common measures of 12, 18, 30. Two or more numbers are prime to each other when they have no common measure but 1. Thus, 8 and 9 are prime to each other; so also are 15 and 28 prime to each other. These numbers, while prime to each other, are not themselves prime numbers. The result obtained by multiplying a number by an integer is called a multiple of the number. Thus, the multiples of 8 are 8, 16, 24, 32, 40, 48, etc. The Least Common Multiple of two or more numbers is the least number which is a multiple of each of the num- bers. In other words, the Least Common Multiple of two or more numbers is the least number which is exactly divisible by each of the two or more numbers. Least Common Multiple is denoted by L. C. M. Example 1. What is the L. C. M. of 8 and 12 ? Writing the multiples of 8 and 12, we have: 8 16 24 32 40 48 64 72 80 12 24 36 48 60 72 84 Notice that 24 is a common multiple of 8 and 12. So also are 48 and 72 common multiples. The L. C. M. is 24. Example 2. What is the L. C. M. of 12 and 18 ? Here the second multiple of 18 contains 12 as a factor. Hence 2 times 18 is the L. C. M. Example 3. A man buys two kinds of sugar dorie up in 4-pound bags and in 5-pound bags. What is the least number of pounds he can buy so as to have the same number of pounds of each kind ? Here the answer is obviously a multiple of 4 and 5. The L. C. M. of 4 and 5 is 20. Hence he buys 20 pounds of each kind. 26 ADVANCED BOOK OF ARITHMETIC EXERCISE 16 1. A person has equal sums of money in dimes and in 25-cent pieces. Find the least amount he can have. 2. Fence posts in two fences are respectively 14 feet and 21 feet apart. What is the smallest distance corre- sponding to an exact number of feet in both fences ? 3. A man earns $4 a day. How many days must he work so as to be paid in 10-dollar notes ? 4. If a person earns $6 a day, how many days must he work to be paid in $ 20 bills ? 5. A housewife puts her flour into 10-pound and 6- pound sacks, and has the same quantity in the 10-pound sacks as she has in the 6-pound sacks. What is the least quantity of flour she can have ? 6. A man buys two grades of sheep at $4 and $6 a head respectively. He spends the same amount in the purchase of the two grades of sheep. What is the smallest amount he can spend ? How many sheep can he buy? 7. A boy buys oranges at 3^, 4^, 5^ apiece. He spends the same amount on each kind of oranges. What is the least amount he can spend on each kind ? How many oranges does he buy ? 8. A person spends the same amount of money on eggs at 15^ a dozen and at 20^ a dozen. What is the smallest amount he can spend on each kind ? 9. Three bells toll at intervals of 4, 5, and 6 seconds respectively. If they start at the same time, after how many seconds will they toll again at the same instant ? 10. Find in feet and inches the least distance that will be measured exactly by a 15-inch and an 18-inch rule. . TESTS OF DIVISIBILITY 27 TESTS OF DIVISIBILITY A number is exactly divisible by 2 when its units' figure is exactly divisible by 2. Thus, 196 is exactly divisible by 2 since 6 is divisible by 2. A number is exactly divisible by 5 if its units' figure is 5 or 0. A number is exactly divisible by 3 when the sum of its digits is exactly divisible by 3. Thus, 735 is exactly divisible by 3 since the sum of its digits, 15, is exactly divisible by 3. A number is exactly divisible by 6 when it is even and the sum of its digits is divisible by 3. Thus, 624 is ex- actly divisible by 6 since it is an even number and the sum of its digits, 12, is a multiple of 3. A number is exactly divisible by 9 when the sum of its digits is exactly divisible by 9. Thus, 765 is exactly divisible by 9 because the sum of its digits, 18, is a multi- ple of 9. A number is exactly divisible by 11 when the differ- ence between the sums of its digits in the even and odd places is or a multiple of 11. Thus, 94,853, is exactly divisible by 11 since the difference between the sums 3 4- 8 + 9 and 5 -h 4 is a multiple of 11. A number is exactly divisible by 25 when the number formed by its two right-hand digits is exactly divisible by 25. Thus, 1,275 is exactly divisible by 25 because 75 is exactly divisible by 25. A number is exactly divisible by 8 when the number formed by its three right-hand digits is exactly divisible by 8. Thus, 19,256 is exactly divisible by 8 because 256 is divisible by 8. The same rule holds for 125, 28 ADVANCED BOOK OF ARITHMETIC NOTATION 2 2 is a short way of writing 2x2. 2 3 is a short way of writing 2x2x2. 2 4 is a short way of writing 2x2x2x2. 2 5 is a short way of writing 2x2x2x2x2. The result of taking a number any number of times as factor is called a power of the number. Thus, 7 4 = 7 x 7 x 7x7=2,401. 2,401 is the 4th power of 7. The 4 written to the right of 7 and slightly above it is called the index or exponent of the power. Example l. Resolve 1,001 into its prime factors. 1,001 * s no ^ Divisible by 2 because its units' figure is not exactly divisible by 2. It is not divisible by 3 because the sum of the digits, 1, 1, is not divisible by 3. It is not divisible by 5 because its units' figure is not or 5. 7 is contained in 1,001, 143 times. 143 is divisible by 11 because the sum of the digits, 1, 3, equals 4, the digit in the even place. Hence the prime factors of 1,001 are 7, 11, 13. Hence, 1,001 = 7 x 11 x 13. Example 2. Resolve 5,040 into its prime factors, and express 5,040 as the product of prime numbers. 5040 2520 Divide by 2 as often as possible. Since 315 1250 ends in 5, 5 is a factor of 315. Divide next by 3 as often as possible. The prime factors of 5,040 are 2, 2, 2, 2, 3, 3, 5, 7, 5040 = 2x2x2x2x3x3x5x7 = 2 4 x 3 2 x 5 x 7. 630 63 - PRIME FACTORS 29 EXERCISE 17 Resolve into prime factors and express each number as the product of its prime factors : 1. 8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 39, 40, 42. 2. 45, 48, 49, 50, 56, 60, 65, 69, 72, 75, 77, 80, 84, 88, 92. 3. 98, 99, 111, 117, 119, 120, 124, 128, 132, 133, 135, 140, 144. 4. 240, 720, 343, 512, 216, 729, 736, 608, 544. 5. 1,331, 11,011, 1,309, 858, 1,274, 891, 3,575. 6. Write all the measures of each of the following numbers: 36, 360, 200, 567, 576, 448. 7. Write all the common measures of: (a) 36, 24 ; (6)18,27; (c?)48,72; (d)21,63; 0) 32, 96; (/) 18, 72. When several numbers are to be taken as a whole and made the subject of an operation, they are inclosed in a sign, or symbol, known as a parenthesis, ( ). Thus, 3 + (9 2) signifies that 3 is to be added to the difference of 9 and 2. A number written immediately to the left of a parenthesis denotes multiplication. Thus, 7x4+ 3(8 + 5) means 7 times 4 is to be added to 3 times the sum of 8 and 5. A composite number can be resolved into only one set of prime factors. Thus, the prime factors of 36 are 2, 2, 3, 3. 36 = 2 2 x 3 2 . The product of no other prime num- bers will give 36. If a number is prime to each of two other numbers, it is prime to their product. ILLUSTRATION. If 7 is prime to 207 and to 8, then 7 is prime to 8 x 207. For 7 does not appear among the prime factors of the product. 30 ADVANCED BOOK OF ARITHMETIC Example 1. Find the G. C. M. of 48, 120, 168. Expressing these numbers as products of their prime factors, 48 = 2 4 x 3. 120 = 2 3 x 3 x 5. 168 = 2 3 x 3 x 7. 2 is contained 3 times as a factor in 168, 3 times as a fac- tor in 120, and 4 times as a factor in 48 ; 3 is contained once as a factor in each of the numbers. Hence, the G. C. M. =23 X 3 = 24. To find the G. C. M. of two or more numbers, express each of the numbers as the product of its prime factors, then take the product of the prime factors common to all the numbers, each factor being taken the least number of times it occurs in any of the numbers. EXERCISE 18 Find the G. C. M. of: 1. 16, 24. 12. 26, 117. 23. 64, 80, 96. 2. 24, 32. is. 57, 76. 24. 63, 84, 105. 3. 18, 27. 14. 115, 161. 25. 64, 96, 224. 4. 24, 36. 15. 144, 264. 26. 72, 108, 180. 5. 45, 60. 16. 140, 252. 27. 88, 132, 220. 6. 75, 90. 17. 20, 30, 40. 28. 126, 189, 252. 7. 54, 72. is. 30, 75, 105. 29. 144, 240, 336. 8. 108, 180. 19. 36, 60, 84. so. 162, 270, 378. 9. 84, 96. 20. 39, 65, 91. 31. 168, 224, 392. 10. 120, 156. 21. 60, 84, 132. 32. 252, 420, 588. 11. 91, 105. 22. 54, 90, 108. 33. 264, 360, 600. LEAST COMMON MULTIPLE 31 LEAST COMMON MULTIPLE Since the L. C. M. of two or more numbers is exactly divisible by each of the numbers, it follows that the L. C. M. contains all the prime factors of each of the given numbers. This fact suggests a method of finding the L. C. M. of two or more numbers. Example. Find the L. C. M. of 48, 60, 72. 48 = 24 x 3. 60 = 2 2 x3x5. 72 = 2 3 x 3 2 . Any multiple of 48 must contain 2, 4 times as a factor. Any multiple of 72 must contain 3 twice as a factor. Hence, the number 2 4 x 3 2 x 5 = 720 contains all the fac- tors of the three numbers 48, 60, 72. Therefore the L. C. M. of 48, 60, 72, is 720. To find the L. C. M. of two or more numbers, resolve each of the numbers into its prime factors, then find the product of all the prime factors of the given numbers, taking each factor the greatest number of times it occurs in any of the numbers. Another Method Example l. Find the L. C. M. of 48, 60, 72. 2 2 2 3 48 60 72 24 30 36 12 15 18 6 15 9 253 L. C. M. =3x5x2x3x2x2x2 = 720. 32 ADVANCED BOOK IN ARITHMETIC Step 1. Arrange the numbers in a horizontal row. Step 2. Divide by a prime factor common to two or more of the numbers. Set down the quotients and the undivided numbers. Step 3. Treat the second horizontal row in the same manner, and so on until a horizontal row is obtained which contains numbers prime to one another. If, at any stage of the process, a horizontal row contains a number which is a factor of some other number in that row, then strike out such factor. The continued product of the numbers in the last row and of the divisors will be the L. C. M. EXERCISE 19 Find the L. C. M. of: 1. 16, 20. 16. 60, 96. 31. 15, 20, 25. 2. 21, 14. 17. 84, 108. 32. 12, 18, 20. 3. 18, 60. is. 55, 77. 33. 45, 63, 70. 4. 18, 45. 19. 54, 90. 34. 14, 35, 40. 5. 21, 49. 20. 72, 108. 35. 12, 16, 18. 6. 28, 70. 21. 75, 125. 36. 14, 24, 40. 7. 42, 56. 22. 36, 54, 72. 37. 15, 24, 25. 8. 36, 54. 23. 36, 90, 60. 38. 77, 143, 22. 9. 34, 51. 24. 48, 64, 36. 39. 18, 20, 45. 10. 48, 72. 25. 12, 15, 18. 40. 30, 70, 105. 11. 96, 120. 26. 14, 21, 35. 41. 30, 40, 48. 12. 28, 30. 27. 30, 35, 21. 42. 21, 28, 35. 13. 32, 80. 28. 28, 42, 70. 43. 12, 18, 27. 14. 26, 39. 29. 32, 35, 150. 44. 12, 15, 16, 18. is. 48, 84. 30. 30, 45, 48. 45. 36, 40, 45. FRACTIONS 33 FRACTIONS If 23 be divided by 4, the process is indicated %- ; the quotient is 5-|. This means 4 is contained in 23, 5 times and 3 remains to be divided by 4. From questions of this character the term fraction (Latin fractus, broken in pieces) has arisen. -^-, 5f, f, are all fractions. A Fraction is an indicated Division. For purposes of instruction fractions are regarded from another point of view. If the rectangle ABOD is divided into four equal parts by lines having the same direction as AB, one of these parts is called one fourth of the whole rectangle ; two of the parts are called two fourths of the rectangle ; three of the parts are called three fourths of the rec- tangle ; and four of the parts are called four fourths of the rectangle. In general, if any one thing is divided into four equal parts, one of the parts is called a fourth ; two of the parts are called two fourths ; three of the parts, three fourths, etc. Similarly, if any thing is divided into five equal parts, one of the parts is called one fifth ; two of the parts are called two fifths ; three of the parts, three fifths, etc. In the above rectangle, if the line EF is drawn so as to divide AB and CD each into two equal parts, the whole figure will be broken up into eight rectangles; one of these rectangles is one eighth of the whole ; two of them are two eighths ; three of them, three eighths, etc. Divide AE and also EB into three equal parts and draw through the points of division lines parallel to BO. What part of ABOD is one of the small rectangles ? two of them ? etc. 34 ADVANCED BOOK OF ARITHMETIC NOTATION OF FRACTIONS 1 eighth is written -|. 2 eighths is written f . 3 eighths is written f . 4 eighths is written |. 5 eighths is written f . 6 eighths is written |, etc. How many thirds are in 1 thing ? How many fourths are in 1 thing ? How many sevenths are in 1 thing ? How many eighths are in 1 thing ? How many tenths are in 1 thing ? In the notation of fractions, what does the number be- low the line indicate ? What does the number above the line indicate ? If a unit quantity is divided into any number of equal parts, one of these parts is called a fractional unit, or unit fraction. A fraction is one fractional unit, or two or more fractional units of the same denomination. A fraction is expressed by two numbers, one number being written above a horizontal line and the other num- ber being written below the same horizontal line. The number above the horizontal line is called the numerator, because it numbers the parts taken, i.e. tells how many fractional units there are. The number below the line is called the denominator : it names the fractional unit, and indicates how many frac- tional units there are in the unit quantity from which the fractional unit is derived. Thus, | signifies the unit quantity is broken into 4 equal parts and 3 of these parts are taken. Here the fractional unit is \ (one fourth) ; 3 is the numerator, and 4 is the denominator. FRACTIONS 35 The numerator and denominator are called the terms of the fraction. A proper fraction is one whose numerator is less than its denominator. Thus, is a proper fraction because 4 is less than 7. An improper fraction is one whose numerator is greater than or equal to its denominator. Examples : -^-, -. A mixed number is a number, part integral and part fractional. Thus, 4| is a mixed number. Read and explain what each represents : * i f < I' f ' I' I' f' I --> > fr *. l< f ' -' f < J. *. V- f f , f v-, ' |, l, |, |, V-. --' iV A- H' ft- Is ^ of 1 week equal to ^ of 2 weeks ? Is |^ of 1 week equal to ^ of 3 weeks? Is ^ of 1 week equal to ^ of 4 weeks? Isf of $1 equal to i of $2? Is | of $1 equal to of f 3? Is | of $1 equal to 1 of $4? Is | of 1 foot equal to \ of 3 feet ? The above illustrations show that a fraction may be read in two ways. For example, f may be read two fifths, or two divided by five ; ^ may be read four sevenths, or four divided by seven. REDUCTION OF FRACTIONS Is If feet = | of 1 foot ? Is 1| = | ? Is f of 1 hour = 41 hours ? Is | = 4 ? Is -|- of an apple = | of an apple ? Is ^ = f ? Changing the form of fractions without changing their values is called reduction of fractions. 36 ADVANCED BOOK OP ARITHMETIC EXERCISE 20 Reduce to integers or mixed numbers : 1. Jf. 10. -%-. 19. -2f . 28. ff. 2. -\ 3 -. 11. -g 9 -. 20. f|. 29. -. 3. \t. 12. 1|. 21. ||. 30. \-. 4. ^. 13. If 22. ff. 31. If. 5. ^g 1 -. 14. -4^. 23. ff. 32. ||. 6. \ 2 -. 15. 1. 24. If. 33. -Ug 8 -- 7. ^-. 16. -1^-. 25. ff. 34. If-. 8. ^-. 17. \ 2 2-' 26 ' if 35 H^' 9. Y-- 18 ' T|- 27> '/ 36 - -W- Reduce to an improper fraction 4|. 1 = 6 sixths. 4 = 24 sixths. 4| = 29 sixths = 2 /-. Another Method What number divided by 6 gives 4 as a quotient and 5 as a remainder ? The answer is 6 times 4 -f the remainder, 5. There- fore, 4 1 = %-. EXERCISE 21 Reduce to improper fractions : 1. 3|. 7. 8|. 13. 19f 19. 201f 25. 2. 3|. 8. 5|. 14. 18|. 20. 16^f. 26. 3. 5f. 9. 9 T \. 15. 27 T \. 21. 14|. 27. 14|. 4. 9f. 10. IQlf 16. 38-^. 22. 15f. 28. 37f. 5. 9f. 11. 11 T L. 17. 39f 23. 19|. 29. 44 T \. 6. 10 T \. 12. 10^. 18. 19f. 24. 291. 30. FRACTIONS 37 How many squares are in the rectangle ABCD? How many are in | of it ? in -J- of it ? In | of it? in \ of it? in f of it? In 1 of it? in \ of it? in \ of it? 555 In | of it? in \ of it? in f of it? In T L of it? in ^ of it? in -^ of it? In T 9 o of it? in ^ of it? in ^ of it? How many squares are in -$- of the rectangle ABCD? How many squares are in -| of the rectangle AB CD ? How do the fractions f and -f$ compare ? How may the fraction -f$ be obtained from ^ ? How many squares are in % of the rectangle ABCD ? How many squares are in J|- of the rectangle ABCD? How do !"! and ^ compare ? How may ^ be obtained from J-| ? From the above rectangle can it be shown that : The terms of a fraction may be multiplied or divided by the same number and the value of the fraction remains unchanged. 38 ADVANCED BOOK OF ARITHMETIC Example. Reduce f- to fourteenths. f = -%. The result is obtained by multiplying the terms of f by 2. EXERCISE 22 Reduce : 1. | to 9ths, to 15ths; to 24ths; to SOths; to 36ths. 2. f to 8ths; to 16ths; to 24ths; to 32ds; to 40ths. 3. | to lOths; to 20ths; to 25ths; to 35ths; to 40ths. 4. | to 14ths; to 21sts; to 28ths; to 35ths; to 49ths. 5. f to 16ths; to 24ths; to 40ths; to 48ths ; to 64ths. 6. f to 32ds ; to 48ths ; to 56ths ; to 72ds ; to SOths. 7. I to 18ths ; to 27ths ; to 45ths ; to 63ds ; to 72ds. 8. -^ to 20ths; to 50ths; to 70ths; to SOths; to 90ths. 9. ^ to 24ths; to 48ths; to 72ds; to 84ths; to 96ths. 10. 3 to 7ths ; to lOths ; to 12ths ; to 20ths. 11. 5 to 4ths; to 9ths; to 14ths; to 25ths. 12. 7 to 3ds ; to 8ths ; to lOths ; to 12ths. Reduce f to 28ths. f = f|. Do this by multiplying the terms of the fraction ^ by 7. Conversely, reduce ||^ to | by dividing the terms of the fraction |-^ by 7. A fraction is said to be in its simplest form when its terms are integers prime to each other. A fraction, in its simplest form, is also said to be in its lowest terms. Example. Reduce to lowest terms 1^0 = If = yf = . Dividing the terms of the fraction by 2, the result is |~|. Dividing the terms of this fraction by 4, the result is if. Dividing the terms of ^| by 3, the result is -|. FRACTIONS 39 EXERCISE 23 Reduce to lowest terms : i- T 6 2< A> ii' If A' if I*. fi I f 2. if. iMMMMMt' t- 3. M, If, *f , If, f I, It If, i\V *. Afe T 4 oV T 6 35> tt< At At '|, ffl* ADDITION Example l. Add J, J. In adding fractions, select, as a matter of convenience, the lowest denomination common to the fractions. 2 6 " 12 3~6- Hence, i + i = | + | = |. Example 2. Add |, |. Here the lowest denomination common to both fractions is 12ths. Notice, 12 is L. C. M. of 4, 6. f = T 9 2- f-lf Hence, f + f = & + {% = || = 1 A- Example 3. Add 4^, 2f, 1^. First add the fractions J, f , y 5 ^. To do 12 ~ this, find the L. C. M. of the denominators. : The L - c - M - f 2 ' 6 ' i2 ' is 12 - the sum is f|- = | = lf. Write | in the sum and carry 1. Add next the integers 4, 2, 1, and the 1 carried. The sum is 8f . 40 ADVANCED BOOK OF ARITHMETIC To add fractions, first reduce them to equivalent fractions having the same denominator. Then add the numerators, and underneath the sum write the common denominator. If the resulting fraction is not in its simplest form, reduce it to its simplest form. To add mixed numbers, first add the fractions, and to this sum add the sum of the integers. EXERCISE 24 Find the sum of : 1. 1 ,J,1. 15. J,J,i,ft. 29. f,^, f. 2. M,f. I*' M.J.&- 30. M.&.H- 3. l,i,f. 17. T VM- 31, 1J,2J,8|. 4. 1 f, |. 18. I|< f, f 32. 21 5J, 7&. 5. J, |, J_. 19. f, T Vlf 33. 31, 21 2f 6. $,,. 20. f, T \,M- 34. 7. 1 ^i- 21. |,^,H- 35 ' 9t.7|,10 8. J 5 , |, J. 22. J, |, ft. 36. 10&, 9f, 9- f iJ- 23. i,|, T VH. 37. 91 10. f,i,iV 24. |,^, If 38. 11. iVi'f 25. J,J,Tk&. 39. 12. T V|,1 26. |,|,|. 40. 41,51,711. 13. |,f. 27. |,1,^. 41. 14. 5 V,^,f. 28. |,|, T V 42. 43. A boy has $ and f |. What part of $ 1 has he ? 44. How much is | of an hour ? ^ of an hour ? ^ of an hour? 45. Which is the largest and which the smallest of the three fractions, f f > f ? FRACTIONS 41 SUBTRACTION Example 1. From | take f . _ is Tl tf-tt-* Example 2. From 17^ take 12|. Reduce the fractions to 20ths. |-| cannot be taken from $. Take it from 1 2 % ; that is, from ^t . Carry ^jU I. 1 and 12 are 13 ; 13 and 4 are 17. The remainder is 4H- EXERCISE 25 Find the value of : 1. f- i 15. 5-3f. 29. 31Jf-20 T V. 2. | - i. 16. 6 - 4^. 30. 40f - 30|i. 3. f-1 17. 11 -f 31. 41 -If. 4. - |. 18. 13 - |. 32. 7| - 3|. 5. &-J. 19. 14-51. 33. 91-5 T V 6. | - f . 20. 15 - B-^. 34. 101 - 9f . 7. T %-f. 21. 28-21 T V 35. 16|-9|. 8. 11 -|. 22. 17|-101. 36. 41-1^. 9. ^-f 23. ISf-llf 37. 9f-2 T V 10. y-|. 24. 5|-2 T V 38. 81-3-^. II. if-f. 25. 29|-24f. 39. 9f-4li. 12. |- T V 26. 33f-171. 40. 18J-9f. 13. 3 - 11. 27. 9^ - 3f 41. 28J - 8|. 14. 4 -If. 28. 32||-30 T V 42. 17i-9 2 V 42 ADVANCED BOOK OF ARITHMETIC 43. What number must be added to 1| to make 7-|? 44. A man buys a suit of clothes for $12-| and gives 3 five-dollar bills in payment. How much change should he receive? 45. From a piece of cloth containing 17f yards 14| yards are sold. How many yards are left? 46. A boy buys two books costing $ f and $-|. How much change should he get out of a $ 2^ gold piece ? MULTIPLICATION OF FACTORS. CANCELLATION Is 2x3x4x9= (2 x 3) (4 x 9) = (2 x 9) x (3 x 4) ? The product of any number of factors, no matter how the factors are grouped, is the same. This is the Associative Law. Is 5 x (2 x 3 x 4 x 9) == 10 x 3 x 4 x 9>= 2 x 15 x 4 x 9 = 2x3x20x9 = 2x3x4x45? Is (2x3x4x9) -2 = 2x3x^2x9 = 3x4x9? A continued product is multiplied by a number if one of its factors is multiplied by the number. A continued product is divided by a number if one of its factors is divided by the number. Cancellation is the shortening of the process of division by dividing dividend and divisor by the same factor or factors. Find by cancellation the quotient : 18x27xl6 = 2x9x2 9x8x3 Ixlxl* Dividing 9 into 18, 8 into 16, and 3 into 27, the quotient is 2 x9 X 2 = 36. Ixlxl FRACTIONS 43 EXERCISE 26 Find by cancellation the quotient : 4 x 18 x 3 x 24 22 x 88 x 15 9 x 4 x 144 132 x 4 18 x 90 x 105 1760 x 99 2. 9. 14 x 25 x 3 4 x 88 x 165 27 x 64 x 8 5280 x 14 o. ^^ - 1O. 108 x 32 176 x 84 343 x 125 1728 x 34 35 x 35 27 x 136 x 8 16 x 16 x 8 x 81 640 x 5200 64 x 32 x 3 125 x 512 x 13 225 x 216 2380 x 104 o. 75 x 9 x 12 119 x 8 x 13 108 x 27 x 121 111 x 39 x 12 22 x 33 x 18 " 74 x 27 x 13 * 15. A farmer exchanged 320 acres of land worth $50 an acre for 25 city lots. Find the price of a lot. (price of a lot) x 25 = $50 x 320. Hence, price of lot = $5Qx32 = $640. 2o 16. How many cows at $40 a head cost as much as 15 horses at $64 a head? 17. How many dozen eggs at 35^ a dozen must be sold to pay for 7 barrels of apples at $2.10 a barrel ? 18. A laborer receives $3.20 a day. How many days must he work to pay for 6 tons of coal at $8 per ton ? 19. A bicyclist rides at the rate of 9 miles an hour. How long will it take him to travel as far as a train goes in 6 hours at the rate of 33 miles an hour ? 20. How many cattle at $42 a head must be sold to pay for 11,200 bushels of wheat at 75 j* a bushel ? 44 ADVANCED BOOK OF ARITHMETIC MULTIPLICATION Multiplication of a fraction by an integer. Example 1. Multiply f by 18. I x 18 = 3 fourths x 18 = 54 fourths = * = 13| = 131- Example 2. Multiply 3 T 3 ^ by 14. 3 T 3_ x 14 = (3 x 14) + (^ x 14) =42 + 4i = 46f To multiply a mixed number by an integer, first multiply the fractional part of the mixed number by the multiplier, next multiply the integral part by the multiplier ; add the two results for the final product. EXERCISE 27 Find the value of : 1. I x 18. 9. T 3 x 24. 17. If x 10. 25. 7 T \ x 33. 2. | x 14. 10. T 5 x 40. is. 1| x 12. 26. 8 T \ x 19. 3. I x 7. 11, T 9 g x 34. 19. 3f x 10. 27. 9 T 5 2 x 80. 4. -& x 18. 12. 11 x 42. 20. 5^ x 54. 28. 6ii x 102. 5. T ^ x 16. is. if x 44. 21. 6| x 36. 29. 7 T 7 e x 60. 6. f X 12. 14. If X 50. 22. 91 X 16. 30. 11^ X 15. 7. |x!9. is. ix45. 23. 7f x44. 31. 12^x18. 8. | X 28. 16. If X 9. 24. 10 T *g X 24. 32. lOf X 13. 33. Find the cost of a dozen cans of baking powder at 37^ cents a can. 34. A pail of mackerel cost 21 dollars. Find the cost of 20 pails. 35. Find the cost of a barrel of sugar weighing 325 pounds, if the cost per pound is 411/. 36. When starch sells for 3^ a pound, find the price of 15 pounds of starch. FRACTIONS 45 37. When wheat is 79| cents a bushel, how much will 164 bushels bring ? 38. Find the cost of a 75-pound chest of Hyson tea at 42| ^ per pound. 39. If a dozen cakes of yeast cost 42|^, find the cost of 9 dozen cakes of yeast. 40. Pepper sells for 14|^ a pound. Find the cost of 2 bags, each containing 120 pounds. 41. A pound package of chocolate costs 31J^. Find the cost of 25 such packages. 42. A square rod equals 30^ square yards. Reduce 160 square rods to square yards. 43. A link of a surveyor's chain is 7|| inches. If the chain contains 100 links, how many inches long is the chain ? 44. When silk sells at $ |-J a yard, what is the cost of 14 yards of silk ? 45. A degree on a meridian of the earth's surface is about 69^ miles. How many miles are in 15 degrees ? in 40 degrees ? v 46. A person fails for $ 9,800. His creditors receive $| on every dollar that he owes. How much in all do they receive ? 47. A mass of copper and lead weighs 2,240 pounds; f of the mass is copper. How much copper and how much lead is in the mass ? 48. A man invests $2,300. At the end of a year his gain is ^ of his investment. Find his gain and the value of the investment at the end of the year. 46 ADVANCED BOOK OF ARITHMETIC MULTIPLICATION OF A FRACTION BY A FRACTION Multiplication of fractions extends the meaning of the term "multiplication." ^ X f , or | of , means 2 times ^ of -|. J X f , or | of |, means 3 times -| of J. Example l. What is the area of a rectangle whose length is |- of an inch and width | of an inch ? D _ E C Take AB 1 inch. Let it be divided into five equal parts. Construct upon it a square ABCD. Divide AD into three equal parts. Draw lines through the points of division. KNED will have for its dimensions |- of an inch and J of an inch. By counting the small rectangles in KNED the number is found to be eight (4 x 2), and the number in ABCD is fifteen (5x3). Hence, the area of KNED is T 8 ^ of a square inch. 1 of f = iV Therefore, 2 times J of | = ^. (Show by figure.) PBOCESS. x = -' Example 2. Find the area of a rectangle 3^ by 2 T 3 ^ feet. To multiply a fraction by a fraction, take the product of the numerators for the numerator of the product, and the product of the denominators for the denominator of the product. To multiply mixed numbers, reduce them to improper fractions, and then apply the rule for multiplication of fractions. FRACTIONS 47 EXERCISE 28 Multiply : 1. f f . 15. T %, ||, _ V 29. 7 1 7J, ^. 2. f |. 16. ^, |f |J. 30. 3f 3f &. 3- |, if " if T\r If- 31. 51 5f T \. * f A- 18 - ii & if- 32 - 7 I' 3 ^' & 5. |, If 19- 1, i ^T- 33. If , | . 6. , J|. 20. |, f, Iff. 34. 7|, If ||. 7. 1|, ff 21. If, If. 35. 41, 4J, 3f . 8. H, Vft. 22. 1|, 1J, 1J. 36. If If f. 9. ^, J&. 23. 2f ^, f . 37. 20J, 20}. 10. If, |f. 24. 3|, T \, f 38. 4f 5f 1 T 3 T . U - M' * 25 - 2 f' 2 !< A- 39. 6|, If ^. 12. if, ^ |. 26. 7f 1|, If. 40. 2|, 1 T V If 13. f, if A. 27. 2f, 3f, If 41. 2f, 8f, 14. f 15, | f 28 . 4 1, 2 |, |f 42. 6f 4|, 43. Find the value of 63| acres of land at $55^ an a,cre. 44. Find the cost of 12 J pounds of meat at 15^ cents per pound. 45. Find the price of 4J bushels of wheat at 81^ cents per bushel. 46. A speculator buys 10,000 bushels of wheat at 79| cents per bushel and sells it when wheat is 81^ cents per bushel. Find his profit. 47. Coal cost $9J a ton. Find the price of 5J tons. 48 ADVANCED BOOK OF ARITHMETIC DIVISION AND RATIO In division the product of two numbers and one of the numbers are given, and the other number is sought. Example. Divide 1^ by J. f of the quotient = 1^ = -J^ 6 -. Therefore, | of the quotient = -|- of - 1 ^ 6 -. Therefore, f of the quotient = | of ^ 6 . Therefore, the quotient = f of - 1 / = 2f . Observe that -^ is divided by f by multiplying by |. Hence the rule for division: Invert the terms of the divisor and then proceed as in multiplication. If the product of two numbers is unity, either is called the reciprocal of the other. ILLUSTRATIONS 7 x 7 = 1. The reciprocal of 7 is ^, and of ^- is 7. | x f = 1. The reciprocal of 4J is |, and of | is 4J. 1 8 5 x ~V~ = ! The reciprocal of T 8 ^ is -^-, and of -^ 5 -, or 1|, Hence the rule for division may be briefly stated : Multiply the dividend by the reciprocal of the divisor. By the ratio of one number to another number is meant the quotient of the first number by the second. Thus the ratio of 9 inches to 12 inches is 9 mches = | . 12 inches The ratio of 9 inches to 12 inches is briefly indicated 9 in.: 12 in. Example. Find the value of the ratio 4 days : 7^ hours. 4 days = 4 x 24 hours = 96 hours. 96 hours -f- 7-J hours = 96 -?- 7 =12.8. FRACTIONS 49 EXERCISE 29 Divide: 1. if by 27. 25. 12by|. 49. 33 by ^ T . 2. 1 by A- 26. 9A by 45. 50. 7 T 9 3 by 80. 3. 1 by T V 27. 4* by 9. 51. 1 by 7f 4. f o by 16. 28. 14 by f 52. 40 by ^ 5. 3f by 10. 29. 16| by 63. 53. 5lf by 50. 6. T 3 oby T V 30. 4f by If. 54. 1 by 9f . 7. & by 12. 31. 16 by f 55. 60 by f . 8. 21 by 3. 32. 181 by 26. 56. 8| by 46. 9. A by if 33. 8f by 2f-. 57. lOf by 2f . 10. if by 16. 34. 18 by f 58. 11 by |. 11. 3f by 8. 35. 17f by 75. 59. A by f . 