UC-NRLF kl\l. 1.11 i^>^ii-.vi'Vf?§v;^fi>^^ I IN MEMORIAM FLORIAN CAJORl INTERMEDIATE BOOK ^M& THE MACMILLAN COMPANY NEW YORK • BOSTON • CHICAGO • DALLAS ATLANTA • SAN FRANCISCO MACMILLAN & CO., Limited LONDON • BOMBAY • CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, Ltd. TORONTO SCHOOL ARITHMETICS INTERMEDIATE BOOK ,» » » . > BY FLORIAN CAJORI THE MACMILLAN COMPANY 1915 AU rights reserved OOPYBIGHT, 1915, By the MACMILLAN COMPANY. Set up and electrotypcd. Published July, 1915. CAJORf J. S. Gushing Co. — Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE As in the Primary Arithmetic, so in this Inter- mediate Arithmetic, the aim is to render the sub- ject attractive to the pupil, without sacrifice of serious intent. The pupil's self -activity is encour- aged. By the selection, so far as possible, of prob- lems bearing on the practical life of to-day, the pupil is made to feel that he is engaged in studies that are truly worth his while. Our constant aim has been to lay emphasis upon fundamental operations. Frequent reviews enable the pupil to hold in mind the new knowledge he has acquired. As in the Primary Arithmetic, so here, the tech- nique of arithmetic is simplified, with the aim of securing greater economy of effort. Thus the sub- ject of ratio is robbed of some of its terrors by its identification with a "common fraction." A pro- portion expresses the "equality of two common fractions." There is no need of the terms " ante- cedent " and " consequent." Again, there is given, as an alternative, a simplified method of reading decimal fractions. After the theory of decimal fractions is understood, .425 is read " Point, four, Q^^^PfH.^ Vi PREFACE two, five." It is recognized as a great convenience to omit the denominator in writing decimal frac- tions ; why not enjoy the same convenience in reading decimal fractions ? The author takes pleasure in acknowledging the help he has received in the preparation of this series of texts from several teachers in the public schools of Colorado Springs, particularly from Mrs. L. D. Coffin, Mrs. S. J. Lewis, Miss Minnie L. McCall, and Miss Edna Kinder. FLORIAN CAJORI. CONTENTS PART I PAOB Notation and Numeration 1 Review 1 Fundamental Operations 6 Review 6 Properties of Numbers 29 Cancellation 34 Common Fractions 37 Review 87 Quantity and Cost 99 Decimals 109 Review . . . 132 Denominate Numbers 140 Analysis and Solution of Problems .... 172 Approximations 181 Averages 183 Direct Method of Solution 186 Problems Illustrated by Graphs 189 PART II Percentage IW Application of Percentage 214 Interest = • 224 vii viii CONTENTS PAGE Bills and Checks .... = ... 233 Problems on Industry 237 Review 250 Positive and Negative Numbers 253 General Review 283 Tables 297 Tests 298 INTERMEDIATE BOOK • • • 5 INTERMEDIATE BOOK PART ONE NOTATION AND NUMERATION Review — Oral 1. 1. Eead the following numbers : 125 3,604 40,587 987,500 360 1,234 18,356 198,799 405 1,034 97,876 897,747 987 1,204 99,999 534,689 2. Tell how many units there are in units' place, in tens' place, in hundreds' place, in thousands' place, in ten-thousands' place, in hundred-thou- sands' place, in each of the numbers in this exercise. Written Exercise 2. Write as one number : 10 + 7 200 + 40 + 5 3000 + 200 + 50 + 8 20 + 8 300 + 60 + 7 4000 + 500 + 60 + 7 30 + 9 400 + 90 + 7000 + 600 + 50 + 4 40 + 6 700 + 00 + 8 8000 + 900 + 10 + 70 + 7 100 + 10 + 1 9000 + 000 + 90 + 9 2':; '"'. INTERMEDIATE BOOK r.f Written Exercise 3. 1. Make a number chart like the illustration. Orders of Whole Numbers Orders of Decimals Millions Thousands Units or Ones Thousandths to c o 1 ■a c 3 X M C o 1 k C i (0 ■a c CJ u> 3 o JZ *-» ■6 0) i- ■a E 3 I C ti CO 3 E ■M c 1- «0 £ m to 3 o to 1 T3 C 3 I to c 0) «o o c o 1- o to 'E D to s: •*■> c to s: ■*-> n •a c 3 1 to x: ■M T3 C (« to 3 O JZ \- 3d Period 2d Period 1st Period 1st Period Oral Exercise Based on the Number Chart 4. What place is always occupied : 1. By the figure that stands for ones? 2. By the figure that stands for tens ? 3. By the figure that stands for thousands ? 4. By the figure that stands for hundred-thou- sands? 5. By the figure that stands for hundreds ? NOTATION AND NUMERATION 3 6. By the figure that stands for millions ? Name : 7. The order of whole numbers. 8. The order of decimals. Oral Exercise 5. 1. How many ones make 1 ten ? 2. How many tens make 1 hundred ? 3. How many hundreds make 1 thousand ? 4. How many times greater is each one than the one in the next place or order to the right ? 5. How many times greater is one of thousands* order than one of tens' order ? 6. How many times greater is one in millions' place than one in thousands' place ? 7. How many times greater is one in ten-thou- sands' place than one in hundreds' place ? 8. How many ones of the thousands' order are equal to one of the hundred-thousands' order? Since each number is ten times greater than the next unit to the right, the system of writing num- bers is called the decimal system. The word deci- mal comes from the Latin word meaning ten. It is easier to read numbers if a comma is placed between the orders of hundreds and thousands and the orders of hundred-thousands and millions. 4 INTERMEDIATE BOOK Oral Exercise 6. Separate the following numbers into their orders and read the numbers : 1. 4,327 Process and Explanation 4,327 = 4 thousands + 3 hundreds + 2 tens -f 7 ones = 4000 +300 +20 +7 2. 125 3. 360 4. 405 5. 987 6. 542 7. 1,234 8. 4,034 9. 7,204 10. 9,560 11. 5,610 12. 18,356 i3. 78,347 14. 92,701 15. 72,079 le. 40,679 i7. 198,765 18. 819,675 19. 576,198 20. 918,567 21. 765,951 Written Exercise 7. Write as one number : 10 + 7 200 + 40 + 5 3,000 + 1,600 + 300 + 50 + 8 90 + 8 100 + 90 + 6 40,000 + 5,000 + 700 + 90+4 60 + 9 300 + 63 + 4 30,000 + 6,000 + 200 + 70 + 5 Written Exercise 8. Fill in the missing orders and write the numbers : 1. 2 hundreds + 5 ones. Process and Explanation 2 hundreds + 5 ones = 2 hundreds + tens + 5 ones = 205 NOTATION AND NUMERATION 5 2. 2 ten-thousands -f 7 thousands + 6 hundreds -f- 7 ones. 3. 3 hundred-thousands -h 9 hundreds. 4. 5 millions + 6 hundred-thousands -h 4 ten- thousands. 5. 3 hundred-millions H- 2 ten-millions H- 5 mil- lions + 5 tens. 6. 1 million + 4 ten-thousands. 7. 6 ten-millions + 5 hundred-thousands -f- one. 8. 4 hundred-millions -h 5 ten-thousands -h 6 tens -f one. Oral Exercise 9. Read the following numbers : 7,123 10,000 673,854 4,275 67,431 705,239 7,155 10,500 900,432 1,279 30,006 821,006 10,000,000 9,234,567 For what does stand in each of these numbers? REVIEW OF FUNDAMENTAL OPERATIONS Addition Oral Exercise 10. 1. Beginning with 8, add by 8 to 96. 2. Beginning with 2, add by 7 to 100. 3. Beginning with 1, add by 9 to 100. 4. Beginning with 101, add by 6 to 200. 5. Beginning with 0, add by 5 to 100. 6. Beginning with 0, 1, 2, 3, 4, 5, add by 6 to 100. 7. Beginning with 0, 1, 2, 3, 4, 5, 6, add by 7 to 100. 8. Beginning with 0, 1, 2, 3, 4, 5, 6, 7, add by 8 to 100. 9. Beginning with 0, 1, 2, 3, 4, 5, 6, 7, 8, add by 9 to 100. Drill on Difficult Combinations 11. Extend each set of operations and drill : 1. 7 7 7 etc. 2. 8 8 8 etc. 8 18 28 § 1? ?^ 3. 9 9 9 etc. 4. 7 7 7 etc. 8 18 28 1 ]1 ^ 6 REVIEW OP FUNDAMENTAL OPERATIONS 7 5. 8 8 8 etc. 6. 9 9 9 etc. 9 19 29 7 17 27 7. 7 17 27 etc. 8. 7 17 27 etc. 7 7 7 8 8 8 9. 7 17 27 etc. lo. 8 18 28 etc. 9 9 9 7 7 7 11. 8 18 28 etc. 12. 8 18 28 etc. 8 J _8 ^ _9 _9 13. 9 19 29 etc. 14. 9 19 29 etc. 7 7 7 8 8 8 15. 9 1 29 etc. 9 9 _9 12. Add rapidly : 1. 65 2. 67 3. 26 4. 37 5. 48 6. 77 . 46 43 75 65 59 54 7. 58 8. 79 9. 46 10. 36 11. 86 12. 74 47 48 98 67 45 . 75 8 INTERMEDIATE BOOK Written Exercise 13. Add and check : Process 1. 103 Explanation. — The columns may 9 737 he added separately and the results ^gg added to find the sum. In this process o ot\A there is no carrying. The process is ' p, somewhat simpler if sums of ten are — ? grouped together. 3 and 7 make a "^^ group of ten. Indicate the other 1*^ groups of ten in these examples. In 21 adding the first column of Ex. 1, say : 17 five, fifteen, twenty-five. 1 29,255 2. 317 3. 875 4. 9,762 5. 2,769 943 465 4,348 8,441 862 878 7,489 4,798 198 760 4,765 7,452 94,628 432 3,484 3,000 6. $ 13.49 7. 9,876 ft. 8. 3,075 1b. 3.71 987 64,072 90.76 7,654 985 87.33 1,063 70,308 108.74 9,616 7,293 98.77 1,007 2,187 In adding dollars and cents, place the decimal points under each other. REVIEW OF FUNDAMENTAL OPERATIONS 9 Written Exercise 14. Copy and add : 1. $475.83 +$437.75 +$789.85. 