12. fbyf 36. 4| by 9|, 60. 14| by 111. 13. it by 15. 37. 21 by f. 61. Ibyi. 14. 4 T \ by 14. 38. lOf by 48. 62. iby T V 15. T 9 *by T V 39. H by 31 63. Mbyi^. 16. ii by 25. 40. 25 by f . 64. Ibyf. 17. 5 ft by 21. 41. 9f by 52. 65. T 9 oby^. 18. A*y&- 42. 4 T V by 17f 66. i|byl T \. 19. IA by 21. 43. 26 by f 67. ibyA- 20. 9^ by 46. 44. 9^ by 75. 68. ibyf. 21. A by if 45. 3*byf 69. 9J T by 111 22. M by 36. 46. 26 by if. 70. lby,V 23. 9^by77. 47. 81 by 15. 71. fbyf 24. 22lf by 8. 48. 1 by 4|. 72. 7{ by 41 50 ADVANCED BOOK OF ARITHMETIC EXERCISE 30 1. What is the ratio of 6 inches to 12 inches? 2. What is the ratio of 1 foot to 1 yard ? 3. What is the ratio of 1 square foot to 1 square yard ? 4. There are 30| square yards in 1 square rod. What is the ratio of 1 square yard to 1 square rod ? 5. What is the ratio of 3 weeks to 10 days ? 6. What is the ratio of 1 hour to 1 minute ? 7. What is the ratio of 4 days to 15 hours? of a minute to an hour ? 8. What is the ratio of 325 to 100 ? Find the values of the following ratios: 9. 2:J. 15. 6:f. 21. 3f:2|. 27. lOf : 71. 10. 3:1. i6. 2l:3f. 22. 9|:31J. 28. 7^: 6J. 11. 4 : \. 17. 7-| : 21|. 23. 5J : 7f . 29. 41 : 7 \. 12. 4 : f . 18. 5| : 4|. 24. 4 : 21. 30. 7| : 7^. 13. 5 : 1. 19. 9| : 7. 25. 51 : 11. 31. 81 : If. 14. 7:f. 20. 3f:5f. 26. 61:31. 32. 6-/ T : 41. Example l. Find the price of 2,000 pounds of wheat at 84 / a bushel. To solve this question there are two steps to take. Step 1. Find the number of bushels by dividing the number of pounds by the number of pounds in one bushel. Step 2. Multiply the price of one bushel by the num- ber of bushels. SOLUTION. Number of bushels = 60 14 Price of the wheat 84X x * 60 FRACTIONS 51 Example 2. Three fifths of a man's money is $2,437. How much money has he? | of his money = 12-437. I of his money = ^|^I O r of $2437. f of his money = x 5 or | of $2437. 3 Hence, his money = $4,061.67 (to nearest cent). This method of solving a problem is known as the analytical method. It is called also the unit method, because the value of the unit of the quantity under con- sideration is first sought and from this the value of any number of units is then obtained. Note that the answer is obtained by multiplying $ 2437 by f , the reciprocal of -|. In this problem there are given the product of two factors and one of the factors. The other factor is sought. The problem is therefore one of division. EXERCISE 31 1. Find the price of 78 acres of land if 25 acres are worth $1,375. 2. When 18 pounds of sugar sell for $1, find the cost of 45 pounds. 3. When 7 bushels of wheat sell for $5.95, how much can a person get for 255 bushels ? 4. If 5 bushels of barley sell for 12, how much will 343 bushels sell for ? 5. If 6 barrels of flour are sold for $45, at this rate how much will 84 barrels sell for ? 52 ADVANCED BOOK OF ARITHMETIC 6. Seven barrels of pork sell for $80.50. Find the cost of 50 barrels of pork. 7. Nine barrels of salt cost $11.70. Find the cost of 19 barrels of salt. 8. Eleven bushels of oats are sold for $4.51. Find the value of 168 bushels. 9. Six barrels of lard bring $115. How much will 46 barrels bring ? 10. When 7 yards of sheeting cost 50 fa find how much must be paid for 98 yards. 11. Six yards of cambric sell for 75^. How much must be given for 34 yards of cambric ? 12. Four yards of flannel cost $1.16. How much will 29 yards of flannel cost ? 13. Eight yards of gingham cost 60 ^. How much will 103 yards cost ? 14. Nine yards of cotton fabric cost 75^. How much will 69 yards cost ? 15. Six yards of cotton cheviot cost $1. How much will 81 yards cost ? 16. Five-eighths of a man's money is $75. How much money has he ? 17. Three-fourths of the length of a pole is 81 feet. Find the length of the pole. 18. The eighth and the twelfth of a number are 15. What is the number ? 19. A dealer sold -| of his coal and had 170 tons left. How many tons had he at first ? 20. The fourth part and the sixth part of a number are 25. What is the number ? DECIMALS 53 DECIMALS It is well to fix in mind the following facts: Tenths occupy the first place to the right of the decimal point ; hundredths, the second place ; thousandths, the third place ; ten-thousandths, the fourth place ; hundred- thousandths, the fifth place ; millionths, the sixth place. Read 22.234. Twenty-two and two hundred thirty -four thousandths. Write twenty-four tenths. Write 24 as if it were an integer. Tenths occupy the first place to the right of the decimal point. Hence, 24 tenths is written 2.4. Write 2,304 hundredths. Write 2,304 as if it were an integer. Beginning at the right, point off two places for hundredths. Hence, 2,304 hundredths is written 23.04. If ^ff^- be reduced to a mixed number, it becomes 23^^ ; that is, 23.04. Write 11 hundred-thousandths. Write 11 as if it were an integer. Beginning at the right, point off five places for hundred-thousandths. Hence, 11 hundred-thousandths is written .00011. Ob- serve that places having no digits are filled in with ciphers. Write five hundred and five thousandths. First write five hundred, and then write five thousandths. Hence, five hundred and five thousandths is written 500.005. Write seven hundred eight thousandths. Here the number of units is 708; the denomination is thousandths. As thousandths occupy the third place to the right of the decimal point, hence 708 thousandths is written .708. 54 ADVANCED BOOK OF ARITHMETIC MULTIPLICATION AND DIVISION BY POWERS OF TEN Consider the two numbers, () 320.12, (6) 3,201.2. Both are expressed by the same figures written in the same order. The number (5) can be obtained from the number (a) by moving each figure one place to the left. But moving a digit one place to the left makes its value ten times as great, and, hence, moving several digits each one place to the left makes the number they represent ten times as great. The number (5) can also be obtained from (a) by mov- ing the decimal point in (a) one place to the right. Also (a) can be obtained from (5) by moving the decimal point in (5) one place to the left. To multiply a number by 10, move the decimal point in the number one place to the right. To divide a number by 10, move the decimal point in the number one place to the left. Consider the numbers, (a) 320.12, (J) 32,012. The number (5) is obtained from (a) by moving each digit in (a) two places to the left. This multiplies each digit by 100. (6) may also be obtained from (a) by moving the deci- mal point in (a) two places to the right; also (a) from (6) by moving the decimal point two places to the left. To multiply a number by 100, move the decimal point in the number two places to the right. DECIMALS 55 To divide by 100, move the decimal point in the dividend two places to the left. Consider the numbers, (a) 320.12, (6) 320,120. (b) is here obtained from (a) by moving each digit in (a) three places to the left. It can also be obtained from (#) by moving the decimal point in (a) three places to the right. To multiply a number by 1,000, move the decimal point in the number three places to the right. To divide a number by 1,000, move the decimal point in the number three places to the left. The rules for multiplying by 10,000, 100,000, are left for the reader to determine. Example 1. Multiply 86.4 by 10,000. Moving the decimal point four places to the right, the number be- comes 864,000. Example 2. Divide 12.3 by 100,000. Moving the deci- mal point five places to the left, the number becomes .000123. EXERCISE 32 Multiply by 10 : 1. 120, 14.2, .1431, .00012, 1.7320, .01234. Multiply by 100 : 2. 173, 172.8, 19.23, .001237, 8,654, 17.1. Multiply by 1,000 : 3. 1156, 32.5, 7.123, .93891, .01275, .00011. 56 ADVANCED BOOK OF ARITHMETIC Multiply by 10,000 : 4. 345, 34.25, 5.1739, 6.001, .01793, .12. 5. Divide each of the following numbers by 10; by 100; by 1,000; by 10,000 ; by 100,000 : 32,734 9,285. 773 3,745.3 325. 298 928.49 127 72,173.5 325 12.792 17 99,999.9 18. 326 3,728.3 7. 294 12.7564 670 1,201 1,000 3,450 7,100 Find the values of the following ratios : 6. 22.3 : .223. 22. .001: 10. 7. 3.74 : .374. 23. .005 : 100. 8. 173.2 : 1.732. 24. 9.265 : 926.5. 9. 7.3 : .073. 25. 12.325: 1,232.5. 10. 1.25 : .0125. 26. 1.534 : 153.4. 11. 9.28 : .00928. 27. 1,001 : .1001. 12. 11.34 : .01134. 28. 54 : .054. 13. 7.04 : .0704. 29. 792 : .0792. 14. 100 : .01. 30. 113: .0113. 15. 1,000 : .001. 31. 79.28 : .7928. 16. .012 : .12. 32. 6.45 : 6,450. 17. 1.24 : 124. 33. 99.29 : 99,290. 18. 9.53 : 9,530. 34. 7.35 : 73,500. 19. 7.1 : 7,100. 35. 9.24 : 92,400. 20. 6.5 : 65,000. 36. 8.123 : .008123. 21. 11.79 : 11,790. 37. .04567 : 45.670. DECIMALS 57 ADDITION" Find the sum of 3.4, 2.38, 5.005, 6.2374, 11.1. 3.4 Write the numbers so that units of the same 2.38 denomination stand in the same vertical column. 5.005 Then add as integers are added. 6.2374 Write the decimal point in the sum in the same 11.1 vertical line with the decimal points in the 28.1224 addends. EXERCISE 33 Add: 1. 2.2, .025, 37.3, 5.284, 6.294, 538.1, 77.77. 2. 3.5, 7.12, .339, 47.35, 39.28, .123, 54.275. 3. 9.28, 11.18, .999, 39.28, 7.451, 94.354, 98.76. 4. 12.49, 1.492, 38.75, 53.41, 98.69, 845.5, 892.9. 5. .009, 5.976, 40.99, 6.385, 9.278, 8.239, 64.271. 6. .098, 9.853, 19.47, 17.392, 28.394, 8.01, 77.47. 7. .285, 11.95, 29.99, 94.931, 1.732, 64.6, 78.75. 8. 11.4, 17.5, 99.37, 15.273, 9.394, 71.3, 92.95. 9. 1.21, 12.1, .121, 8.295, 7.777, 68.7, 78.28. 10. 15.9, 9.158, 91.58, 9.158, 2.293, 84.5, .139. 11. 98.5, 11.667, 66.66, 8.394, 9.928, 76.8, 9.359. 12. 77.8, 88.88, 99.99, 6.325, 7.384, 94.9, 1.798. SUBTRACTION Find the difference between 4,001 and 1.7003. Arrange the numbers so that units of the 4001.0000 same denomination stand in the same vertical 1.7003 column. Ciphers may be inserted after the 3999.2997 decimal point in the minuend. Proceed next as in the subtraction of integers. 58 ADVANCED BOOK OF ARITHMETIC EXERCISE 34 Find the remainder and verify your answer in each case: 1. 7.73-6.78. 14. 10.1-7.325. 2. 9.29-3.47. 15. 9.24-5.3481. 3. 6.34-1.95. 16. 8.73-4.4444. 4. 9.82-7.78. 17. 12.32-5.6741. 5. 7.45-3.59. 18. 19.33-6.2734. 6. 10.71-7.79. 19. 9.271-4.3847. 7. 8.94-3.95. 20. 3.213 -.9875. 8. 5.012-2.9. 21. 4.321 -.73201. 9. 10.943-7.97. 22. 5.204-1.3256. 10. 8.325-4.378. 23. 8.731-5.4557. 11. 8.924-5.938. 24. 9.21-7.2349. 12. 7.312-2.7. 25. 7.29-3.4551. 13. 9.419 - 5.57. 26. 6.001 - 5.112. 27. From seven hundred four thousandths take two hundred five ten-thousandths. 28. From five hundred ten thousandths take five hun- dred ten-thousandths. 29. From two thousand take two thousandths. 30. How much does one thousandth exceed one hundred- thousandth ? 31. Find the difference between a hundred and a hun- dredth. 32. From 39 tenths take 39 thousandths. 33. From 100 hundredths take 100 ten-thousandths. 34. How much must be added to one and five-tenths to make ten ? DECIMALS 59 35. By how much does 175 hundredths exceed 175 hundred-thousandths? What is the ratio of the first number to the second? 36. By how much does $1 exceed 1 mill ? 37. By how much does $2 exceed 15 mills? MULTIPLICATION Example l. Multiply 3.23 by 25. 3.23 = 323 hundredths. 323 hundredths x 25 = 8075 hundredths = 80.75. Example 2. Multiply 3.23 by .25. Since the multiplier is y^g- of 25, the product 3.23 x .25 =1^ of 3.23 x 25. ^ of 80.75= .8075. The mechanical work of multiplying may be performed as follows : 3 23 Multiply as if both numbers were integers, and ' point off in the product, commencing at the right, as many places as there are decimal places in both .8075 multiplicand and multiplier. Example 3. Multiply .32 by .018. Point off five places. Another ^Explanation x T Uo = nfiflW = - 00576 - To square a number means to multiply the number by itself or to take the number twice as a factor. To cube a number means to take the number three times as a factor. 60 ADVANCED .BOOK OF ARITHMETIC EXERCISE 35 1. Find .04 of $108; .05 of $274; .06 of $720; .07 of 1144. 2. Find .09 of $34.50; .3 of $75.30; .08 of $75.80; .07 of $84.70. 3. Find .4 of $29.75; .5 of $69.48; .6 of $68.32; .1 of $328.50. 4. Find .125 of $80.80; .75 of $54; .6 of $300.50; .25 of $98.84. 5. Find .625 of $688; .875 of $792.80; .375 of $900.80. 6. Find .375 of 84 acres; .0625 of 64 acres; .3125 of 96 acres. 7. Find .1 of .1; .3 of .4; .3 of .3; .01 of .2 ; .01 of 1.2. 8. Multiply 27.9 by 18. 23. 1.18 x .1695 = ? 9. Multiply 1,327 by 1.6. 24. .97 x .97 = ? 10. Multiply 3,927 by .46. 25. .68 x .68 = ? 11. Multiply 120.01 by 3.6. 26. .373 x .373 = ? 12. Multiply 25 by .017. 27. .901 x .901 = ? 13. Multiply 37.5 by .07. 28. .803 x .803 = ? 14. Multiply 11.9 by 2.4. 29. .693 x .693 = ? is. Multiply 182.54 by 1.49. 30. .1 x .1 x .1 = ? 16. Multiply .286 by 1.96. 31. .3 x .3 x .3 = ? 17. Multiply 92.24 by 2.7. 32. .4 x .4 x .4 = ? is. .148x1.15 = ? 33. .7x.7x.7 = ? 19. .82x.51 = ? 34. 1.04x1.04x1.04 = ? 20. 1.875 x. 32=? 35. 1.06x1.06x1.06 = ? 21. 1.78x1.89 = ? 36. 1.08x1.08x1.08 = ? 22. 18.24 x. 95 = ? 37. .25x.25x .25 = ? DECIMALS 61 38. .7645 of the asphalt found in West Virginia is com- posed of carbon, .0783 is hydrogen, .1346 is oxygen, and the remainder is ash. How much of each constituent is in 254 tons of asphalt ? Check your answers. 39. .7217 of the asphalt found in Oregon is composed of carbon, .079 of hydrogen, .1461 of oxygen, and the remainder of ash. Find the amount of each in 385 tons of asphalt. Check your answer. 40. Multiply the square of 14 by .7854. 41. The area of the surface of a sphere is obtained by multiplying the square of the diameter by 3.1416. Find the area of the surface of the earth, taking the diameter to be 7,920 miles. Compare your answer with the area given in your geography. 42. The moon is nearly 2,200 miles in diameter. Find the area of its surface in square miles. 43. The velocity of the earth in its orbit is 18.5 miles per second. How far does it go in 1 minute? in 1 hour? 44. A hurricane moves at the rate of 146.6 feet per second. How far does it travel in 1 minute ? in 1 hour ? 45. One meter = 39.37 inches. Find in inches the dif- ference between 64 meters and 70 yards. DIVISION Before undertaking Division, it may be well to lay stress on the fact that numbers in the decimal system of notation may be read in many ways. Thus, 32.25 may be read, (a) 32 and 25 hundredths ; (5) 3,225 hundredths ; O) 32,250 thousandths; (d) 322,500 ten-thousandths; (e) 322.5 tenths; (/) 3.225 tens. 62 ADVANCED BOOK OF ARITHMETIC Example l. Divide 1.293 by 8. 8")1 293000 ^ * nt ^ t en ths gives 1 tenth, with a re- 161625 mainder 4 te nths. 4 tenths = 40 hundredths ; 40 hundredths and 9 hundredths = 49 hun- dredths. 8 into 49 hundredths gives 6 hundredths, with a remainder 1 hundredth. Change 1 hundredth to thou- sandths, and proceed as before. Example 2. Divide .01234 by 4. 4). 012340 The work calls for no explanation. .003085 EXERCISE 36 Divide: 1. 73.21 by 8. 9. 8.218 by 7. 17. 5.472 by 6. 2. 3.45 by 4. 10. 3.942 by 6. is. 8.2548 by 9. 3. 19.362 by 6. ii. 6.475 by 7. 19. .34794 by 9. 4. 1.791 by 9. 12. 9.143 by 8. 20. .67356 by 9. 5. 4.564 by 5. 13. .1234 by 5. 21. .999999 by 7. 6. 3.927 by 8. 14. .73206 by 6. 22. 7.3745 by 7. 7. .015 by 5. 15. 1.1466 by 7. 23. 6.2676 by 6. 8. 8.846 by 6. 16. 6.2751 by 8. 24. 1.7346 by 7. Find the difference between .07858 and .078; also find the difference between .07858 and .079. Hence .07858 is nearer to .079 than .07858 .07900 it is to .078. If, therefore, one is .078 .07858 asked to give the value of .07858 .00058 .00042 correct to three figures, write for an- swer .079. Express .73948 correct to three figures. Ans. .739. Express .25764 correct to three figures. Ans. .258. Whenever asked to give a decimal correct to any num- ber of figures, discard the remaining figures if the first DECIMALS 63 one of them is less than 5; if it is 5 or more than 5, in- crease the last figure by 1. Example l. Divide .0732 by .8. Make the divisor an integer by moving the decimal point one place to the right. Make a corresponding change in the dividend. This change is equivalent to multiplying divisor and dividend by 10. 8). 7320 .0915 Example 2. Divide 12 by .125. Move the decimal point in the divisor and in the dividend three places to the right, i.e. multiply each by i' 000 ' 96 125)12000 1125 750 750 Example 3. Divide 3.274 by 6.25. .523 + 625)327.400 3125 1490 1250 2400 1875 525 "Whenever the divisor is a decimal, make it an integer by moving the decimal point to the right. Make a corresponding change in the dividend. After doing this, proceed in exactly the same manner as in long division of integers. Write the decimal point in the quotient in the same vertical line -with the decimal point in the dividend transformed. 64 ADVANCED BOOK OF ARITHMETIC EXERCISE 37 Divide: 1. 2.34 by .8. 26. 5 by .004. 2. .012 by .5. 27. .1 by .0001. 3. 3.475 by .4. 28. .04 by .0008. 4. 1.2348 by .6. 29. .32 by .00128. 5. .1798 by .5. 30. .45 by .0018. 6. 3.144 by 1.2. 31. .078 by .00312. 7. 5.96 by 1.6. 32. .067 by .0268. 8. 3.2903 by 1.3. 33. .01 by .8. 9. .27 by .2. , 34. .002 by 1.6. 10. 5.376 by 1.6. 35. .018 by 45. 11. 9.4851 by 1.5. 36. .54 by 81. 12. 3.2 by 6.4. 37. .243 by 1.944. 13. 20 by .5. 38. .216 by 1.44. 14. 10 by .16. 39. 5.12 by .16. is. 40 by .32. 40. 7.29 by 270. 16. 56 by 1.12. 41. 34.7231 by .713. 17. 84 by 5.6. 42. 31.8791 by 3.97. is. 392 by 7.84. 43. .267584 by 2.96. 19. 100 by .625. 44. .348336 by .492. 20. 100 by .008. 45. .190256 by .188. 21. 400 by .05. 46. 59.4204 by 5,860. 22. 144 by .288. 47. 55.9911 by 108.3. 23. 15.4 by .616. 48. .575484 by 54.6. 24. .096 by .192. 49. .461071 by 122.3. 25. 1 by .001. 50. 4.50775 by 123.5, DECIMALS 65 EXERCISE 38 The mileage and valuation by counties in Texas of the St. Louis and San Francisco Railway as given by the Texas Railroad Commission for the year 1906 are as follows: COUNTY MILEAGE VALUATION 1. Collin 19.51 $346,538.13 2. Dallas 2.7 53,300.16 3. Denton 9.99 188,311.64 4. Grayson 27.44 843,427.59 5. Hardeman 8.68 183,997.77 6. Tarrant 4.56 191,208.29 7. Wilbarger 12.77 192,843.01 Find the valuation per mile in each of the above counties. 8. On July 16, 1907, a contract for paving Broadway, Denver, Colorado, was awarded on the following itemized specifications and prices: 3,050 ft. 6" x 18" stone curb @ $ 1.05* 2,750 yd. brick gutter @ $ 2.25 22,900 yd. street asphalt pave- ment @ $ 2.25 704 ft. oak header @ $ .50 945 ft. 27" pipe sewer @ $ 2.40 580 ft. 24" pipe sewer @ $ 2.00 580 ft. 21" pipe sewer @ $ 1.75 580 ft. 15" pipe sewer @ $ 1.10 398 ft. 12" pipe sewer @ $ .86 516 ft. 10" pipe sewer @ $ .75 12 manholes @ $45.00 17 catch basins @ 165.00 10 M ft. lumber @ $30.00 Find the total cost. * 6" x 18'' means 6 inches by 18 inches. 66 ADVANCED BOOK OF ARITHMETIC REDUCTION OF FRACTIONS TO DECIMALS AND REDUC- TION OF DECIMALS TO FRACTIONS Example l. Reduce -| to a decimal. 8)7.000 .875 Example 2. Reduce -fa to a decimal. 11)7.00000 .63636+ Example 3. Reduce -$fa to a decimal. =.016125. Divide numerator and denominator by 1,000 by moving the decimal point three places to the left; then divide the numerator by 8. A fraction in its lowest terms having for denominator a number whose prime factors are 2's or 5's or both can always be exactly expressed as a decimal. A fraction in its lowest terms having for denominator a number containing prime factors other than 2's and 5's will give rise to a decimal which never terminates. EXERCISE 39 Reduce to decimals: 1 3 5 JL _9_ 11 13 15 3_ * 8' ' 16' 16' 16' 16' 16' 16* 2- A' A' ii- it' if' lV ii- 3 - T 3 0> iVk T^OU' TITOO Q 11 9Q 1Q Q*7 Q1 ^Q 4 ' 0' H' ft' 2-0' f 0' 60' It' e- f f iV' ^ ft' A' ii' if- 7< t' it' ill' 9 4 9 0' DECIMALS 67 Example l. Reduce .0625 to a common fraction. .0625 is read 625 ten-thousandths ; ^ff f ^ is read in the same way. 0625 = - EXERCISE 40 Reduce to common fractions: 1. .3, .8, .25, .125, .1875. 2. .07, .0125, .00875, .0625, .0075. 3. .009, .0225, .1125, .0275. 4. .072, .0104, .035, .0119, .0375. 5. .144, .0504, .0768, .162, .0112. 6. .288, .0176, .0325, .0175, .425. 7. .2875, .3375, .5125, .7375. EXERCISE 41 1. A man walks 3 miles an hour. At this rate, how long will it take him to walk 12 miles ? 2. A train goes 25 miles an hour. How long will it take it to go 300 miles at this rate ? 3. A bicyclist travels at the rate of 9 miles an hour. How long will it take him to go 60 miles ? 4. How would you find the time to go any given dis- tance, if you knew the distance gone in a unit of time ? 5. A man walks 3.5 miles an hour. At this rate, how long would it take him to go 49 miles ? 6. The distance from London to Glasgow is 401.5 miles. An express train goes this distance in 8 hours. Find its rate per hour. 7. From London to Edinburgh is 393.5 miles. The daily mail train takes 7.75 hours to go this distance. Find its rate per hour. I 7 68 ADVANCED BOOK OF ARITHMETIC 8. The Empire State Express goes from New York City to Buffalo, a distance of 440 miles, in 8.25 hours. Find its rate per hour. 9. The mail train from Paris to Bayonne goes 486.25 miles in 8.983 hours. Find its rate per hour. 10. The distance from New York City to Cleveland is 568 miles. A train goes this distance in 19.5 hours. Find its average speed. 11. A steamer goes from New York City to Bremen, a distance of 4235 miles, in 7.75 days. Find its rate per day. Also its rate per hour. 12. The earth moves in its orbit at the rate of 1110 miles a minute. How many times faster does the earth move than a train which goes 54 miles an hour ? 13. A city lot is worth $1800. If this sum is .75 of the value of the house on it, what is the value of the house ? 14. If .7 of a sum of money is $ 196, what is the sum of money ? 15. Cast iron is 7.2 times as heavy as water. How many cubic feet of cast iron weigh as much as 6120 cubic feet of water ? 16. Coal is 1.3 times as heavy as water. How many cubic feet of coal weigh as much as 546 cubic feet of water? 17. There are 231 cubic inches in a gallon. How many gallons are in 1 cubic foot ? (1 cu. ft. = 1728 cu. in.) 18. If 2000 pounds of coal cost 18.75, find the price of 8750 pounds of this kind of coal. 19. If 3.5 yards of cloth cost $12.25, find the price of 7.5 yards of this cloth. 20. If 1.6 yards of velvet cost $2.88, find the price of 9.75 yards of velvet. FRACTIONS 69 EXERCISE 42 1. What fraction of a yard is 1 foot? What fraction of a yard is 2 feet ? 2. What fraction of 1 foot is 1 inch? 3 inches? 4 inches ? 5 inches ? 7 inches ? 8 inches ? 9 inches ? 10 inches ? 3. What fraction of 1 yard is 1 inch ? What fraction of a yard is 2 inches ? 3 inches ? 4 inches ? 5 inches ? 6 inches ? 9 inches ? 12 inches ? 16 inches ? 17 inches ? 19 inches ? 24 inches ? 27 inches ? 4. There are 8 quarts in 1 peck. What fraction of a peck is 1 quart ? What fraction of a peck is 2 quarts ? 3 quarts ? 4 quarts ? 5 quarts ? 6 quarts ? 5. What fraction of a square yard is 2 square feet ? 3 square feet ? 4 square feet ? 5 square feet ? 6 square feet ? 7 square feet ? 6. What fraction of 10 is 2 ? What fraction of 10 is 7 ? 7. What fraction of 11 is 4 ? What fraction of 13 is 9 ? 8. What fraction of 100 is 80 ? 9. Which of the four fundamental rules enables us to solve a problem of this character : What fraction of a number is some other number ? 10. If 4 men can do a piece of work in 7 days, how long will it take 1 man to do the same work ? 11. If a team of horses can plow a 40-acre lot in 16 days, how long will it take 4 teams, working together, to plow the same lot ? 12. If a man can do a piece of work in 9 days, what fraction of the work can he do in 1 day ? in 2 days ? in 3 days ? in 4 days ? in 6 days ? 70 ADVANCED BOOK OF ARITHMETIC COMPLEX FRACTIONS A complex fraction is a fraction one or both of whose terms contain one, or more than one, fraction. 2i 3 1+1-4. Ihus, -i jyi 22' are complex tractions. ' ^ 5 "5 T" 3" 21 Example 1. Simplify ^-|- SOLUTION. 2f + If = f x T 6 T = Or, multiply numerator and denominator by any number which will make the terms of the fraction integers. 2| __ 2| x 6 _ 16 _ If If x6 11" ir Example 2. Simplify ' 2 J ~ * ~~~ Step 1. Simplify the numerator. Step 2. Simplify the denominator. Step 3. Divide the result of Step 1 by the result of Step 2. EXERCISE 43 , m. 10 . js. 13 . ?Jtzi 8 u 9| ' 40| 9 ' -~ ' I- + 11 1 4 + | of 11 16. L_t_i. ,, ^1^. Z8. - 19 . HiH+l2A. 20. i^l}l.xljx: AREAS OF RECTANGULAR FIGURES 71 AREAS OF RECTANGULAR FIGURES EXERCISE 44 1. The dimensions of a room are 40 feet by 30 feet, and 18 feet high. How many square yards are in its walls and ceiling ? 2. Find the area, in square yards, of the walls and ceiling of a room 24 feet by 16 feet, and 12 feet high. 3. ABQD is a rectangular plot of ground 400 feet by 160 feet. Surrounding it is a road 15 feet wide. Find the area of the road. 4. A rectangular park, 600 feet long by 560 feet wide, has a road surrounding it. Find the area of the road if its width is 24 feet. Suppose the road is fenced in, how many feet of wire will it take to go once round ? 5. A rectangular grass plot 252 feet by 180 feet has a walk around it. The width of the walk is 9 feet. How many flags, 9 inches square, will be required to flag the walk? 6. Find the area of each of the following rectangles, in square feet, correct to two decimal figures: (a) 136 feet 8 inches by 115 feet 4 inches. (J) 225 feet by 93 feet 10 inches, (c) 78 feet 5 inches by 56 feet 6 inches. 25 feet 9 inches by 50 feet 2 inches. 72 ADVANCED BOOK OF ARITHMETIC (e) 104 feet 2 inches by 153 feet 11 inches. (/) 203 feet by 53 feet 9 inches. (#) 223 feet 10 inches by 78 feet. (A) 618 feet 1 inch by 130 feet 7 inches. HINT. Reduce the inches in each example to the fraction of 1 foot. 7. Find the area of the following rectangles, giving the results in square yards, correct to two decimal figures : (0) 84. 5 feet by 76.75 feet. (6) 90.67 feet by 84.33 feet. 0) 96. 34 feet by 85. 28 feet. (d) 177.33 feet by 82.54 feet. 0) 129.55 feet by 79.63 feet. 8. A cornfield is 213^ rods long and 96 rods wide. How many bushels of corn will it produce at 32 bushels to an acre ? Find the value of the crop at $ .48| per bushel. 9. A city block is 110 yards long by 90 yards wide. How many acres are in a park which extends 7 blocks one way and 5 blocks the other way ? 10. A 'street is 1,760 yards long and 20 yards wide. How many thousand bricks, 8 inches by 4 inches, will be needed to pave it ? 11. How many square tiles, 4 inches on a side, will be required to tile a hall 60 feet by 16 feet ? 12. The dimensions of a room are 16 feet by 12 feet, and 10 feet high. How many square yards are in the four walls of the room ? How many square yards are in the walls and ceiling ? 13. When the pressure per square foot of a hurricane is 19.47 pounds, find in tons the total pressure exerted against the side of a building 50 feet long 45 feet high. COMPUTATION 73 COMPUTATION ON THE BASIS OF 100, 1,000, AND 2,000 Example l. Find the cost of transporting 5 bales of cotton weighing respectively 510 lb., 515 lb., 508 lb., 496 lb., 487 lb., at 46^ per 100 lb. 510 + 515 + 508 + 496 + 487 = 2516 = 25.16 x $ .46 = $11.5736. Am. $ 11.57. Example 2. Find the cost of shipping 7 head of cattle, average weight 1,089 lb., at 97 ^ per 100 lb. 7 * J^ 89 x $ .97 = 7 x 10.89 x $ .97 = $73.9431. Ana. $73.94. 10.89 The shortest way to multiply by .97 is 7 to take .03 of the multiplicand from itself. 76.23 2.2869 =.03x76.23 73.9431 Example 3. Find the value of a car load of coal weigh- ing 43,275 lb. at $4.80 per ton of 2,000 lb. x $4.80 = x $4.80 = 43.275 x$ 2.40 2000 2 = $103.86. Example 4. How much will it cost a man a year to in- sure his life for $8,750 if the annual premium is $32.80 per 1 1,000 ? 8750 1000 x $ 32.80 = 8.75 x $ 32.80 = $287.00. NOTE. In the above examples the sign x is to be interpreted as mean- ing times. 74 ADVANCED BOOK OF ARITHMETIC EXERCISE 45 The following rates in cents per 100 Ib. are taken from the annual Report of the Railroad Commission of the state of Texas for the year 1906. Find the cost of shipping: 1. 5 bales cotton, average weight 503 Ib., @ 45^. 2. 12 bales cotton, average weight 496 Ib., @ 48^. 3. 15 bales cotton, average weight 490 Ib., @ 8^. 4. 130 bbl. flour, 200 Ib. to the barrel, @ 16 f. 5. 124 bbl. flour, 200 Ib. to the barrel, @ 17 i. 6. 1 carload grain, weighing 27,500 Ib., @ 14^. 7. 256 sacks flour, 98 Ib. to the sack, @ 12 8. 32,800 Ib. grain @ 7j 9. 1 carload cotton seed products, weighing 23,800 Ib., 10. 1 carload cotton seed hulls, weighing 28,600 lb.,@ J* 11. 1 carload cotton seed meal, weighing 42,000 Ib., @ 12. 1 carload cotton seed oil, weighing 43,600 Ib., @ 5^. 13. 1 carload brick, weighing 45,000 Ib., @ 5| ^. 14. 1 carload fire brick, weighing 27,000 Ib., @ 14|^. 15. 1 carload common brick, weighing 47,000 Ib., @ t. 16. 1 carload mules, weighing 29,000 Ib., @ 23^. 17. 1 carload cattle, weighing 25,000 Ib., @ 14^. 18. 1 carload sheep, weighing 15,500 Ib., @ 15/. 19. 1 carload crude petroleum, weighing 42,000 Ib., @ COMPUTATION 75 20. 1 carload asphaltum, weighing 27,000 lb., 21. 1 carload melons, weighing 20,500 lb., @ 22. 5,880 lb. molasses @ 7j 23. 19,200 lb. sugar @ 48^. 24. The freight rate on coal in cents per ton of 2,000 lb. from Eagle Pass to the points named is : Weimer 138^ Flatonia 127^ Columbus 140^ Beaumont 217^ Gonzales 121^ Schulenburg 134^ Find the cost of shipping 1 carload of coal, weighing 39,000 lb., from Eagle Pass to each of these points. 25. Find the cost of shipping 105,000 lb. gravel from Austin to San Antonio at 60^ per ton of 2,000 lb. 26. Find the cost of shipping 116,000 lb. crushed rock from Clay Quarry to Houston at 67-| ^ per ton of 2,000 lb. 27. Find the cost of shipping 130,000 lb. crushed rock from Jacksboro to Fort Worth at 50 / per ton of 2,000 lb. 28. Find the cost of shipping a carload of sand, weigh- ing 50,000 lb., from Kingsbury to San Antonio at 40^ per ton of 2,000 lb. 29. Find the premium on a $5,500 life insurance policy at $21.50 per 11,000. 30. Find the premium on a life insurance policy for $4,500 at $25.30 per $ 1,000. 31. What is the premium on a life insurance policy of $6,500 at $19.92 per $1,000 ? 32. Find the premium on a life insurance policy for $10,500 at $29.80 per $1,000. 33. Find the premium on a life insurance policy for $8,500 at $51.20 per $1,000. 34. A man insured his life for $9,450. Find the annual premium at $62.40 per $1,000. 76 ADVANCED BOOK OF ARITHMETIC PERCENTAGE Per cent means by the 100, or on the 100. Thus, 6 per cent means 6 on 100, or 6 out of 100. 25 per cent means 25 on 100, or 25 out of 100. 6 per cent is written 6%; 25 per cent is written 25%. If a man invests $100 and gains on his investment $100, he makes a profit of 100%. Therefore, 100% of a number = the number. 50 % of a number = J of the number. 25 % of a number = ^ of the number. 20 % of a number = ^ of the number. 16| % of a number = ^ of the number. 7 % of a number = ^fa of the number. The following per cent equivalents should be remem- bered : 100 % = 1 50 % = 1 331 % = 1 20 % = | 40 % = f 60 % = f 80 %=| 16|% = i 12-| 6' A' ITP lV 7< 2. 2f,3 2 4,4ix. 8. 3 - i> iV' A' iV 9 - I' I 3 !' TO' 4. If, iWi'liV 10. ^,3 T 6. 2i|, 2||, 4 2 V 12. 2 T V, 3if, 21 if . Find the difference between : 13. 4 T \ and If. 22. J| and if 14. l^and T W 23. H and lf 15. l^and T V 24. 7f and 311 16. -3*5- and T ^. 25. 8 1 and 711. 17. 1 and -^L. 26. 9 T 5 2 and 6if . 18. 1 and ^g-. 27. 5 T 7 g and 2||. 19. -^ and f . 28. 6 T ^ and 20. fandf 29. 21. 11 and if 30. 12^- and 8|f. 31. What number must be added to 3f to give 5f ? 32. A man's capital amounts to $ 1727J. By how much must he increase his capital so that it may amount to $3000? 33. What number must be taken from 30| to leave 14|? 34. By how much does 84^ exceed 17|? 35. Of the weight of the earth's atmosphere y% 3 oV ^ s oxygen. What fraction of the weight of the earth's atmosphere do its other constituents aggregate ? REVIEW OF FRACTIONS 121 Simplify : 1. foffof2f 2. f of T ^of4f 3. I of 1| of $. b4f EXERCISE 72 TT x T 3 T x4|. 4. 5. 6. 7. 8. 9. 10. 2|x3^x 11 ^1 V ^1 V -I.J.. t-JS" /N '^9' ^ 12. 2|x2|x 13. 74-xiSx 14. 9^ 15. 1\ 16. (i 18. 19. of 4| of If 20. v \1 v 1 1 O1 S\ ^^ /N J-"5-5". i> i> iV A' iV- 26. Express each of the following decimals as a per cent: .04, .08, .121, .0165, .002, .006, .0024. 27. The composition of a piece of coal taken from the Texas and Pacific Coal Company's mine is given as fol- lows: moisture, 5.46% ; combustible matter, 35.66% ; fixed carbon, 49.17% ; ash, 9.71%. Find the amount of each constituent in 2000 Ib. of coal. Check your answer. 28. The analysis of a specimen of lignite is given as follows: moisture, 29.07%; combustible matter, 28.96% ; fixed carbon, 24.47% ; ash, 17.50%. Find the amount of each constituent in a ton of lignite. Check. 29. Distilled water is composed of two gases, H|% by weight being hydrogen, and 88f % by weight being oxy- gen. Find the weight of each gas that can be obtained from 10 Ib. of water. 30. A bookkeeper receives a salary of $ 1800 per annum. If he spends 62 1% of his salary, and saves the remainder, how much does he spend ? How much does he save ? 31. An auriferous ore contains .5% of gold. How many pounds avoirdupois of gold would 2240 Ib. of this ore yield ? 32. A copper ore contains 5J% of copper. Find the number of pounds of copper in 500 Ib. of this ore. 33. A owns 16|% of a boat valued at $12,300. What is the value of A's share of the boat ? 124 ADVANCED BOOK OF ARITHMETIC 34. The estimated value of the exports of the United States for 1899 was $1,275,000,000. The percentages of exports from United States ports for that year are given as follows: New York, 37.4%; Boston, 10.43%; Phila- delphia, 5.05% ; Baltimore, 8.9%; New Orleans, 6.4%; Galveston, 7.17%. Find the value of the exports from these cities. Given a number as a per cent of some other number, to find the other number. Example 1. If 15.5% of a number equals 22.785, what is the number ? 15.5% of the number = 22. 785 22.785 15.5 1 % of the number = .-. 100 % of the number = x 100 = 147. 15.5 /. the number is 147. Example 2. If 3| % of a number is 2934, what is the number ? . 100 400 80 -g^j- of the number = 2934. .*. -g 1 ^ of the number = 978. .-. fj. of the number = 78,240. /.the number is 78,240. EXERCISE 75 1. If 5 % of a number equals 185, what is the number ? 2. If 12| % of a man's salary is $156, what is the man's salary ? 3. If 37^ % of a man's property is valued at $324, what is the value of his property ? PERCENTAGE 125 4. The number 360 is equal to 5% of what number? 6% of what number? 8% of what number? 9% of what number ? 5. The number 120 is equal to 3% of what number? 4% of what number? 5% of what number? 6% of what number? 8% of what number? 12% of what number? 6. $750 is 6% of what sum of money? 6|% of what sum of money? 7-|-% of what sum of money? 12|% of what sum of money ? 7. $108 is 30% of what sum? 25% of what sum? 331% of what sum? 44|% of what sum? 8. $450 is 6^% of what sum? 6|% of what sum? 8J% of what sum? 12^% of what sum? 16 f% of what sum ? 9. $420 is 37^% of what sum? 62|% of what sum? 87^% of what sum? 10. In 1900 the commercial value of silver was 3% of the value of gold. Find the number of ounces of silver equivalent in value to 126 ounces of gold. 11. Of the population of the United States in 1900, 27,849,760 were married. This number was 36.5% of the population. Find the population in 1900. 12. The census of 1900 gives the number of married men in the United States as 14,003,798. This number was 35.9% of the number of males. Find the male popu- lation. 13. The census of the same year gives the number of married women in the United States as 13,845,963. This number was 37.2% of the entire number of females. Find the female population. 126 ADVANCED BOOK OF ARITHMETIC 14. Of the number of illiterates above 10 years of age in the United States 15.5%, according to the census of 1900, can neither read nor write. This number is 955,840. Find the number of illiterates above 10 years of age in the United States in 1900. 15. Of the average value of the raw cotton exported from the United States in 5 years 50.4% went to Eng- land. If the export of raw cotton to England amounts in value to 1107,500,000, find the average value of the raw cotton exported. 16. Of the sheep exported from the United States 86% are shipped to England. If the value of the sheep shipped to England in a certain year was $1,685,800, find the total value of the export of sheep for that year. 17. Of the pupils attending school in a certain city .6% are in the senior class of the high school. If the senior class numbers 21, find the number of pupils attending school in that city. 18. In a certain city the number of pupils promoted at the end of the scholastic year was 2765. This number was 79% of the number of pupils in school. Find the number of pupils attending school in that city. 19. The foreign -born population of a city is 10,668. This number is 3^% of the population of the city. Find the population of the city. 20. Of the water of the Dead Sea 22.8?T% is saline material. If a quantity of Dead Sea water is evaporated, and the saline material left behind weighs 914.28 lb., what is the weight of the water before evaporation ? PERCENTAGE 127 To find -what per cent one number is of another. Example 1. What per cent of 12 is 5 ? 5 is of 12. Example 2. The value of the property of the Fort Worth and Denver City Railroad in Texas, as ascertained by the Railroad Commission of Texas, for the year 1906 was $5,771,600, and the income from its operation was $1,178,040. Find the rate per cent of income from opera- tion. 4=20.4,, To find what per cent one number is of another, find what fraction the first number is of the second and multiply by 100%. EXERCISE 76 1. What per cent of 15 is 12? 2. What per cent of 20 is 4 ? is 7 ? is 11 ? is 13 ? 3. What per cent of 40 is 2 ? is 8 ? is 12 ? is 17 ? 4. What per cent of 90 is 9 ? is 12 ? is 27 ? is 11| ? 5. What per cent of 120 is 15 ? is 18 ? is 45 ? is 80 ? 6. What per cent of 480 is 28.8? is 33.6? is 40 ? 7. What per cent of 1728 is 345.6 ? is 155.52 ? is 288 ? 8. What per cent of 231 is 21 ? is 77 ? is 34.65 ? 9. What per cent of 5280 is 88 ? is 440 ? is 330 ? 10. What per cent of a bushel is a quart ? 11. What per cent of a mile is 8 rd. ? 12. What per cent of an acre is a square rod ? 13. What per cent of a chain is a yard ? 14. What per cent of 1 sq. ch. is 1 sq. rd, ? 128 ADVANCED BOOK OF ARITHMETIC 15. What per cent of 1 gal. is 1 pt.? 16. What per cent of 1 sq. rd. is 7 sq. yd. 5 sq. ft. 9 sq. in.? 17. What per cent of a rod is 1 yd. 2 ft. 6 in.? 18. What per cent of 1 mi. is 1 knot? 19. What per cent of 1 mi. is an arc of 1' measured on the 40th parallel of latitude ? (!' = 4670 ft.) 20. What per cent of 1 mi. is 22 yd.? is 176 yd.? 21. What per cent of 1 sq. mi. is the S. W. 1 of N.E. -| of a section of land ? 22. What per cent of a common year is 73 da. ? is 219 da.? is 292 da.? 23. A meter is 39.37 in. What per cent of a meter is 1 yd. ? What per cent of 1 yd. is 1 meter? 24. A kilometer is 1000 meters. What per cent of 1 mi. is 1 km.? 25. What per cent of 1 Ib. avoirdupois is 1 Ib. troy? 26. What per cent of 1 oz. avoirdupois is 1 oz. troy? 27. What per cent of the area of each of the following states consists of irrigated land? ACRES IRRIGATED AREA IN SQUARE MILES (a) California . 1,446,114 158,360 (b) Colorado 1,611,270 103,925 (c) Louisiana . 201,685 48,720 (d) Montana . 951,054 146,080 (e) Nevada 501,168 110,700 (/) Oregon . 388,110 96,030 (. Find the cost price to the purchaser. 6. Find the cash value of a bill of $320 with discounts of 15/o, 10/o, and 5/o. 7. Suppose you were offered a single discount of 45/>, or two discounts of 30 /> and 20 Jfc, which would you take ? What would be the difference in a bill of $1000 ? 8. A dealer buys a quantity of goods marked $550, with a discount of 20/>. If he sells the goods at 6jfc above the marked price, what is his gain per cent ? 9. If I buy goods listed at $150, w r ith a discount of 10/>, and sell them at 12 fi above the marked price, find my gain per cent. 10. A bookseller buys 100 books marked $1.50 each, at a discount of 20 /> and sells them at the marked price. What is his gain per cent ? 11. A bookseller bought 100 books at $1.25 each. He make a profit of 20 /> after giving a discount of ^. At what price did he mark each book, and what was his profit ? 12. Find the cost price in each case, if the list price and the rates of discount are as follows : LIST PRICE DISCOUNTS O) $480 20/o, 80 #, 10 /o (6) $1000 40/o,5^,4/o 0) $775 25/o, I (i) $880 12i/o,4/o 00 $720 , 16/o (/) $960 12i?o, 12 Jb (#) $1200 10^, 8/0, 5 Jfe Qi) $1760 PROFIT AND LOSS 133 PROFIT AND LOSS In actual business, gains and losses are reckoned as a per cent of the cost price. Example l. How much does a person gain by buying 360 yd. of cloth at $1.30 per yd. and selling it at a profit of 15 % ? What is the selling price per yard ? SOLUTION. 360 x $1.30 x 15 % = $70.20, gain. $1.30 .13 =10% of $1.30 .065= 5% of $1.30 $1.495 = selling price per yd. Observe, selling price per yd. is 115 % of cost price. Example 2. A dealer buys apples at $1.75 per barrel, and sells them at $2.10 per barrel. Find his gain per cent. SOLUTION. $2.10 - $1.75 = $.35. j|p~ = 1. The gain is of the cost. | = 20%. EXERCISE 79 1. Find the selling price of articles, the cost prices and rates per cent of profit being given as follows : COST PRICE RATE PER CENT OP PROFIT (a) $150 12% (i) $75 25% 00 $31 7i% (cT) $215 16|% O5 $540 27% GO $318 S% (<7) $512 18|% (A) $234 15% (0 $457 134 ADVANCED BOOK OF ARITHMETIC 2. Find the rate per cent of profit or loss, if the cost prices and selling prices are given as follows : COST PRICE SELLING PRICE (a) $ .20 9 .25 (5) $ .22 $ .20 0) $ .90 $1.50 (df) $2.10 $1.40 0) $125 $160 Given selling price and rate per cent of profit or loss, to find cost price. Example l. A piano was sold for $450 at a profit of 121 % . Find the cost price. (100 % + 12i %) of cost = 1121 cf of cost = f of cost. | of cost = $450. Therefore, cost =$450 *- = $400. Example 2. A dealer sells goods for $200.56 at a loss of 8%. Find the cost price. (100 % - 8 % ) of cost = 92 % of cost. 92% of cost = $200. 56. ... cost = ~ 1218.00. EXERCISE 80 1. Find the cost price, the selling price and rate per cent of profit being given as follows : SELLING PRICE RATE PER CENT OF PROFIT (a) $63 25% (6) 1143 8% O) $54 8% (e) $205 (/) $315 5% PROFIT AND LOSS 135 2. Find the cost, if the selling price and the rate per cent of loss are given : SELLING PRICE RATE PER CENT OF Loss (a) $41.30 10% (8)' $87.50 20% ( iV 6T- 6. From of (1 _ i + 1) ta ke ( J + | - ). 7. Express in pounds the difference between .0125 of a ton and | cwt. 8. What fraction having 48 for denominator is equiva- lent to .1875? 9. Find in feet the value of l| of a mile. 10. Express f yd. as the decimal of a rod. 11. Express f of 95 Ib. as the decimal of 1^ cwt. 12. Express 27| Ib. as the decimal of a ton. 13. Take -| T. from 1J T., and express your result in pounds. 14. From | of a right angle take 33 45'. 15. From ^ of a circumference take ^ of the circum- ference and express your answer in degrees. 16. There are two numbers in the ratio of 3 to 5. What fraction of their. sum is their difference? REVIEW 159 17. An estate is left to A, B, and C. A gets |- of the estate, B, ^ of the estate, and C, the remainder. What part of the estate does C get ? If C's share is $450, what is the value of the estate ? Find A's share and B's share. 18. A man spends f of his salary on board, ^ on cloth- ing, -^ on rent. He saves the remainder, amounting to $125. Find his salary. 19. Subtract -| from 2.1, and divide the remainder by .25. 20. The dividend is 3.562 and the quotient is .3125. Find the divisor. 21. Express $5.24 as a decimal of $100. 22. After giving away |, -j^, and J of his money, a man has left $392.95. How much money had he at first ? 23. The third part of a number exceeds the fifth part of the same number by 15. What is the number ? 24. The sixth part and the eighth part of a number together make 66 J. What is the number? 25. If | and ^ of a farm are together worth $1650, what is | of the remainder worth? 26. Subtract the product of | and f from their sum. 27. What number divided by 3| gives ly 1 ^ for the quotient ? 28. Find the value of % of 1^ of $19.80. 29. A man owns J- of a boat and sells f of his share for $750. At this rate, find the value of the boat. 30. If 6 men do a piece of work in 9 days, how long will it take 4 men to do the same work ? 31. If 15 men pave a street in 16 days, how long will it take 40 men to pave the same street ? 160 ADVANCED BOOK OF ARITHMETIC 32. If 18 men remove an embankment in 12 da., how long will it take 24 men to remove the embankment ? 33. Two trains start at the same time from two sta- tions 840 mi. apart, and travel toward each other, one train going at the rate of 35 mi. an hour, and the other of 25 mi. an hour. In how many hours will they meet ? 34. If a man performs ^ of a piece of work in 15 da., in how many days more will he complete the work ? 35. Add |, -, , -j 6 g, and ^. Express the sum as a decimal. Check by reducing to decimals and adding. 36. What is the smallest number which, added to the sum of ^, ^, and -|, will make the final result an integer ? 37. If -| of a barrel of sugar is sold and afterward 40 Ib. are sold, how many pounds of sugar were in the barrel originally, supposing it still contains 90 Ib.? 38. By selling a piano at -^ of its cost, a dealer loses $98. Find the cost of the piano. 39. If 45 sq. rd. of land cost $ 18, find the cost of 1 A. Find also the cost of 3| A. 40. If | of a clerk's salary per year is $675, find his salary per month. If his expenses average $48.89 per month, how much will he save per year ? How long will it take him to save $652.75 ? 41. If 31 A. of land are worth $119, find the value of 2|- A. How much is a rectangular strip of this land | of a mile long and 33 feet wide worth ? 42. How much is a plot of ground 80 ft. by 60 ft. worth, if an acre is worth $55 ? If an acre is worth $121 ? REVIEW 161 43. Find the weight of a piece of coal in the shape of a rectangular solid, if its dimensions are 1J ft. by 1J ft. by 10 in. A cubic foot of coal weighs 81-| Ib. 44. A man's property is assessed at $4550. If he pays 40^ on every $100 for school tax, how many dollars school tax does he pay? 45. How much taxes will be paid on real estate worth $9580, if the tax is at the rate of $1.27 on $100? 46. A man invests $4800 and gains $540. How much does he gain on every dollar invested? How much does he gain on every $100? 47. If an investment of $9600 produces a gain of $1056, find the gain on $1 ; also on $100. 48. When $8400 produces a profit of $1092, how much does $ 1 produce ? $ 100 ? 49. The school tax in a city is 2 mills on the dollar. The assessed valuation of the property is $17,294,000. Find the total tax levied for school purposes. If .95 of this total is collectible, find the amount 3ollected. 50. Find the tax on $33,254,000 at 4 mills on the dollar. 51. A man insures his dwelling for $5450. If he pays $11.50 on every $1000, how much does he pay altogether? 52. Find the insurance on a house valued at $7840, if the rate of insurance is $1J on every $100? 53. If $47.50 is paid to insure a boat valued at $9500, how much is paid on $1? on $100. 54. The distance from New York City to Plymouth, England, is 2962 knots. The steamship DeutscUand sailed from Plymouth to New York in July, 1900, in 5 da. 15 hr. 45 min. Find, in knots, its rate per hour. Find also its rate in miles per hour. 162 ADVANCED BOOK OF ARITHMETIC 55. In March, 1902, the steamship La Savoie made the voyage from Havre to New York in 6 da. 10 hr. The distance from Havre to New York is 3170 knots nearly. Find, in knots, the rate per hour. Express the rate also in miles per hour. 56. The steamer Kronprinz Wilhelm made, in Septem- ber, 1902, a voyage from Cherbourg to New York in 5 da. 11 hr. 57 min. Find its rate per hour, the distance from Cherbourg to New York being 3184 knots. 57. In May, 1900, a passenger train ran from Burling- ton to Chicago, 205.8 mi., in 3 hr. 8 min. 30 sec. Find its rate per hour. 58. The fastest time on record by a passenger train for a distance over 450 miles was made in October, 1895, on the Lake Shore and Michigan Southern Railroad, from Chicago to Buffalo, a distance of 510 mi., in 8 hr. 1 min. Find its rate per hour. 59. The run from London to Edinburgh, 393 J mi., has been made in 7 hr. 45 min. Find the speed per hour. 60. The market quotations, Feb. 19, 1903, were : wheat, 78|^ per bushel; corn, 45 1^ per bushel; oats, 34^ per bushel. Find the price of 100 Ib. of each of these commodities. 61. Market quotations of live stock sales are in dollars per 100 Ib. Find the cost of : (a) 44 cattle, average weight 1121 Ib., @ $4.80. (6) 132 cattle, average weight 1018 Ib., @ $4.80. O) 24 cattle, average weight 915 Ib., @ $4.20. (d) 23 cattle, average weight 1060 Ib., @ $3.90. (e) 133 heifers, average weight 862 Ib., @ $4.00. (/) 69 calves, average weight 201 Ib., @ $5.00. (jgf) 82 hogs, average weight 188 Ib., @ $6.20. CHAPTER III GENERAL REVIEW BY TOPICS r ADDITION To add numbers is to find a single number equivalent to the numbers jointly. The result is the sum. Only numbers of the same kind can be added. Thus, 5 yd. and 7 yd. may be added; but 5 yd. and $ 7 cannot be added. 5 ft. and 7 in. may be added provided the 5 ft. is changed to inches, or the 7 in. changed to feet. The sum in one case is 67 inches, in the other case, 5^ feet. Example 1. Add 279, 514, 928, 763. The process is 3, 11, 15, 24 ; 24 units = 2 tens 279 " and 4 units. 514 Write the 4 units and carry the 2 tens. 928 2, 8, 10, 11, 18; 18 tens=l hundred and 8 763 tens. 2484 Write 8 and carry 1. 1, 8, 17, 22, 24 ; write 24. Example 2. Add $2.79, 15.14, 19.28, $7.63. 12.79 5.14 9.28 The process is the same as in Example 1. 7.63 $24.84 163 164 ADVANCED BOOK OF ARITHMETIC Example 3. Add 6 ft. 8 in., 3 ft. 6 in., 5 ft. 4 in. FT. IN. 6 8 The process is 4, 10, 18 ; 18 in. = 1 ft. 366 in. Write 6 in., carry 1 ft. 5 4 1, 6, 9, 15. Write 15 ft. 15 6 ^ Example 4. Add 6f , 3J, 51. Change f , |, and J to equivalent fractions having 12 for denominator. 21 l_8 + 6+4_18 - 3 + 2 + 3~ ~12~ "12 *' 15 2 Write |-, carry 1. 1, 6, 9, 15. Write 15. Observe the same principle pervades the four examples. Units of the same name are placed in the same column, and the columns added separately. An example in addition may be checked by adding the columns in reverse order. EXERCISE 93 Add: (i) (2) (3) 00 (5) 4792799 7998992 5998476 5879824 6876894 8399384 8409499 9208503 9473473 9219479 7207383 7294792 8392393 7777888 7328337 9476583 8514798 6555777 6456789 8474494 4728737 9998777 7918924 9875874 9318327 9219777 6666784 5729998 8294295 6438444 6444673 8542728 6329888 7318392 7473478 8299299 5293294 7774673 9299497 9888777 9218288 4445454 6428427 6713729 6666679 7666729 7778898 9444779 9873876 9218289 ADDITION 165 6. Add vertically and horizontally, arid finally sum the vertical and the horizontal totals : 392 876 543 878 929 965 329 707 538 928 873 393 427 925 599 307 448 388 394 222 399 937 542 234 7. The mileage of railroads in operation in the several states is given for the years 1903, 1904, and 1905 as follows : GROUP AND STATE 1903 1904 1905 NEW ENGLAND Maine ...... 2,004.79 2,029.89 2,091.12 New Hampshire .... Vermont ..... Massachusetts .... Rhode Island .... Connecticut . . . 1,191.42 1,057.84 2,117.41 209.84 1,025.90 1,191.77 1,056.96 2,110.81 209.84 1,020.12 1,191.77 1,063.20 2,104.87 209.84 1,020.12 Total . . . . . MIDDLE ATLANTIC New York ..... 8,180.85 8,167.21 8,212.12 New Jersey Pennsylvania . Delaware ..... Maryland ..... District of Columbia 2,242.56 10,784.54 333.63 1,368.98 24.70 2,266.64 10,991.97 334.86 1,364.45 24.70 2,269.61 11,161.45 333.60 1,406.81 24.70 Total CENTRAL NORTHERN Ohio Michigan Indiana Illinois ...... Wisconsin ..... 9,023.61 8,459.65 6,834.75 11,502.38 6,921.40 9,163.97 8,467.76 6,863.03 11,742.10 7,014.78 9,243.26 8,521.46 7,046.90 11,959.09 7,188.18 Total 166 ADVANCED BOOK OF ARITHMETIC GROUP AND STATE 19O3 19O4 1905 SOUTH ATLANTIC Virginia 3,833.09 3,823.67 3,862.11 West Virginia .... 2,565.49 2,820.82 2,966.05 North Carolina .... 3,790.73 3,913.86 4,015.58 South Carolina .... 3,112.48 3,146.24 3,184.19 Georgia 6,109.21 6,298.97 6,516.61 Florida 3,469.92 3,585.83 3,635.38 Total GULF AND MISSISSIPPI VALLEY Alabama ..... 4,442.69 4,590.89 4,758.57 Mississippi ..... 3,156.56 3,367.23 3,541.04 Tennessee ..... 3,355.19 3,484.92 3,606.88 Kentucky ..... 3,193.31 3,261.56 3,355.07 Louisiana 3,419.38 3,592.68 3,764.17 Total SOUTHWESTERN Missouri 7,316.62 7,797.18 7,859.57 Arkansas ..... 3,651.28 3,946.54 4,165.72 Texas . . . . 11,308.05 11,614.13 11,949.02 Kansas ...... 8,810.50 8,841.09 8,874.58 Colorado 4,852.44 4,989.85 5,093.20 New Mexico 2,450.02 2,441.93 2,596.64 Indian Country .... 2,320.02 2,585.69 2,686.47 Oklahoma ..... 2,359.52 2,635.64 2,836.19 Total Find the total mileage for each group of states as indicated. Add: 891 Q3 ^5 ' %> "*' 6* 9. 10. 3i,2|,5f 2 . 11. 54, 9A, 2A. 12. If, 51 21. 13. 7f,6&,&. 14. 9&,5|, 7|. is. 12|, llf , 61 ADDITION 167 16. 2 IV 50", 7 24' 30", 9 27' 37", 128 14' 43". 17. 2 ft. 9 in., 7 ft. 3 in., 9 ft. 11 in., 15 ft. 7 in. 18. 8 qt. 1 pt., 9 qt. 1 pt., 15 qt., 12 qt. 1 pt. 19. 4 gal. 2 qt., 7 gal. 3 qt., 9 gal. 1 qt., 8 gal. 3 qt. 20. 5 pk. 7 qt., 9 pk. 3 qt., 12 pk. 5 qt., 13 pk. 4 qt. 21. 3 bu. 3*pk., 9 bu. 2 pk., 7 bu. 1 pk., 4 bu. 3 pk. 22. 12 hr. 15 min., 15 hr. 8 min., 17 hr. 42 min., 5 hr. 13 min. 23. 5 da. 12 hr., 18 da. 17 hr., 13 da. 18 hr., 5 da. 3 hr. 24. 15 yd. 2 ft., 25 yd. 1 ft., 32 yd. 2 ft., 9 yd. 1 ft. 25. How many times does a clock strike in 24 hours? 26. If 4 jars contain 3.92 liters, 7.84 liters, 9.57 liters, and 6.3 liters respectively, how many liters are in the four jars? 27. The dimensions of a table are 8 ft. 3 in. by 3 ft. 7 in. How many feet are in its perimeter? 28. The Galveston Sea Wall was constructed by Gal- veston County and the United States Government; the former built 3.5 miles, and the latter .87 mile. Find the total length of the sea wall. In its construction there were used 1150 carloads of cement, 6100 carloads of crushed rock, 1400 carloads of round piling, 475 carloads of sheet piling, 4300 carloads of riprap, and 6 carloads of reenforcing rods. How many carloads of material were used in its construction ? How many miles would the cars extend if placed end to end, allowing 39.6 ft. to a car? 29. The decapod locomotives operating between Clarion Junction and Freeman, Ohio, weigh 268,000 Ib. each. Express this weight in tons. 168 ADVANCED BOOK OF ARITHMETIC SUBTRACTION Subtraction is the inverse of addition. To subtract 7 from 16 is to find a number which added to 7 will make 16. Example l. Subtract 63 from 92. % 92 no PROCESS. 3 and 9 are 12 ; write 9, carry 1. 1 and on 6 are 7, 7 and 2 are 9 ; write 2. Example 2. Subtract 6.3 from 9.2. 9.2 6.3 The process is the same as in Example 1. 2^9 Example 3. Subtract 6 hr. 3 min. from 9 hr. 2 min. HR. MIN. PROCESS. 3 min. and 59 min. make 1 hr. and 2 2 min.; write 59 min., carry 1 hr. 1 hr. and 6 _ 6 hr. are 7 hr. 7 hr. and 2 hr. are 9 hr. ; write 2 59 2hp- Example 4. Subtract 6| from 9|. 92. PROCESS. | and -| are 1^ ; write -|, carry 1. 1 and -~ 6 are 7, 7 and 2 are 9 ; write 2. Example 5. From 75,218 take the sum of 4799, 3928, 9476, 8873. 75218 PROCESS. 3, 9, 17, 26 ; 26 and 2 are 28. Write 4799 2, carry 2. 2, 9, 16, 18, 27 ; 27 and 4 are 31. Write 3928 4, carry 3. 3, 11, 15, 24, 31 ; 31 and 1 are 32. 9476 Write 1, carry 3. 3, 11, 20, 23, 27; 27 and 8 8878 are 35. Write 8, carry 3. 3 and 4 are 7. Write 48142 4 The remain( ier is 48,142. This example shows the practical value of this method of subtraction. (Austrian Method.) SUBTRACTION 169 EXERCISE 94 l. Exports of domestic manufactures from the United States for the years ending June 30, 1897, and 1907 : ARTICLE 1897 1907 Iron and steel, manufactures of ... $57,497,872 $181,530,871 Copper, manufactures of 31,621,125 88,791,225 Wood, manufactures of 35,679,964 79,704,395 Oils mineral, refined 56,463,185 78,228,819 Leather and manufactures of .... 19,161,446 45,476,960 Cotton, manufactures of 21,037,678 32,305,412 Agricultural implements 5,240,686 26,936,456 Naval stores 9,214,958 21,686,752 Carriages, cars, and other vehicles . . 9,952,033 20,513,407 Chemicals, drugs, dyes, and medicines . 8,792,545 18,220,630 Instruments and apparatus 3,054,453 14,661,455 Paper and manufactures of 3,333,163 9,856,733 Paraffin and paraffin wax 4,957,096 9,030,992 Fibers, manufactures of ...... 2,216,184 3,308,112 India rubber, manufactures of .... 1,926,585 7,428,714 Furs and skins 3,284,349 7,139,221 Books, maps, engravings, etc. .... 5,647,548 5,813,107 Tobacco, manufactures of 5,025,817 5,735,613 Brass and manufactures of 1,171,431 4,580,455 Gunpowder and other explosives . . . 1,555,318 4,082,402 Paints, pigments, and colors 944,536 3,391,988 Soap 1,136,880 3,806,097 Musical instruments 1,276,717 3,252,063 Nickel and manufactures of 726,789 3,218,862 Clocks, watches 1,770,402 3,160,272 Coke 547,046 3,013,088 Glass and glassware 1,208,187 2,604,717 All other articles 19,799,642 47,295,739 Find the increase in the exports of each of the above articles, or group of articles, and verify your work. 170 ADVANCED BOOK OF ARITHMETIC Find the difference between : 2. 200 and .02. 10. $403.05 and 192.89. 3. 400 and 1.37. 11. 160.52 and $23.87. 4. $75 and 73^. 12. 100 and .01. 5. $700 and $2.84. 13. 6.29 and 2.9924. 6. $100 and $1.75. 14. 5.001 and 4.0073. 7. $1000 and 5^. 15. 7.2 and 2.77. 8. $324.80 and $100.99. 16. 11 and 1.5. 9. $70.73 and $19.94. 17. 17.3 and 11.9. 18. The square of 6.715 and the square of .285. 19. 7f and 41J. 25. 9 T ^ and 3J. 20. 18f and 7f . 26. 6 T 9 g and 3|. 21. 9| and 4|. 27. 10 T \ and 1\. 22. 21 1 and 11 U. 28. 19| and 84. o 1 o o o 23. 7 T 3 T and 2J. 29. 12^ and 9^-. 24. 8 T 3 g and 5 T %. 30. 23{ and 12 Jf . 31. 5 ft. 7 in. and 4 ft. 9 in. 32. 17 ft. 3 in. and 12 ft. 8 in. 33. 19 ft. 1 in. and 9 ft. 4 in. 34. 27 ft. 3 in. and 18 ft. 4 in. 35. 