2. $1,198.95 +$364.20 + $375.98. 3. $754 +$7,689.50 +$5,000. 4. $376.95 + $1,234.25 + $1,990.05. Written Exercise 15. Add by columns and by lines : 1. 9,234 + 8,920+ 356 + 9,076 = 8,456 + 7,122 + 3,738 + 5,951 = 7,078 + 6,324+ 394 + 5,230 = ■ 6,910 + 6,256-1- 404 + 4,545 = + + 16,263 + 9,790 + 9,920 + 11,002 = 7,465 + 8,610 + 8,934+ 9,003 = 7,067+ 828 + 6,945+ 8,004 = 6,869 + 7,485 + 5,967+ 7,112 = + + + = 3. $40.12 + $57.28 + $ 4.23 +$35.60 33.14+ 32.93+ 4.45+ 25.78 15.16+ 21.32+ 46.47+ 15.96 7.18 + 3.3 4+ 4.89 + 6.01 + + + 10 INTERMEDIATE BOOK 4. $50.71 + $58.74 +$49.89 + $43.14 = 47.23 + 38.88 + 19.90 + 34.15 = 17.45 + 18.90 + 8.90 + 16.17 = 7.68 + 9.01 + + 6.01 + + 10.19 = + = Subtraction 16. 1. Beginning with 50, subtract by 2 to 0. 2. Beginning with 41, subtract by 2 to 1. 3. Beginning with 60, subtract by 3 to 0. 4. Beginning with 71, subtract by 3 to 2. 5. Beginning with 80, subtract by 4 to 0. 6. Beginning with 81, subtract by 4 to 1. 7. Beginning with 83, subtract by 4 to 2. 8. Beginning with 90, subtract by 5 to 0. 9. Beginning with 96, subtract by 5 to 1. 10. Beginning with 94, subtract by 5 to 4. Continue the exercise. Drill on Difficult Combinations 17. Drill until habits of accuracy and rapidity are established. 1. 17 27 37 47 57 etc. 7 7 7 7 7 REVIEW OF FUNDAMENTAL OPERATIONS 11 2. 17 27 37 47 57 etc. _8 _8 ^ J _8 3. 17 27 37 47 57 etc. _9 _9 _9 _9 _9 Do likewise with 18 and 7, 19 and 7, 18 and 8, 19 and 8, 19 and 9. 18. Subtract rapidly : 1. 67 2. 60 3. 75 4. 92 5. 80 25 28 46 43 34 6. 55 7. 45 8. 99 o. 91 10. 83 27 32 35 57 45 11. 74 12. 95 13. 52 14. 73 15. 45 32 26 23 24 39 Written Exercise 19. Subtract and check : p Ij /~v pt "pi ca Q .^ -or Explanation. 4 and 1 are 5; 1. 4U,7d& g ^^^ y ^^.^ ^3 . ^ ,^^^ 2 ^j ^j^^ j^ ^^^^^^ are 7 ; 9 and 1 are 10. 1,171 Ans. 2. 107,864 3. 987,603 4. 367,890 72,895 367,809 176,805 5. 134,578 6. 3,764,001 7. 500,897,431 90,387 1,987,373 176,354,897 12 INTERMEDIATE BOOK 8. $754.37 9. $7,689.50 lo. $7,989.50 376.95 1,234.25 1,990.35 Written Problems 20. 1. A man had $ 1,000 in a bank. He drew out the following amounts: $95.50, $180.65, $75.05, $96.75, $7.85, $1.25, $60.73. How much money remained in the bank ? 2. A man earned $ 30.45 in January, $ 40.26 in February, $42.19 in March, $61.34 in April, $53.50 in May, $39.29 in June, $27.00 in July, $47.45 in August, $40.10 in September, $ 50.24 in October, $ 55.75 in November. During the month of December he was idle. If his expenses during the year were $ 240, how much did he save ? 3. The Panama Canal is about 50.4 mi. long. The first 8 mi. are a sea-level channel. The next 24 mi. are through a lake above sea level. From this point the channel passes 7^ mi. through a cut in the Culebra Hill. The channel then passes through a lake 5 mi. long. The rest of the distance is a sea-level channel. How long is the last section of the canal ? 4. The aggregate population of 25 cities of the United States in 1910 was 11,042,500. In 1900 they had an aggregate population of 8,273,482 ; in 1890 they had 6,213,583. What was the increase REVIEW OF FUNDAMENTAL OPERATIONS 13 in population from 1890 to 1900 ? What was the increase in population from 1900 to 1910 ? How does the increase between 1890 and 1900 compare with the increase between 1900 and 1910? Which is the greater ? How much greater is it? Multiplication Okal Drill 21. How much is 1. 7x8 2. 6x9 3. 9x7 5. 9x4 6. 9x6 7. 7x9 9. 6x12 10. 12x7 11. 12x9 13, 7x11 14. 11x8 15. 9x11 4. 8x8 8. 8x7 12. 8x12 16. 11x6 Oral Exercise 22. Multiply the numbers in the upper line by each number in the lower line : 3 7 6 5 2 4 11 8 12 10 9 X 4 7 9 3 12 6 5 11 8 2 3 7 6 5 2 4 11 8 12 10 9 X 10 100 1000 Make devices that provide drill upon the prod- ucts in which you fail. 14 INTERMEDIATE BOOK Drill — Magic Circle 23. 1. If the 8 vacant spaces are properly filled, we get circles, called " magic circles." Lines radi- ating from the center divide the circles into 8 parts. Fill the empty spaces so that the sum of all 8 numbers in 4c/ each part, plus the number in the middle, is 360. 2. If the empty spaces are properly filled, then the sum of the 8 numbers in any one ring, together with the number in the middle, is 360. Add and show that this is true. 3. Show that the sum of the numbers in any half ring, above the line AB or below it, together with half the middle number, is 180. 4. Show that the sum of any four numbers, each next to the other three, together with half the middle number, is equal to 180. Thus, 20, 67, 75, 12 are four such numbers, also 14, 25, 72, 63. REVIEW OF FUNDAMENTAL OPERATIONS 15 Written Exercise 24. Find the product : 1. 648 14 Process Explanation. — To multiply 648 by 648 14 14 means to multiply 648 b}^ 4 and 10. The process of multiplying 648 by 4 is to multiply by 4 ones. It remains to multi- ^ ply 648 by 1 ten. This gives 648 tens, ^^^ Since the number is tens, the first figure, 9072 8, is written in tens* place. The vacant place is sometimes filled in with a 0. This is not necessary. 4. 295 5. 798 32 16 2. 754 3. 423 24 36 6. 581 7. 649 24 52 10. 953 11. 825 21 6^ 14. 558 15. 275 28 ■ 27 18. 589 19. 643 76 87 22. 7,245 58 23. 9,398 69 8. 959 9. 764 27 24 12. 647 13. 939 68 44 16. 821 17. 478 97 84 20. 824 21. 7,468 48 47 24. 8,556 25. 5,729 76 87 16 INTERMEDIATE BOOK 26. 5,749 27. 3,128 28. 1,847 29. 3,985 89 96 97 98 30. 9,346 31. 9,098 32. 9,009 33. 9,999 99 78 89 86 Written Problems 25. Find the cost of : 1. 64 gallons of oil at 17 ^ per gallon. 2. 47 pounds of tea at 48 ^ a pound. 3. 6 dozen penknives at $ 8.75 a dozen. 4. 25 barrels of flour at $ 4.25 a barrel. 5. 50 dozen eggs at 27^ a dozen. 6. 620 pounds of chicken at 19 ^ a pound. 7. 72 packages of crackers at 15 ^ a package. 8. 6 firkins of butter, each containing 100 pounds, at 26 ^ a pound. Mxiltiplication by more than Two Digits 26. Multiply: 1. 1,854 by 237. Process Explanation. — Multiply by 7 ones. 1 ftf^l Write the product so that its right-hand 'oQ7 ^^^^^ ^^ ^^ ones' column, under the 7. Multiply by 8 tens. Write the product 12978 so that its right-hand digit is in tens' 5562 column. 3708 Then multiply by 2 hundreds. Write i qq qqo the product so that its right-hand digit is in ' hundreds'* column. Add the three products. REVIEW OF FUNDAMENTAL OPERATIONS 17 2. 175 bj 206. Process Explanation. — In this exercise it is nec- -ifrr essary to multiply by six ones and two hun- 9nA dred. Multiplying by two hundred may be — - — done by multiplying by two and moving the product two places to the left. The vacant ^^^ places are shown under the 5 and the cipher. 31,550 3. 312 by 620. Process 312 Explanation. — Multiply first by two 620 tens^ then by six hundreds. Fill in the first 5240 place with a cipher. There are no ones. 1872 '^^^ ^^^^ partial product is 312 times 2 tens. 193,449 Written Exercise 27. Multiply : 1. 123 by 123 2. 497 by 132 3. 568 by 231 4. 759 by 322 5. 897 by 432 6. 575 by 405 7. 612 by 740 8. 1,019 by 305 9. 765 by 600 10. 896 by 500 11. 1,324 by 901 12. 897 by 700 13. 678 by 509 14. 437 by 987 Division 28. Give the answer : 1. 56^8 2. 63-^ • 9 3. 72-^8 4 48-4-6 18 INTERMEDIATE BOOK 5. 54^9 6. 64-^8 7. 63-^7 8. 49-^7 9. 81-^9 10. 84-4-7 11. 45H-9 12. 55^5 13. 60-^12 14. 72^12 15. 84-4-12 le. 99-i-ll Oral Exercise 29. Divide each number in the upper line by each number in the lower line. 2. 60 24 36 72 48 144 96 -^ 2 3 6 4 12 9 1,000 10,000 17,000 90,000 -r- 10 100 1,000 30. Divide 1. 43,641 Process 614^1 71)43641 426 104 71 331 284 47 Division by Two Digits by 71. Explanation. — Indicate above the dividend by a small cross mark, x, the last or right-hand figure in the first partial dividend. 436 is the first partial dividend. (Try 436^70.) 436^71 = 6 and a remainder. Place the 6 above the last figure in the partial dividend. 6 x 71 = 426 ; 436 - 426 = 10, the remainder. Bring down the 4, the next figure in the dividend. 104 is the second partial REVIEW OF FUNDAMENTAL OPERATIONS 19 dividend. (Try 104 -- 70.) 104^71 = 1 and a remain- der. 1 X 71 = 71 ; 104 - 71 = 33, the remainder. Bring down the 1, the next figure in the dividend. (Try 331^70.) 331^71 = 4 and a remainder. 4x71 = 284; 331 - 284 = 27, the last remainder. The quotient is 614 and the remainder 47, or 614|^. Ans, 2. $ 158.22 by 54. Process Explanation. — The first partial $ 2 93 dividend is 158. Trial division indi- ^ Gates that 54 is contained in 153, 3 54)$ 158.22 ^ij^gg^ 3x54 = 162. 162 is greater IQ^ than the first partial dividend. 1 58 -^ 54 502 = 2 and 50 remainder. The second 486 partial dividend is 502. 