9 Ib. 2 oz. and 4 Ib. 7 oz. 36. 17 Ib. 6 oz. and 5 Ib. 11 oz. 37. 33 Ib. 2 oz. and 18 Ib. 8 oz. 38. 12 hr. 10 min. and 9 hr. 24 min. 39. 90 and 34 14' 15". 40. 180 and 115 4' 50". 41. 180 and the sum of 56 16', and 92 18'. 42. 15 pk. 3 qt. and 3 pk. 7 qt. SUBTRACTION 171 43. 23 pk. 5 qt. and 13 pk. 6 qt. 44. From 40,000 take the sum of 3211, 4711, 5283, 9438. 45. From 50,580 take the sum of 19,311, 12,218, 1273, 5559. 46. From 18,900 take the sum of 3419, 3428, 4584, 2293. 47. A man owns two houses worth respectively $2390 and $4575 ; he has deposited in the bank $3280 ; he owes two notes for $783 and $870. How much is he worth ? 48. The area of the British Isles is 120,975 square miles; the area of Texas is 265,780 square miles. By how many square miles does the area of Texas exceed the area of the British Isles ? 49. The population of the Chinese Empire is 433,553,000; of the British Empire, 363,900,000 ; of the Russian Em- pire, 141,000,000; of the United States, exclusive of colonial possessions, 84,150,000 ; of Germany, 60,478,000. How many more people are in the United States than in Germany ? In the British Empire than in Russia, United States, and Germany combined? By how many does the population of China exceed the population of Russia, United States, and Germany together? 50. The areas of Maine, New Hampshire, Vermont, Massachusetts, Rhode Island, and Connecticut in square mrles are respectively : 33,040, 9305, 9565, 8315, 1250, 4990. The area of California is 158,360 square miles. By how many square miles does the area of California exceed the area of the six New England states? 51. In going from Galveston to Chicago by rail, a dis- tance of 1410 miles, a man travels the first day 345 miles ; the next day, 201 miles ; the third day, 290 miles. How far is he from Chicago at the end of the third day ? 172 ADVANCED BOOK OF ARITHMETIC MULTIPLICATION If one factor of the product is multiplied by a number, and the other factor divided by the same number, the product will be unchanged. Thus, 84 x 20 = 1680. 420 x 4 = 1680. Here 84 is multiplied by 5, and 20 is divided by 5. Example 1. Multiply 3782 by 234. 3782 or 3782 234 234 15128 = 4 x 3782 7564 = 200 x 3782 11346 = 30 x 3782 11346 = 30 x 3782 7564 = 200 x 3782 15128 = 4 x 3782 884988 = 234 x 3782 884988 = 234 x 3782 To multiply integers, write multiplicand and multiplier so that units of the same name stand in the same column, then multiply the multiplicand by each digit of the multiplier, placing the first figure of each partial product directly under the digit of the multiplier producing it, and add the partial products. .Example 2. Multiply 17.32 by .47. 17.32 PROCESS. The numbers are multiplied as if .47 both were integers ; then beginning at the right 12 124 of the product four places are pointed off, that is 69 28 the number of decimal places in multiplicand and 8.1404 multiplier combined. This may be readily seen by multiplying the multiplier by 100 and dividing the multiplicand by 100. Compare with explanation page 59. MULTIPLICATION 173 Example 3. Multiply 4J by 2|. 2|=| o o 13 Therefore, 4| x 2| = 3$. x f = ^ ^| = 13. EXPLANATION. If the first factor is multiplied by 8, the result is 39, and if the second factor is multiplied by 3, the result is 8. Hence, 8 x 3 x required product = 39 x 8. Therefore, required product = o X o Compare with explanation on page 46. Example 4. Multiply 5 gal. 2 qt. 1 pt. by 9. PROCESS. 9 times 1 pt. = 4 qt. 1 pt. ; write GAL. QT. PT. 5 2 1 * pk* carr y 4 qt. 9 times 2 qt. = 18 qt. 18 9 qt. + 4 qt. = 22 qt. = 5 gal. 2 qt. ; write 2 5Q 2 I *$"> carr y 5 gal. 9 times 5 gal. = 45 gal. 45 gal. -f- 5 gal. = 50 gal. PARTICULAR SHORT METHODS OF MULTIPLICATION 5 = 1 of 10 75 = 100 - \ of 100 25 = of 100 875 = 1000 - of 1000 125 = | of 1000 99 = 100-1 .16| = | 97 = 100-3 Example 1. Multiply 97.3 by 125. 125 x 97.3 = \ of 1000 x 97.3 = \ of 97300 = 12162.5 Example 2. Multiply 29.374 by 993. 29374. =1000x29.374 205.618= 7x29.374 29168.382= 993x29.374 174 ADVANCED BOOK OF ARITHMETIC EXERCISE 95 1. Multiply each of the following numbers by 10 : 234, 350.2, 25.07, .127, .0788, 1.003. 2. Multiply the following numbers by 100 : 505, 67.5, 27.28, 5.347, .07954, .00392. 3. Multiply the following numbers by 1000 : 728, 96.4, 12.87, 1.732, .0139, .00782. 4. Multiply the following numbers by 10,000 : 318, 25.4, 19.96, 18.832, 27.796, .012. 5. Find in the shortest possible way the following products : O) 2780 x 99 ; 9218 x 998 ; 7215 x 999. (5) 2.79x25; 3.18x125; 243x875. O) 78 x .16|; 90 x .331; 297 x 9998. 6. Multiply 5280 by 5280 ; 1020 by 1020. 7. Multiply 7309 by 256 ; 9417 by 735. 8. The estimated production and value of the following cereal crops as given in the Annual Report of the Depart- ment of Agriculture for the year 1906 are as follows : CEREALS YIELD PER ACRE VALUE PER BUSHEL Corn bushels 30.3 cents 39.9 Wheat 15.5 66.7 Oats 31.2 31.7 live . 16.7 58.9 Barley 28.3 41.5 Buckwheat 18.6 59.6 Find the value of the yield per acre of each of these cereal crops. MULTIPLICATION 175 9. The number of bales of cotton produced in Texas in the season 1904-05 was 2,598,949, and in 1903-04, 3,214,133. Allowing 500 Ib. to a bale, how many more pounds of cotton were produced in the latter year than in the former? 10. The estimated production and value per ton of the hay crop for the year 1906 are as follows : STATE YIELD PER ACRE PRICE PER TON New Hampshire .... tons 1 15 $12 50 Massachusetts .... 1 31 17.00 Connecticut 1.17 15.00 New York 1 28 12.10 New Jersey 1.32 15.95 Pennsylvania 1 30 13.40 .Maryland 1 26 13.50 Virginia 125 15.50 South Carolina 146 1525 Georgia 1 65 15.75 Alabama 1.95 13.30 Louisiana 1.93 11.50 Tennessee 1 51 13.45 Kentucky 1.35 13.25 Illinois .98 12.50 1.70 5.50 Kansas 1.28 6.25 Colorado 2.50 9.50 Utah 4.00 7.50 Idaho 2.95 8.00 1.85 11.25 Find the value of the yield per acre in each of the above states. 176 ADVANCED BOOK OF ARITHMETIC 11. A piece of coal taken from the mine at Coos Bay, Oregon, had the following composition by weight: Moisture =.1042 Combustible matter = .4221 Fixed carbon = .4318 Ash = .0419 Find the amount of each in 87 tons of this coal: in 783 tons. Check your answers. 12. Find to the nearest cent the value of each of the following articles: (a) 25J bu. corn @ 42| ^ per bu. (6) 12 T ^ bu. wheat @ 69 J^ per bu. 0) 28| bu. oats @ 25J per bu. (d) 16f bu. rye @ 60| $ per bu. O) 201 bu. barley @ 47f ^ per bu. (/) 4| Ib. wool @ 61 ^ per Ib. (#) 497 Ib. cotton @ llf $ per Ib. (A) 512 Ib. cotton @ 10{ f per Ib. 13. The inside dimensions of the floor of a box car are 40 ft. -| in. by 8 ft. 6 in. Find the perimeter of the floor. 14. The inside dimensions of the floor of a refrigerator car are 28 ft. 9| in. by 8 ft. 1^ in. Find its perimeter. 15. Multiply 5 yd. 2 ft. by 8 ; 9 ft. 8 in. by 7. Find the product of : 16. 9 Ib. 4 oz. by 5 ; 16 Ib. 11 oz. by 9. 17. 3 hr. 20 min. 30 sec. by 6 ; 7 hr. 17 min. by 9. 18. 53 12' by 10 ; 68 12' 18" by 5. 19. 4 pk. 7 qt. by 6; 7 pk. 3 qt. by 8. 20. 5 gal. 2 qt. by 9; 9 gal. 3 qt. by 12. DIVISION 177 DIVISION Division is the inverse of multiplication. To divide 84 by 7 means to find a number which multi- plied by 7 gives 84. If the divisor and dividend are both multiplied by the same number, the quotient remains unchanged. Thus, 96 - 8 = 12. (96 x 6) - (8 x 6) = 12. Example l. Divide 2483 by 7. 7)2483 PROCESS. 7 is contained in 24 hundreds 3 354|- hundred times, remainder 3 hundreds ; 3 hun- dred = 30 tens. 30 tens and 8 tens = 38 tens. 7 is con- tained in 38 tens 5 tens times, remainder 3 tens ; 3 tens = 30 units. 30 units and 3 units = 33 units. 7 is contained in 33 4 times, with a remainder of 5. Example 2. Divide .437 by 1.92. .2276+ PROCESS. Move the decimal point two places to the right in divisor and dividend ; this multiplies both by 100. Then write each quotient figure directly above the right-hand figure of the partial dividend which produces it. Write the decimal point in the quotient above the decimal point in the dividend. 192^43 7 QQ 4 1 344 Example 3. Divide 3| by 6. | =(3| x 3 x 2) (61 x 3 x 2) = Multiply divisor and dividend by 3 x 2. divisor and dividend both whole numbers. This makes 178 ADVANCED BOOK OF ARITHMETIC Example 4. To how many long tons are 3.30693 short tons equivalent? 100 3.30693 x nn _ 330.693 __ 112 To check an example in division, multiply the quotient by the divisor. EXERCISE 96 l. The estimated acreage, production, and value of the potato crop by states for the year 1905 are as follows : STATE ACREAGE PRODUCTION FARM VALUE New Hampshire acres 19,700 bushels 2,367,000 dollars 1,704,000 Rhode Island 6,490 811,200 722,000 7,680 714,000 421,200 25,900 1,993,000 1,355,000 Florida 4,110 308,200 369,900 5,860 644,900 548,200 34,400 3,025,100 1,754,500 242,000 16,203,000 9,073,700 149,000 11,186,000 7,494,600 86,100 7,059,000 3,882,614 25,400 2,415,000 917,800 Nevada 2,800 336,700 276,100 Find the number of bushels yielded per acre in each state, and the average price per bushel in cents. Divide correct to four decimal places: 2. 128.016 by 420. 6. .02734 by .044. 3. 2.3774 by 7.8. 7. .035936 by .0888. 4. 10.4987 by 3.2. 8. 1.57899 by .639. 5. .77087 by .479. 9. 60.247 by 78.8. 10. 5.0748 by 3.88. DIVISION 179 Divide : 11. 21 by 1^ T . 16. 93f by 62f . 12. 48 by 2f . 17. 17f by 9fi. 13. 42byl T V 18. 14. 72 by 3f. 19. 15. 72lby4f 20. 2ilby2ff. Find the value of : SOLUTION. fxfx|xf= 22. X If -- 2 X T V 24. | X 2f X 1* -- If. 23. | X If -*- 1J -H 8f 25. 1^-8J-!.3JXJ. 26. If -I- (I -8- 41) X 4f . 27. | Of If Of 31 - I Of If HINT, i x f x I -i- if. Observe the divisor is f of 1-J. 28. | of If of 2| - f of If. 29. 11 of If of T V^ A off. 30. |of4|X2 9 o-^f|. 31. If Of 3f Off-*-! Off. Express in long tons : 32. 3.36 T., 6.6139 T., .00992 T., 4.4092 T. Express in short tons : 33. 1.9684 long T., 6.8894 long T., .004921 long T. Express in troy pounds : 34. 5 Ib. avoirdupois, 13.228 Ib. avoirdupois, 6613.87 Ib. avoirdupois. 180 ADVANCED BOOK OF ARITHMETIC Express in avoirdupois pounds : 35. 7 Ib. troy, 13.396 Ib. troy, 5.358 Ib. troy. 36. Express 1 ft. as a decimal of 1 mi. 37. Express 1 rd. as a decimal of 1 mi. 38. Express 1 sq. rd. as a decimal of 1 A. 39. Express 1 A. as a decimal of 1 sq. mi. 40. Express 1 sq. yd. as a decimal of 1 A. 41. Express 1 Ib. as a decimal of 1 ton. 42. A lot is 40 by 120 feet. How many such lots make 40 acres? 43. How many barrels of 31 J gallons each will a rec- tangular tank 12 ft. by 8 ft. and 5 ft. deep hold ? (Allow 7^ gal. to a cubic foot.) 44. The weight of a half dollar is 12J grams. How many half dollars can be made out of 7500 grams of stand- ard silver ? 45. Find the cost of boring an artesian well 1400 feet deep at $4 a linear foot for the first 900 feet, $4.50 per linear foot for the next 200 feet, $ 5 per linear foot for the next 100 feet, $ 5.50 per linear foot for the next 100 feet, and $ 6 per linear foot for the remainder. 46. The rails of the Great Western. Railway, England, weigh 97| Ib. per yard. Find in tons the weight of the rails required to construct 1 mile of this railway. 47. Express in the ordinary decimal notation : 3.27 x 10 6 , 17.45 x 10 9 , 9.4 x 10 3 , 7.3 x 10 6 . 48. Given 10-i = T V 10-2 = _i_, 10 -3 = _i_ = TWO 0> 10 ~ 5 = TFoVoT' 1~ 6 = TOOOOOO- Express in the ordinary decimal notation : (a) 3.2 x 10- 3 , 4.71 x 10~ 6 , 9.83 x 10' 3 . (6) 4.98 x 10- 5 , 9.371 x 10" 6 , 4.329 x 10- 4 . LONGITUDE AND TIME 181 LONGITUDE AND TIME A meridian is an imaginary line running due north and south from pole to pole. Longitude is the distance, expressed in circular arc measure, east or west from the prime or standard me- ridian. The meridian through any particular place may be used as the prime meridian. The meridians through the ob- servatories of Greenwich, Washington, Paris, Madrid, Rome, Stockholm, Pulkova, and Lisbon have been used as prime meridians by the nations to which these cities belong. The International Geodetic Congress, which met at Washington in 1884, recommended that the meridian passing through the observatory at Greenwich, a suburb of London, be the prime meridian. This recommendation is now generally adopted by the great nations of the world. The meridian of Greenwich is taken as prime meridian in this book. Longitude is reckoned in either direction halfway around the earth from the prime meridian. The greatest longi- tude a place can have is 180 E. or 180 W. The meridian 180 E. or 180 W. of the prime meridian is a continuation of the prime meridian on the other side of the earth, and forms with the prime meridian what is called a great circle passing through the poles. The earth rotates on its axis from west to east. Con- sider two places not on the same meridian ; for example, New York City and St. Louis. New York being farther east will come the sooner under the influence of the sun's rays. Therefore, when it is noon in New York City it is before noon in St. Louis. Since the earth's motion is uniform, and furthermore, since 182 ADVANCED BOOK OF .ARITHMETIC in 24 hr. the earth rotates 360, /.in 1 hr. the earth rotates 15; .'.in 1 min. the earth rotates 15' ; /.in 1 sec. the earth rotates 15". A difference of 15 of longitude corresponds to a differ- ence of 1 hr. of time. A difference of 15' of longitude corresponds to 1 min. of time. A difference of 15" of longitude corresponds to 1 sec. of time. Hence, to convert difference of longitude into difference of time, divide by 15. EXERCISE 97 1. When it is noon at London, what is the time at New Orleans, 90 W. ? 2. When it is 9 o'clock A.M. on the meridian 75 W., what is the time on the meridian 90 W. ? 3. The longitude of Denver is 105 W. When it is 3 o'clock P.M. in Denver, what is the time in London? 4. Two places differ in longitude by 20. What is their difference in time ? 5. A person travels east 15. What change must he make in the time indicated by his watch so that it may indicate local time ? Supposing he goes the same distance west, what change must be made in the time indicated by his watch ? 6. When it is noon, in London, what is the longitude of the places in which it is 4 o'clock P.M. ? 5 o'clock A.M.? 7. When it is 2 o'clock P.M. in Washington, what is the time in places 30 W. of Washington ? in places 75 E. of Washington? 8. What is the difference in longitude between places which differ in time by 2 hr. 30 min. ? by 4 hr. 10 min. ? LONGITUDE AND TIME 183 9. If a person travels from Denver to New York, will his watch be fast or slow when he reaches New York, and how much ? 10. To how many hours "does a difference of 80 in longitude correspond? 11. What difference in longitude corresponds to a dif- ference of 4 hr. 20 min. in time ? 12. A person living on the 90th meridian W. wishes to send a telegram to a bank in New York City, directing the bank to pay on the same day a sum of money. Up to what hour in the afternoon may he do this, allowing 30 minutes for the transmission of the telegram, taking the longitude of New York as 75 W. ? (New York banks close at 3 P.M.) 13. At places on the same parallel of latitude the sun rises at the same instant local time. How many minutes earlier does the sun appear to a person who travels 1E.? LONGITUDES OF CITIES REFERRED TO IN THIS CHAPTER Austin, Baltimore, Bangor, Bismarck, 97 76 68 100 44' W. 37' W. 47' W. 47' W. Galveston, Havana, Honolulu, Louisville, 94 47' 82 21' 157 52' 85 46' W. 30" W. W. W. Boston, 71 3' 50" W. Melbourne, 144 58' 32" E. Brisbane, 153 2' E. Manila, 120 58' 3 n E. Buenos Ayres, 58 22' 14" W. Mexico City, 99 6' 39 n W. Charleston, 79 52' 58" W. Montreal, 73 33' 4 n W. Chicago, 87 40' W. New Orleans, 90 3' 28 11 W. Cincinnati, 84 24' W. New York > 74 0' 24 n W. Constantinople, 29 0' 50" E. Norfolk, 76 17' 22 n W. Detroit, 83 3' W. Paris, 2 20' 15 n E. Dublin, 6 20' 30" W. Pekin, 116 29' E. 184 ADVANCED BOOK OF ^ARITHMETIC Pensacola, 87 16' 6" W. St. Petersburg, 30 19' 40" E. Philadelphia, 75 9' 3"W. San Francisco, 122 24' 32" W. Portland, 122 40' W. Savannah, 81 5'25"W. Providence, 71 24' 20" W. Tientsin, 117 11' 44" E. Borne, 12 28' 40" E. Tokyo, 139 44' 30" E. St. Louis, 90 16' W. Washington, 77 0' 36" W. In recent years scientific publications often give longi- tudes in terms of time, the -f sign denoting west and the sign denoting east. H. M. S. H. M. S. Harrisburg, +5 7 32 Adelaide, - 9 14 2 Milwaukee, + 5 51 37 Omaha, -f- 6 23 46 Example 1. Find the difference between the longi- tudes of Austin and Honolulu. SOLUTION. Honolulu, 157 52' W. Austin, 97 44' W. 60 8' .-. Honolulu is 60 8' farther west than Austin. Example 2. Find the difference between the longi- tudes of Galveston and Constantinople. SOLUTION. Galveston, 94 47' W. Constantinople, 29 0' 50" E. Here, the places are on opposite sides of the prime meridian. By going east from Galveston 94 47', one arrives at the prime meridian, and by going 29 0' 50" still farther east, he arrives at the meridian of Con- stantinople. Hence, the difference between the longitudes is (94 47' + 29 0' 50") = 123 47' 50". To find the difference in the longitudes of two places : (l) Subtract their longitudes, if the places are on the same side of the prime meridian. (2) Add their longitudes, if the places are on opposite sides of the prime meridian. LONGITUDE AND TIME 185 EXERCISE 98 Find the difference in longitude between : 1. Baltimore and Bismarck. 2. Bangor and Detroit. 3. Boston and Havana. 4. Buenos Ayres and Chicago. 5. Charleston and Constantinople. 6. Cincinnati and Honolulu. 7. Cincinnati and Melbourne. 8. Havana and Rome. 9. Louisville and St. Petersburg. 10. Constantinople and Tientsin. 11. Paris and Pekin. 12. Norfolk and Paris. 13. Montreal and Mexico City. 14. Pensacola and Portland. 15. St. Louis and St. Petersburg. 16. Savannah and Dublin. 17. San Francisco and Dublin. Example 1. Find the difference in local time between Boston and Portland, Ore. Portland 122 40' W. Boston 71 3' 50" W. 15)50 36' 10" 3 22 25 Ans. 3 hr. 22 min. 25 sec. Example 2. Find the difference in local time between Washington and Manila. Washington 77 0' 36" W. Manila 120 58" 3" E. 15)197 58' 39" 13 11 54.6 Ans. 13 hr. 11 min. 54.6 sec. 186 ADVANCED BOOK OF ARITHMETIC EXERCISE 99 Find the difference in the local time of : 1. Mexico City and Montreal. 2. Philadelphia and San Francisco. 3. Philadelphia and Dublin. 4. Norfolk and Tientsin. 5. Chicago and Tokyo. 6. St. Louis and Rome. 7. Austin and St. Petersburg. 8. Savannah and Paris. 9. Washington and Brisbane. 10. Cincinnati and Manila. 11. Havana and Louisville. 12. Rome and Manila. 13.. New Orleans and Portland. 14. Providence and St. Petersburg. 15. Montreal and Tokyo. 16. Bangor and Melbourne. 17. Baltimore and Buenos Ayres. Example l. When it is noon, February 22, in St. Louis, it is 15 min. 6 sec. past three o'clock A.M., Feb. 23, in Adelaide, Australia. Find the longitude of Adelaide. SOLUTIOIST. The time difference between St. Louis and Adelaide is 15 hr. 15 min. 6 sec. Multiply by 15, 15 228 46' 30' ' .-. Adelaide is 228 46' 30" E. of St. Louis. Longitude of St. Louis is 90 16' W. .-. longitude of Adelaide = (228 46' 30" - 90 16") E. = 138 30' 30" E. LONGITUDE AND TIME 187 EXERCISE 100 Calculate the longitude of each of the following cities, the time difference between New York City and each of them being given : 1. Berlin, 5 hr. 49.5 min. 2. Brussels, 5 hr. 13.4 min. 3. Calcutta, 10 hr. 49.2 min. 4. Edinburgh, 4 hr. 43.2 min. 5. Hamburg, 5 hr. 35.8 min. 6. London, 4 hr. 55.9 min. 7. Madrid, 4 hr. 41.1 min, 8. Vienna, 6 hr. 1.2 min. 9. The time difference between London and Amherst, Mass., is 4 hr. 50 min. 3 sec. Find the longitude of Amherst. 10. Find the difference in the time of sunrise between two points in the same latitude and which differ in longi- tude by 39 20'. oJfe^ncT/^ B *l3Sfc* K -r ir""iS^^^^BSfi^ 188 ADVANCED BOOK OF ARITHMETIC STANDARD TIME Standard time is the time of a fixed meridian, generally a multiple of 15. It was established in the United States in 1883 primarily for the convenience of railroads. It is now adopted generally throughout the civilized world. STANDARD MERIDIANS AND PLACES USING THEM 0. Great Britain, Spain, Belgium, Holland. 15 E. Germany, Austria, Italy, Denmark, Norway. 30 E. South Africa, Egypt, Turkey. 821 E. British India (since July 1, 1905). 971 E. Burma (since July 1, 1905). 120 E. West Australia, eastern coast of China, Phil- ippine Islands. 135 E. Japan. 142J E. South Australia. 150 E. Victoria, Queensland, New South Wales. 172| E. New Zealand. 60 W. Newfoundland and Eastern Canada. 75 W. Eastern belt of the United States. 90 W. Central belt of the United States. 105 W. Mountain belt of the United States. 120 W. Pacific belt of the United States. 135 W. Alaska. 150 W. Tahiti. 1571 W. Hawaiian Islands. France uses Paris time, Ireland uses Dublin time. STANDARD TIME 189 EXERCISE 101 1. Mariners carry on board ships chronometers which keep Greenwich time. When it is noon, local time, the chronometer indicates 4 hr. 48 min. P.M. What is the longitude of the ship ? 2. When it is 10 o'clock P.M., March 2, in Washing- ton, what is the standard time in Manilla? Melbourne? Berlin? 3. When it is 2 o'clock A.M., standard time, in Den- ver, what is the standard time of London? Manchester? Glasgow? Tientsin? Constantinople? 4. A telegram is sent from Madrid to Washington at 9 o'clock A.M. Allowing 1 hr. for transmission, when will it reach Washington? 5. At noon, local time, a chronometer indicates 11 o'clock P.M. What is the longitude? 6. A telegram is sent from Galveston to London at 10 o'clock P.M. When will it be received, allowing 2 hr. for transmission? 7. When it is 2 o'clock A.M. in Washington, standard time, what is the time in New Zealand? Tahiti? British India? 8. The San Francisco earthquake occurred April 18, 1906, at 5 A.M. When should the news have reached London? Berlin? Tokyo? Adelaide? (allowing 1 hour for transmission). 9. When it is noon in Paris, France, what is the time in Denver? Natal? Calcutta? Wellington (New Zealand)? 10. When it is 9 o'clock A.M. in Madras, what is the time in St. John's, Newfoundland? Chicago? Sitka? 11. When it is noon in the Hawaiian Islands, what is the time in Cairo (Egypt) ? Perth (Western Australia) ? 190 ADVANCED BOOK OF ARITHMETIC APPROXIMATIONS. CONTRACTED PROCESSES. GENERAL METHODS OF SOLUTION In business problems results of computation are gen- erally required to be correct to not more than two decimal places. For example : The interest on $79.50 for 4 months at 7% is $1.855. From a business point of view the answer is $1.85. In all practical measurements of length it requires skill and long practice to get results correct to more than three figures. For example : A surveyor measures the length of a field and finds it to be 3729 feet. It is extremely probable that the last figure in this result is not correct. As the results of measurement are correct to only three or four figures, hence it is useless in computation to give results to more than three decimal places. Before undertaking to show how results may be obtained correct to any given number of figures, it is well to fix in mind the following facts : Tenths multiplied by tenths give hundredtks. Tenths multiplied by hundredths give thousandths. Tenths multiplied by thousandths give ten-thousandths. Hundredths multiplied by hundredths give ten-thousandths. Example 1. Multiply .0537928 by 43.27. Move the decimal point one place to the .537928 left in the multiplier. 4.327 Move the decimal point one place to the 2.151712 right in the multiplicand. 1613784 These changes make no change in the 1075856 product. 3765496 Suppose it be required to get the product 2.327614456 correct to two decimal figures. The answer would be 2.33. APPROXIMATIONS 191 Write the units' figure of the multi- CONTRACTED PROCESS pii er under the third decimal figure of .537982 the multiplicand. Multiply 4 by 7 4.327 and carry 4 from 4 multiplied by 9 be- 2.152 cause 36 is nearer 40 than 30. Multi- 161 ply the remaining figures to the left by 11 4 in the usual manner. 4 As tenths multiplied by thousandths 2.328 give ten-thousandths, multiply 3 by 3 2.33 to the left of 7 and carry 2 from 3 times 7. 3 times 5 are 15 and 1 make 16. As hundredths multiplied by hundredths give ten-thou- sandths, multiply 2 by 5 and carry 1 from 2 times 3. As thousandths multiplied by tenths give ten-thousandths, 7 is multiplied by no figure of the multiplicand, 4 is carried from 7 times 5. Compare the two processes. Example 2. Multiply 253.7 by .079 correct to two deci- mal figures. Move the decimal point in the multi- 2.537 plier two places to the right. Move the 7.9 decimal point in the multiplicand two 17.759 places to the left. These changes make 2.283 no change in the product. Multiply by 20.04 7 in the usual manner. Multiply by 9 beginning with 9 times 3, and adding 6 to the product which is the figure carried from 9 times 7. Move the decimal point in the multiplier so that it con- tains one integral figure. Move the decimal point in the multiplicand the same number of places in an opposite direc- tion. Place the units' figure of the multiplier under the third place of the multiplicand, if a product to two decimal figures 192 ADVANCED BOOK OF ARITHMETIC is required. If a product to three decimal places is required, place the units' figure of the multiplier under the fourth deci- mal place of the multiplicand. Then multiply as indicated in the above examples. Example 3. Multiply .732 by .864 correct to two deci- mal figures. Begin multiplying by 8 by taking .0732 the product 8x3, carrying 2 from 8 .0732 8.64 x 2. Begin the multiplication by 468 .586 6 with 6x7, carrying 2 from 6 x 586 44 3. Begin the multiplication by 4 44 3 with 4x0, carrying 3 from 4x7. 3 .63 The arrangement in the right mar- .63 gin conserves energy, for the multi- plication by each figure of the multiplier is begun with the figure directly above it. Example 4. Divide 120.005 by 17.293 correct to three decimal figures. 6.939+ 17293)120005 103758 162470 155637 68330 51879 164510 155637 8873 The answer correct to three decimal figures is 6.940 as the next will be 5. APPROXIMATIONS 193 CONTRACTED PROCESS j n this example the quotient is 6.939 + required correct to four figures. 17293)120005 The divisor contains five figures. 103758 Whenever the divisor contains one 16247 or more figures than are required in 15564 the quotient, a figure may be struck 683 off the divisor in place of annexing 519 or taking down a figure, as is usu- 164 ally done in getting each figure of 155 the quotient. 9 Compare the two processes. GENERAL METHOD Example a. If 3 acres of land are worth $ 129, how much are 5 acres worth at the same rate per acre ? Example b. If 8 masons build a wall in 18 days, how long would it take 9 masons to build the wall ? Example (a) The cost of 3 acres = $ 129. The cost of 1 acre = $ of $ 129. The cost of 5 acres = f of 9 129 = f 205. Example (5) The time 8 masons take = 18 da. The time 1 mason takes = 8 x 18 da. The time 9 masons take = | x 18 da. = 16 da. The answer in Example (#) is a fraction of $129. The answer in Example (6) is a fraction of 18 days. The solution of examples of this character consists in multiplying the quantity of the same kind as the answer by a fraction. If the answer is to be greater than the given quantity, form the fraction so that the numerator is greater than the denominator. If the answer is to be less than the given quantity, form the fraction so that the numerator is less than the denominator. 194 ADVANCED BOOK OF ARITHMETIC Example l. If a dealer sells a piano for $425, thereby losing 15 %, what should he have sold it for to make a profit of 15 % ? In this example 85 % of cost is given, and 115 % of cost is sought. The answer will be obviously more than $425. Hence, iff- x $425 = $ 575, Ans. Example 2. A kilometer is very nearly equivalent to ^ of a mile. Express a mile in kilometers. 5 eighths of 1 mile is given and 8 eighths is sought. Hence, f of 1 kilometer = 1.6 kilometers. EXERCISE 102 Solve by the above method. 1. If 6 horses plow a field in 9 days, how long will it take 9 horses to plow the same field ? 2. If a train runs in 3| hours between two stations at the rate of 18 miles an hour, how long will it take a train whose speed is 30 miles an hour to make the same run? 3. If 5 acres of land sell for $ 423, at this rate what will be the selling price of 7 acres ? 4. If 22 yd. of cloth are bought for a sum of money, how many yards may be bought for the same sum when the price falls 12 % ? 5. Eight horses consume a quantity of corn in 24 days. How long should the same quantity of corn last 12 horses? 6. The minute hand of a clock goes 360 in 1 hour. How many degrees does it go in 22 minutes ? 7. An arc of 75 is 4 ft. 6 in. How many feet are in the circumference of the circle ? APPROXIMATIONS 195 8. If 2^ of the number of miles from Paris to Turin is 27|, what is the entire distance separating the cities? 9. If ^ of the number of miles from New York City to Panama is 1727, how far is Panama from New York? 10. Given .9 of the distance from London to Constan- tinople as 1827 mi., how many miles is it from the former to the latter? 11. If |~| of the distance from Hamburg to Vienna is 143 mi., find the distance between these cities. 12. In the year 1902, l^ of the United States internal revenue receipts from tobacco amounted to $22,852,687. Find the total internal revenue receipts from tobacco for that year. 13. In the year 1902, ^ of the excise tax in the United States on gross receipts under the War Revenue Law of 1898 amounted to $117,221. Find the total tax on gross receipts in 1902. 14. In the year 1902, -fa of the United States internal revenue receipts from the tax on oleomargarine amounted to $1,325,021.40. Find the total receipts from this source. 15. The mark is the unit of money in Germany ; f^ of its value in our currency is 42 mills. Express the value of a mark in dollars. 16. The yen is the standard of value in Japan ; -^ of its value is equivalent to 4 cents and 2 mills. Express in dollars the value of the yen. 17. In Venezuela, the monetary unit is the Bolivar ; | of its value is equivalent to $.1158. Find its value in cents. 18. Thirty-two thirty-fifths of a meter is very nearly equivalent to 1 yd. Express the value of a meter in yards. 196 ADVANCED BOOK OR ARITHMETIC THE LANGUAGE OF MATHEMATICS, RATIO, PROPOR- TION, PARTNERSHIP By mathematics is understood those branches of knowl- edge which deal with quantity. Arithmetic, algebra, geometry, surveying, etc., are included in the term mathe- matics. Mathematics has a language of its own. The word eight conveys a definite idea to the mind ; the sign or symbol 8 conveys the same idea. The words eight squared convey a definite idea to the mind ; the symbol 8 2 conveys the same idea. The words three fourths of sixteen convey an idea ; the symbols f X 16 convey the same idea. Similarly, the words the quotient of seventy- two divided by eight convey an idea ; the symbol ^- con- veys the same idea. Letters may represent numbers. Thus, a, 6, c, x, y, z, etc., may each represent any number whatever. The product of the numbers represented by letters is indicated by writing the letters in succession, one after the other. Thus, abc implies the continued product of a, 6, and c. lip stands for principle, r for rate, t for time in years, and i for interest, the rule for computing interest is given by the relation prt = { In like manner the rule for computing the area of a rec- tangle may be expressed by the relation F=la, F stands for area, b for base, and a for altitude. A num- ber written before a letter indicates multiplication. Thus, 5 a means 5 times a. 5 a is then a short way of writing a + a + a + a + a. 46 is a short way of writing b -f b + b + b. THE LANGUAGE OF MATHEMATICS 197 2 is a short way of writing a x a or aa. 3 is a short way of writing a x a x a or aaa. 4 is a short way of writing 0x0x0x0 or aaaa. 5 is a short way of writing 0x0x0x0x0 or aaaaa. The number denoting how many times a number is added is called a coefficient. The coefficient of the expres- sion 9 b is 9. The expression a + b stands for the sum of any two numbers. The expression a b stands for the number which when added to b gives 0, or in other words, the remainder obtained when b is subtracted from a. Example 1. If a = 5, b = 3, what is the value of + 6 ? a -b = ? 4a = ? 36 = ? 20-36=? SOLUTION. + 6 = 5 + 3 = 8. 0-6 = 5-3 = 2. 40 = 4x5 = 20. 36 = 3x3=9. 20-36 = 2x5-3 x3 = l. Example 2. If a = 7, what is the value of a 2 ? 3 ? 3 a 2 ? 4 a 3 ? a2 = axa== 7 x 7 = 49. a B ==axaxa== j x 7x7 =343. 3 2 = 3x0x0= 3x7 x7=147. 40 3 = 4x0x0x0 = 4x7x7x7 = 1372. EXERCISE 103 If 0=4, what is the value of 40? 70? 110? 130? 170? 190? 270? |0? ^0? If 0=5, what is the value of 2 ? 3 ? 4 ? 20 2 ? 30 2 ? 20 3 ? 40 2 ? + 2 ? 2 + 3 ? If = 3, 6 = 2, what is the value of + 6? 6? 20-f6? 0+26? 20-6? 20-36? 30-26? If = 6, 6 = 3, what is the value of 60 + 36? 30 + 56? 60-36? 50-106? 70-56? 30 2 ? 2 +6 2 ? 2 -6 2 ? If # = 5, y = 6, what is the value of xy? 2xy? oxyl *V? xy? Ja? 198 ADVANCED BOOK OF^ ARITHMETIC If x = 9, y = 4, what is the value of 2 x* - y* ? ? 2z 2 +3 2/ 2 ? ?/ 2 -^? If a = 10, 6 = 7, find the value of ab, 5 6, a6 + a 2 , 2 a6 + 6 2 , ab + 2b*. EXERCISE 104 1. What is the sum of two times a number and three times the same number? What is the sum of 2x and 3#? 2. What is the sum of 4 x and 3 x? of 8 a and 3 a? of 56 and 26? of 66 and 4 6? 3. What is the difference between 8x and 3x? 6x and 2x? 116 and 76? Saanda? 4. Add 5 x and 7 # ; 4 # and 9 # ; 96 and 66; 10 y and 6 y ; 12 x and 4 #. 5. Subtract 4 # from 9 x ; 8 # from 14 #; 9#froml6#; 7 x from 13 x ; 5 a6 from 8 ab. 6. Find the difference between 11 y and 2y; 5a6 and a6 ; 7 ab and 4 a6 ; 12 a6 and 2 #6. Every sentence conveys a thought. (1) The sum of 3 and 4 is 7. This sentence is expressed in the language of mathematics as follows : 3 + 4 = 7. (2) Write another sentence : The difference between 18 and 7 is 11. This sentence, written in mathematical language, is 18 7 = 11. (3) Write a third sentence : Two thirds of 27 is 18. In mathematical language this sentence is written | x 27= 18. The statements (1), (2), (3), are called equations. An equation is a statement in symbols that two expressions are equal to each other. The part of an equation to the left of the sign of equal- ity is called the first member of the equation ; the part of an equation to the right of the sign of equality is called the second member of the equation. THE LANGUAGE OF MATHEMATICS 199 What is the product of a and a? a x a = a 2 . What is the product of a and a 2 ? a x a 2 = a x a x a = a 3 . What is the product of a 2 and a 3 ? # 2 = ax# a x a x a. .-. a 2 x a 3 = (a x a) x (a x a x a) = a 5 . What is the product of a 4 and a 3 ? a 4 =axaxaxa. a 3 = a x a x a. .'. a 4 x a 3 = (a X a x a x a) x (a x a x a) = a 7 . What is the product of 4 a 2 and 4 a 2 = 4 x a x a. .'.4a 2 x5a 3 =:4xaxax5x#x x a =4x5xaxa x a x a x a = 20 a 5 . (Associative Law.) EXERCISE 105 1. a x 2 a = ? 7. 3 a x 5 a 3 = ? 13. 4 6 2 x 3 b = ? 2. 2axa 2 =? 8. 9a 2 x2a 3 =? 14. 55 3 x45 3 = ? 3. 3 a 2 x a = ? 9. 4 a 3 x a 2 = ? 15. 2 b 2 x 5 6 = ? 4. 3a 2 x2a 2 =? 10. 5^x4z = ? 16. 4*/ 2 x3# 3 =? 5. 4 a x a 3 = ? 11. 6 x 2 x :r 3 = ? 17. 7 z 2 x 5 a = ? 6. 7 a 2 x 2 a 2 =? 12. 5 5 x 6 2 = ? 18. 4 y x 8 5 3 = ? What is the quotient when a 3 is divided by a? o a 3 a x a x a o a d -^ a = = -- = a x a = a*. a a Here cancellation is utilized. What is the quotient of 8 a 3 by 2 a? 4 o 8 a 3 $ x a x a x a 4 2a= --- = r - = 4xaxa 2a $ xa 200 ADVANCED BOOK OF ARITHMETIC EXERCISE 106 Find the following quotients: i * 6 ?1^ u 22o_ 6 16 5^ 60^ 2 a' 7 a 2 ' 2 a 2 ' Sz 3 ' 10 fc 5 ' 9a 28a* 16 a 39^ gg 26^ 3 ' ' 14 a 2 ' 4a 3 ' 13 a;' 13 a;' 12 a 18 a* 32a 42 33ft 2 3. 8. -. 13. . 18. s 2d. . 4 o a 6 Ib a* ba^* 116 16 a 2 24 a 4 24 a 5 45 z 4 48 J 4 400 Q3 O 1^J7 tt o a o d \j x J-O o 12 a 3 11 a 4 25 4 50 x 5 46 5 5 5. 10. . 15. -= x-. 20. -. 25. 23 6 2 ' RATIO The ratio of one number a to another number b is the quotient obtained by dividing a by b. The ratio of a to b is written a : b. When the quotient a -r- b is written -, the expression - is a fraction. b b The ratio b : a is called the inverse ratio of a to b. EXERCISE 107 1. What is the ratio of 2 ft. to 6 ft.? 2. What is the ratio of 4 in. to 1 yd.? of 3 in. to 1 yd.? of 1 yd. to 1 rd.? of l rd. to 1 rd.? 3. What is the ratio of 80 A. to 1 sq. mi.? of 120 A. to 1 sq. mi.? of l A. to 2 A.? 4. What is the ratio of the distance traveled by two trains in the same time, if the rate of the first train is 20 mi. per hour, and the rate of the second train is 30 mi. per hour ? THE LANGUAGE OF MATHEMATICS 201 5. If A walks at the rate of 2J mi. per hour, and B walks at the rate of 5 mi. per hour, what is the ratio of A's time to B's time in going any given distance ? 6. What is the ratio of the time that 8 men take to do a piece of work to the time that 6 men take to do the same piece of work ? 7. If you ride in a carriage at the rate of 7 mi. an hour and walk back the same distance at the rate of 3 mi. an hour, what is the ratio of the time in the carriage to the time walking ? 8. What is the ratio of the price of 7 Ib. of sugar to the price of 10 Ib. of sugar of the same kind ? 9. What is the ratio of the work done by 6 men to the work done by 9 men ? 10. What is the ratio of the time that 9 men take to do a piece of work to the time that 6 men take to do the same work ? 11. I can buy two kinds of matting for 40^ and 50^ a yard respectively. If I spend the same amount of money in the purchase of the two kinds of matting, what is the ratio of the number of yards of matting of the first kind to the number of yards of the second kind bought ? 12. Divide 15 in the ratio 2 : 3. 13. Divide 20 in the ratio 3 : 7. 14. Divide f 1 in the ratio 18 : 7. 15. Divide 1 mi. in the ratio 7 : 9. 16. Divide 1 in the ratio 9 : 11. 17. Divide 1 gal. in the ratio 1 : 3. 18. Divide $'1 in the inverse ratio 9 : 16. 19. Divide 22 yd. in the inverse ratio 3 : 8. 20. Divide $1000 in the inverse ratio 3 : 5. 202 ADVANCED BOOK OF ARITHMETIC PROPORTION A statement indicating that two ratios are equal is called a proportion. Illustrations, 2:3 = 4:6. (1) 9:15 = 12:20. (2) Statement (2) is a proportion because the value of the first ratio is f , and the value of the second ratio, i.e. 12 : 20, is also ^. Statement (2) may read 9 is as large compared with 15 as 12 is compared with 20. The first and fourth terms of a proportion are called the extremes, and the second and third terms are called the means, of the proportion. In a proportion the product of the extremes is equal to the product of the means. Let a : b = c : d be any proportion whatever. Then ad = be. PROOF. 7 = -;- Multiply each member by Id and get b d abd cbd *, ,-, ,. 7 z ; = . *. by cancellation ad = be. b d This property of a proportion enables us to find any term of a proportion, if three of the terms of the propor- tion are known. The proportion a:b = c:d is sometimes written a: b : :c: d. The double colon used as a sign of equality is now rapidly becoming obsolete. Example 1. Find x in the proportion x : 4 = 9 : 6. SOLUTION. The product of the extremes is equal to the product of the means. ..63=36. 3=6. PROPORTION 203 Example 2. Find x in the proportion 10 : 35 = x : 42. SOLUTION. Since the product of the means is equal to the product of the extremes, 35 x = 10 x42. Two numbers which vary directly are said to be directly proportional. Two numbers which vary inversely are said to be inversely proportional. EXERCISE 108 If x stands for the unknown term in each of the follow- ing proportions, find it : 1. 2:3=6: x. 13. 57 : 133 = x : 126. 2. 3: 4= 6: a?. 14. 68 : 85 = x : 75. 3. 15:25 = 12:z. 15. 36 : x = 52 : 65. 4. 12:20 = 18:z. 16. 28:^=36:63. 5. 14:21 = ^:27. 17. 27: a = 15: 50. 6. 21: 27 = a;: 45. 18. 15:^=21:77. 7. 35: 84 = z: 72. 19. 28: a =36: 81. 8. 20:48 = z:96. 20. 25: x = 45: 72. 9. 16:24 = ^:33. 21. 35:^=30:48. 10. 20: 32 = x: 72. 22. x: 81 = 16: 72. 11. 25:45 = ^:99. 23. x: 99 = 26: 117. 12. 45 : 126 = x : 154. 24. x : 65 = 24 : 52. 25. x : 112 = 45 : 144. Example 1. If 7 bu. of wheat cost $5.25, find the cost of 11 bu. of wheat at the same rate. SOLUTION. It is reasonable to assume that the price of 11 bu. of wheat is greater than the price of 7 bu. of wheat. .-. the price of 11 bu. of wheat = -U of $5.25= $8.25. 204 ADVANCED BOOK OF^ ARITHMETIC Example 2. If 12 men pave a street in 15 da., how long will it take 9 men to pave a street of the same area ? SOLUTION. It will take 9 men longer than it takes 12 men. /. the time 9 men take = -^ of 15 da. = 20 days. To solve a problem in proportion, find first the relation of the answer to the quantity of the same kind as the answer given in the problem. Second, multiply this quantity by a fraction, proper or improper, according as the answer is less or greater than it. EXERCISE 109 1. If 20 men earn $450 in a given time, how much will 30 men earn in the same time ? 2. If 15 bu. of corn cost $7.20, what will 48 bu. of corn cost? 3. If 12 A. of land cost $456.90, what will 16 A. of the same land cost ? 4. If 4 men can do a piece of work in 15 da., how long will it take 6 men to do an equal amount of work ? 5. If 18 head of cattle cost $1450, what will 27 head of cattle cost at the same rate ? 6. If a train goes 400 mi. in 12 hr., how long will it take to go 560 mi. ? 7. If 8 masons build a wall in 15 da., how long will it take 6 masons to build a wall of the same size ? 8. If 18 horses consume 14 bu. of corn in a week, how much will 24 horses consume in the same time ? 9. If 18 horses plow a tract of land in 13 da., how long will it take 26 horses to plow the same tract ? 10. How long will it take 126 sheep to eat a quantity of feed which will last 105 sheep 30 da. ? COMPOUND PROPORTION 205 11. A garrison consisting of 1200 men has provisions for 16 da. How many men must be sent away so that the provisions may last 24 da. ? 12. A garrison consisting of 1400 men has provisions for 27 da. If the garrison is reenforced by 400 men, in how many days will the provisions be consumed? 13. If I can buy a dozen turkeys for $20.50, how many turkeys can I buy for $30.75? 14. If the interest on $750 for 4 mo. is $12.50, what is the interest on $39.60 for the same time? 15. If an arc of 12" on the 40th parallel of latitude is 933.92 ft., find the length of 1 on the 40th parallel of latitude. 16. If an arc of 30' on the circumference of a wheel is 1^ in., find the length of the circumference of the wheel. 17. A fly wheel 63 ft. in circumference makes 150 revo- lutions per minute. Find the velocity of its rim per second. 18. A train is running at 50 miles an hour. This speed is 25% greater than usual. Find its usual speed. COMPOUND PROPORTION If the product of the corresponding terms of two or more ratios are taken, the ratio of the resulting products is called the ratio compounded of these ratios. For example, the ratio compounded of the ratios 2 : 3, 4 : 5, 7:8, is the ratio 2x4x7:3x5x8, or 56 : 120, or 7 : 15. A proportion in which the final result depends upon a ratio compounded of two or more ratios is called a compound proportion. A concrete example may give a clearer conception of compound proportion than any formal definition. 206 ADVANCED BOOK OF ARITHMETIC Example l. If 15 men mow 90 A. in 12 da., how many acres will 12 men mow in 14 da.? SOLUTION. The 12 men in a given time will mow less than 15 men in the same time. .*. the 12 men in 12 da. will mow If of 90 A. But the 12 men in 14 da. will mow more than this quantity. /. 12 men in 14 da. will mow if of if of 90 A. = if of 90 A. = 84 A. Example 2. If 24 men build a house in 18 da. of 10 hr. each, how many men will it take to build the same house in 30 da. of 8 hr. each? SOLUTION. Step. 1. It will take fewer men to build a house in 30 da. than it will take to build it in 18 da. of the same length. .*. the number of men it will take to build the house in 30 da. of 10 hr. each = if of 24 men. Step 2. More men are needed when they work 8 hr. a day than when they work 10 hr. a day. .*. the number of men, in 30 da. of 8 hr. each, required to build the house = -^ of ^| of 24 men= 18 men. EXERCISE 110 1. If 12 horses plow 84 A. in 6 da., how many acres will 16 horses plow in 4^ da.? 2. If 14 men pave a street 200 ft. long in 8 da., how many feet will 12 men pave in 7 da. ? 3. If a man earns $117 in 3 mo. working 6 hr. a day, how much will he earn in 5 mo. working 8 hr. a day ? 4. A garrison of 3650 men consumed in 30 da. 82.3 T. of food. How much food would be required for 7500 men for 1 yr. at the same rate ? COMPOUND PROPORTION 207 5. If 8 masons build in 2 da. a wall 40 ft. long and 6 ft. high, what height of wall 30 ft. long can they build in 5 da. ? 6. If 21 men complete a piece of work in 8 da. of 7|- hr. each, in how many days of 10 hr. each can 18 men do the same work ? 7. A wall is to be built in 10 da. by 30 men. After 2 da. 10 men are dismissed. In what time will the remaining 20 men finish the work ? 8. If 4 men or 6 boys dig a trench in 12 da., in what time can 2 men and 9 boys dig it ? 9. If 12 men mow 30 A. in 3 da. of 8 hr. each, how many hours a day must 16 men work to mow 48 A. in 4 da. ? 10. If the interest on $100 for 1 yr. is 16, find the interest on 1840 for 2 yr. 3 mo. 11. If 12 men working 7 hr. a day earn 1227.50 in 20 da., how much will 15 men earn in 20 da., working 9 hr. each? 12. If 6 men mow f of a meadow in 4J da., how long will it take 8 men to mow the remainder ? 13. In 10 da. of 8 hr. each 9 horses can plow f of a field. In how many days of 9 hr. each can the remainder of the field be plowed by 15 horses ? 14. A marble block 3 ft. by 4 ft. and 5 ft. in length weighs 5.1 T. Find the weight of a marble block 7 ft. by 3 ft. and 10 ft. long. 15. A mason can build 3 yd. of a wall in 15 hr. How long will it take 9 masons to build 24 yd. of a wall whi a is one-third higher than the other wall ? 208 ADVANCED BOOK OF ARITHMETIC PARTNERSHIP NOTE. Partnerships are rapidly becoming a thing of the past. Those partnerships that still survive are conducted on somewhat different principles from the partnerships that existed prior to the introduction of the telegraph, telephone, and modern means of rapid transit. Example. A, B, and C enter into partnership. A puts in $840, B puts in $350, and C puts in $2000. A with- draws from the concern in 5 mo., C in 7 mo., and at the end of 8 mo. the profits are divided. If the entire profit is $450, how shall this be divided among A, B, and C? SOLUTION. A has $840 in the concern for 5 mo. This is equivalent to $4200 for 1 mo. B has $350 in the concern for 8 mo. This is equiva- lent to $2800 for 1 mo. C has in the concern $2000 for 7 mo. This is equiva- lent to $14,000 for 1 mo. The profits will be divided in proportion to the num- bers 4200, 2800, 14,000, or in proportion to the numbers 3, 2, 10, since 1400 divides each of them. 3 + 2 + 10 = 15. .. A's share = -^ of the profits = -f% of $450 = < B's share = T 2 ^ of the profits = T 2 ^ of $450 = I C's share = ^ of the profits = jf of $450 = $300. EXERCISE 111 l. A, B, and C enter into partnership with capitals of $3000, $3750, and $4500 respectively. At the end of the year they divide among themselves a profit of $3000. Find each person's share. PARTNERSHIP 209 2. Two partners, A and B, invest $600 and $1125. A's money remains in the business 6 mo., and B's 8 mo. If they make a profit of $2100, find each person's share. 3. Two men rent a pasture for $171 ; one puts in the pasture 30 cattle for 30 da., and the other 45 cattle for 18 da. How much rent should each pay ? 4. A and B enter into partnership. A's capital is $200 more than B's. Out of a profit of $640, B gets $280. Find A's and B's capital. 5. A and B enter into a partnership, A contributing $6400 and B $7200. At the end of 3 mo. A withdraws $1600, and at the end of 5 mo. B withdraws $1440. C then enters into the partnership with a capital of $4800. Seven months later a gain of $2154 is divided among them. Find each person's share. 6. A, B, and C enter into partnership. A puts in $1000, B $1200, and C $1800. At the end of 3 mo. C withdraws, and at the end of 10 mo. B withdraws. At the end of a year the profits are divided. If C gets $135, how much do A and B receive ? 7. Two men form a partnership. Their capitals are in the ratio 2 : 3. After 6 mo. the first man increases his capital by J of itself, and the second man diminishes his capital by J of itself. After 6 mo. more they divide their profits, amounting to $1450. Find each partner's share. 8. A cistern 66' x 27' 6" x 10' will hold enough water to irrigate 2J A. of land to the depth of 2 inches. How many acres will a cistern 77' x 41' 3" x 12' 6" irrigate to the depth of 3J inches ? 9. If A pays |- of the cost of irrigation when the rate charged is $4 an acre to the depth of one inch, find his share of the cost. 210 ADVANCED BOOK OF ARITHMETIC PERCENTAGE A per cent of a number implies a fraction of the num- ber having 100 for denominator. Thus, 5 per cent, 5%, yj-g-, and .05 are four ways of expressing the same fact. The per cent equivalents of the following fractions should be thoroughly fixed in mind : i' i* f> i> I' i' t> t' i> i> t> f' s> iV- Example l. The total value of imports into this coun- try through the Atlantic ports for the year 1906 was $974,562,800; of this 75.35% came through New York City. Find the, value of the imports through this city. SOLUTION. $974,562,800 x 7 -^~ = $9,745,628 x 75.35 = $734,333,069.80, value of imports through New York. As 75.35% is correct to four figures only, the result is not likely to be correct to more than four figures. To get four figures multiply 97.456 millions by 7.535 by the contracted process explained on page 191. EXERCISE 112 1. Write the equivalent per cents of the following decimals : .04, .08, .075, .0525, .1666f. 2. Express as decimals the following per cents : 41%, 15%, 121%, 621%, 6 i%, 3|%. 3. Find 5 % of each of the following numbers : 2151, 366.7, 689.5, 7.188, 12.469. 4. Find 6 % of each of the following numbers : 5262, 520.7, 2.66, 3.097, 6.41, .783. 5. Find 4| % of each of the following numbers : 4150, 1418, 7120, 43.43, 53.17, 2.42. PERCENTAGE 211 6. The Engineer's Year Book for the year 1906 gives the cost of railway construction in England as $194,660 per mile. The per cents of cost were as follows : Land 10 Permanent way 11| Fencing 1J Sidings 3 Earthworks 24 Junctions 1 Tunnels 12 Stations 6-| Viaducts and bridges 17 Maintenance J Accommodation works 2 Legal and engineer- Culverts 5 ing expenses 6 Find the cost of each of the above items of expense. 7. The value of the total imports to the United States for the year 1906 was $1,226,560,000. Of this value 79.45% came through the Atlantic ports, 4.42% through the Gulf ports, 1.38% through the Mexican border ports, 5.41% through the Pacific ports, 7.97% through the northern border ports, 1.37% through the interior ports. Find the value of the imports through each of these divisions. 8. The value of the total exports of the United States for the year ending June 30, 1906, was $1,743,860,000. The per cents of total value by principal customs districts were as follows : New York 34.81 Savannah 3.72 Boston 5.66 Puget Sound 2.82 New Orleans 8.63 Detroit 2.02 Galveston 9.54 Buffalo Creek 1.72 Mobile 1.25 Philadelphia 4.73 Newport News 1.15 Baltimore 6.31 Wilmington 1.06 San Francisco 2.29 Pensacola 1.06 Find the values of the exports through these cities. 212 ADVANCED BOOK OF ARITHMETIC Given a quantity, to find its value when decreased by a per cent of itself. Example 1. In the year 1906 the state of Ohio produced 11,562,500 Ib. of wool ; this shrunk 50 % from scouring. Find the number of pounds of scoured wool. SOLUTION. 100 % - 50 % = 50 % = J. 11,562,500x1=5,781,250. Am. 5,781,250 Ib. EXERCISE 113 l. The wool production and per cent of shrinkage from scouring for the year 1906, as given by the Bulletin of National Association of Wool Manufacturers, for the states named are as follows : STATE NUMBER POUNDS UNWASHED PER CENT OF SHRINKAGE Michigan 9,450,000 50 2,450,000 52 568,750 40 35,815,000 65 Wyoming 32,849,750 68 Idaho 16,905,000 67 Oregon 15,300,000 70 13,125,000 67 Utah 12,350,000 65 New Mexico 15,950,000 62 Colorado 9,450,000 67 Arizona 4,420,000 66 Texas 9,360,000 66 4,887,500 70 Find the number of pounds of scoured wool produced in each of the states. PERCENTAGE 213 Given a per cent of a number, to find the number. Example l. During the month of January the average daily attendance of a school was 414. This number was 92 % of the school enrollment. Find the number enrolled. SOLUTION. 92 % of enrollment is given. 100 % of enrollment is sought. .-, enrollment = 414 x -^ = 450, or if x stands for enrollment, 414 .9 a = 414, therefore x = - ^ = 450. . y Example 2. A dealer sells an article for $ 522 at a gain of 16%. Find the cost price. SOLUTION. 116 % of cost price is given. 100 % of cost price is sought. /. cost price = $522 x = 1450, or 1.16s = $522. .'.x = $522 -s-1.16 = $450. EXERCISE 114 1. Find the number of which 79 is 4 % . 2. In a certain town 60 % of the grown people are mar- ried. If there are 2394 married people, how many grown people are in the town ? 3. A man spends $320 for board. This sum is 40% of his income. Find his income. 4. A man spends 83% of his salary and saves 1170. What is his salary ? 5. A lot is sold for $3380 at a gain of 12f %. Find the cost of the lot. 6. After a discount of 16f % is given, a man pays $84 for a bill of goods. Find the amount of the bill. 214 ADVANCED BOOK OF ARITHMETIC 7. The total levies of ad valorum taxes and tax rate per cent of assessed valuation are as follows in the states named : STATE LEVY RATE PER CENT Maine . $ 6,855,776 1.95 Pennsylvania 58,269,455 1.49 South Carolina .... ... 3 736 344 1 91 Kansas 14 847 136 4.09 Tennessee . . 7,626 068 1.88 9,002,727 3.45 Texas 13,683,526 1.34 Find the assessed valuation of property in each of these states. To express one number as a percentage of another number. Example 1. The foreign population of Danish extrac- tion according to the United Census of 1890 and 1900 was 132,543 and 153,805. Find the increase per cent during the ten years. SOLUTION. 153805 - 132543 = 21262, increase. 21262 132543 21262 132543 = fraction the increase is of population in 1890. ,=16.04%. In- 16.04 crease per cent. 132^)2126200 132543 80077 79526 551 530 As the divisor contains 6 figures and the quotient is required to 4 figures, for each quotient figure cut off one from the divisor instead of annexing a cipher. PERCENTAGE 215 Example 2. A dealer buys goods at a discount of 40 % off the list price, and sells them at 16 % off the list price. Find his gain per cent. SOLUTION Cost price to dealer = 60 % of list price (100 % - 40 %). Selling price of dealer = 84 % of list price (100 % - 16 %). Gain = 24 % of list price. |-^ x 100 % = rate per cent of gain. Ans. 40 / . EXERCISE 115 1. The railway mileage of the world January 1, 1906, as given by a German statistician was as follows : COUNTRY MILES COUNTRY MILES Europe 192,251 North America 253,098 Asia 50,593 South America . . 32,859 Africa 16,538 Australasia .... 17,441 Find the per cent of the total railway mileage in each of the six continents. 2. The foreign-born population of the United States by countries for the years 1890 and 1900 was as follows : COUNTRY 1890 1900 COUNTRY 1890 1900 Austria . . England . . France . . 123,270 909,090 113,174 275,910 840,513 104,197 Germany Ireland . . Scotland 2,785,000 1,871,500 242,200 2,663,000 1,615,500 233,500 Find the rate per cent of increase or decrease. 3. A dealer buys goods at a discount of 40 % off the list price, and sells them at 2 % below the list price. What per cent of profit does he make? 216 ADVANCED BOOK OF ARITHMETIC 4. Eggs are bought at the rate of 5 for 4^, and sold at the rate of 4 for 5^. What per cent of profit is made? 5. A lot is sold for $1560 at a profit of $120. Find the rate per cent of profit. 6. Meat is sold at 18^ per pound at a profit of 20%. Find the cost price per pound. 7. If the butcher has to pay 1^ per pound more for the meat, how must he sell it to make a profit of 25 %? 8. A piano is sold for $470 at a loss of 6%. What would the gain per cent have been if the piano had been sold for $520? 9. A tradesman buys at a discount of 10%, and sells at an advance of 15 % on the nominal cost price. Find his rate per cent of profit. 10. A book costs the publisher 60^ for printing and publishing. At what price should he sell the book in order that he may make a profit of 20%, after paying the author 10% on the selling price? 11. What should be the selling price of an article which costs $15, so that a profit of 20 % may be made after giv- ing the dealer a discount of 10 % ? 12. A tradesman marks his goods at 25 % above cost, but allows the customer 6 % discount. What per cent of profit does he make ? 13. Tea is sold at 60^ per pound at a profit of 33J%. If the total gain is $15, how much tea is sold ? 14. A man buys a house for $4000 which he rents for $40 per month; his taxes are 3 % on a valuation of $ 3000. What per cent does his money yield ? 15. A merchant marks his goods 20 % above cost. What discount does he give if he sells at cost ? INTEREST 217 INTEREST Example 1. Find the interest on $ 670 at 5% from Jan. 14 to Aug. 10. . MO. DA. SOLUTION. $670 Aug. 10 = 8 10 .05 Jan. 14 = 1 14 6 mo. = of 1 yr. 2)133 50 = int. for 1 yr. 6 26 16.75 = int. for 6 mo. 20 da. = 1 of 6 mo. 1.861 = int. for 20 da. 5 da. = I of 20 da. .465 = int. for 5 da. 1 da. = | of 5 da. .093 = int. for 1 da. $19.17 = int. for 6 mo. 6 da. EXERCISE 116 Find the interest and amount of : 1. 1728 for 1 yr. 6 mo. at 5%. 2. $670 for 1 yr. 6 mo. at 7 %. 3. $1260 for 1 yr. 3 mo. at 8%. 4. $385 for 1 yr. 4 mo. 12 da. at 1%. 5. $2750 for 1 yr. 8 mo. at 3 %. 6. $3345 for 1 yr. 4 mo. at 6%. 7. $783 for 1 yr. 1 mo. 10 da. at 4 %. 8. $597 for 1 yr. 4 mo. 24 da. at 5 %. 9. $3000 for 3 mo. 6 da. at 7%. 10. $940 for 1 yr. 4 mo. at 3%. 11. $1800 for 1 mo. 15 da. at 4%. 