502-^54=9 162 ^^^ 16 remainder. The third partial 252 dividend is 162. 162 -- 54 = 3. $2.93 Ans. 3. 2,790 by 29. Process 96^ Explanation. — 279 is the first par- 9QT97^ tial dividend. Use 30 for trial divisor. ^9)^790 279-^30 = 9. 9x29 = 261; 279-261 "^^^ =18 remainder. The next partial divi- 180 dend is 180. 180 -^ 30 = 6. 6 x 29 = 174 ; 174 180 - 174 = 6 remainder. 96/^ Ans. 6 When the ones' figure is 9, as 29, 39, etc., it is generally easier to use 30, 40, etc., as a trial divisor. 20 INTERMEDIATE BOOK Written Exercise 31. Divide : 1. 4,875 by 31 2. 2,849 by 42 3. 18,785 by 23 4. 54,632 by 77 5. 64,751 by 78 6. 60,543 by 89 7. 49,790 by 96 s. 38,000 by 67 9. 50,000 by 65 lo. $ 657.50 by 24 11. $ 785.60 by 26 12. $ 6000.00 by 33 Division by more than Two Digits 32. Divide : 1. 6,084 by 234. Process Explanation. — 234 is contained in 26 608, 2 times, and a remainder. 1_ Write the 2 of the quotient. 234)6084 234 into 1404 goes 6 times. 468 Write the 6 of the quotient. There 1404 is no remainder. 1404 The answer is 26. 2. 8,499 by 293. Process Explanation. — Since 293 is nearly 29 300, it is convenient to use 300 as our ^ trial divisor to find the first figure in 293)8499 the quotient. 586 300 is contained in 849, 2 times. 2639 Write the 2 of the quotient over the 9, 2537 the last figure of the partial dividend, 849. 2 ^^^ ^^^^ 2639 goes nearly 9 times. Try 9. REVIEW OF FUNDAMENTAL OPERATIONS 21 293 X 9 is less than 2639. Hence 9 is the second figure of the quotient. The quotient is 29, the remainder is 2. Written Exercise 33. Divide and check : 1. 19,623 by 211 2. 17,347 by 209 3. 40,260 by 915 4. 52,288 by 817 5. 15,022 by 406 6. 33,744 by 703 7. 20,262 by 614 8. 16,302 by 429. 9. 37,700 by 725 10. 50,255 by 529. 11. 19,277 by 663 12. 28,644 by 682 13. 20,000 by 607 14. 50,000 by 600 15. 60,000 by 705 16. 75,000 by 850 Divisors 10, 20, 100, 600 34. Divide: 1. 87 by 10. Process Explanation. — 87 = 8 tens + 7 ones. ^ro 10 is contained in 8 tens + 7 ones 8 times lj9)8|7 with 7 as a remainder or S^. A short method of dividing a number by 10 (when the dividend does not end with a 0) is to separate the tens from the ones by a vertical line and divide. 2. 7,300 by 100. Process Explanation. — 7,300 = 73 hundreds. H-o 1 hundred is contained in 73 hundreds mWm 'VsTL quotient. 22 INTERMEDIATE BOOK A short method of dividing by 100 a number that ends with two O's is to cancel the two O's at the right of the divisor and the dividend, and divide. 2. 3,660 by 200. Process Explanation. — Write the explana- 18-^ tion. 200)36|6O Explain a short method. 3. 4,575 by 300. Process Explanation. — Write the explana- 15^ tion. 300)45175 Canceling a cipher at the right of a number divides the number by what ? Canceling two ciphers at the right of a number divides the number by what? Written Exercise 35. Divide : 1. 30)127 2. 40)4,326 4. 60)4,563 5. 70)7,564 7. 90)9,046 8. 30)2,147 10. 80)3,064 11. 90)6,847 13. 100)8,400 14. 200)7,800 16. 500)64,500 17. 600)9,600 19. 800)89,700 20. 900)54,600 3. 50)6,534 6. 80)6,574 9. 40)8,435 12. 70)9,037 15. 400)97,600 18. 700)8,890 21. 600)93,310 22. 700)33,970 23. 800)828,240 24. 900)565,830 REVIEW OF FUNDAMENTAL OPERATIONS 23 Oral Problems 36. 1. What is the cost of 20 lb. of beef at 23^ a pound ? 2. A clerk in a store sold 30 sets of books at $ 6 per set. How much did he receive altogether? 3. How far will a train travel in 12 hours at the rate of 40 miles an hour ? 4. At 4 miles per hour, how long will it take a man to walk 52 miles ? 5. How many inches are there in 30 feet ? In 40 feet? 6. How many feet are there in 240 inches ? 7. At 6^ a quart, how many quarts of milk can be bought for 78^? 90^? 8. What is the cost of eggs a dozen, if 7 dozen cost $2.80? 9. What is the cost of 9 dozen eggs at 40^ a dozen ? 10. How many square feet in 288 sq. in.? Written Problems 37. 1. Find the number of minutes in 24 hours. 2. How many hours in the month of July ? 3. Find the number of hours in 365 days. 4. How many feet in 76 miles ? 1 mi. = 5280 ft. 24 INTERMEDIATE BOOK 5. What is the weight in pounds of 15|- tons of coal ? 1 T. = 2000 lb. 6. How many street cars are necessary to carry 901 passengers, if 53 passengers are put into each car? 7. A train runs 984 miles in 24 hours. How many miles does it run an hour ? 8. On a city street there are 139 houses. Each house contains 4 families and each family 6 persons. How many persons live on the street ? 9. A booklet has 65 pages. There are 30 lines on each page and in each line 43 letters. How many letters are there in the book? 10. 4,368 oranges are to be packed in 56 boxes of equal size. How many oranges must be put into each box? 11. What number multiplied by 73 will give 4,526? 12. How many bushels of potatoes will 54 acres yield, if each acre yields 243 bushels? 13. 35 acres yield 8,295 bushels of potatoes. How many bushels is this per acre? 14. It is 85 miles from Chicago to Milwaukee. How many rods is this? 15. From New York to Buffalo it is 411 miles. How many miles does a man travel in going 18 times from one city to the other? REVIEW OF FUNDAMENTAL OPERATIONS 25 16. A locomotive has been run 75 times between Chicago and New York. The distance between these cities is 908 miles. How many miles has this locomotive traveled? 17. If 29 acres of land cost $ 3,828, what is the cost of 1 acre? 18. If an automobile travels 23 miles an hour, how far will it go in 78 hours? 19. At 27 miles an hour, how long will it take an automobile to go 351 miles? 20. At 57^ a bushel, what is the cost of 125 bushels of potatoes? 21. The President of the United States has a salary of $ 75,000 per year. How much does he receive per month? Per day, counting 365 days to the year? 22. A man started on a journey of 620 miles. After he had traveled 15 hours at the rate of 39 miles an hour, how far was he from his journey's end? 23. How many pounds of coffee, at 60^ a pound, will cost as much as 120 gallons of molasses at 57^ a gallon? 24. A merchant makes five payments of $ 1,275 each on a debt and finds that he still owes $ 785. How much was the debt? 26 INTERMEDIATE BOOK 25. A store has 475 boxes of soap, each box containing 175 cakes. What is the entire number of cakes? 26. A factory made 2,748 suits during a season. At $ 19 a suit, how much, was received for them? 27. A man sold a farm for $ 5,775 and gained $1,200. What would have been the selling price if he had gained $ 1,354? 28. If 525 gallons of milk sell for $ 84, what is the rate per gallon? Oral Exercise 38. 1. Let the drawing represent 1 sq. yd. 2. Then the 9 smaller divi- sions represent what? 3. How many square feet in 1 sq. yd.? 4. What does each small di- vision in A rep- resent? 5. How many small squares in one row in ^? 6. How many rows of small squares in A are there? :::::j5::::ii REVIEW OF FUNDAMENTAL OPERATIONS 27 7. How many small squares in J.? 8. Make the table of square measure. Construction Exercise 39. 1. Draw a square, one inch long and one inch wide. Call it a square inch. 2. Draw a rectangle 4 in. long and 2 in. wide. The picture shows the rectangle smaller than it really is. How many square inches are there in one row? In the two rows? What is the area of the rectangle? Tell how to find the area of a rectangle. 3. How many square inches are there in a square that is 12 in. long and 12 in. wide? How many square inches make a square foot? 4. How many square feet are there in a square, 3 feet long and 3 feet wide ? How many square feet make a square yard? The area of a rectangle is ohtained hy multiplying its length hy its width. Written Exercise 40. 1. How many square feet in 139 sq. yd.? 2. How many square inches in 15 sq. ft.? 3. How many square yards in 1 sq. mi.? 4. How many square yards in 180 sq. mi.? 5. How many feet in 108 mi.? 28 INTERMEDIATE BOOK Written Problems 41. 1. Find the area of a floor, the dimensions of which are 6 yd. by 5 yd. 2. Find the number of square yards in the sur- face of a wall 9 ft. high and 15 ft. long. 3. A garden is 48 ft. by 120 ft. There is a gravel walk 3 ft. wide around the outside of the garden. Make a drawing of the garden (scale, 1 in. = 1 ft.) and show the walk. 4. What is the area of the garden in square feet? 5. What is the area of the walk in square feet? 6. What is the area of the garden and walk? 7. A piece of land, 16 rd. long and 10 rd. wide, is divided into 4 equal parts by lines 4 rd. apart. What are the dimensions of each part? 8. How many square rods in each part ? PROPERTIES OF NUMBERS Factors 42. A factor of a whole number is an integer that is an exact divisor of that number. Thus, the number 24 has, besides itself and 1, the factors 2, 3, 4, 6, 8, 12. A prime number is an integer that has no fac- tors, except itself and 1. 2, 3, 7 are examples of prime numbers. It is evident that 7 is not exactly divisible by any integer, except itself and 1. Hence it is a prime number. The same is true of 2 and 3. A prime factor is a factor that is a prime number. 2 and 3 are prime factors of 24. Oral Exercise 43. 1. Is 8 a prime number? Why? 2. Is 11 a prime number? Why? 3. There are 4 prime numbers between 1 and 10. Find them. 4. There are 4 prime numbers between 10 and 20. Find them. 5. Find the prime numbers between 20 and 30. 6. Name the prime factors of 4, 6, 8, 9, 10, 12. 29 30 INTERMEDIATE BOOK Divisibility of Numbers 44. A whole number is exactly divisible by 2, if the digit in ones' place is 2, 4, 6, 8, or 0. A whole number is exactly divisible by 5, if it ends in 5 or 0. A whole number is exactly divisible by 3, if the sum of its digits is divisible by 3. For example, 432 is exactly divisible by 3, because the sum of its digits (4 + 3 + 2) is 9, and 9 is exactly divisible by 3. A whole number is exactly divisible by 6, if it is even and exactly divisible by 3. Thus, 168 is even; 1 + 6 + 8 = 15; hence 168 is exactly divisible by 6. A whole number is exactly divisible by 4, if the number made up of the two right-hand digits is so divisible. Thus, 3148 is exactly divisible by 4, because 48 is so divisible. A whole number is exactly divisible by 8, if the number made up of the three right-hand digits is so divisible. Thus, 94,128 is exactly divisible by 8, because 128 is so divisible. Oral Exercise 45. 1. What numbers multiplied together give the following products : 33 16 26 27 32 30 34 35 36 42 44 45 48 49 PROPERTIES OF NUMBERS 31 2. What whole numbers will exactly divide: 44 46 63 64 54 56 72 81 84 88 96 108 3. Name two factors of 12. P HOC ESS 1 9 _ Q A Explanation. — 3 and 4 are called factors of 12. ^^ 2 and 6 are also factors of 12. 2x6 4. Name two factors of each of the following: 65 35 42 50 72 48 24 18 36 45 33 27 Oral Exercise 46. 1. Name two factors of each of the follow- ing: 6 10 15 21 26 33 35 39 49 55 25 22 2. Name three factors of each of the following: 8 12 16 18 24 27 30 36 42 45 48 54 3. Name all the factors of each of the following; 33 56 64 49 72 81 63 54 96 108 120 144 32 INTERMEDIATE BOOK 4. Which of the following numbers are exactly divisible by 2? By 3? By 4? By 5? By 6? By 8? 15, 18, 24, 30, 42, 45, 48, 62, 170, 212, 312, 330, 450, 790, 1,012, 3,618, 7,544, 6,908, 4,345, 7,545, 10,000. 5. The prime factors of 12 are 2, 2, and 3. What are the prime factors of 15? Of 18? Of 24? 6. Name the prime factors of 14; of 28; of 30; of 42; of 45; of 49; of 50. Finding Prime Factors 47. 1. Find the prime factors of 36. Process Explanation. — 36 is exactly 2)36 divisible by 2. Why ? 18 is exactly divisible by 2. ^)_ 9 is exactly divisible by 3. 3) 3 The prime factors of 36 are 36 = 2x2x3x3. 2x2x3x3. 2. Find the prime factors of 450. Process Explanation.— 10, 9, 5, and 3 are 5 )450 all factors of 540. 5)90 In finding the prime factors it is 3)18 better to factor by the prime factors. Therefore divide 450 by 5, 5, 3, 3, and 2. 3)_6 2 1 o\~2~ ^^'^^ prime factors of 450 are: ^ " 2x3x8x5x5. PROPERTIES OF NUMBERS 33 Written Exercise 48. Find the prime factors of 1. 72 75 81 2. 120 125 128 3. 201 195 333 4. 340 570 690 96 108 144 150 444 234 640 729 CANCELATION 49. 1. Divide 12 x 15 by 8 x 10. Process by Cancelation Explanation. — o o Take the common fac- 12 X 15_ ;i^ X U (Dividend) ^^''.^ °"* °* ^^ in the "o — T7r~~o — TnT /t\' • \ dividend and 8 in 8x10 8x10 (Divisor) ,, -,. . , . no the divisor, leaving 3 3 the factors 3 and 2. Take the common fac- l^^l^ = ^^^ = ^= 21 tor 5 out of 15 in the f>xlp 2x2 4 * dividend, and 10 in ^ ^ the divisor leaving the factors 3 and 2. Find the product of 3 x 3, the remaining factors in the divi- dend, and 2x2, the remaining factors in the divisor and divide. • 12 X 15 2. Divide, using cancelation, . 10 X 18 Explanation. — Re- Process 3®^^ t^^® common factors 6 and 5 from both the 2 3 -t% 7 « a dividend and the divisor. IfAi^ = MAM = 5=1 Divide the product of 10 X 18 ;I0 X ;i^ 6 the factors in the divi- ^ *^. dend by the product of the factors in the divisor. 34 CANCELATION 35 12 X 15 X 42 3. Divide, usino; cancelation, — ' 8x9x24 Process Explanation. — ^ w -| . Reject the common io -I ^ ^o fiji Ttf 4 0i factors 4, 3, and 3 I a"\, - ^l^^f^f - f-» both the divi. 8x9x24 |X^>^^^ dend and the divisor. Reject the common 7 factors 3 and 2 from ^ 5 Jif: the new dividend and jLfXjLpx ff _ ^ _. 43. divisor. ^ X ^ X ;^^ 8 ^ Divide the product ^ ^ 8 of the remaining fac- tors of the dividend by the product of the remaining factors of the divisor. PRINCIPLE TO BE REMEMBERED Dividing both dividend and divisor by the same factor does not change the value of the quotient. Written Exercise 50. Divide, using cancelation : 1. 24 X 49 X 18 by 12 x 21 x 36 2. 25 X 35 X 56 by 15 x 28 x 49 3. 32 X 108 X 100 by 64 x 36 x 25 4. 39 X 28x72 by 35x52x24 5. 16x40x24 by 20x8x48 36 INTERMEDIATE BOOK 30 X 32 X 36 X 40 350 x 120 x 72 ^50x16x20x18 '^* 500x63x6 625x49x81 1728 x 99 x 100 °* 75x210x140 ^* 1440x108x25 625x1728x121x1000 10. 2500 X 720 X 99 X 24 f 7 8 1 U 1* tf M* f ¥ ¥ -V- w- W m H 4| 5| 9* 25i 135j^ 97ii COMMON FRACTIONS Review 51. Read the fractions. Which number is the numerator ? Which the denominator ? 1. 2. 3. In which of these fractions is the numerator less than the denominator ? In which of these fractions is the numerator greater than the denominator ? One or more of the equal parts of a unit is called a fraction. The denominator shows into how many equal parts the unit is divided. The numerator shows how many of the equal parts have been taken. The numerator and denominator are sometimes called terms of the fraction. A proper fraction is one whose numerator is less than its denominator. For example, ^, f, f. An improper fraction is one whose numerator is equal to or greater than its denominator. For example, |, |, f . A mixed number is an integer and a common fraction taken together. 3f is a mixed number. 37 38 INTERMEDIATE BOOK Oral Exercise 52. Name the unit in each of these fractions : 1. fbu. I- mi. 1^0 lb. f ft. 2. f of an apple ; | of a circle ; | of a rectangle. 3. Into how many equal parts has each of these units been divided ? 4. What term of the fraction shows into how many equal parts the unit has been divided ? 5. How many equal parts of the unit have been taken in each of these fractions ? 6. What term of the fraction shows how many of the equal parts have been taken ? 7. f means that 1 unit is divided into 6 equal parts and that 4 parts are taken. I may be explained also as meaning 6)4. The dividend being smaller than the divisor, we indicate the division by writing |. The simpler form is |. 53. 1. 2. Drill Exercise 100^= $1 25^ = $i 75^ = $f 12 in. = 1 ft. 6 in. = 1 ft. 3 in. = i ft. lin. = ii2ft- 2 in. = 1 ft. 8 in. = 1 ft. COMMON FRACTIONS ^ 39 3. 16oz. = llb. loz. = 3iglb. 8oz. = ^lb. 2oz. = i-lb. 4 oz. = ilb. 12oz. = |lb. Oral Exercise 54. 1. What part of a dollar is 1 dime ? 2. What part of a dollar are 5 dimes ? 2 dimes ? 3. What part of a dollar is 1 nickel? What -part are 2 nickels ? 4 nickels ? 5 nickels ? 10 nick- els? 4. What part of a dollar are 2 nickels and 1 dime? 5. What part of a dollar are 2 nickels and 4 dimes ? 6. What part of a dollar are 2 dimes and 1 nickel ? 7. What part of a foot are 3 inches and 1 inch ? 8. Three books, each 1 inch thick, are placed upon a library shelf. What part of a foot of shelf- room do they occupy ? Reduction of Fractions 55. 1. Examine the drawing of the ruler. 3 2 1 2. How many half inches in one inch ? 3. How many fourths of an inch in one inch ? 40 INTERMEDIATE BOOK 4. How many eighths of an inch does f in. lack to make 1 in. ? 5. How many fourths make one half ? 6. Compare ^ in. in length with J in. 7. How many eighths in one fourth ? 8. Examine the draw- ing of the square. 9. How many thirds of the whole square are shaded ? 10. How many ninths of the whole square are shaded? The illustration shows that | = f. 11. Divide both terms of the fraction | by 3, thus: ? = -. To divide both terms of the fraction ^ 3 3 f by 3, cancel the common factor 3 and reduce f to lower terms, f . ^ is reduced to ^ by canceling the common factor 1 4, thus: 1=1. 