12. $ 2100 for 2 yr. 9 mo. at 4%. 13. $960 for 8 mo. 17 da. at 1%. 14. $2911.25 for 1 yr. 7 mo. 16 da. at 4%. 15. $1857 for 1 yr. 5 mo. 18 da. at 5%. 16. $2775 from May 1 to Dec. 19 at 4 %. 218 ADVANCED BOOK OF ARITHMETIC 17. $1770 from Jan. 10 to Oct. 5 at 4J %. 18. $1975.14 from Feb. 8 to Nov. 1 at 19. $1218 from March 6 to Nov. 1 at 20. $1788 from Feb. 14 to Dec. 20 at 7-| %. EXACT INTEREST Interest reckoned on the basis of 365 days to the year is called exact interest. Exact interest is used by the United States government and sometimes in business transactions. Example. Find the exact interest on $2384.50 from Jan. 12 to July 5 at 5%. SOLUTION. From Jan. 12 to July 5 there are (19 -f 28 + 31 + 30 + 31 + 30 + 5). days = 174 days. $2384.50 x .05 x'if = exact interest. $2384.50 x. 05x174 - - = $56.84, nearly. obo EXERCISE 117 Find the exact interest on : 1. $913 from Jan. 4 to Feb. 4 at 5%. 2. $731.11 from Jan. 14 to Jan. 28 at 7%. 3. $52.50 from Jan. 1 to April 28 at 7 %. 4. $2745 from Feb. 1 to April 6 at 5 %. 5. $1095.80 from March 6 to June 7 at 5%. 6. $1911.17 from March 1 to May 11 at 7 %. 7. $1464.98 from Jan. 4 to May 30 at 6 %. 8. $10565.65 from May 13 to June 25 at 4 % 9. $834 from Feb. 5 to July 12 at 11 %. 10. $3561.50 for 81 da. at 5%. INVERSE QUESTIONS IN INTEREST 219 INVERSE QUESTIONS IN INTEREST Example l. What principal will produce $78.75 interest in 75 days at 1\%1 Let x denote the principal. .. $# will produce 178.75 x -y~- in 1 year. = 78.75 x Example 2. In what time will $840 produce $57.40 in- terest at 5 % ? Int. on $840 for 1 yr. at 5 % = $42. $57.40 11 of 1 yr. = 11 of 12 mo. = 4f mo. I of a mo. = | of 30 da. = 12 da. The time is 1 yr. 4 mo. 12 da. Example 3. At what rate percent will $720 produce $42.50 interest in 1 yr. 2 mo. 5 da. ? $720 will produce $42.50 -*- (1 + 1 + ^|^) in 1 yr. $42 50 $ 1 will produce -in- -*-(!++ 3 to) in * J r - $100 will produce 100 x'^-(l + H si o) inl J r - = * 5 - .. the rate is 5%. Example 4. What principal will amount to $136.27 in 1 yr. 3 mo. 15 da. at 5 %? The interest on $1 for 1 yr. 3 mo. 15 da. is $.0645|. .-. the amount of $1 for 1 yr. 3 mo. 15 da. is $1.0645f. . \ the number of dollars in principal = $136.27 -*- $1.0645f = $128, nearly. 220 ADVANCED BOOK OF ARITHMETIC EXERCISE 118 What principal will produce: 1. $60 in 11 yr. at 8%? 2. $120 in 2yr. at 5 % ? 3. 1135 in 1 yr. 6 mo. at 9%? 4. $36 in 3 yr. at 5%? 5. $144 in 1 yr. 4 mo. at 4J%? 6. $12 in 1 yr. at 4%? 7. $21 inl yr. at 3| %? 8. $84 in 3yr. 6 mo. at 3%? 9. $16.90 in 2 yr. 2 mo. at 4 % ? 10. $ 42 in 2 yr. 4 mo. at 4 % ? 11. $25.50 in 6 mo. at 5 %? 12. $5.40 in 4 mo. at 5 %? 13. $6.75 in 9 mo. at 3%? EXERCISE 119 In what time will: 1. $1088.75 produce $87.10 interest at 8 % ? 2. $144 produce $21.60 interest at 5 % ? 3. $215 produce $6.45 interest at 5 % ? 4. $ 1160 produce $278.40 interest at 6 % ? 5. $810 produce $56.70 interest at 7 % ? 6. $312.50 produce $43.75 interest at 8 % ? 7. $2220 produce $216.45 interest at S%? 8. $1400 produce $78.75 interest at 5%? 9. $480 produce $85.50 interest at 9| % ? 10. $3835 produce $345.15 interest at 8%? 11. $1380 produce $88.55 interest at 3|%? 12. $5400 produce $267.75 interest at 7 % ? 13. $7630 produce $1335.25 interest at 6 % ? INVERSE QUESTIONS IN INTEREST 221 EXERCISE 120 Find the rate per cent when the interest on 1. $750 for 1 yr. is $45. 2. 1928 for 1 yr. is 1 64. 96. 3. $880 for 1| yr. is $79.20. 4. $945 for 6 mo. is $37.80. 5. $828 for 8 mo. is $38. 64. 6. $1200 for 1 yr. 3 mo. is $90. 7. $1800 for 9 mo. is $67.50. 8. $2400 for 8 mo. is $64. 9. $2500 for 9 mo. 18 da. is $100. 10. $3000 for 7 mo. 12 da. is $90. 11. $3750 for 4 mo. 15 da. is $112.50 12. $2754 for 2 mo. 20 da. is $55.08. 13. $4846 for 6 mo. 20 da. is $121.15. 14. $1440 for 7 mo. 10 da. is $52.80. Find the rate per cent when 15. $1080 amounts to $1123.20 in 8 mo. 16. $1200 amounts to $1270 in 10 mo. 17. $1600 amounts to $1640 in 6 mo. 18. $2460 amounts to $2574.80 in 9 mo. 10 da. 19. $92 amounts to $102.12 in 2 yr. 20. $324 amounts to $333.72 in 8 mo. EXERCISE 121 What principal will amount to 1. $840 in 1 yr. at 5% ? 4. $903 in 1| yr. at 5 % ? 2. $749 in 1 yr. at 7% ? 5. $414 in 3 yr. at 5% ? 3. $645 in 1 yr. at 7-| % ? 6. $255.30 in 2 yr. at 222 ADVANCED BOOK OF ARITHMETIC 7. $12,540.45 in 2 yr. 3 mo. at 4 %? 8. $168.35 in 7 mo. 6 da. at ( % 9. $618.67 in 1 yr. 4 mo. at 5%> 10. $646.80 in 8 mo. at 4%? 11. $776.07 in 1 yr. 1 mo. at 4| %? 12. $481.50 inlyr. at 7 %? 13. $432.55 in 11 mo. at 8 %? 14. $282.75 in 2 yr. 4 mo. 15 da. at 7| %? 15. $2090.07 in 1 yr. 1 mo. 15 da. at 8 %? 16. $2067.75 in 1 yr. 5 mo. at 10|%? 17. $268.28 in 1 yr. 7 mo. at 6%? 18. $254.25 in 7 mo. 15 da. at 9|%? 19. $25,346.25 in 1 yr. 3 mo. at 10%? 20. $843.70 in 8 mo. 3 da. at 7|%? EXERCISE 122 REVIEW 1. Find the interest on $4000 for 13 mo. 2 da. at 9%. 2. Find the interest on $256.30 for 4 mo. 9 da. at 1%. 3. Find the interest on $30.85 for 11 mo. 6 da. at 5%. 4. Find the interest on $653 for 2 mo. 16 da. at 4%. 5. Find the interest on $2105.60 for 84 da. at 5%. 6. Find the amount of $805 for 10 mo. at 8%. 7. Find the amount of $507 for 1 yr. 12 da. at 8%. 8. What principal will produce $20.83 interest in 5 mo. at 5%? 9. What principal will produce $17.50 interest in 9 da. at 5%? 10. Find the rate of interest when $500 produces $2.92 interest in 1 mo. REVIEW 223 11. Find the rate of interest when $250 produces 17.30 in 5 mo. 12. How much must I invest at 5% interest to have an annual income of $1200 from my investment ? 13. A man buys a house and lot and rents it for $40 a month. Taxes and insurance cost him $120 a year. If his net receipts give him a profit of 6% on his investment, find the cost of the house and lot. 14. For how long a time must $3000 be loaned at 5% to produce $20 interest? 15. If I borrow $2400 at 1% interest and pay in princi- pal and interest $2456, how long did I keep the money? 16. For how long a time must a sum of money be loaned at simple interest at 8% to produce in interest -1 of itself ? 17. For how long a time must a sum of money be loaned at simple interest at 6% to produce in interest jV of itself? 18. Find the exact interest of $1200 for 292 da. at 5%. 19. Find the exact interest of $7300 for 146 da. at 7%. 20. The exact interest of $10,800 at 5% is $324. Find the time. 21. In what time will $260 amount to $262.60 at 5% ? 22. What sum must be deposited in a savings bank which pays 3-|% interest to produce semiannually $8.75? 23. A man deposits his money in two banks. In one bank he has $572 which pays 3^%. The other bank gives 4% interest. If he receives as interest the same amount from both banks, how much money has he all together? 24. If I invest half my money at 6% and the remainder at 4%, and derive an income of $650 annually, how much money have I invested ? 224 ADVANCED BOOK OF ARITHMETIC REVIEW QUESTIONS 1. Define principal, rate, per cent, interest, amount. 2. How does exact interest differ from interest accord- ing to the common use of the term? What is the dis- tinction between simple interest and annual interest ? 3. How do you find the interest of a sum of money at a given rate and for a given time ? 4. If you were given the interest, the rate, and the time, how would you find the principal ? 5. Given the principal, interest, and time, how would you find the rate per cent? How would you find the rate ? 6. Given the principal, the rate, and the interest, how would you find the time ? 7. Given the principal, the amount, and the rate, how would you find the time ? 8. Given the principal, the amount, and the time, how would you find the rate ? 9. Given the amount, the rate, and the time, how would you find the principal ? 10. If you knew the interest of a sum of money for a given time at 6%, how would you find the interest for the same sum for the same time at 5% ? at 4% ? at 8% ? 11. If you knew the interest at 4%, how would you find the interest of the same sum at 1% ? at 3% ? at 5%|? at 3i%? 12. If }^ou were given the interest of a sum of money for a number of days, how would you determine from this the exact interest of the same sum for the same number of days? PROMISSORY NOTES 225 PROMISSORY NOTES A written promise by one person to pay another person on demand, or after a specified time, a sum of money is called a promissory note. The following are promissory notes written in standard form : Galveston, Texas, ?Wcvi&h 7, 1907. ^t&v dat& c/ promise to pay to the order of --&/iis&& fvwndA&d, &LqhX/u ________ Dollars I / 100 at ______ t/i& ofi/i&t ofau>w&C joa/wk ________________________ Value received, w-UJ^ imteAs&^t at 6 %. No. Dallas, Texas, Tn^eA V, 1907. dewuwvcL c/ promise to pay to the order of Dollars at _____________ tfi Value received, w(A iM,t&^&&t at 7 % . Due. The first of the above promissory notes is called a time note; the second is called a demand note. Q 226 ADVANCED BOOK OF ARITHMETIC The person who promises to pay is called the maker. John Mosley is the maker of the first note above. The person to whom the money is to be paid is called the payee. The person who has legal possession of a note is called its holder. The sum specified in a note is called its face. A time note is legally due on the date indicated. In some states 3 days more than are indicated in the note are allowed before the note is legally due. These days are called days of grace. The day on which a note is legally due is called the day of maturity. A note made payable to the order of a person, or a note made payable to the bearer is negotiable, i.e. it may be transferred from one person to another person. A note made payable to the payee only is non-negotiable. When a note payable to the order of the payee is trans- ferred, every holder before parting with it must indorse it, i.e. write his name on the back of it. Every indorser thus becomes liable for the payment of the note, if the maker fails to pay it. The holder in whose possession the note is at maturity presents it to the maker for pay- ment. If the maker refuses to pay it, the holder engages a Notary Public to give to the indorser, or indorsers, a written notice of its non-payment. This notice is called a protest. A protest must be sent on the date of maturity ; otherwise the indorsers are not held responsible for the payment of the note. An indorser who writes over his signature the words ivithout recourse is not held responsible for the payment of the note. A note made payable to the bearer is negotiable without indorsement. In some states a note must contain the words value received in order to be legal. BANK DISCOUNT 227 If the words with interest are not in a note, no interest is charged. If, however, the note is not paid on the date of maturity, interest at the legal rate may be charged. If a note contains the words with interest, and no rate is speci- fied, it is then understood that the note bears the rate of interest usually charged in the state where it is made. When the time of payment is indicated in months, cal- endar months are understood. A note drawn March 6, and payable two months after date, matures on May 6 in states where days of grace are not allowed, and on May 9 in states where days of grace are allowed. About one half of the states and territories allow 3 days of grace. BANK DISCOUNT New Orleans, La., <$c,6.. /P, 1903. W.OO. c/t/^y cLaAf& a^tsA* cla,t& c/ promise to pay to the order of ________________ fotefiA, (^o-an _________________ fauncU&ci fvfty __________________________ Dollars Value received. No. 33. Due ftfalt 15 1 18, The above time note is negotiable when indorsed. Supposing the payee, Joseph Coan, needs money, he can sell the note to a bank. The sum the bank gives him for the note is called the proceeds of the note. The differ- ence between the proceeds and the face of the note is called the bank discount. 228 ADVANCED BOOK OF ARITHMETIC The bank discount is always a rate per cent of the value of the note 011 its day of maturity, reckoned from the date of the sale of the note to the day of maturity. The bank discount is then the interest on the maturity value of the note computed from the date of discount to the date of maturity. This time is called the term of discount. The maturity value of the note minus the bank discount is the proceeds of the note. From the computer's point of view the essential features of a note are the face, maturity value, date of drawing, date of sale, or date of discount, rate of interest the note bears, rate of interest charged, known as rate of discount, and date of maturity. BANKERS' INTEREST Banks charge interest for the exact number of days between dates, allowing 30 days to a month. Banks usually draw notes for 30, 60, or 90 days. Example. Find the discount and the proceeds of the above note, if it was discounted at 8%, March 1, 1903. SOLUTION. The bank charges discount from March 1, to April 18. From March 1 to April 18 is 48 days. $350. = maturity value. .08 28.00 = int. for 1 yr. 45 da. = of 1 yr. 3.50 = int. for 45 da. 3 da. = T ^ of 45 da. .23 = int. for 3 da. $3.73 = int. for 48 da. = bank discount. $350 - $3.73 = $346.27 = proceeds of the note. COMPUTING DISCOUNT 229 EXERCISE 123 Find the bank discount and the proceeds of the follow- ing indicated notes, allowing 3 days of grace in examples 1, 2, 10, 11, 12, 13, and no grace in the remaining: DATE TIME FACE Dis- RATE OF C'TI SD DISC'T 1. Jan. 12, 60 da., $600, Feb. 13, 8%. 2. July 4, 60 da., $800, Aug. 3, 8%. 3. Mar. 3, 90 da., $500, Mar. 3, 6%. 4. April 5, 60 da., $700, April 5, 6%. 5. May 7, 60 da., $600, May 10, 10%. 6. May 9, 20 da., $900, May 24, 8%. 7. May 30, 90 da., $750, July 1, 6%. 8. June 5, 60 da., $450, July 5, 6%. 9. Aug. 11, 30 da., $800, Aug. 11, 6%. 10. Sept. 9, 30 da., $350, Sept. 12, 10%. 11. Oct. 4, 60 da., $800, Oct. 7, 10%. 12. Oct. 14, 60 da., $500, Nov. 16, 12%. 13. Nov. 10, 90 da., $600, Nov. 25, 8%. 14. Dec. 4, 90 da., $750, Feb. 5, 6%. 15. Jan. 10, 45 da., $650, Feb. 9, 7%. 16. Jan. 5, 60 da., $850, Feb. 5, 9%. 17. Jan. 30, 75 da., $950, Feb. 28, 9%. 18. July 10, 3 mo., $380, July 12, 9%. COMPUTING DISCOUNT ON INTEREST-BEARING NOTES Find the bank discount and the proceeds of a 90-day note for $250, dated Portland, Me., June 9, 1907, bearing interest at 6%, and discounted July 8, 1907, at 8%. 230 ADVANCED BOOK OF ARITHMETIC Step 1. Find the maturity value of the note. Maine allows 3 days of grace. Hence, the interest will be com- puted for 93 days. $250 ~ 3.87= interest at 6% for 93 days. $253.87 = maturity value of the note. Step 2. Find the term of discount (exact number of days from July 8, to Sept. 10, the date of maturity). Step 3. Find the bank discount. This is reckoned on the maturity value of the note. Int. on $253.87 for 64 days at 8% =$3. 61. $254.87 - $ 3.61 = $250.26, proceeds of note. EXERCISE 124 Find the discount and the proceeds of the following in- dicated notes, allowing 3 days of grace in examples 2, 3, 6, 9, 10, 13, and no grace in the remaining : FACE DATE TIME J.W1.J. K, OF INT. J.V.A. J. Hi \J DISC'T DISC'T 1. $350, Jan. 1, 45 da., 10%, 12%, Feb. 1. 2. $395, Jan. 10, 3 mo., 6%, 10%, Jan. 20. 3. $450, Feb. 1, 30 da., 6%, 8%, Feb. 18. 4. $600, Mar. 1, 60 da., 8%, 12%, Mar. 31. 5. $500, Apr. 2, 60 da., 6%, 8%, Apr. 17. 6. $900, Apr. 10, 30 da., 6%, 8%, Apr. 10. 7. $1000, Apr. 15, 60 da., 8%, 10%, Apr. 15. 8. $750, May 4, 60 da., 6%, 6%, May 5. 9. $800, July 10, 30 da., 6%, 8%, July 26. 10. $400, Aug. 15, 60 da., 6%, 6%, Aug. 15. 11. $850, Nov. 11, 90 da., 1%, 10%, Nov. 12. 12. $900, Dec. 12, 45 da., 6%, 6%, Jan. 13. 13. $1200, Dec. 5, 3 mo., 6%, 10%, Jan. 15. COMMERCIAL DISCOUNTS 231 COMMERCIAL DISCOUNTS Commercial, or Trade Discount is a reduction from the list price of goods, or the amount of a bill. If two or more discounts are allowed, the first is reckoned on the list price, the next on the remainder after deducting the first discount, the third is reckoned on the second remainder, etc. Example. Find the cost price of a bill of goods, if the list price is $690 and discounts of 25%, 10%, and 5% are allowed. SOLUTION. $690 .$172.50 = 25% of $600. $517.50 .= first remainder. $ 51.75 = 10% of $517.50. $465.75 = second remainder. $ 23.287 = 5% of $465.75. $442.46 = cost of the goods. EXERCISE 125 1. Find the cost if the list price is $350, and the discount 20%. 2. Find the cost when the list price is $823, and the discount 12|%. 3. What is the cash value of a bill of goods listed at $937.50 when a discount of 16|% is given? 4. A suit of clothes is marked $17.50, and is sold for $12.00. What is the rate of discount ? 5. A bookseller buys 60 books marked $2.10 each at a discount of 16|%, and sells them at the marked price. What is his gain per cent, and how much profit does he make on the sale of the books? 232 ADVANCED BOOK OF ARITHMETIC 6. A dealer buys goods at a discount of 16| % and sells them at 5% above the list price. What is his gain per cent? 7. Find the cost price in each case, if the list prices and rates of discount are as follows : COST DISCOUNTS (a) $700 20%, 121%, 10% (5) $7000 10%, 20%, 30% () 97330 bu. wheat at 7fd. In examples 5 and 6 take 1 = $4.80. 244 ADVANCED BOOK Ol? ARITHMETIC FOREIGN EXCHANGE The settling of outstanding indebtedness between par- ties in different parts of this country by means of drafts is known as domestic exchange. The settling of debts between parties in this country and parties in foreign countries by means of drafts is called foreign exchange. Foreign drafts are known as bills of exchange. These are issued in duplicate, i.e. two are drawn exactly alike. The two bills are sent by different mails. As soon as one is paid, the other is void. Usually exchange is a little above or a little below the par of exchange, i.e. intrinsic value of foreign coins. Exchange on Great Britain is quoted at the number of dollars to i s called the radical sign. It is a degenerate form of the first letter of the word radix, the Latin word for root. The exponent |- is also used as a sign for the square root. V49, 49* are the two ways of indicating the same process, namely, the extraction of the square root of 49. The number written under the radical sign is called the radicand. Students should fix firmly in mind : 10 2 = 100 40 2 = 1600 70 2 = 4900 20 2 = 400 50 2 = 2500 80 2 = 6400 30 2 = 900 60 2 = 3600 90 2 = 8100 Example l. What is the square root of 5329? SOLUTION. By trial the square root of 5329 is more than 70 and less than 80. .'.5339 = 4900 +140 x+x* Subtract 4900, and get, 140 x + x 2 = 439. .'. (140 -f- z> = 429. Since 140 is contained in 429 3 times, try 3 as a value of x. (140 + 3)3 = 429. .-. V5329 = 70+ 3=73. 260 ADVANCED BOOK ^OF ARITHMETIC Example 2. V9025 = ? SOLUTION. V9025 = 90 + x. .-. 8025 = 8100 + 180 x + x*. .-. 180 x + a^ = 925. .-. (180 + s> = 925. Since 180 is contained in 925 5 times, try 5 for the next figure of the root. (180 + 5)5 = 925. .-. V9025 = 90+5 = 95. In practice, the work is contracted as follows : Beginning at the decimal point, point off the figures of the number in periods of two figures each. By trial find 9 5 the greatest digit whose square is contained 90.25 in the number denoted by the period to the 81 left. Write it as the first figure of the root, 185 9 25 and write also its square. Subtract the latter _9_2t> from the period to the left and bring down the next period. Double the part of the root just found for trial divisor. Find next the number of times the trial divisor is contained in the number denoted by the remainder and the period brought down. Write this result in the quotient and in the divisor and then multiply. EXERCISE 142 Find the square root of : 1. 169. 7. 2809. 13. 7056. 19. 7921. 2. 441. 8. 3844. 14. 7569. 20. 6084. 3. 625. 9. 4225. 15. 8464. 21. 4761. 4. 961. 10. 5329. 16. 9216. 22. 3481. 5. 1024. 11. 5776. 17. 9604. 23. 2401. 6. 1849. 12. 6724. 18. 9801. 24. 3364. EVOLUTION 261 Example. 3 5 12 74 49 9 65 3 74 3 25 4949 hence \/127449 = ? SOLUTION. Divide the figures of the number into periods of two as in the pre- vious exercises. Then proceed to extract the square root of the number denoted by the two periods to the left. The answer is obviously 350 + some number. Let x represent this number; 350 + x = V127449. .-.(350 + *) 2 = 127449. .-. 122500 + 700 x + x* = 127449. .-. 700 x+x* = 4949. ... (700 + x)x = 4949. Since 700 is contained in 4900 7 times, try 7 as a value of x. (700 + 7)7 = 4949. .-. V127449 = 350 + 7 = 357. Notice the trial divisor is twice the part of the root found. Therefore in a problem in square root where the radicand is an integer consisting of five or six figures, proceed in exactly the same way as has been done in a problem consisting of four figures. Beginning once more, the solution in its contracted form stands as follows : 357 12 74 49 9 The trial divisor is always twice the part of the root already found. 65 707 3 74 325 4949 4949 262 ADVANCED BOOK OF ARITHMETIC EXERCISE 143 Extract the square root of: 1. 100,489. 7. 229,441. 13. 474,721. 2. 110,224. 8. 277,729. 14. 501,264. 3. 120,409. 9. 310,249. 15. 654,481. 4. 171,396. 10. 354,025. 16. 772,641. 5. 190,096. 11. 391,876. 17. 819,025. 6. 199,809. 12. 456,976. 18. 826,281. Memorize : (.I) 2 = .01 (.5) 2 =.25 (.9) 2 =.81 (.2)2 =.04 (.6)2 =.36 (.ll) 2 =.0121 (.3) 2 =.09 (.7)2 = .49 (.12)2 =.0144 (.4)2 = .16 (.S) 2 =.64 (.Ol) 2 = .0001 To extract the square root of a decimal, begin at the deci- mal point, and proceeding to the right, point off the figures in periods of two ; next proceed as if the number were an integer. Thus, in taking the square root of .0225, first point off in periods of two figures each. This gives .02 25. Next extract the root of the number de- noted by the figures 225. The result is 15. Hence, the required root is .15. To extract the square root of a number part integer and part decimal, begin at the decimal point, and proceeding to the left, point off the integral part in periods of two figures each; next point off the decimal part in periods of two figures each, beginning at the decimal point. If there are not enough figures in the decimal part to make an exact number of periods, annex a cipher or as many ciphers as are necessary to make the required number of periods. EVOLUTION 263 Example. Extract i\ 1. 3 3 8 4 1. 70 00 00 00 00 1. ie square root of 1.7. Double 1 for the first trial divi- sor. Double 13 for the next trial divisor. Then find the next figure of the root is 0. Write it in the root and in the trial divisor. Then annex two more ciphers, and find that the next figure of the root is 3, and so on. 23 70 69 2603 26068 26076 10000 7809 219100 208544 4 1055600 1043056 EXERCISE 144 Extract the square root of : 1. .150932. 2. .246016. 3. .3448. 4. .2909. 5. .2632. 6. .4616. 7. .5319. 8. .61575. 9. .784. 10. .083. 11. .062. 12. .0037. To extract the square root of a fraction when its numer- ator and denominator are perfect squares is a simple matter. Thus, the square root of || is f; the square root of IW, or |4, is |, or 1J-. b4:~ 84' o" O To get the square root of a fraction, take the square root of the numerator, and the square root of the denominator, and then write the former result for numerator and the latter result for denominator. The fraction thus found is the required square root. Example 1. What is the square root of -|| ? SOLUTION. Vl7 = 4.123; V36 = 6. 264 ADVANCED BOOK OF ARITHMETIC Example 2. What is the square root of || ? SOLUTION. V23 = 4.796; This is a roundabout way to take the square root of |~|. A shorter and better way is to reduce the fraction to an equivalent decimal, and then to extract the square root of this decimal. To extract the square root of a fraction whose denomi- nator is not a perfect square, reduce the fraction to an equivalent decimal and then extract the square root of this decimal. EXERCISE 145 Extract, to three decimal figures, the square root of : 1. 1.2. 4. 5.2. 7. 3i. 10. 41 13. f. O 4 O 2. 4.25. 5. 3.3. 8. 9J. 11. 2|. 14. f. 3. 1.1. 6. 5|. 9. If. 12. T V 15. T V EXERCISE 146 PROBLEMS INVOLVING SQUARE ROOT 1. The area of a square field is 1 A. Find the length in yards of one of its sides. 2. The area of a square field is 12 A. Find the length in yards of one of its sides. 3. The dimensions of a rectangle are 289 yd. and 196 yd. Find the side of an equivalent square. 4. The dimensions of a rectangle are 1| mi. and .7 mi. Find, correct to four decimal figures, the side of an equiva- lent square. EVOLUTION 265 5. Find in rods the perimeter of a square field whose area is ^ of a square mile. 6. The area of a rectangle whose length is twice its breadth is 10 A. Find its dimensions in yards. HINT. Draw a diagram ; divide it into two equal parts by a line paral- lel to its width. Notice what each part is. 7. The area of a rectangle whose length is three times its width is 20 A. Find its dimensions in yards. 8. A square and a rectangle have the same area, namely 40 A. If the length of the rectangle is twice its width, find in rods the difference between their perimeters. The side of a right triangle opposite the right angle is called the hypothenuse. The other two sides are called the legs of the right triangle. One of the legs is called the base of the right triangle, and the other leg is called the altitude of the right triangle. In a right triangle the square on the hypothenuse is equal to the sum of the squares on the two legs. This is the famous Pythagorean Theorem. Designate the sides of the right triangle AB by the letters #, 5, c. (a -f 5) 2 = a 2 + 52 + 2 ab. But (a -f- 6) 2 = c 2 -f- 4 triangles, each having for its base and altitude a and b. AMNR = c 2 + 4 x J ab -f 5 2 + 2 ab = c 2 + 2 ab. 266 ADVANCED BOOK > OF ARITHMETIC Example. In a right triangle the legs are 7 and 24. Find the hypothenuse. SOLUTION. a 2 + b* = c 2 . .-.49 + 576 = (! At compound interest, The amount of $1 at 4% for 20 yr. is f (1.04) 20 . The amount of $1 at 5% for 20 yr. is 9(1.05)*. The amount of 91 at 6% for 20 yr. is f (1.06) 20 . The labor of raising 1.06 to the 20th power is considerable. A student who has a working knowledge of logarithms can do this by the aid of a table of logarithms in less than a minute. 294 ADVANCED BOOK OF ARITHMETIC The following table gives the amount of pound interest : at com- YRS. 2% 2*% 3% 4% 5% 6% 1 1.020000 1.025000 1.030000 1.040000 1.050000 1.060000 2 1.040400 1.050625 1.060900 1.081600 1.102500 1.123600 3 1.061208 1.076891 1.092727 1.124864 1.157625 1.191016 4 1.082432 1.103813 1.125509 1.169859 1.215506 1.262477 5 1.104081 1.131408 1.159274 1.216653 1.276282 1.338226 6 1.126162 1.159693 1.194052 1.265319 1.340096 1.418519 7 1.148686 1.188686 1.229874 1.315932 1.407100 1.503630 8 1.171659 1.218403 1.266770 1.368569 1.477455 1.593848 9 1.195093 1.248863 1.304773 1.423312 1.551328 1.689479 10 1.218994 1.280085 1.343916 1.480244 1.628895 1.790848 11 1.243374 1.312087 1.384234 1.539454 1.710339 1.898299 12 1.268242 1.344889 1.425761 1.601032 1.795856 2.012196 13 1.293607 1.378511 1.468534 1.665073 1.885649 2.132928 14 1.319479 1.412974 1.512590 1.731676 1.979932 2.260904 15 1.345868 1.448298 1.557967 1.800943 2.078928 2.396558 16 1.372786 1.484506 1.604706 1.872981 2.182875 2.540352 17 1.400241 1.521618 1.652847 1.947900 2.292018 2.692773 18 1.428246 1.559659 1.702433 2.025817 2.406619 2.854339 19 1.456811 1.598650 1.753506 2.106849 2.526950 3.025599 20 1.485947 1.638616 1.806111 2.191123 2.653298 3.207136 Example l. Find the amount of 12500 at compound interest for 12 yr. at 5 % . SOLUTION, a = $2500 (1.05) 12 = $2500 x 1.T95856 = $4489.64. Example 2. Find the amount of $5000 for 8 yr. at 4 % compound interest, the interest being compounded semi- annually. SOLUTION. $5000 x (1.02) 16 = $5000 x 1.372786 = $6863.93. \ MISCELLANEOUS TOPICS 295 EXERCISE 161 With the aid of the above table, find the amount at compound interest of : 1. $2000 for 8 yr.at 4%. 3. $5000 for 12 yr. at 4%. 2. $3000 for 10 yr. at 3%. 4. $6000 for 10 yr. at 5%. 5. $8000 for 8 yr. at 5%, interest compounded semi- annually. 6. $2250 for 6 yr. at; 4%, interest compounded semi- annually. MISCELLANEOUS TOPICS WORK AND TIME Example l. A can do a piece of work in 6 da., and B can do the same piece of work in 8 da. In what time can A and B do the work together? ANALYTICAL SOLUTION. A does l of the work in 1 da. B does | of the work in 1 da. .-. A and B together do (l -f |) of the work in 1 da. /. A and B do ^ of the work in 1 da. .*. A and B do ^ of the work in ^ of 1 da. .-. A and B do || of the work in ^ of 1 da. .. A and B do the work in 3^ da. Example 2. A cistern has three pipes. The first pipe fills the cistern in 12 hr., the second, in 15 hr., and the third empties it in 10 hr. In what time will the cistern be filled, if all three pipes run together, and the cistern is empty when the pipes start running? ANALYTICAL SOLUTION. The first pipe fills -^ of the cistern in 1 hr. The second pipe fills ^ of the cistern in 1 hr. The third pipe empties ^ of the cistern in 1 hr. 296 ADVANCED BOOK OF ARITHMETIC /. the three pipes fill CiV + T5~~lV) ^ ^ ne cistern in Ihr. .