12. How is I reduced to its equal, ^ ? Explain by diagram. COMMON FRACTIONS 41 Changing the forms of fractions without chang- ing their value is called reduction of fractions. PRINCIPLE TO BE REMEMBERED Dividing both the numerator and the denomi- nator of a fraction by the same number does not change the value of the fraction. This process reduces the fraction to lower terms. Reduction of Fractions to Lower Terms 56. Reduce to lower terms : 1. 6 9* Process Explanation. — Cancel from the nu- merator and the denominator the common 2 9 3 factor 3. This leaves the factor 2 in the numerator and 3 in the denominator. The fraction is thus reduced to lower terms. «4 'a3 4.4 r2 fi2 2. 3 3. 3-5 4. g 5. ^ 6. 3-2 7. 3^ 8. 3^ 9. If 10. I 11. f 12. I 13. I 14. ^ 15. 3-^2 16. i\ Written Exercise 57. 1. Reduce to lowest terms. Cancel from the numerator and the denominator all the common factors. 10 10 10. 10. _8_ ' 12. 14 18 12 16 20 16 16 16 _8_ 14 16. 18. 20 1_2. 12. 12 32 42 42 32 22 18 14 2 2 2 2 6. 2.8 2 5 4 2 28 48" 88 39 56 3"0 5F 42 INTERMEDIATE BOOK 2. Cancel from the numerator and the denomi- nator all the common factors. 25 ¥ n a H ¥ u n M u- M A fraction is in its lowest terms when the numer- ator and the denominator do not contain a common factor. Reduction of Improper Fractions 58. 1. How many bushels in ^- bu. ? How many whole bushels ? What fraction is left over ? i/bu. =2| bu.= = 2i bu. In the same manner reduce 2. ^ in. 3. y-yd. 4. 1 lb. 5. ttbu. 6. f da. 7. Vtir. 8. Vib. 9. 11 T. 10. I^mi. 11. ffyr. 12. ffmo. 13. -V- bbl. 14. ^wk. 15. y- sec. 16. ¥ 17. U 18. ft 19. M 20. fi 21. ih To what whole number and what fraction is each equivalent ? Oral Problems 59. 1. John has 21 dimes in the bank. How many dollars h'as he ? 2. In a bin there are 9 pk. of wheat. How many bushels are there in the bin ? COMMON FRACTIONS 43 3. A board is 38 in. long. What is its length in feet ? 4. A roll of paper is 43 ft. long. How many yards long is it ? 5. How many strips of cloth, each 10 in. wide, may be cut from a piece of cloth 36 in. wide ? 6. A straight section of road is 4 rd. wide and 100 rd. long. How many square rods in the sec- tion ? 7. A roll of paper is f of a yard wide. How many feet wide is it ? 8. A roll of paper is 27 in. wide. How many strips each 8 ft. long must be taken from the roll in order to paper a wall 16 ft. long and 8 ft. high ? To Reduce a Fraction to a Mixed Number 60. Reduce to a mixed number : 1 121 ■^- 13 • Process Explanation. — In one unit there ^Y 3 ^1"^ if- 1^1 the fraction ^^ there are 13)121 as many units as 13 is contained in 117 121' c»r 9 and 4 remainder. The re- J mainder is written as a fraction ^. o 385 o496 ^578 k T 5. 6 8_9J_ 7 901 « 8_91 q 1050 **• 2 5 ^-2 6 ^-2 7 ^-6 7 in -67 5 tt837 i« 912 no 64 7 44 INTERMEDIATE BOOK To Reduce Whole Numbers to Improper Fractions Oral Exercise 61. 1. How many halves in 1 unit ? 2. How many halves in 2 units ? 3. How many halves in 5 units ? 4. How many thirds in 1 unit ? 5. How many thirds in 2 units ? 6. How many thirds in 4 units ? 7. How many fourths in one ? 8. How many fourths in two ? 9. How many fourths in five ? 10. Tell how to reduce a whole number to a fraction having a given denominator. Written Exercise 62. Reduce : 1. 16 to halves. Jp ID /-v i^ ■p* a c -t _2. Explanation. — In one unit there 1A 2 are two halves. In 16 units there are lb = lb X 2 ^g ^^ 2 halves, or 32 halves, or ^^. — 32 ' 2 "" 2 2. 24 to halves. 3. 25 to halves. 4. 27 to halves. 5. 29 to halves. 6. 36 to halves. 7. 39 to thirds. 8. 38 to thirds. 9. 45 to thirds. 10. 47 to thirds. ii. 49 to thirds. COMMON FRACTIONS 45 12. 56 to fourths. 13. 57 to fourths. 14. 58 to fourths. 15. 59 to fourths. 16. 64 to fourths. 17. 75 to eighths. 18. 87 to tenths. 19. 99 to twelfths. 20. 117 to elevenths. 21. 100 to tenths. Oral Problems 63. 1. How many quarters will a person re- ceive in change for a $ 2.00 bill ? 2. How many dimes will be received in change for a $5.00 bill? 3. How many eighths of an inch are there in 12 in.? 4. How many sixteenths of a pound are there in 2 lb.? 5. How many eighths of a gallon are there in 2 gal. ? To Reduce a Mixed Nximber to an Improper Fraction Oral Exercise 64. 1. How many halves in one ? In 1| ? 2. How many thirds in one ? In 2f ? 3. How many fourths in one ? In 3|- ? 4. How many fifths in one ? In 2| ? 5. How many eighths in one ? In 4|- ? 46 INTERMEDIATE BOOK Written Exercise 65. Reduce to an improper fraction : 1. 4f. Explanation. — The denomi- nator of the fraction in the mixed number 4| is 5. Reduce 4 to fifths. Add the fraction f . The ^^ ~ "5" "^ 5^ ~ "5~ mixed number 4|^ = -2^. Process 43 = 4 + 3 4 = 4xf = -2g^ 6. 10. 14. 101 3. 21 7. 6f 11. 101 15. 20^ H 8. 7| 12. 30| 5 41 9. 8i- 13. 60| 16. 4O3-V 17. 73-% Written Exercise 66. Change to improper fractions : 1. " " 6. 11. Change to sixths : 25, 63, 75, 76J^. 12. Change to eighths : 14, 24, 48, 321 151 2. 120f 3. 107f 4. 631 5. 131 89A 7. 601 8. 63f 9. 125f 10. 75,^ To Reduce a Fraction to Higher Terms Oral Exercise 67. 1. Examine the drawing of the square. 2. Show by diagram that J = f . 3. Make a diagram to show fVij,fi_3. 1—4 1__5_ tuctu 2"~6' 2~"8' 2~'10* COMMON FRACTIONS 47 4. To change | to sixths, make the denominator of the new fraction 6. To make the denominator of the fraction 6, multiply both terms of ^ by 3. Thus: ix|=f. 5. By what number must both terms of ^ be multiplied to change the fraction to eighths ? Ix?^ ? 2 X ? 8* Oral Exercise 68. Give the answers : 1. 3. 5. 7. 9. 13. 2 4 6 8-10 2. 1 __ ? _ ? _ ? _ ? 3 6 9 12 IS" 2 _ ? _ ? _ ? _ ? 3 6 9 12 IS 4. 19 9 9 9 4 8 12 16-20 3 _?_?_?_ ? 4-8-T6-T2-24 6. 2 _?_?_?_ ? 5 -10-15 -20-2^ 3 _?_?_?_ ? 5-10- 20-12-15 8. 4 9 9 V 9 5"— to — 2^— 3^— IT 6-12-24-18-30 10. 5 _?_?_?_ ? 6 — 12 — 24 — 18 — 3T5" ■^ ? ? V ? 8 16 ~ 32 24 — 40 12. 3 _?_?_?_ ? 8 — 16 — 32 —40"- 24 5 _?_?_?_ ? 8 — 24 "32 — 40 — 16 14. 7 _?_?_?_ ? 8 —40 — 24— T6"— 32 Written Exercise 69. Change to the required denomination : 1. ^, |-, I, 1, 2 to sixths. 2- h h h h h ^ to twelfths. 3- i h h h h 1 to twelfths. *• h h h h h 4 to eighths. s- h A» h h f to twenty-fourths. 48 INTERMEDIATE BOOK Fractions having the same denominator are called similar fractions. In order to compare the vahies of fractions, to add or to subtract fractions, we must first reduce the fractions to similar fractions. To Reduce a Fraction to Higher Terms Oral Exercise 70. 1. Change f to eighths. Explanation. — To change Process fourths to eighths, make the de- 3 3x2 6 nominator of the new fraction 8. 4. ~ 4 y 9 ~ ft ^^ make the denominator 8, mul- tiply both terms of the fraction by 2. 2. Change 2J to tenths. Process Explanation. — Reduce the o^ _ ]i mixed number to an improper 11 ' 11x2 22 ^^^^^^^"- — = = — To change fifths to tenths, O O X Z lU njake the denominator of the new fraction ten. Then mul- tiply both terms by the factor that will give the re- quired denominator. AN IMPORTANT PRINCIPLE IN FRACTIONS Multiplying both terms of a fraction by the same number does not change the value of the fraction. COMMON FRACTIONS. 49 Written Exercise 71. Reduce : 1. 1 1 1 li 2| 2. i 1 4 i H 2i 3. \ i f 1 4| 4. \ i 2 5 f f 5. 1* 3i If 3f 5^ 6. \ i 1 1 4 1 7. 4i 121 5f 16i 10| 8. i i i * A 9. 10. tV lOf 4i 3| 1^ lA A 11. A 7 10 A A i& 12. i i 1 J i 13. #0 2lr A A A 14. 4 25 7 23" M *§ H to sixths, to eighths, to twelfths, to tenths, to tenths, to sixteenths, to sixteenths, to twentieths, to twentieths, to hundredths, to hundredths, to hundredths; to hundredths, to hundredths. Least Common Denominator 72. Heretofore easy fractions were considered, the least common denominator of which could be told at sight. Now we proceed to explain the process of dis- covering the least common denominator when it cannot be readily recognized at sight. 1. Find the least common denominator (Led.) of A 5 _i_ ^-^ 4' 8? 12- 60 INTERMEDIATE BOOK Find the prime factors of each denominator. Thus: ^^2-2 8=2-2-2 12 = 2-2 .3 The Led. must contain the prime factor 2 three times, or it would not be exactly divisible by 8. The 1. c. d. must contain the factor 3 once, or it would not be exactly divisible by 12. Hence the 1. c. d. is 2 x 2 x 2 x 3 = 24. Factor each denominator and take each prime factor the greatest number of times it occurs in any denominator. 2. Find the 1. c. d., if the given denominators are 4, 6, 16. We have : 4=2.2 6 = 2 16=2.22. 2 The 1. c. d. must contain the factor 2 four times. The Led. must contain the factor 3 once. Hence the 1. c. d. is 2 x 2 x 2 x 2 x 3 = 48. 