*. the three pipes fill ^ of the cistern in 1 hr. /. the three pipes fill |- of the cistern in 20 hr. .*. the cistern is filled in 20 hr. Example 3. A and B do a piece of work in 3^ hr. ; A alone, in 7 hr. In what time does B do the work? SOLUTION. A and B together do - of the work in 1 hr. ; i.e. A and B together do -|J of the work in 1 hr. But A alone does ^ of the work in 1 hr. /. B alone does (^| j-) of the work in 1 hr. /. B alone does -| of the work in 1 hr. /. B does the work in 8 hr. Example 4. A and B can mow a field in 21 hr. A can do | as much work as B. Find the time in which each 4 <, does the work. SOLUTION. Represent B's work by the rectangle mnst. Represent A's work by the rec- tangle pqxy. A will do in J hr. the work A and B do together in 1 hr. /. A will do in Jhr. x 21 the work A and B do together in 21 hr. .'. A will do in 49 hr. the work A and B do together in 21 hr. B will do in | hr. the work A and B do together in 1 hr. .*. B will do in | hr. x 21 the work A and B do together in 21 hr. m V /t-k a? MISCELLANEOUS TOPICS 297 /. B will do in 36 1 hr. the work A and B do together in 21 hr. A's time, 49 hr.; B's time, 36| hr. EXERCISE 162 1. If a person can do a piece of work in 7 da., what is his day's work ? If he can do the work in 4^ da., what is his day's work ? 2. If B can copy a manuscript in 5|hr., how much can he copy in 1 hr. ? 3. John travels T 5 g of the distance between two cities in 1 hr. How many hours will it take him to travel the remainder of the distance ? 4. A can do a piece of work in 3 hr., B in 4 hr., and C in 5 hr. How long will it take the three working together to do the work ? 5. A can do a piece of work in 4 hr., B in 5 hr., and A, B, and C together in 1J hr. How long would it take C alone to do the work ? 6. A, B, and C can do a piece of work in 6, 8, and 10 da., respectively. If they begin the work together, what part of the work remains to be done at the end of the second day ? 7. A, B, and C can build a fence in 10, 15, and 20 hr. re- spectively. They work together for 4 hr., when B quits. In what time can A and C finish the work ? 8. A cistern has two pipes. One can fill it in 20 min., and the other can empty it in 30 min. If the cistern is empty, in what time can it be filled, if both pipes begin to flow at the same instant ? 298 ADVANCED BOOK OF ARITHMETIC MOTION IN THE SAME DIRECTION, OR IN OPPOSITE DIRECTIONS Example 1. A starts to overtake B, who is 100 yd. ahead of him. A travels 11 yd. to B's 9 yd. How far must A travel in order to overtake B ? SOLUTION. A gains on B 2 yd. in every 11 yd. he goes. .'. A gains on B 1 yd. in every 5J yd. he goes. .*. A gains on B 100 yd. in every 550 yd. he goes. Ans. 550 yd. Example 2. A freight train moving at the rate of 18 mi. an hour is 78 mi. ahead of a passenger train moving in the same direction at the rate of 80 mi. an hour. Find the distance the passenger train must run to overtake the freight train. SOLUTION. In 1 hr. the passenger train gains on the freight (30 - 18) mi., i.e. 12 mi. .*. in (78 -f- 12) hr. the passenger train will overtake the freight train, i.e. in 6| hr. In 6^ hr. the passenger train goes 30 mi. x 6-|- = 195 mi. EXERCISE 163 1. Two ships leave New York for Glasgow, one on Mon- day morning at 9 o'clock, and the other on the following morning at 9 o'clock. Their. rates are 15 and 21 miles an hour respectively. How far from New York City will the second ship overtake the first ? 2. Dallas and Galveston are 315 mi. apart. A train leaves Dallas for Galveston at 8 o'clock A.M. at the rate of 30 mi. an hour. At the same time a train leaves Galves- ton for Dallas at the rate of 33 mi. an hour. How far will the trains be from Dallas when they meet ? CIRCULAR MOTION: CLOCKS 299 3. Paris, Texas, is 584 mi. from St. Louis. A passen- ger train leaves Paris for St. Louis at 6.50 P.M. Three hours later a freight train leaves St. Louis for Paris. When and where will they meet, the rates being respec- tively 24 mi. and 16 mi. per hour ? 4. A man walking at the rate of 4 mi. an hour is overtaken by a train 88 yd. long, and is passed in 10 sec. Find the rate of the train. 5. A train going at the rate of 40 mi. an hour passes in 6 sec. a man walking in the same direction at the rate of 4 mi. an hour. What is the length of the train? 6. Two trains start from the same station and travel in the same direction. The first train leaves at 7 A.M., and the second train at 9 A.M. How many miles from the station will the second train overtake the first if the rate of the first train is 30 mi. per hour and the rate of the second train is 45 mi. per hour? CIRCULAR MOTION: CLOCKS Example l. At what time between 4 and 5 o'clock are the hands of a clock together? SOLUTION. At 4 o'clock the hour hand is 20 minute spaces ahead of the minute hand. In 1 hr. the minute hand goes 60 minute spaces. In 1 hr. the hour hand goes 5 minute spaces. .-. ratio of rates of motion of minute hand and of hour hand is 60 : 5, or 12 : 1. 300 ADVANCED BOOK OF ARITHMETIC .*. in 12 min. the minute hand gains on the hour hand 11 minute spaces. .'.in I T *J min. the minute hand gains on the hour hand 1 minute space. .'.in l^y min. x 20 the minute hand gains on the hour hand 20 minute spaces. But IJy min. x 20 = 21^ min. /. the hands are together 21y 9 ^ min. after 4 o'clock. Example 2. At what time after 7 o'clock do the hour and minute hands first point in opposite directions ? SOLUTION. They will point in op- posite directions whenever they are 30 minute spaces apart. At 7 o'clock the hour hand is 35 minute spaces ahead of the minute hand. .'. as soon as the minute hand gains on the hour hand 5 minute spaces, they will point in opposite directions. But the minute hand gains on the hour hand 5 minute spaces in l^y min. x 5, i.e. in 5 T 6 T min. (See Example 1.) Ans. 5^\ min. past 7 o'clock. Example 3. When, between 5 and 6 o'clock, will the hour and minute hands be at right angles ? SOLUTION. The hour and min- ute hands will be at right angles when they are 15 minute spaces apart. FlG - i- /. they will be at right angles between 5 and 6 o'clock CIRCULAR MOTION: CLOCKS 301 FIG. 2. when the minute hand gains on the hour hand (25 15) minute spaces, i.e. 10 minute spaces. l^min. x 10 = 1014 min. Ans. 101-J- min. past 5 o'clock. They will also be at right angles when the minute hand gains on the hour hand (25 + 15) minute spaces, i.e. 40 minute spaces. (See Fig. 2.) l^Y min. x 40 = 43 T r T min. the hands will be at right angles to each other at 10^ min. past 5 o'clock and at 43^ min. past 5 o'clock. To the young learner the following suggestions may be of use : To solve a question in circular motion : first, draw a dia- gram showing the things that move ; second, find the ratios of the rates of motion of the things moving ; third, having found the relative rates of motion of the objects, proceed as in a simple exercise involving motion in a straight line. EXERCISE 164 1. What angle do the hour and minute hands of a clock make with each other at 1 o'clock ? at 2 o'clock ? at 3 o'clock? at 4 o'clock? at 5 o'clock? at 6 o'clock? at 12 o'clock ? 2. What angle do the hour and minute hands of a clock make when they point to positions 8 minute spaces apart? 12 minute spaces apart? 19 minute spaces apart? 23 minute spaces apart? 27 minute spaces apart? 50 minute spaces apart? At what time between 1 and 2 o'clock do the hands of a clock make an angle of 36 ? 302 ADVANCED BOOK OF ARITHMETIC 3. How many minute spaces apart do the hands of a watch indicate when they make an angle of 36? 66? 84? 114? 126? 144? 162? 174? 4. Find at what time between the hours of 2 and 3 o'clock the hands of a clock are together ; between 3 and 4 o'clock ; between 5 and 6 o'clock ; between 7 and 8 o'clock; 9 and 10 o'clock ; 10 and 11 o'clock; 11 and 12 o'clock. 5. At what time do the hands of a watch point in opposite directions between (a) 1 and 2 o'clock? (cT> 9 and 10 o'clock? (5) 3 and 4 o'clock? (V) 11 and 12 oclock? 0) 8 and 9 o'clock ? (/) 12 and 1 o'clock ? 6. At what time after 3 o'clock do the hands of a watch first point in opposite directions ? When after 6 o'clock do they first point in opposite directions? When after 10 o'clock ? 7. At what time or times are the hands of a watch at right angles between (a) 2 and 3 o'clock? (6) 3 and 4 o'clock ? ( ^, T \-, and ^ to equivalent fractions having 100 for denominator. 108. Find the difference between " - and 16 .16 109. Reduce 16| to an improper fraction having 16 for a denominator. 110. A rectangular field which is 18 rd. wide contains 6 A. How much will it cost to fence it at 75^ a rod ? 111. What decimal of 4 ft. 2 in. is 9 ft. 6 in. ? 112. Find the least fraction which added to J, |, ^, ^, and |- will make the sum an integer. 113. Divide 27.8 of a yard by .00125 of a foot. 114. 8 cwt. 20 Ib. of sugar cost $41.42. What will 1 T. cost at the same rate ? 115. Find the least length which is a multiple of 1 ft. 3 in., 1 ft. 8 in., 2 ft. 1 in., and 2 ft. 6 in. 116. Twelve tenths of a number equals 42. Find it. 117. Divide 54,218 by 64, using the factors of 64. 118. A train 165 yd. long passes a telegraph pole in 12 sec. Find the rate of the train in miles per hour. 119. A city lot 42 ft. by 120 ft. is sold for $840. At this rate, find the value of 1 A. of land in that city. 120. Find the greatest number which, when divided into 1958 and 2741, will give for remainders 8 and 11 respectively. 121. By buying eggs at 25^ per dozen and selling them at 60^ a score, a dealer makes a profit of $10.01. How many eggs does he sell ? 122. Reduce $$ to its lowest terms. 123. If a sum of money which will pay A's wages for 41| da. will pay B's wages for 55| da., for how long will it pay both ? 312 ADVANCED BOOK OF ARITHMETIC 124. If gold weighs 19.3 times as much as water, and copper 8.9 times as much as water, how much heavier than water is an alloy consisting of 16 parts of gold and 3 of copper ? 125. A rectangular tank is 18 ft. 8| in. long, 11 ft. 3f in. wide, and contains 41 cu. yd., 6 cu. ft., and 34^ cu. in. Find its depth. Find the area of each of its faces. 126. A tennis court is 42 yd. long and 20 yd. wide. It has a walk around it 6 ft. wide. Find the cost of paving the walk at $ 1.25 per square yard. 127. Telegraph poles along a certain railroad are 132 ft. apart. Find the rate of a train, in miles per hour, which passes 18 poles in 24 sec. LONGITUDE AND TIME 128. A train leaves New York City at 9 A.M., Apr. 1, 1903, and arrives in Carson City, Nev., in 109 hr. 15 min. Find the hour of the day, and day of the month, Standard time, that it reaches its destination. 129. The time of mail transit between Chicago and Santa Fe, N. M., is 60 hr. 55 min. " The California Limited " leaves Chicago at 10 P.M. At what time, by the clocks in Santa Fe, should " The California Limited " pass Santa Fe ? 130. The longitude of Cairo, Egypt, is 31 21' E., and the longitude of Savannah, Ga., is 81 5' 30" W. Find the differ- ence in time. 131. The longitude of Toulon is 5 56' E. The time differ- ence between Toulon and Halifax, N. S., is 4 hr. 38 min. 4 sec. Find the longitude of Halifax. 132. The time difference between Toulon and Point Barrow, Alaska, is 10 hr. 48 min. 44 sec. Find the longitude of Point Barrow. 133. The time difference between Osaka and Point Barrow is 19 hr. 26 min. 48 sec. Find the longitude of Osaka. (See previous problem.) MISCELLANEOUS EXAMPLES 313 134. (a) The difference between Standard and local time of Portland, Me., is 19 min. Find the longitude of Portland, Me. (6) The difference between Standard and local time of Fort Wayne, Ind., is 20 min. Find the longitude of Fort Wayne. (c) Cleveland, 0., uses Central time ; the difference between its local and Standard time is 33 min. Find the longitude of Cleveland, 0. PERCENTAGE 135. The total sugar production of the world for the year 1902 was 9,635,000 T. The amount of sugar consumed in the United States the same year was 2,372,000 T. What per cent of the world's production was the amount consumed in the United States? 136. The foreign-born population of New Orleans, according to the census of 1900, was 30,325 ; of this number, 1262 came from England, 4428 from France, 8733 from Germany, 5398 from Ireland. What per cent of the foreign-born population of New Orleans came from England? from France? from Germany ? from Ireland ? 137. The number of Canadians in Detroit, according to the census of 1900, was 25,400; this number was 26.3% of the foreign-born population. Find, correct to 100, the number of foreign-born population of Detroit. 138. A horse is sold, at a loss of 15%, for $127.50. Find the cost of the horse. 139. By selling silk at $1.60 per yard, a dealer makes a profit of 25%. What would the selling price be if he made a profit of 12 \% ? 140. When cloth is sold for $1.04 per yard, a clothier makes a profit of 30%. What would his profit be if he sold the cloth at 96^ per yard ? 141. A wholesale dealer makes a profit of 10% on canned goods. The retail dealer makes a profit of 25%. Find the original cost of canned goods which cost the consumer $11. 2i 314 ADVANCED BOOK OF ARITHMETIC 142. A coal merchant buys coal by the long ton at $ 4.50 a ton, and sells it at the rate of $ 5 a short ton. Find his gain per cent. 143. How much water must be added to a 25% wine mixture to make it a 20% mixture? 144. A sells goods to B at a profit of 20% ; B sells them to C at a profit of 20% on his outlay; C sells them to D for $ 180, thereby losing 16|%. How much did the goods cost A ? 145. A merchant buys goods at 20%, and 10% off list price, and sells them at the list price. Find his per cent of gain. 146. When 20 Ib. of tea are sold for what 22^ Ib. cost, what is the gain per cent ? 147. (a) A vessel contains 31 gal. of wine and 17 gal. of water. What per cent of the mixture is wine and what per cent is water ? (5) How many gallons of water must be added to this mixture to make a mixture containing 60% wine? 148. The following table gives the distances from Atlantic to Pacific ports by the present routes : New York to San Francisco 13,244 mi., nautical New York to Sydney 14,560 mi., nautical Charleston to San Francisco 13,180 mi., nautical Charleston to Valparaiso 8,296 mi., nautical New Orleans to San Francisco .... 13,644 mi., nautical New Orleans to Melbourne 15,535 mi., nautical Galveston to San Francisco 13,826 mi., nautical Galveston to Wellington 14,182 mi., nautical Liverpool to San Francisco 13,844 mi., nautical Hamburg to Callao . 10,702 mi., nautical Bordeaux to San Francisco 13,691 mi., nautical The following table gives the distances from Atlantic to Pacific ports via the Panama Canal route : New York to San Francisco 5299 mi., nautical New York to Sydney 9852 mi., nautical Charleston to San Francisco 4898 ini., nautical MISCELLANEOUS EXAMPLES 315 Charleston to Valparaiso 4229 mi., nautical New Orleans to San Francisco .... 4698 mi., nautical New Orleans to Melbourne 9826 mi., nautical Galveston to San Francisco 4800 mi., nautical Galveston to Wellington 8392 mi., nautical Liverpool to San Francisco 8038 mi., nautical Hamburg to Callao 6527 mi., nautical Bordeaux to San Francisco 7938 mi., nautical What per cents of the distances by the old routes are saved by the Panama Canal route ? INTEREST 149. Find the simple interest on $78 for 93 da. at 8%. 150. Find the simple interest on $98 for 63 da. at 7%. 151. Find the amount of $179 for 123 da. at 6%. 152. Find the simple interest on 324 7s. 9d. from June 12 to Dec. 7 following at 5%. 153. Find the simple interest on 1169 6s. Sd. from Jan. 25 to June 18 following at 9%. 154. What principal will produce $19.50 in 1 yr. at 6% ? 155. What principal will produce $180 interest in 3 mo. at 5% ? 156. What principal will amount to $ 412.50 in 7| mo. at 5 % ? 157. What principal will amount to $ 1219 in 3 mo. 5 da. at 6% ? 158. What principal will produce $29.17 in 5 mo. at 7% ? 159. At what rate will $1000 produce $23.33 interest in 4 mo. ? 160. What principal will produce 75 ^ interest in 9 da. at 6 % ? 161. Find the exact interest on $73.15 from June 18 to Aug. 1 at 1%. 162. A note for $3500 bearing interest at 8% and dated Jan. 2, 1900, was indorsed as follows: June 1, 1900, $450; 316 ADVANCED BOOK OF ARITHMETIC Aug. 2, 1900, $208; Jan. 2, 1901, $500; July 7, 1901, Oct. 4, 1901, $500; Jan. 11, 1902, $300; Aug. 4, 1902, $700. Calculate, by the United States Rule for partial payments, the amount due on this note on Jan. 1, 1903. 163. A demand note dated Jan. 5, 1902, and drawn for $575 was paid 6 mo. 18 da. later. Find the date of payment and the amount of the note, the rate of interest being 7%. BANK DISCOUNT 164. A note for 60 da. is drawn on Jan. 10, 1903. Find the proceeds of this note, if its face is $ 150, the date of discount Feb. 5, and the rate 6%. (Neglect days of grace.) 165. A note for $900, dated Mobile, Ala., Jan. 8, 1903, and drawn for 90 da., is discounted March 1. Find the proceeds. 166. A 60-day note bearing interest at 8%, drawn Feb. 1, 1903, for $1000, is discounted Feb. 27 at 9%. Find the pro- ceeds of this note. 167. A demand note was drawn Oct. 1, 1902, for $ 800, and paid 5 mo. 10 da. later. Find the date of payment and the amount of the note; rate of interest, 1%. 168. The proceeds of a note is $ 450 when the term of dis- count is 93 da., and the rate of interest is 8%. What is the maturity value of the note ? MENSURATION 169. Find the area of a parallelogram if its base is 100 yd. and its altitude is 75 yd. 170. Find the area of. a trapezoid if its parallel sides are 60 yd. and 80 yd., and its altitude is 50 yd. 171. A tract of land is sold for $ 3943.84 ; the land cost as many dollars per acre as there were acres in the tract. Find the cost per acre. 172. Find the number of square yards in the walls and ceil- ing of a room 36 by 23, and 16 ft. high. MISCELLANEOUS EXAMPLES 317 173. Find the perimeter of a square which contains 40 A. 174. A tract of land in the shape of a rectangle contains 320 A. ; its length is twice its width. Find its dimensions. 175. Find the area of an equilateral triangle one side of which is 100 ft. 176. Find the area of a regular hexagon each side of which is 50 ft. 177. Find the circumference of a circle whose radius is 56.5 in. 178. Find the area of a circle if its diameter is 20 in. 179. Find the surface of a sphere whose diameter is 20 in. 180. Find the volume of a cube one of whose dimen- sions is 1 ft. 3 in. 181. The surface of a cube is 221.0694 sq. in. Find the length of one edge of this cube. COMPARISON OF PRICES Express : 182. 6 francs per kilogram as dollars per pound Avoir- dupois. 183. 5 francs per meter as dollars per yard. 184. 1.369 francs per liter as dollars per gallon. 185. 9 francs per hektoliter as dollars per bushel. 186. 7 marks per kilogram as dollars per pound Avoidu- pois. 187. 4 marks per meter as dollars per yard. 188. 2 marks per liter as dollars per U. S. gallon. 189. 9 marks per hektoliter as dollars per bushel. 190. 13.785 marks per meter as dollars per yard, 191. $40 per acre as francs per hektar. MISCELLANEOUS EXAMPLES (B) (TAKEN FROM VARIOUS EXAMINATION PAPERS) 1. What fractional part of f of a gallon is --% of a pint ? 2. The difference in time between two places is 2 hr. 33 min. - Find the difference in longitude. 3. A bicycle wheel measuring 88 in. in circumference r^ust make how many revolutions a minute to run eighteen mileL ail hour ? 4. A coal bin 16^- ft. long and 8 ft. 9 in. wide must be how deep to contain 10 T. of coal, if one ton of coal occupy 40 cu. ft. of space ? 5. Eeduce 2 yr. 21 da. to years and decimals of a year. 6. Eeduce .09625 bbl. to integers of lower denominations. 7. Find the value of a piece of land 64 ch. by 13^ ch. at $48^ an acre. 8. Required the cost of 18 2^ in. plank 16 ft. long and 10 in. wide, and 33 pieces of scantling 2 in. by 4 in. 16 ft. long, at $ 22 per M, board measure. 9. The average yield per bushel of wheat is 14 bu. 1 pk. What will 7 bu. 3 pk. 2 qt. yield ? 10. What is the difference in weight, expressed in Avoirdu- pois pounds, between 100 Ibs. Troy and 100 Ibs. Avoirdupois ? 11. Eeduce to simplest form 3_i-i-j 2 . 7-1 2 + * 12. If it cost $510 to fence a rectangular field 98 rd. by 72 rd., what will it cost to fence a square field of the same area? 13. Express f as a decimal fraction. 318 MISCELLANEOUS EXAMPLES 319 14. What is the ratio of 32 ft. to 6 yd. ? Express the result decimally. 15. What is the length of a plank 11 in. thick, 1 ft. 6 in. wide, containing 36 board feet ? 16. When it is 12 M. in New York City (74 W.), what is the time in Manila (120 E.) ? 17. If the value of | of f of an estate is $ 4500, what is the value of T \ of ^ of it ? 18. At $ 16 per M, board measure, find the cost of 20 plank 2 in. by 8 in. 18 ft. long, and 30 plank 1-J in. by 6 in. 10 ft. long. 19. A can do a piece of work in 6 da. and B can do the same work in 8 da. How long will it take B to finish after they have worked together two days ? 20. 20f is the product of three factors. Two of these factors are If and 4|. Find the other factor. 21. How many bushels of wheat will a box 6 ft. by 3 ft. by 2 ft. 8 in. hold ? 22. How many yards are in .04675 mi. ? 23. If the dividend is 807 and the quotient 34^, what is the divisor ? 24. How many rods of fence will inclose a square field whose area is 20 acres ? 25. Coal sells at $ 5.75 per ton. What will be the cost of 2315 Ib. at this rate ? 26. How many gallons of water will a tank 5 ft. by 2 ft. by 2 ft. hold ? 27. What is the length of one side of a square piece of land whose area is 538,756 sq. rd. ? 28. A room is 27 ft. by 22 ft. 6 in. How many yards of carpet 27 in. wide will be required to carpet this room ? 29. A man is hired to dig a cellar 20 ft. by 15 ft. by 5 ft. How much money will he receive at 30^ per cu. yd. ? 320 ADVANCED BOOK OF ARITHMETIC 30. How many days are there between Aug. 14 and Dec. 29 ? 31. Find the value of a car load of wheat, estimated at 21,643 lb., at 92^ per bushel. 32. Two persons travel in opposite directions from the same point at the rate of 4^ and 7f mi. per hour, respectively. How far apart are they after traveling 37 J hrs. ? 33. A man was born Nov. 22, 1861. What is his age to-day ? 34. Factor the following numbers and from these factors determine the G. C. M. : 42, 112, 140, 308. 35. What will 75 boards 2 in. by 4 in. by 16 ft. long cost at $ 12 per M board measure ? 36. 160 rd. of fence will inclose how many acres in the form of a square ? 37. The difference in longitude between two places is 7 42' 30". Find the difference in time. 38. How wide is a rectangular field containing 5 A., the length of the field being 7 ch. 25 1. ? 39. A pavement is 5^ rd. long and 8 ft. 6 in. wide. What did it cost at $ 1.40 per sq. yd. ? 40. Three men, A, B, and C do a piece of work ; A works 3 da. of 5 hr. each, B, 2 da. of 6 hr. each, and C, 7 da. of 3 hr. each. At the same rate of wages, how should they divide $ 120, the total amount received for doing the work ? 41. A miller charges y 1 ^- for toll. How many bushels of wheat must one take to mill to get 12 bbl. of flour, each con- taining 196 lb., if a bushel of wheat makes 40 lb. of flour ? 42. The annual rainfall in a certain locality is 30 in. How many tons of water fall on an acre of land in this locality, if a cubic foot of water weighs 1000 oz. ? 43. How much does a man gain or lose on the sale of two houses at $ 1200 each, if he gains ^ of the cost price on one, and loses ^ of the cost price on the other ? MISCELLANEOUS EXAMPLES 321 44. What is the ratio of 7 Ib. Troy weight to 10 oz. Avoir- dupois ? 45. The divisor is 357, the quotient is 6f ; what is the divi- dend? 46. A farmer had 28 A. of land left after selling \ of his farm to one neighbor, f of it to another, and f of the remainder to another. How large was his farm ? 47. Multiply 8.035 by .0035, add 3, and divide the sum by .000625. 48. Divide $459.25 into three parts that shall be to one another as f, f, and 3 respectively. 49. When it is two o'clock P.M. in Jerusalem, what is the time in Cincinnati? The longitude of Jerusalem is 35 12' E., and of Cincinnati, 84 26' W. 50. Find the exact number of days between Dec. 23, 1902, and to-day. 51. A man's farm is mortgaged for f of its cost ; he sells it for $6000 which is 25% above its cost. How much money will he have after paying the mortgage ? 52. A note for $ 600, dated Oct. 24, 1902, and due in 8 mo., with interest at 6% per annum, is discounted at bank Dec. 20, 1902. Find the proceeds. 53. A man sold two lots each for $ 600, gaining 20% on one, and losing 20 % on the other. What was his gain or loss ? 54. A man buys a book the list price of which is $ 7.20, at a discount of 16|%, and sells it for $7.50. What is his gain per cent ? 55. What principal at interest for 1 yr. 3 mo. will amount to $506, the rate of interest being 8% per annum? 56. Twelve per cent of 90 is what per cent of 100 ? 57. At the following rates per annum of simple interest, what time is required for the accruing interest to equal the principal: 6%, 8%, 9^% ? 322 ADVANCED BOOK OF ARITHMETIC 58. What is the exact interest on $10,000 from Jan. 18, 1903, to May 6, 1904, at 3% ? 59. A 30-day note, without interest, is discounted at a bank at 8% for $350. What is the face of the note ? 60. Bonds bearing 5% interest are bought at 120. What is the rate of income on these bonds ? 61. An agent buys sugar at 4|^ per pound ; his commission at |-% is $25. How many pounds of sugar does he buy ? 62. The discount of a note, discounted at bank, for 3 mo. 18 da. at 5% is $4.20. Find the proceeds. 63. What single discount is equivalent to trade discounts of 25%, 10%, and 5% on the list price of an article ? 64. The property in a school district is assessed at $ 196,000. What rate of taxation would be required to provide about $ 800 annually for the improved maintenance of the schools ? What annual tax would a man pay on this account whose prop- erty is assessed at $ 1200 ? 65. A man sold two horses at $ 80 each. On one he gained 20%, on the other he lost 20%. Find the gain or loss. 66. What must I ask for an article worth $36 that, after falling 20%, I may gain 25% on the value ? 67. A school district advertised for bids to build a school- house, the lowest bid being $21,049. If it costs 3% to collect the money, how large a levy should be made, supposing 29% of it to be non-collectible ? 68. W T hat must a man pay for 4% stock to get 5% on his investment ? 69. If you buy United States 3's at 110, what per cent per annum would your investment pay ? 70. A merchant's expenses average 10% of his sales. At what per cent advance on cost must he sell his goods to clear 20% profit? MISCELLANEOUS EXAMPLES 323 71. A merchant sold goods to the amount of $ 760.95, thereby losing 11%. What did he pay for the goods ? 72. A ship is insured for half its value for $374. If the rate is 2f %, what is the value of the ship ? 73. A carriage dealer sold 16 buggies at $ 200 each ; on one half he gained 10%, and on the other half he lost 10%. Find his net gain or loss. 74. What principal will amount to $1253.86 in 2 yr. 11 mo. 13 da., interest at 5% ? 75. How do you find the rate per cent per annum when the principal, interest, and time are given ? 76. How do you find the principal when the rate per cent per annum, time, and interest are given ? 77. How do you find the time when the principal, rate per cent per annum, and interest are given ? 78. The list price of office desks is $ 15, but 12 desks are sold for $ 126. What rate of discount is allowed ? 79. In a certain time $650 will amount to $713.05 at 6% simple interest. Find the time. 80. A note for $ 500, due in 3 months, is discounted at bank at 6%. Find the proceeds. 81. 2361 is what per cent of 78 ? 82. A man insures his life, paying a premium of $ 28, which is at the rate of -|% on the amount of his insurance. Find the face of the policy. 83. If 25% of the selling price of an article is profit, what is the per cent of gain on its cost ? 84. A man fails in business; his assets amount to $2100, his liabilities to $ 6000. What per cent will his creditors receive ? 85. What is the interest on $ 475 for 1 yr. 3 mo. 24 da. at 324 ADVANCED BOOK OF ARITHMETIC 86. A man bought four loads of hay, each weighing 2750 lb., at $ 20 per ton ; he gave in payment his note, without interest, at 60 da. What are the proceeds of this note, discounted at a bank at 6% ? 87. What per cent of -J- is -|- ? 88. An agent's commissions at 5% amount to $37.65. Find the amount of his sales. 89. The tax on property assessed at $ 8500 is $ 48.37. What is the rate on $ 1000 ? 90. Find the date of maturity of a note made and dated Sept. 11, 1902, and payable 90 da. after date. 91. Find the cost of 87 shares of stock at 76J, brokerage | per cent. 92. A New York sight draft was sold in Atlanta, Ga., for $3542, exchange being at f% premium. What was the face of the draft ? 93. What per cent of 5 lb. is 3 oz. Avoirdupois ? 94. An agent's commission for renting a house is $13.25; his rate of commission is 2^%. What is the yearly rent of the house ? 95. A man pays a premium of $ 150 for insuring his house for -| of its value ; the rate of premium is 1^ per cent per annum. What is the value of the house ? 96. A building worth $6000 is insured for f of its value at 75 ^ on the $ 100. In case of the destruction of the building by fire, what will be the owner's loss, including premium ? 97. What per cent of 1 bu. is 3 qt. ? 98. A merchant can buy flour on six months' credit at $ 8 per barrel, or for cash at $ 7.50 per barrel. He buys 100 bbl., paying cash, but borrows the money at 8% to pay for it. Is this better than to buy on credit, and how much better ? 99. A man sells 16 shares of bank stock at 127f, brokerage i-%. How much does he receive for his stock ? S EXAMPLES 325 ocks at 20 % premium and discount. What per cent > ^s books show sales during me montn ot March amounting to f 1000. One half of his sales are at a profit of 25 % on the cost, and the other half a loss of 16| % on the cost. Find the cost of the gcods sold during the month. 102. A merchant failing in business paid his creditors $3874.75, which was at the rate of 55^ on every dollar of his indebtedness. Find his indebtedness. 103. The list price of a mower is $38 ; the retail dealer is allowed discounts of 20%, 5%, and 3%. What does he pay for mowers ? If the retailer sells these mowers at a profit of 50 %, what does the farmer pay for these mowers ? 104. A certain stock, selling at 121|, pays a semiannual dividend of 4%. What is the rate per cent per annum on an investment in this stock ? TABLES APOTHECARIES' WEIGHT 20 grains (gr.) = 1 scruple (3) 3 scruples = 1 dram (3) 8 drams = 1 ounce ( ) 12 ounces = 1 pound (fib) LIQUID MEASURE 4 gills (gi.) = 1 pint (pt.) 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) 31| gallons = 1 barrel (bbl.) 2 barrels = 1 hogshead (hhd.) LONG MEASURE 12 inches (in.) =1 foot (ft.) 3 feet = 1 yard (yd.) 5^ yards = 1 rod (rd.) , or pole 40 rods = 1 furlong 8 furlongs =1 mile (mi.) TROY WEIGHT 24 grains (gr.) = 1 pennyweight (pwt.) 20 pennyweights = 1 ounce (oz.) 12 ounces =1 pound (Ib.) 326 ADVANCED BOO DRY MEASURE 2 pints (pt.) 8 quarts 4 pecks = 1 quart (qt.) = 1 peck (pk.) = 1 bushel (bu.) NUMERICAL MEASURE 12 articles = 1 dozen 12 dozen = 1 gross 12 gross = 1 great gross 20 articles = 1 score AVOIRDUPOIS WEIGHT 16 drams (dr.) = 1 ounce (oz.) 16 ounces = 1 pound (Ib.) 25 pounds 100 pounds CIRCULAR ivu&Asuivr, 60 seconds (") = 1 minute (') 60 minutes = 1 degree () 30 degrees 12 signs = 1 sign (S.) = 1 circle (C.) or circumference = 1 circumference 20 cwt. 2240 pounds = 1 quarter = 1 hundredweight (cwt.) = 1 ton (T.) = 1 long ton 360 degrees NAUTICAL MEASURE 6 feet = 1 fathom 608 feet = 1 cable length 10 cable lengths = 1 nautical mile (6080 feet) The following denominations are also used : 1.152 statute miles = 1 geographic mile, or knot 3 geographic miles = 1 league 60 geographic miles, or ) 69.1 statute miles } = 1 Ae ^* f latltude on a meridlan 360 degrees = the circumference of the earth 4 inches = 1 hand SURVEYORS' AND LAND MEASURE 9 inches = 1 span 7.92 inches = 1 link (1.) 21.888 inches = 1 sacred cubit 25 links = 1 rod 3 feet = 1 pace 4 rods = 1 chain (ch.) TIME MEASURE 10 square chains = 1 acre 60 seconds (sec. ) = lminute(min.) 640 acres = 1 square mile 60 minutes = 1 hour (hr.) 625 square links (sq. 1.) = 1 pole (P.) 24 hours = 1 day (da.) 16 poles = 1 square 7 days = 1 week (wk.) chain 4 weeks = 1 lunar month 30 days = 1 commercial APOTHECARIES' FLUID MEASURE month 60 minims (m.) = 1 fluidrachm(f 3) 12 months = 1 year 8 fluidrachms 1 fluidounce(f J ) 365 days = 1 common year 16 fluidounces = 1 pint (O) 366 days = 1 leap year 8 pints = 1 gallon (Cong.) APPENDIX 327 CUBIC MEASURE 1728 cubic inches (cu. in.) 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) 16 cubic feet, or i , , A , A \ > = 1 cord of wood (cd.) 8 cord feet / 24} cubic feet = 1 perch of stone or masonry (pch.) SQUARE MEASURE 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet 1 square yard (sq. yd.) 30 square yards = 1 square rod or perch (sq. rd. or sq. pch.) 40 square rods = 1 square rood (sq. R.) 4 roods = 1 acre (A.) 640 acres 1 square mile (sq. mi.) SPANISH LAND MEASURE In Texas, California, New Mexico, and other parts of this country which were formerly parts of the Spanish empire, the vara, the unit of linear measure, is still used in connection with original grants of land. In Texas, the value of the vara is 33^ in. In California and New Mexico it is usually considered 33 in. 1,000,000 square varas = 1 labor = 177.136 acres 25,000,000 square varas = 1 league 4428.4 acres 3,612,800 square varas 1 square mile = 640 acres 1,806,400 square varas = J square mile = 320 acres 903,200 square varas = J square mile = 160 acres 451,600 square varas = J square mile = 80 acres 225,800 square varas = T J ? square mile = 40 acres 5645 square varas = 1 acre ANSWERS Exercise 4. North Atlantic Division, 5,499,620, 3,843,908, 2,866,074, $ 105,332,839. South Atlantic Division, 3,521,920, 2,324,906, 1,503,917, $15,907,956. South Central Division, 5,002,836, 3,235,121, 2,074,304, $19,870,733. North Central Division, 7,878,448, 5,895,631, 4,188,517, $ 107,663,687. Western Division, 1,125,924, 956,472, 685,444, $24,441,012. Totals, 23,028,748, 16,256,038, 11,318,256, $273,216,227. Exercises. 1. $2,187,217.51. 2. $2,749,532.73. 3. $8,593,793.27. 4. $733,501.99. 5. $4,330,637.88. 6. $9,145,702.01. 7. $773,768.16. 8. $6,124,815.32. 9. $7,750,185.14. 10. $3,429,967. 11. $2,001,002.77. 12. $3,135,817.92. 13. $10,029,526.43. 14. $659,959.65. 15. $2,004,181.95. 16. $853,917.00. Exercise 6. 1. $ 75, $ 200, $ 900, $ 1225. 2. $ 2.08, $ 4.68, $ 24.44. 3. 100,320ft.; 401,280ft. 4. 273 da. 5. $6084. 6. 2136 hr. 7. 1598 mi. 8. 62,720 A. 9. 1173 mi. 10. 487,956 Ib. 11. $1008. 12. 85,008 cu. in. 13. 103,680 pages. 14. $956.80. 15. $1736. Exercise 7. 1. 9000 sq. yd. 2. 7056 sq. yd. 3. 102,400 sq. rd. 4. 11, 760 sq. yd. 5. 22,848 sq. yd. 6. 167,200 sq. yd. 7. 4725 sq. rd. 8. 20,160ft. 9. 252 sq. in. 10. 36 sq. mi. 11. 432 sq. mi. 12. 40,480 sq. yd. 13. 2200 sq. yd. 14. 5332 sq. ft. 15. 2880 sq. in. 16. 1296 sq. in. Exercise 8. 1. $639.52. 2. $225.40. 3. $476.94. 4. $898.03. 5. $3594.51. 6. $5455.08. 7. $8023.18. 8. $2136.42. 9. $5046.93. 10. $2285.44. 11. $359.31. 12. $555.03. 13. $239.76. 14. $1076.28. 15. $1866.95. 16. $6399.52. 17. $7402.50. 18. $10,986.58. 19. $9345. 20. $6594.75. 21. $6498.38. 22. $3269.10. 23. $93.60. 24. $159.60. 25. $187.65. 26. $322.56. 27. $195.44. 28. $7.02. 29. $19.50. 30! $6075. Exercise 9. 1. 841.75 mi. 2. 614.2 mi. 3. 809.93 mi. 4. 710.4 mi. 5. 1401. 56 mi. 6. 1280.2 mi. 7. 754.8 mi. 8. 1032.3 mi. 9. 1191.4mi. 10. 1,879.6 mi. 11. 1742.7 mi. 12. 1894.4 mi. 13. 2,217,648. 14. 2,230,095. 15. 4,821,600. 16. 1,380,768. 17. 2,423,546. 18. 1,881,250. 19. 4,167,520. 20. 6,297,900. 329 330 ANSWERS Exercise 10. 1. 4834, rem. 10. 2. 4408, rem. 14. 3. 4892, rem. 14. 4. 4416, rem. 2. 5. 4166, rem. 16. 6. 2957, rem. 23. 7. 2485, rem. 13. 8. 2668. 9. 2760, rem. 32. 10. 1706, rem. 28. 11. 1234, rem. 39. 12. 2380, rem. 28. 13. 1431, rem. 10. 14. 2519, rem. 25. 15. 17,530, rem. 43. 16. 14,616, rem. 31. 17. 14,792, rem. 22. 18. 8261, rem. 35. 19. 8741, rem. 42. 20. 3377, rem. 42. 21. 2133, rem. 74. 22. 8214, rem. 4. 23. 9701, rem. 37. 24. 8217, rem. 3. Exercise 11. 1. 14 Ib. 2. 29 hr. 3. 375 pk. 4. 29 hr., 4 mi. rem. ; 94 mi. 5. 384 T. 6. 179 da. 7. 225 da. 8. 2083, 4 in. rem. 9. 52 wk. 10. 14 doz. 11. 16. 12. 37 boxes. 13. 417 bbl. 14. 24 hr. 15. $ 125. 16. 161 sheep. 17. 159 coins. 18. 45 hogs. 19. 245 boxes. 20. 167 bbl. 21. 19 Ib. 22. 36 horses. Exercise 12. 1. 42 bags. 2. 39 bbl. 3. 79 bbl. 4. 2777yd., 28 in. rem. 5. 241 bu. 6. 125 da. 7. 36. 8. 364 sq. yd. 9. 36 gal. 10. 33 chests. 11. 325 bbl. 12. 114 hr. 13. 1000 A. 14. 1250 sq. mi. Exercise 13. 1. 68. 2. 69. 3. 427. 4. 5320. 5. 245. 6. 385. 7. 198. 8. 17. 9. 2125. 10. 717 sacks. 11. 810. 12. 563 bbl. 13. 17,503. 14. 396. 15. 39. Exercise 14. 1. 196.2. 2. 629.8. 3. 167.6. 4. 189.8. 5.289.6. 6. 303.8. 7. 324.8. 8. 445.2. 9. 16.3. 10. 95.6. 11. 356.1. 13. .5, .75, .8, .3125, .24, .21875, .171875, .6171875. 14. .85, .38, .1025, .415, .536, .348. 15. 2875, .425, .45, .74, .206, .02725. 16. .333, .1429, .111, .0909, .0833, .0769, .0714, .0667, .0588. 17. .1852, .4783, .3214, .1290, .4571, .0959. Exercise 15. 1. $58.48. 2. $136.15. 3. $ 16.35. 4. $31.42. Exercise 16. 1. $1. 2. 42ft. 3. 5 da. 4. 10 da, 5. 60 Ib. 6. $24, 5 sheep. 7. 60^ on each kind, 47 oranges. 8. 60^. 9. 60 sec. 10. 7ft. 6 in. Exercise 18. 1. 8. 2. 8. 3. 9. 4. 12. 5. 15. 6. 15. 7. 18* 8. 36. 9. 12. 10. 12. 11. 7. 12. 13. 13. 19. 14. 23. 15. 24. 16. 28. 17. 10. 18. 15. 19. 12. 20. 13. 21. 12. 22. 18. 23. 16. 24. 21. 25. 32. 26. 36. 27. 44. 28. 63. 29. 48. 30. 54. 31. 56. 32. 84. Exercise 19. 1. 80. 2. 42. 3. 180. 4. 90. 5. 147. 6. 140. 7. 168. 8. 108. 9. 102. 10. 144. 11. 480. 12. 420. 13. 160. 14. 78. 15. 336. 16. 480. 17. 756. 18. 385. 19. 270. 20. 216. 21. 375. 22. 216. 23. 180. 24. 576. 25. 180. 26. 210. 27. 210. 28. 420. 29. 16,800. 30. 720. 31. 300. 32. 180. 33. 630. 34. 280. 35. 144. 36. 840. 37. 600. 38. 2002. 39. 180. 40. 210. 41. 240. 42. 420. 43. 108. 44. 720. 45. 360. ANSWERS 331 Exercise 21. 1. |. 2. y. 3. y. 4. -V 5 . 5. -*f-. 6. -V T 3 -. 7. Y- 15. - 3 T T 2 - 16- W- 17- ^*-. 18. V- I 9 - -T- 20. ^. 21. 1J*. 22. -if*. 23. I}*. 24. *fi. 25. - 3 T 2 ^. 26. J^. 27. 1^. 28. *f*. 29. -Vr 8 - 30. - 3 T V. Exercise 22. - 1. f , j, J|, if It- 2. f , Jf , if, If, 3. T %, JJ, if, H, 4, A, i 2 , J}, H, jj. 5. I?, if, f 5 , fj, f|. 6. H, if, J}, }, If J' if , if *f M- 8- 18. if ?f if tt^ - if if *f I!' if Exercise 23. 1. J, f |, |, f, f, f, f, f. 2. |, f, |, f 3. f , f f , f, f, f, f f - 4. f, f, f, A,. A, Ai A, A- 5. T 9 6, i & A- Exercise 24. 1. l r V 2. 1JJ. 3. 1 T V 4. 1}. 5. If. 6. 1J. 7. f. 8. 1 T V 9. If. 10. If. 11. If. 12. 1J. 13. If. 14. 1J}. 15. 1}J. 16. 2-}i. 17. 1A- 18- 2A- 19. 2^. 20. 1}}. 21. 2 T %. 22. 1}. 23. 1 T %. 24. 1}}. 25. |f. 26. 2 A- 27. 1J}. 28. 1J}. 29. 2J. 30. 1J}. 31. 7A- 32. 15J. 33. 8f. 34. 25^. 35. 26}J. 36. 22f 37. 17f 38. 12J}. 39. 19^. 40. 17}. 41. 25^. 42. 14}|. 43. $ A- 44. 1-A hr. 45. f is largest ; is smallest. Exercise 25. 1. J. 2. J. 3. }. 4. }. 5. |. 6. }. 7. ^. 8. }. 9. J. 10. A- 11- i- 12 - } 13- 1}. 14. 2i. 20. 11 A- 26. 16}. 32. 3i|. 33. 38. 4|. 44. $2}. 45. Exercise 26. 7. 27. 8. 55. 9. 3. 10. 5. 11. 2. 12. 4. 13. 20. 14. 2. 16. 24 cows. 17. 42 doz. 18. 15 da. 19. 22 hr. 20. 200. Exercise 2 7. 1. 12. 2. 10}. 3. 5f 4. 16J. 5. 11}. 6. 4}. 7. 11}. 8. 24}. 9. 5f ^ 10. 12}. 11. 19}. 12. 38}. 13. 35}. 14. 47*. 15. 38J. 16. 15}. 17. 16f 18. 22}. 19. 33}. 20. 301}. 21. 247}. 22. 152. 23. 335}. 24. 247}. 25. 243. 26. 160A- 27. 753}. 28. 705}. 29. 446J. 30. 166}. 31. 221|. 32. 132|. 33. $4}. 34. $42}. 35. $15.23 T V 36. 52} j*. 37. $129.76}. 38. $31.87}. 39. $3.84f. 40. $35,40, 41. |7.81J. 42. 4840 sq. yd. 43. 792 in. if. 16. IA- 51. 6 T V 22. 7}. '. 6}. 28. 2&- 17. 10}. 18. 12}. 23. 7A- 24. 3}. 29. 11}}. 30. 10&. 19. 8f. 25. 51. 31. 2f \. 3}}. 34. A 35. 6 A 36. 2A- 37. 6A- 4f 40. 8H- 41. 19f 42. ?A. 43. 5f i. 2i}. 46. $1.15. 1. 1. 1. 2. 162. 3. 4. 4. 35. 5. 27. 6. 6. 332 ANSWERS 44. $11.90. 45. 1036 J mi.; 2764* mi. 46. $7350. 47. 1792 Ib. copper, 448 Ib. lead. 48. Gain $805, investment at end of year $3105. Exercise 28. 1. J- 2 J- 3. T r . 4. 5- iVo- 6. 7. ft. 8 . f. 9. f. 10. A 11. f. 12. f. 13. A- 14. i- 15. A- 16. f. 17. TV 18. T V 19- rVV 20. i- 21. 2}- s 52. 3f. 23. f. 24. f. 25. f. 26. 7. 27. 9. 28. lOf. 29. 15. 30. 8J. 31. 22. 32. 26H- 33. 1- 34. 14A- 35. 64 |. 36. 1. 37. 420. 38. 311. 39. 7. 40. 2. 41. 272*J$. 42. $398|f. 43. $35241. 44. $ 1.9 3|. 45. $3.65f. 46. $162.50. 47. , $52J. Exercise 29. 1 . A- 2. * 3. If. 4. T*8- 5. 1- 6. & 7. A- 8- - 9. 10. A- 11. f- 12. 13. A' 14. A- 15. 1A. 16. A- 17. if. 18. 3i. 19. *V 20. A- 21. 1- 22. A- 23. 7 %. 24. 2 25. 18. 26. A- 27. A- 28. 49. 29. T 4 3- 30. 2f 31. 28. 32. A- 33. iff. 34. 21. 35. A- 36. ^ IT?- 37. 28. 38. &. 39. 2A- 40 , 30. 41. A 42. /f 43. 321. 44. A- 45. 4f. 46, , 28. 47 . f. 48. T9- 49. 601. 50. A- 51. A- 52. 96. 53. 17- 54. A 55. 671. 56. A- 57 . 4. 58. 49^. 59. f. 60. 7 61. 2. 62. f. 63. i- 64. 1J. 65. I Jf. 66. H- 67. 51. 68. |f. 69. ft. 70. 2f. 71. A- 72. If Exercise 30. 9. 4. 10. 9. 11 . 16. 12. 5}. 13. 25. 14. 81. 15. 9. 16. |. 17. 1. 18. 11. 19. li. 20. i If- 21. If 22. JW- 23. 1- 24. 5 2A- ' 25. *^ 26. 1 H- 27. IfJ-. 28. 1 T V 29. T V 30. 1J. 31. 6. 32. Exercise 31. 1. $4290. 2. $2|. 3. $216.75. 4. $137f 5. $630. 6. $575. 7. $24.70. 8. $68.88. 9. $881|. 10. $7. 11. $4.25. 12. $8.41. 13. $7.72f 14. $5.75. 15. $13J. 16. $120. 17. 108ft. 18. 72. 19. 255 T. 20. 60. Exercise 33. 1. 666.973. 2. 151.987. 3. 261.304. 4. 1943.232. 5. 135.148. 6. 160.687. 7. 282.238. 8. 317.187. 9. 176.483. 10. 212.728. 11. 281.308. 12. 377.077. Exercise 35. 1. $4.32, $13.70, $43.20, $10.08. 2. $3.105, $22.59, $6.064, $5.929. 3. $11.90, $34.74, $40.992, $32.85. 4. $10.10, $40.50, $180.30, $24.71. 5. $430, $693.70, $337.80. 6. 31.5 A., 4 A., 30 A. 7. .01, .12, .09, .002, .012. 8. 502.2. 9. 2123.2. 10. 1806.42. 11. 432.036. 12. .426. 13. 2.625. 14. 28.56. 15. 271.9846. 16. .56056. 17. 249.048. 18. .1702. 19. .4182. 20. .6. 21. 3.3642. 22. 17.328. 23. .20001. 24. .9409. 25. .4624. 26. .139129. ANSWERS 333 27. .811801. 28. .644809. 29. .480249. 30. .001. 31. .027. 32. .064. 33. .343. 34. 1.124864. 35. 1.191016. 36. 1.259712. 37. .015625. 40. 153.9384. 41. 197,061,258. 42. 15,205,344. 43. 1110 mi., 66,600 mi. 44. 8796 ft., 527,760 ft. 45. .32 in. Exercise 36. 1. 9.16125. 2. .8625. 3. 3.227. 4. .199. 5. .9128. 6. .490875. 7. .003. 8. 1.47433. 9. 1.174. 10. .657. 11. .925. 12. 1.142875. 13. .02468. 14. .12201. 15. .1638. 16. .7843875. 17. .912. 18. .9172. 19. .03866. 20: .07484. 21. .142857. 22. 1.0535. 23. 1.0446. 24. .2478. Exercise 37. 1. 2.925. 2. .024. 3. 8.6875. 4. .2058. 5. .3596. 6. 2.62. 7. 3.725. 8. 2.531. 9. 1.35. 10. 3.36. 11. 6.3234. 12. .5. 13. 40. 14. 62.5. 15. 125. 16. 50. 17. 15. 18. 50. 19. 160. 20. 12,500. 21. 8000. 22. 500. 23. 25. 24. .5. 25. 1000. 26. 1250. 27. 1000. 28. 50. 29. 250. 30. 250. 31. 25. 32. 2.5. 33. .0125. 34. .00125. 35. .0004. 36. .00666+. 37. .125. 38. .15. 39. 32. 40. .027. 41. 48.7. 42. 8.03. 43. .0904. 44. .708. 45. 1.012. 46. .01014. 47. .517. 48. .01054. 49. .00377. 50. .0365. Exercise 38. 1. $17,762.08. 2. $19.740.80. 3. $18,850.01. 4. $30,737.16. 5. $21,197.90. 6. $41,931.64. 7. $15,101.25. 8. $69,022.28. Exercise 39. 1. .375, .625, .4375, .5625, .6875, .8125, .9376, .1875. 2. .2667, .4667, .7333, .8667, .9333, .5833, .9167. 3. .3, .79, .087, .0183, .2779. 4. .0375, .1375, .3625, .95, .926, .6167, .8833. 5. .03125, .09375, .15625, .28125, .40625, .59375, .96875. 6. .2857, .7143, .0769, .3077, .2143, .6429, .7857, .9286. 7. .5556, .1919, .1471, .0544, .0274, .0569. Exercise 40. -1. &, f, J, J, A- 2. T J^, ^ *fo A dhr- 3 - To 9 o 9 911 A 9 13 7 119 3 K 1 8 fi 3 48 1 00"' S7> 407' * T25 T2^> ^tf T~QQ^Qi SO' ' T25' Ittfr 6^3' 5frO> vh- 6. T \ 6 62 a. 7. $30,918.75. 9. $875; no brokerage. 10. 5f%. 11. $200, $5093.76. 13. Central. 14. 438 shares. Exercise 137. 1. $62.40. 2. $500. 3. $2.16. 4. $26.28. 5. 43 4s., or $210.23. 6. $1.81. 7. $1.46. 8. $256.80. 9. $58. 10. $72, $8.19 each. 11. $91.80. 12. $17.50, $81. 13. $65. 14. $552, $2.208. 15. $750. 16. $28.80. 17. $94.50. 18. $125. 19. 5. 20. $82.12,29.4^. Exercise 141. 1. .01, .04, .09, .16, .25, .36, .49, .64, .81. 2. 1, J, i re? A A A> A 8T> Trioi T?T> ii?> ri-g? rirs> si? siff *}? *ii *ii ^o- 3. t, A, it, A, j& ifj, m ^v 4. 2j, 5|, if i, IA, SA, 3ft, 4fj, 39 A- 6. 1, J, sV, ^, T i 5 , ^ rfs^sh. 7l* ioW TsVp T?V^' ^1^7' ^TT? "SlVs' 4^^' "^r"J' 3FS^ ^S^' S'TjW' ^' 3025. 8. (a) 3, (6) 85.6735, (c) 29.1708, (d) 66.0806, (e) .4670, (/) .5790, (0) .8991, (ft) .5, (0 .6, (j) .9. Exercise 142. 1. 13. 2. 21. 3. 25. 4. 31. 5. 32. 6. 43. 7. 53. 8. 62. 9. 65. 10. 73. 11. 76. 12. 82. 13. 84. 14. 87. 15. 92. 16. 96. 17. 98. 18. 99. 19. 89. 20. 78. 21. 69. 22. 69. 23. 49. 24. 58. Exercise 143. 1. 317. 2.332. 3.347. 4.414. 5.436. 6. 447. 7. 479. 8. 527. 9. 557. 10. 595. 11. 626. 12. 676. 13. 689. 14. 708. 15. 809. 16. 879. 17. 905. 18. 909. Exercise 144. 1. .388. 2. .496. 3. .587. 4. .539. 5. .513. 6. .679. 7. .729. 8. .785. 9. .885. 10. .288. 11. .249. 12. .0608, ANSWERS 347 Exercise 145. 1. 1.095. 2. 2.062. 3. 1.049. 4. 2.28. 5. 1.817. 6. 2.291. 7. 1.768. 8. 3.028. 9. 1.173. 10. 2.121. 11. 1.696. 12. .764. 13. .816. 14. .632. 15. .7977. Exercise 146. 1. 69.57yd. 2. 241yd. 3. 238yd. 4. 1.025 mi. 5. 739 rd. 6. 311.13 yd., 155.56 yd. 7. 538.89 yd., 179.63 yd. 8. 19.41 rd. Exercise 147. 1. 10. 2. 13. 3. 17. 4. 29. 5. 10(3. 6. 101. 7. 145. 8. 89. 9. 149. 10. 68.5. 11. 425. 12. 305. 13. 433. 14. 305. 15. 50ft. 16. 14.14 rd. 17. 30.232 rd. Exercise 148. 1. 152. 2. 184. 3. 280. 4. 2.17. 5. .2P1. 6. .319. 7. .2. 8. .748. 9. .455. Exercise 149. 1. 126. 2. 180. 3. 264. 4. 840. 5. 522. 6. 9240. 7. 150,769. 8. 8.625 A. 9. 5.775 A. 10. 1.638 A. 11. 1.68 A. 12. 43.301 sq. rd. 13. 1082.53 sq. rd. 14. 182.25 sq. in. 15. 2592 sq. ft. 16. 57.42 ft. Exercise 150. 1. 69.12. 2. 144.51. 3. 471.24. 4. 515.22. 5. 615.75. 6. 420.97. 7. 540.35. 8. 22.62. 9. 37.07. 10. 45.87 11. 102. 12. 152. 13. 7.8. 14. 13.27. 15. 60. 16. 8.5. 17. 9.6. 18. 6.7. 19. 480 times. Exercise 151. 1. 615.8. 2. 1520.5. 3. 4071.5. 4. 69.4. 5. 132.73. 6. 232.35. 7. 295.59. 8. 4778.4. 9. 3217. 10. 7238.2. 11. 4300.8. 12. 6647.6. 13. 795.8. 14. 484.15. 15. 548.2. 16. 688.3. 17. 6.023. 18. 3.789. 19. 7.643. 20. 9.282. Exercise 152. 1. 31. 2. 17. 3. 33. 4. 42. 5. 2.6. 6. 7.2. 7. 9.2. 8. 9.8. 9. 11.8. 10. 13.4. 11. 7.6. 12. 9.3. 13. 99. 14. 850. 15. 650. 16. 39.25yd. 17. 288,576,452.4. 18. 329.9 sq. in. 19. 1661.9 sq. in. 20. 259.8. 21. 139.1 sq. in. 22. 259.8 sq. in., 225 sq. in. 23. Circle. Exercise 153. 1. 46.5 in. 2. 49.22 in. 3. 1.01ft. 4. 36. 5. 3 36'. 6. 31' .2. Exercise 154. 1. 20ft. 2. 176ft. 3. 110 mi. 4. 7J mi. 5. 22.36 ft. 6. 7.07 ft. 7. 16 : 121. 8. 4:9. 9. 186.96 sq. mi. Exercise 155. 1. 576 sq. ft. 2. 680ft. 3. lOf ft. 4. 7J ft. 5. 2080 sq. ft., $52. 6. 630 sq. ft. 7. $26.40. 8. 147 sq. yd. 9. 53.45 sq. ft. 10. 4021 sq. in. 11. 67 yd. 12. 2984.5 sq. ft. 13. 15,456.6 sq. in. 14. 8392.70. 15. 120,687 sq. in. 16. 5541.7 sq. in. 17. 28,842,700 sq. mi. 18. 186,265,000 sq. mi. 1'9. (1) Jupiter, 23,235 million sq. mi. ; (2) Uranus, 3217 million sq. mi. ; (3) Neptune, 348 ANSWERS 3421 million sq. mi. ; (4) Saturn, 16,741 million sq. mi. 20. 58 in. Exercise 156. 1. 5280 cu. in. 2. 700 cu. in. 3. 4071. 5 cu. in. 4. 15,708 cu. in. 5. 2598 cu. in. 6. 33,510 cu. in. 7. 2144.7 cu. ft. 8. 4094 gal. 9. 4562 gal. 10. 3:2. 11. 1:2. 12. 52 cu. ft. 1269 cu. in. 13. 3 ft. 14. 4189 cu. in. 15. 40 ft. 16. 1,367,631. 17. 48 times. 18. 760 times. 19. 1331 times. Exercise 157. 1. 30 C. 2. 25 C. 3. 95 C. 4. 120 C. 5. 20 C. 6. 12fC. 7. 3iC. 8. - 5 C. 9. - 10 C. 10. -25C. 11. -40C. 12. -67JC. 13. 95 Fahr. 14. 131 Fahr. 15. 77 Fahr. 16. 68 Fahr. 17. 64.4 Fahr 18. 46.4 Fahr. 19. 14 Fahr. 20. - 4 Fahr. 21. 6.8 Fahr. 22. 0.4 Fahr. 23. - 11.2 Fahr. 24. - 459.4 Fahr. 25. Mer- cury, -40; sulphur, 235.4; lead, 618.8; zinc, 779; gold, 1895; cast iron, 2012 to 2102. 26. 39.2. Exercise 158. 1. 3,921,138.813 meters. 2. 107,934,859.86cm. 3. 3,131,587.7mm. 4. 28.434km. 5. 19454m. 6. 140.784m. 7. 960 times. 8. 7.03 times. 9. 5000 times. 10. 112|. 11. 20,000. 12. 62.5 times. 13. 5000. 14. 1200, 480. Exercise 159. 1. 7.8125 ha. 2. 1.8605 ha. 3. 5.7122 ha. 4. 6.82 ha. 5. 11.95 ha. 6. 6.65 ha. 7. .81 cbm. 8. 3.36 a. 9. 1.91 ca. 10. 24,429 c.cm. 11. 119 cm. 12. 1596 cbm. 13. 169.65 ca. 14. 51.08 km. 15. 462.3 qcm. Exercise 160. 1. $1156.43. 2. |622.91. 3. $1650. 4. $1278.12. 5. $983.59. Exercise 161. 1. $2737.14. 2. $4031.75. 3. $8005.16. 4. $9773.37. 5. $11,876.05. 6. $2853.54. Exercise 162. 3. 2J hr. 4. IJf hr. 5. 4 T 8 g hr. 6. J. 7. f hr. 8. 1 hr. Exercise 163. 1. 1260 mi. 2. 5 hr., 150 mi. 3. 10.38A.M. 4. 22 mi. per hr. 5. 105.6 yd. 6. 180 mi. Exercise 164. 1. 48, 72, etc. 2. 6 min. spaces. 3. lO^min. past 2 o'clock, 16^ min. past 3 o'clock, 27 T 3 T min. past 5 o'clock, 38 r 2 T min. past 7 o'clock, 49^ min. past 9 o'clock, 54 T 6 T min. past 10 o'clock. At no time. 5. (a) 38 T 2 T min. past 1 o'clock, (&) 49J r min. past 3 o'clock, (c) 10} min. past 8 o'clock, (d) 16^ min. past 9 o'clock, (e) 27 T 3 r min. past 11 o'clock, (/) 32 T 8 T min. past 12 o'clock. 6. 49^ min. past 3 o'clock, 5j\ min. past 7 o'clock. 7. (a) 27^ min. past 2 o'clock, (&) 32 T * r min. past 3 o'clock, (c) 5j 5 r min. past 4 o'clock, 38 r 2 T min. past 4 o'clock, (d) 16 r 4 T min. past 6 o'clock, 49 T 1 1 min. past o'clock, (e) 27^ min. past 8 o'clock, (/) 32 T 8 r min. past 9 o'clock, ANSWERS 349 ft) 16 T 4 r min. past 12 o'clock. 8. 13^- min. past 4 o'clock, 5 o'clock, 48 min. past 4 o'clock. 9. 38 T 2 r min. past 5 o'clock. MISCELLANEOUS EXAMPLES (A) 8. 8999.991. 9. 274.999225. 10. 8,447,537,940,492. 11. 636,300,000. 12. 751,700,800. 13. 7.1407. 14. 814,585.36. 15. 166.375 mi. 16. $3828.12. 17. $3670.12. 18. .000504. 19. 1.12550881. 20. 278,500,000. 21. (1) $.475, (2) $3.508, (3) $51.877. 22. 372.015. 23. 28,127,000 nearly. 25. 52,800 mi. 26. .341. 27. .8251. 28. .1704. 29. .000439625. 30. 14.461. 81. 8 39' 6". 32. 80.7 X'. 33. 113,400. 34. 250. 35. 80,032,000 nearly. 36. 77. 37. $3.78. 38. $1800. 39. $384.45. 40. $30.80. 41. 12,290. 42. $20.25. 43. .5774. 44. .037. 45. T | T , or .0390625. 46. 144. 47. 2, 3, 6, 9, 18. 48. 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. 49. 1110 = 2 3 5 . 37 ; 777 = 3 . 7 . 37 ; 1001 = 7 11 13 ; L.C.M. 2 . 3 - 5 - 7 11 - 37. 50. 26. 51. 65,520. 52. 25. 53. 19 ; 66,880. 54. 2, 3, 5 2 , 7 2 , 11. 55. 2,784,873. 56. 2* .38 - 11 . 13. 57. 25 A. 58. 80 poles. 59. }, f , JJ. 60. fa. 61. 14 j} 62. T %. 63. 1.09375, .109375, .128. 64. J> ff , ^ T , ^. 65. 3&. 66. 1^. 67. .025. 68. .027. 69. .075. 70. .16583. 72. Upwards of 80,000 yr. 73. (1)39.534, (2) 48.321, (3) 54.197, (4) 63.25, (5) 63.615, (6) 68.234, (7) 72.932, (8) 76.459, (9) 79.68, (10) 82.951, (11) 70.13. 74. $11,687.50. 75. 20,000 bu. 76. $3062.50. 77. $42.40. 78. $92.67. 79. $2.174. 80. $728.42. 81. 36.3 nearly. 82. 5,201,300. 83. 52,100,000 nearly. 84. 56,310,000. 85. 12.44496 ft. 86. 712.5 Ib. 87. 320. 88. 1.292 sec. 89. 4888fcu. yd. 90. .002055 nearly. 91. $6256. 92. $3344. 93. .2565. 94. 19s. 2J<2. 95. .15625. 96. (a) 147,824, (ft) 22,098, (c) 463,180, (d) 823,750. 97. (a) 166,488, (ft) 526.380. 98. 640 A. 99. 2640 revolutions. 100. 112.5 bu. 101. ^. 102. $75. 103. $24,800. 104. 10 A. 105. $ 14f 106. (i) .00}, (ii) .00375. 107. (a) , (ft) , (c) J?t, (d) . 108. 99.99. 109. - 2 r 6 /. 100 ' 100 ' 100 100 110. $107. 111. 2.28. 112. iJ. 113. 66,720. 114. $101.02. 115. 25ft. 116. 35. 117. 847.15625. 118. 28J. 119. $7260. 120. 390. 121. 1092 eggs. 122. ^f. 123. 23f-. 124. 17.66. 125. 5.25ft., 212. 0045 sq. ft., 98.328 sq. ft., 59.427 sq. ft. 126. $330. 127. 67^ mi. 128. 7.15p.M., Apr. 5. 129. 9.55A.M. 130. 7 hr. 29 min. 46 sec. 131. 63 35' W. 132. 156 15' W. 133. 135 27' E. 134. (a) 70 15' W., (ft) 85 W., (c) 81 45' W. 135. 24.62%. 350 ANSWERS 136. (i) 4.16%, (ii) 14.6%, (iii) 28.79%, (iv) 17.8%. 137. 96,600. 138. $150. 139. $1.44. 140. 20%, or 16? per yard. 141. $8. 142. 24f%. 143. }. 144. $150. 145. 38|%. 146. 12J%. 147. (i)64 T V/ , (ii)3fgal. 148. (a) 59.99%, (6)32.34%, (c) 62.84%, (d) 49.02%, (e) 65.57%, (/) 36.75%, (?) 65.28%, (ft) 40.83%, (0 41.94%, 0) 39.01%, (fc) 42.02%. 149. $1.61. 150. $1.20- 151. $182.67. 152. 7 17s. 8d 153. 41 16s. Id. 154. $300. 155. $14,400. 156. $400. 157. $1200. 158. $1000. 159. 7%. 160. $500. 161. 62^. 162. $546.69. 163. July 23, $597.14. 164. $149.15. 165. $891.80. 166. $1004.72. 167. Mar. 11, 1903, $824.89. 168. $459.50. ' 169. 7500 sq. yd. 170. 3500 sq. yd. 171. $62.80. 172. 301|sq. yd. 173. 1 mi. 174. 1 mi. long, J mi. wide. 175. 4330 sq. ft. 176. 6495 sq. ft. 177. 355 in. 178. 314.16 sq. in. 179. 1256.64 sq. in. 180. 1.9531 cu. ft. 181. 6.07 in. 182. $.525. 183. $.882. 184. $1. 185. $.612. 186. $.756. 187. $.871. 188. $1.802. 189. $.755. 190. $3. 191. 512.12 francs. MISCELLANEOUS EXAMPLES (B) 1. T |o. 2. 38 15'. 3. 216. 4. 2.77ft. 5. 2.0575 yr. 6. 3 gal. 1.02 gi. 7. $4176. 8. $20.94. 9. Ill bu. 1 pk. 2J qt. 10. 17f . Ib. 11. IfjjJ. 12. $504. 13. 1.2857142. 14. 1.777+. 15. 16ft. 16. 12.56A.M. 17. $500. 18. $11.28. 19. 3J da. 20. 2ii|. 21. 45J bu. 22. 82.28 yr. 23. 23^. 24. 226.27+ rd. 25. $6.66. 26. 149.61+ gal. 27. 734 rd. 28. 90yd. 29. $16.67. 30. 137 da. 31. $331.86. 32. 4531 mi. 33. Answers will depend on date. 34. 14. 35. $9.60. 36. 10 A. 37. 30 inin. 50 sec. 38. 6ch. 89.6+b. 39. 116.35$. 40. $37.50, $30, $52.50. 41. 62.72 bu. 42. 3403J T. 43. Neither gained nor lost. 44. 9 r %. 45. 2295. 46. 140 A. 47. 4844.996. 48. $75.15, $83.50, $300.60. 49. 6 hr. 1 min. 28 sec. A.M. 50. Answer will depend on day calculation is made. 51. $2800. 52. $604.86. 53. $50 loss. 54. 25%. 55. $460. 56. 10.8%. 57. 16 yr. 8 mo., 12 yr. 6 mo., 11 yr. 58. $453.561. 59. $352.35. 60. 4J-%. 61. 210,526.3 Ib. 62. 8275.80. 63. 35J%. 64. 410 on the $ 100, $4.92. 65. $6.67. 66. $56.25. 67. $30,563.38+. 68. 80. 69. 2 X \%. 70. 33J%. 71. $855. 72. $27,200. 73. $32.32 loss. 74. $1092.56. 78. 30%. 79. 1 yr. 7 mo. 12 da. 80. $492.50. 81. 300%. 82. $3200. 83. 33 J%. 84. 35%. 85. $37.525. 86. $108.90. 87. 25%. 88. $753. 89. $5.69. 90. Dec. 13, 1902. 91. $6644.63. 92. 83515.63. 93. 3|%. 94. $530. 95. $15,000. 96. $1533.75. 97. 9f% 98. Yes, $20 better. 99. $2042. 100. 25%. 101. $1000. 102. $7045. 103. $28.01, $42.02. 104. 6}Jf Tarr and McMurry's Geographies A New Series of Geographies in Two, Three, or Five Volumes By RALPH 5. TARR, B.S., F.G.S.A. CORNELL UNIVERSITY AND FRANK M. McMURRY, Ph.D. TEACHERS COLLEGE, COLUMBIA UNIVERSITY TWO BOOK SERIES Introductory Geography Complete Geography THE THREE BOOK SERIES 60 cents $1.00 FIRST BOOK (4th and 5th years) Home Geography and the Earth as a Whole 60 cents SECOND BOOK (6th year) North America 75 cents THIRD BOOK (jth year) Europe and Other Continents . . . 75 cents THE FIVE BOOK SERIES FIRST PART (4th year) Home Geography 40 cents SECOND PART (sth year) The Earth as a Whole . . . .40 cents THIRD PART (6th year) North America 75 cents FOURTH PART (yth year) Europe, South America, etc. . . .50 cents FIFTH PART (Sth year) Asia and Africa, with Review of North America (with State Supplement) 50 cents Without Supplement 40 cents Home Geography, Greater New York Edition Teachers' Manual of MCMURRY Method in Geography. By CHARLES A. 50 cents net 40 cents net To meet the requirements of some courses of study, the section from the Third Book, treating of South America, is bound up with the Second Book, thus bringing North America and South America together in one volume. The following Supplementary Volumes have also been prepared, and maybe had separately or bound together with the Third Book of the Three Book Series, or the Fifth Part of the Five Book Series : SUPPLEMENTARY VOLUMES New York State ... 30 cents The New England States . 30 cents Utah 40 cents California 30 cents Ohio 30 cents Illinois 30 cents New Jersey .... 30 cents Kansas 30 cents Virginia 30 cents Pennsylvania .... 30 cents Tennessee 30 cents Louisiana 30 cents Texas 35 cents When ordering, be careful to specify the Book or Part and the Series desired, and whether with or without the State Supplement. THE MACMILLAN COMPANY 64-66 FIFTH AVENUE, NEW YORK BOSTON CHICAGO ATLANTA SAN FRANCISCO FIRST BOOK OF PHYSICAL GEOGRAPHY, By RALPH S. TARR, Professor of Dynamic Geology and Physical Geography at Cornell University. tamo. Illustrated. Half leather. $1.10, net. M The style is simple, direct, and the illustrations helpful; the book, indeed, being so attractive that one hopes it will inspire even in the pupil who gives it briefest time a longing to know more of the marvels of our world." Providence Journal. " Although intended for school use, there are few readers who will not be profoundly interested in the volume, which is profusely illus- trated. Technical terms are avoided as far as possible, and where they are used they are clearly explained." Boston Transcript. " This book is packed with information needed by every grammar- school pupil; but what signifies vastly more, the pupil gets this infor- mation in a way that gives thorough discipline in observation, careful reading, discriminating thinking. This book is the best possible proof of the statement that all new science work depends for its value upon being rightly taught. This book is an admirable presentation of prac- tical pedagogy." Journal of Education. "The style of Professor Tarr's book is literary, scholarly, and sane; a pleasing relief from the disjointed paragraphs of some of his con- temporaries. . . . This book will prove a formidable rival to the best physical geographies now in the field. " - Educational Review. " No written description of the book can do justice to it. It will well repay personal examination." New York Education. THE MACMILLAN COMPANY 66 FIFTH AVENUE, NEW YORK. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. OCT' 5 1936 ._.._::.,_ OCT 6 1936 OCT 1.1. '723: 918235(54,03 //73 /707 THE UNIVERSITY OF CALIFORNIA LIBRARY