3. If the denominators are 4, 9, 12, find the Led. ^^2-2 9 = 3.3 12 = 2. 2. 3 Hence the 1. c. d. = 2 x 2 x 3 x 3 = 36 COMMON FRACTIONS 61 Written Exercise 73. Find the 1. c. d. of the following denomina- tors : 4. 2, 6, 36. 5. 3, 4, 16, 20. 6. 9, 24, 36. 7. 10, 20, 25. 8. 15, 8, 30. 9. 20, 15, 30. 10. 7, 49, 6. 11. 8, 6, 9, 36. 12. 12, 16, 36. 13. 16, 18, 12. Written Exercise 74. Find the common denominator : l23.5 o S i 1 o311 ^' 3? 5' 8 ^' 4' Ef 6 ^' 5"? 2? "8" 15.1 553. 5. fi732 ^' 4? 9? 3 ^' 8' 4? 6 ^' 8^ 3jJ? ^S" 7_5_AJL q357 q54 '• 12> 6? 8 **• TO' 6' 2"0' ^- 2T' Tj 10 -^ 3 _27._ n _5 25 80 •^"- 10? 100? 10 "• 1F05 IFOO^? 10000 Oral Exercise 75. 1. What are similar fractions ? 2. Change ^ and J to fractions having a common denominator. 3. Change ^, J, and ^ to fractions having a com- mon denominator. 4. Change ^ and ^ to fractions having a common denominator. 5. Change ^, J, and ^ to fractions having a com- mon denominator. 52 INTERMEDIATE BOOK 6. Change ^, f , and |- to fractions having a com- mon denominator. 7. Change ^, f , and f to fractions having a com- mon denominator. 8. Tell how to change fractions to fractions having a common denominator. Study Exercise 76. 1. Reduce ^, f, and f to similar fractions; that is, to fractions having the same denominator. Explanation. — By inspection we find Process ^j^^t these fractions may all be reduced to 1 = ^ twelfths. t=A fit 2. Reduce to similar fractions : ^, f, -f^^ A Process Find the least common denominator 3 = 3 4 = 2x2 12 = 2x2x3 16=2x2x2x2 1. c. d. = 2 X 2 X 2 X 2 X 3 The least common denominator is 48 Then48H-3=16 1x16^16 3x16 48 3x12 36 4x12 48 5 x4 20 12x4 48 9 x3_ 27 16x3 48 COMMON FRACTIONS 53 48^4 = 12 48-^12 = 4 48^16 = 3 Explanation. — 48 is the 1. c. d. Change the de- nominator of the fraction to 48. Multiply both terms of the fraction ^ by 16. Multiplying both terms of the fraction by 16 does not change the value of the fraction. Proceed in like manner with the fractions f , ^^, and ^. 3. Reduce If, ^q, f to similar fractions. Explanation. — Reduce the mixed number, 1|, to the improper fraction. Process 4* ^'^^^ ^^ inspection or by factoring the least common denominator (1. c. d.). The 1. c. d. = 30. 12.^ 5.^50 ■*-o o — Q n 7 7 2 1 iT ~ To ~ 3^7 The fractions must be reduced to 1 = 1 = 11 thirtieths. 1^ = 1* 4_24 5~30 These fractions have a common denominator; hence, they are similar fractions. Tell how to add similar fractions. Written Exercise 77. Reduce to similar fractions : 1. h h I- ^- h h^ 64 INTERMEDIATE BOOK 5. 7. 9. 11. 13. 15. Oral Exercise 78. Reduce to similar fractions and compare. h h 3 6. A? 8' T^ hhi 8. 15' 12' TTT' ¥ 7 4 8 9' s^ rg" 10. TO' 30"' T^' 2^0" Toy 25' 5" 12. T' A' 1' 1 h h h 1 14. 1^6' 4' 8' A 2 7 5 9 5? 10' 6' 20 16. 2^0' sV' T5-' 12 Which is the larger ? 1. 1 and ^ 2. A and 1 3. i and 1- 4. f andf 5. T2 and 3-6^ 6. |and| 7. f and 1 8. landf 9. fandf Addition of Fractions 79. 1. Add I and |, J and J, ^ and 1 2. What is the common denominator of ^ and J?landl?iandi? 3. Change J and ^ to similar fractions. 4. Change J and ^ to similar fractions. 5. Change ^ and ^ to similar fractions. 6. Change ^ and ^ to similar fractions. 7. Tell how to add J and ^, ^ and J, ^ and J. 8. What is the sum of ^ and ^ ? f and f ? 9. Name four common denominators of ^ and f . 10. What is the least common denominator of ^ andf? COMMON FRACTIONS 55 Written Exercise 80. 1. Show by the dia- gram that I + ^ = f . o 1 _ ? . " Wa m 6 ' 2^3 — 6- 3. Can you show by a diagram that i^ + i^ = f ? 4. Can you show by dia- gram that i+i+^=l? 5. Can you show by diagram that ^ + -3 = 1^? Hl= = A? Written Exercise 81. Add: 1. 1 and I. Pkooess (1) The common denominator is 24. (2) 5 5x4 6 6x4 20 24 7_7x3 8 8x3' 21 24 (3) 24)41 11^ Ans. Explanation. — (1) Find the common denominator by inspection or by factoring. (2) Reduce the fractions. 56 INTERMEDIATE BOOK (3) Add the numerators of the similar fractions. Place the sum over the common denominator. (4) Reduce this fraction to a whole or a mixed number. What other common denominator might have been used? Why is the least common denomi- nator the best denominator to use? Why is it called the least common denominator ? Written Exercise 82. Add, using pencil only v^hen necessary : 1. 5. 13. 83. Add: 1. 4. 10. Written Problems 84. 1. A boy earned $ f one day, $ f another day, and $-| the third day. How much did he earn in the three days ? i+^- 2. l+i 3. i+i 4. i+i 1+4 6. f+f 7. 4+i 8. 4 + T^ f+A 10. l+f 11. i+l 12. i+* f+f 14. i+l 15. i+A 16. i+l Written Exercise i+l 2. *+l 3. l+j^ t + ^'tr + i 5. f+A+i 6- l + A + l M+i4+A 8. Hf + f 9- f+l + xV H+tt+A+ 9 10 11. 1^ + l + tt + f COMMON FRACTIONS 57 2. During the forenoon four observations of the thermometer were made. At the first reading the temperature was 50°; twenty minutes later it showed an increase of 1°; twenty minutes later it had increased |^° more; and at the last reading it had raised |-° more. What was the temperature at the last reading ? 3. If the length of a book is ^ ft. and the width ^ ft., what is the length of both sides and both ends of the book ? 4. A farmer gathers J doz. eggs the first day, f doz. on the second day, and ^ doz. on the third day. How many eggs did he gather in the three days? 5. A clerk cut 3 pieces from a roll of ribbon. The first piece was ^ yd., the second ^ yd., and the third f yd. What was the total length cut from the roll ? Addition of Mixed Numbers s. Add at sight : 1. 10| 2. 301 3. 50f 4. 70f 6* 15| 25i lOf 5. 901 6. 201 7. 301 8. 40*. 10i_ lOi 101 l4 9. 80| 10. 67f 11. 591 12. 361 lOi 15^ 111 14* 58 INTERMEDIATE BOOK 13. 46| 14. 831 15. 144| 16. 760| 5* 81 671 1871 17. 2651 18. 4721 19. 34f 20. 144f 3211 691 768 793 871 9671 251 471 Written Exercise 86. Add: 1. 43fandl9|. Explanation. — Add the frac- Process tions. The least common denomi- nator is 15. 43 1= 43^^ 791= 79+f Add integers l^V Write 3. -^■ 123A Find the sum of the integers and the fractions. 2. 14| 3. 113f 4. 761 5. 123| 5f 19f 47i Hi 6. 3O1V '• 706| 8. lOOf 9. i^m 241 69i 79f 94| 10. 43if 11. lllH 12. 87511 13. 405^ SVcr 22| 691 131 14. 763| 15. 108| 16. 95tV 17. 78| 45* 45i 36 97f COMMON FRACTIONS 59 Written Exercise 87. Add: 1. 75| 2. 79f 3. 761 4. 701| 5. lOlJ 461 63| 401 671 67f 6. 11. 46f 7. 79| 8. 101^ 9. 780| 10. 167i 231 631 67J^ 63f 98 291 12. 65| 13. 49| 14. 987| 15. 787| 381 79f 58| 769| 868| 47f 87f 67tV 898| 979| 56| 98i 86| 654| 6973-'5 Written Problems 88. 1. Mr. Edwards sold IQl bii. of wheat to one man and 4f bu. to another. How many bushels did he sell to both? 2. An agent's expenses were $ 4| the first day, $ 3^ the second day, and $ 3f the third day. What is the total amount of his expenses for the three days? 3. Robert worked f of the day Monday, ^ of the day Tuesday, 1 of the day Wednesday, and | of the day Thursday. How many days did he work ? 4. A girl studied 21 hours Monday, IJ hours Tuesday, 31 hours Wednesday, 2| hours Thursday, and ^ hour Friday. How many hours did she study in all ? 60 INTERMEDIATE BOOK 5. A jointed fishing pole has 3 sections. The first section is 2^ ft. long, the second is 2|- ft., and the third is 3 ft. How long is the pole ? 6. A field is 80f rd. long and 40|- rd. wide. How many rods of fence are required to inclose the field? 7. The ice for a family weighs 20^^ lb., 22| lb., 18i lb., 25f lb. Find the total weight of ice used in four days. 8. A painter, working by the hour, works 6|- hr. the first day, S^ hr. the second day, 8^ hr. the third, 7f hr. the fourth, and 4 hr. the fifth day. How many hours did he work in the five days ? Subtraction of Fractions 89. 1. What is the difference between | and 1? I audi? fandi? f andl? 2. Change ^ and J to similar fractions. 3. Change f and ^ to similar fractions. 4. Change | and ^ to similar fractions. 5. Tell how to subtract J from f . 6. Tell how to subtract ^ from f. 7. Tell how to subtract ^ from f. 8. Tell how to subtract f from f. 9. Explain how to subtract similar fractions. 10. Make a rule for subtracting fractions that are not similar. COMMON FRACTIONS 61 Written Exercise 90. 1. Show by the diagram that | — 1- = ^-. 2. 2.__L . 1 — X . 3 ~ 6 ' 2 "~ 6 ' 2._i_ J. 3 2 ~ 6* 3. Can you show by dia- gram that |-i = J? 4. Can you show by diagram that 1 — i^ — i = i? 5. Can you show by diagram that f — i = ^2 ^ Oral Exercise 91. Subtract: 1. 1-i 2. 1-1 3. 1-i 4. 1-1 5. 1-J 6. 1-1 7. 1-t 8. 1-^ 9- i-i 10. f-J 11. f-i 12. i-i Oral Problems 92. 1. What is the difference between J of an apple and ^ of an apple ? 2. What is the difference between ^ lb. and ilb.? 3. In a bin there are If bu. of potatoes. If J bu. is taken from the bin, how many bushels remain ? 4. From a roll of cloth 12^ yd. long a salesman cut 41 yd. How many yards remained in the roll? 62 INTERMEDIATE BOOK 5. The distance between two villages is 4 mi. If a house is 1| mi. from one village, how far is it from the other ? 6. From a piece of cheese containing 3f lb. a grocer sold 2^ lb. How many pounds remained ? 7. A farmer has two fields. One field contains 20 J A. The other field contains 4^ A. less. How many acres in the smaller field ? 8. From a piece of steak weighing 4^ lb. a butcher cut | lb., J lb., and ^ lb. How many pounds of steak has he left ? 9. From a gasoline tank containing 20 gal., 41 gal., 2^ gal., and 3 gal. were drawn. How many gallons were left in the tank ? 10. A carpenter had a piece of molding 4^ ft. long, from which he cut two pieces, one 2J ft. and the other 1| ft. long. How long is the piece that is left? 1 i — 4 2 R 1 _ 2 R 1-1 3 6 1 _ 1 10. 13. i_x 14. 17. liV-l 18. Oral Exercise imilar i fractions and subtract : i-i 3- i- ■i 4. 2 12 i-i '• i- i. 8. i-i i-iV 11- i- 1 5 12. i-i i-i 15. 1- . _7_ 12 16. li-8^ H-i 19. If -t 20. li-l COMMON FRACTIONS 63 Written Exercise 94. Subtract : 1. 271 121 Process Arrange the work as follows : 27i=27f 121 = 121 Explanation. - 271 = 27| 27f - 12^ = 16i 2. 1051 3. 63^ 4. 57f 5. 67| 40 15-1 191 23f 6. 251 7. 751 8. 48f 9. 5^ 131 691 231 24f 10. 105t 11. 302f 12. 5141 13. 524| 971 67| 86f ^^ 14. 574| 15. 7331 16. lOOX 17. 1000^ 1251 m 601 9011 18. 700-1 5491 19. 487-1 looj 20. 5271' 2001 21. 800/^ 4001 22. 501f 23. lOlOf 24. 16261 25. 2000J 109| 700| 448| llllf 64 INTERMEDIATE BOOK Study Exercise 95. Reduce to similar fractions and subtract : 1 from |. Process (1) 1. c. d. is 6 /ox 2 2x2_ ^^^ 3"3x2" /ox l_lx3_ ^"^^ 2"2x3" 4_3^ 6 6 2. I" from f . Process 3^3x3^^ 4 4x3 12 ? = ?x- = A 3 3 4 12 12 12 12 Explanation. — To reduce to similar fractions multiply both terms of the fraction J by 3 and both terms of the fraction I by 2. Then subtract. Multiplying both terms of a fraction by the same number does not change the value of the fraction. The difference between 4 and # is 4, the answer. Explanation. — The Led. is 12. Multiplying both terms of the fraction by the same number does not change the value of the fraction. 1 ~ 12 3. 15| from 26 1. Process 26| =26^^ = 25^1 15# 15^ T2 loii Explanation. — Reduce § and I to similar fractions. We obtain -^ and ^. We cannot subtract -^^ from ■^^' Take 1 from 26, and add it tOtV; !% + {! 10 ]2- 25f| 15-^^- — lO^J, the answer. COMMON FRACTIONS 65 Written Exercise 96. Subtract: 1. 51 2. 61 3. 81 4. lOf 5. 7i 03 Ql A2 Q4 4.3. 6. 81 7. 31 8. 12^ 9. 101 10. 191 5J If _34 _6i 12| 11. 231 12. 181 13. 141 14. 151 15. 27f 13| 9| lOj 10| nil Written Problems 97. 1. A farmer had 30^ bu. of apples. He sold 17f bu. How many bushels has he left? 2. If I have $ 79 J and spend $ 51f , how many- dollars have I left ? 3. A table is 3^ ft. long and 2^ ft. wide. How much greater is the length than the width ? Find the perimeter. 4. A farmer sold 105f bu. of his potato crop, kept 91 bu. for planting, and used 55J bu. for cook- ing. How many bushels in the crop ? 5. A 4|-in. spike is driven through a 2l-in. board into a post. How far is it driven into the post ? 6. The time required to travel from A to C is 28f hr. The time required to travel from A to B is lOf hr. How much longer does it require to travel from A to C than from A to B ? 66 INTERMEDIATE BOOK 7. Along one side of a field 80 rd. long, there is, for a distance of 37f rd., a stone fence. The remaining distance is fenced with wire. How long is the wire fence ? 8. A loaded truck weighs 2^ T. The load con- sists of two parts. The first part weighs IJ T. the second | T. Find the weight of the truck. Multiplication of Fractions 98. Give answers rapidly : 1. How many fourths are 5 times J ? 2. 7 times | are how many fifths ? 3. 6xf=? 4. 10x| = -230 = 6| 5. 6x| = ? 6. 10xf = ? 7. Jx6 = ? 8. ix7=? 9. 1x2 = ? 10. ix3 = ? 11. -|xi=? 12. 11x4 = ? 13. 8xl|=? 14. 12x11 = ? 15. 10x1-1 = ? 16. 12x11 = ? Oral Problems 99. 1. William earns $f daily, or $ in 6 days. 2. John's step is ^ of a yard. How far does he go in 6 steps ? 3. George paces the width of his tennis court, taking 10 steps. How wide is the court, if his step is |- of a yard long ? COMMON FRACTIONS 67 4. A father gives $ f to each of his 5 children. How much does he give them all together ? 5. A man takes a run of f of a mile every day. How many miles does he run in 6 days ? To Multiply a Fraction by a Whole Niunber 100. 1. Multiply I by 64. Process ^^ oa j.- n r. Explanation. — 64 times | 7 f^/<_ 7x^^ may be written — - — . Cancel - X 04— - 8 ^ the common factor. Why? = 56 2. Multiply 3-^2 by 64. Process 16 — - X 64 = ^ ^^ Explanation. — 64 times -^^ 12 12 64 X 7 \r may be written — — — . Cancel 7 X 16 the common factor and reduce ""3 to a mixed number. — 112 =371 To Multiply a Fraction by a Whole Number Multiply the numerator of the fraction by the whole number and divide the product by the de- nominator of the fraction. Cancel when possible. 68 INTERMEDIATE BOOK Written Exercise 101. Solve, using pencil only when necessary : 1. 22 X ^ 2. 6xf 3. Txt\ 4. 8x A 5. 120 x 1 6. 360x1 7. 72 X f 8. 75xf 9. 840 xf 10. 960 X H 11. 42x| 12. 35xf The sign x may mean times or multiply hy. In this exercise it should be read times. To Multiply a Whole Number by a Fraction 102. Multiply and explain : 1. 64 by J 2. 96byi 3. 64 by 3^ 4. SSby^i, 5. f times 24 6. ^x38 7. 1 times 48 8. /^ times 20 9. 1 times 54 10. 1 times 42 To Multiply a Fraction by a Fraction 103. 1. What part of the circle is shaded? Show ^ of the part that is shaded. What part is this of the whole circle ? 2. Show f of the part that is shaded. What part of the whole is this? |of i = |xi = ? 3. Draw a circle and shade ^ of it. Draw lines from the center of the circle dividing the shaded COMMON FRACTIONS 69 part into 4 equal parts. What part of the whole circle is one of these parts ? 4 of |^ = ? 4. A boy takes f of an apple and cuts each quarter into two equal parts. What fraction of the whole apple is each part ? 5. Make a rule for the multipli- cation of a fraction by a fraction. ip.f3_3 iofi— 1 1 of 1 = 1 2^ 01 4-8" 4:^^2"~sr 3OJ-26 6. In these examples, the word of indicates multiply, i^ of f means ^ x J, or f multiplied The fraction |^ of f is sometimes called a com- pound fraction. Written Exercise 104. 1. Find I of f . Process 4 ^^ F~t ^ F Explanation. — Write the frac- 1 tions in the form for multiplication. _ ^ X 5 Cancel common factors from the 4 X ^ numerator and denominator. Write 2 the product of the remaining factors in the numerator and the denomi- nator. 1x5 4x2 5 8 Explanation. — Write the ex- planation. 70 INTERMEDIATE BOOK 2. Find^-Vofif. Process 1 4 3 3 Find: 3. i of 1 4. i of i 5. 1 of i 6. lofi 7. loff 8. loff 9. iof I 10. f of f 11. I of I Tell how to multiply a fraction by a fraction. Make a rule for the multiplication of a fraction by a fraction. TO MULTIPLY A FRACTION BY A FRACTION Cancel factors common to the numerators and denominators and multiply the remaining factors in the numerators for the new numerator, and the remaining factors in the denominators to- gether for the new denominator. Oral Exercise 105. Solve, using pencil only when necessary 1- foff 2. foff 3. fofl6 4. ^Xf 5. fxV- 6. ^Xf 7. 120 x| 8. 12x1^ 9. 15Xj-V COMMON FRACTIONS 71 10. 63x^1^ 11. IGx^L 12. 7x48 13. f Xi 14. ^-^X^ 15. fx| 16. ffxf 17. -y^-Xf 18. 3^xf 19. tVx^ 20. fxf 21. 2X| 22. 2/xiJ- 23. 5xf 24. fxf 25. fX-V- 26. -V-X72 27. | X 3^ 28. V-Offf 29. f off! 30. |X^ Explain how the multiplication of a whole num- ber by a fraction or a fraction by a whole number may be performed as if it were the multiplication of a fraction by a fraction. Oral Problems 106. 1. How much does a boy earn in 7 days if he makes $ ^ in a day ? 2. A man charges $ ^ an hour. How much does he earn in |- of an hour ? 3. A boy rides his bicycle at the rate of 12 mi. an hour. How far does he go in |- of an hour ? 4. Find the cost of 6 chairs at $ If each. 5. Make a bill for 6 days' wages at $2^ per day. 6. What is the cost of a piece of dress goods containing 7 yd. if the material sells at $ If per yard? 72 INTERMEDIATE BOOK 7. A roll of wall paper is f yd. wide. How many yards can be covered with 6 strips ? 8. A man's expenses were $ 4|^ per day. What were his expenses for 20 da. ? To Multiply a Mixed Number by a Whole Number or a Whole Number by a Mixed Number 107. 1. Multiply ^ by 8. Process 8x4|=8xl^ Explanation.-— Reduce the mixed number to an improper frac- = r ^ -'-^ tion. Proceed as in multiplication F of a fraction by a whole number. 1 = 38 2. Multiply 26 by 4|. Process 42. X S6 = l^ x36 Explanation. — Reduce the ^ ^19 J^ixed number to an improper 1 i oa fraction. Proceed as in multipli- = rt"-^ cation of a fraction by a whole ^ number. = 168 3. Multiply 39 by 4f . Process Explanation. — Reduce the 4^ X 39 = -2^'* X 39 mixed number to an improper 24 X 39 fraction. Proceed as in multipli- ^ ^ cation of a fraction by a whole = 1871 number. COMMON FRACTIONS 73 Written Exercise 108. Multiply, using pencil only when necessary : 1. 2|x4 2. 3|x8 3. 4^x5 4. 12xlf 5. 14x3f 6. 17x4f 7. 25 X 3| 8. 48 X 4,^ 9. 7fxl5 10. 271x20 11. 48f X 40 12. 56fx60 13. 100 X 2| 14. 450x1 15. 600 x 5f To Multiply a Mixed Number by a Fraction or a Fraction by a Mixed Number 109. Multiply 1. iby20|. Process 5x201 _^ 31 i ^ 2. 5. 8. 11. 14. _ 31 = 151 2fx| 3fxi 4|xf 11x101 |x4| Written Exercise Explanation. — Reduce the mixed number to an improper frac- tion and multiply. Cancel where- ever possible. 3. 6. 9. 12. 15. 6ix| 2|xi 7fxf fJxSf ¥ X T| 4. 3fx^ 7. 3fxJ 10. 5fx| 13. i|x6i 16. ¥x20i 74 INTERMEDIATE BOOK To Multiply a Mixed Number by a Mixed Number 110. Multiply: i. 9|byl5i Process 13 23 2 — 299 = 149i 31x41 7|x6t 48fx24| 66f X 331 2. 5. 8. 11. Explanation. — Reduce the mixed number to improper frac- tion and multiply. Cancel where possible. 3. 6. 9. 12. 51x1^ 17fx 11^ ^^3 331 X 3i^\ ixl6| 4. 7. 10. 13. 21x51 191x28^ 161x161 33|^ X 3y% Written Problems 111. 1. Theodore earns $ 4^ a week for 7 weeks. How many dollars does he earn ? 2. James works 13 J hr. at 30^ an hour. How much does he earn ? 3. A plumber charges $^ per hour for his time. How much does he get for 3 hours' work ? 4. An engineer charged $ b^ a day for work that occupied him 13^ days. How much was his bill? 5. What is the cost of 13^ doz. eggs at $;^ a dozen ? COMMON FRACTIONS * 75 6. John is 5^ ft. tall. James is only ^ as tall as John. How tall is James ? 7. For 9 years a boy has spent J of every year in school. How many years has he spent in school ? 8. If pepper sells at 15^^ a pound, find the cost of 1^ bags, weighing 120 pounds each. 9. A boy invests $ 26^ in pigeons. At the end of a year he gains ^ of his investment. What is his gain? 10. When wheat is S5^^ a bushel, how much will 205 bushels bring ? 11. Theodore's wages are | of his father's. What does Theodore receive, if his father earns $ 22 a week ? If his father earns $ 17|- a week ? 12. If a cord of wood cost $ 31, what will 5J cords cost ? 13. William has spent ^ of his weekly allowance. He has $ 2^ left. What is his weekly allowance? 14. At 5^ per square foot what is the cost of painting an advertisement upon a wall, IS^ ft. by 5i ft. ? 15. A grocer having 16|- crates of berries sold f of them, or crates. 16. Find the cost of 125 tons of lignite coal at $ 5|- a ton. 76 INTERMEDIATE BOOK Practical Problems — Area 112. 1. How many square feet in the area of a rectangle lOf ft. long and 3|- ft. wide ? Explanation. — In finding the product of the base and the alti- tude, indicate the operations, and then cancel factors common to the numerator and denominator. Not until after this is done should the multiplications be performed. In this example, the indicated area is *-^ — sq. ft. Cancel the fac- 3x4 tors and multiply. Find the area of a rectangle 7^ ft. long and 4 J ft. long and half as wide. 4. The area of a garden is 30f sq. yd. If an area f as large as the garden be added to it, how many square yards larger will it be ? 5. A rectangle has a base 64 ft. long and an altitude 36^ ft. Find its area. 6. The base of a rectangle is f of a foot, its height is f of a foot. Find its area. 7. How many square feet are there in the sur- face of a trunk 4 feet long, f of a foot wide, and f of a foot high ? Pkocess 8 5 ^^ x;i:^_40 1 X ^ 1 1 = 40 The answer is 40 sq. .ft. 2. Find the ai Hit. wide. 3. A table is How wide is it ? COMMON FRACTIONS 77 8. A border is |- of a yard wide and 4 yd. long. What is its area in square yards ? 9. A room is 25 ft. long and 17|- ft. wide. What is the area of the floor? 10. A room is 44 ft. by 20 J ft. What is the area of the floor ? 11. A room is 8 ft. high, 16 ft. long, and 12 ft. wide. What is the area of the four walls ? What is the area of the floor and ceiling ? Written Exercise 113. 1. Find the product of f, f, and |. Pjrocess Explanation. — In multipli- cation of fractions, it is best merely to indicate the operations % L- c = _ Ans. at first, then to cancel equal ^ X p X y y factors in the numerator and r denominator. 2. f off off 3. fxfofl 4. f 0f|0f| 5. ^^^i-^l 8. Ij0f|0ff0ff 9. 1| of I Of -1/ Of i 10. 12x8 of fix ^ A fraction of a fraction is called a compound fraction, f of ^ and f of f of f are compound fractions. 78 INTERMEDIATE BOOK Written Exercise 114. Find the product without reducing to an improper fraction : 1. 15 times 7f . Process H Explanation 15 10 105 115 Ans. 15x|= 10 15 X 7 = 105 15 X 7| = 115 2. 6| times 18. Pkocess 18 Explanation 108 1 6 6f X 18 = ^4-= 13J xl8= 108 X 18 = 121J 121| Written Exercise 115. Find the products : , 2. 5 7. 6fxl5 8 10. 371x331x161 11 X2. 1078x30f x8j^x203^ 1. 161x24 4. 781x9 731x35 97|xl2 83^x24 3. 7261x8 6. 987fxl5 9. 401x60 121 X 125| X 20| COMMON FRACTIONS 79 In the multiplication of mixed numbers, it is usually the best plan to reduce the mixed numbers to improper fractions. Then multiply, canceling wherever possible. To Divide a Fraction by an Integer 116. 1. Divide | by 3. Process Explanation i-^3=i jof| = i 2. Divide |^ by 5. Process 1 f; _ 1 4^ 1 Explanation "2 "^ 2" ^ 5" = l.x4 iof J = |times J 2 ^F ~ 10 3. In the exercise, |-^3, tell how to obtain the answer. 4. Give the answers : 2.^2 A-^4 -S^-^4 10^5 10^2 ^-^3 • 11 • ^ 11 • ^ 7 • ^ 5. In the exercise, J -s- 5, tell how to obtain the answer. To divide a fraction by an integer, divide the numerator of the fraction by the integer, or multiply the denominator of the fraction by the integer. 80 INTERMEDIATE BOOK In dividing ^^- by 4, we may multiply the de- nominator by 4 and get ^f , or we may divide the numerator by 4 and obtain |. Which is the better way ? Why ? In dividing | by 4, which is the better way? Why? Oral Problems 117. 1. A mother divides J of a cake equally among 3 children. What portion of the whole cake does each receive ? 2. Mary cuts J of a yard of ribbon into 2 equal parts. How long is each part? 3. Three boys are given f of a pound of dates. How much is each boy's share? 4. Four baskets of coal weigh ^ of a ton. What is the weight of 1 basket ? 5. John jumps ^ yd. in 3 jumps. What is the distance covered in one jump ? Oral Exercise 118. Answer at sight : 1. 1 + 2 2. 1 + 3 3. 1 + 2 4. 1+3 5. J-*- 3 6.1 + 2 7. i + 3 8. *-4 9. i + 3 10. f + 2 11. 1 + 3 12. 1-4 13. 1+3 14. 1 + 6 15. 1 + 4 16. f-4 17. ^^2 18. ^3^ + 3 19. 1*-+ 7 20. ¥-11 COMMON FRACTIONS 81 Written Exercise 119. Solve: 1. |l■^12 2. II : 14 3. e^i5 4. ^^96 5. 3V_^48 6. It -^21 7. 11^24 8. e^i4 9. i^sji^3o 10- Wxfl^ •24 11. 125 xi|-x|-^3 To Divide a Mixed Number by an Integer 120. Divide: 1. 5iby3. Process ^1 . Q_26 Q Explanation. — Reduce the mixed ^ * /ft 1 number to an improper fraction. = "5" >< 3 2^^ 3 is the same as J of ^^-. = ff Solve 4 of -\^. 2. 3|^5 3. 171-^13 4. 251^19 5. 12|^5 6. 29|-^9 7. 49|-^17 8. 6833^-^21 9. 791^18 10.481-^39 To Divide an Integer by a Fraction Oral Exercise 121. 1. Draw a line 4 in. long. Divide it into parts, each J in. long. How many parts are there ? How many fourths in 1 ? In 4 ? 2. 4-^l = ? 82 INTERMEDIATE BOOK 3. Draw a line 6 in. long. Divide it into parts each ^ in. long. How many are there ? How many halves in 1 ? In 6 ? 4. 6 + 1 = ? 5. Draw a line 2 in. long and divide it into parts each ^ in. long. How many parts are there ? 6. 2 + i=? 7. Would there be less parts if the divisor were V 8. Tell how to divide an integer by a fraction. To divide an integer by a fraction, invert the fraction and then multiply. Written Exercise 122. Divide : 1. 6byf • Process 6H-|=6xf = ¥ = 6f Explanation. — Multiply. [nvert the divisor -J. 2. 9ft. + | 3. $6^$| 4.5+1 5. 8yd.-f-f 6. 75mi. -^f 7. 98 1b. + 1 8. 128 + 1 9. 200-^3-33- 10. 1000 + 1 11. 400 + 1 12. 2000-^1 13. 5000 + f COMMON FRACTIONS 83 To Divide a Fraction by a Fraction 123. Divide: 1. ibyf. Process 2*3 2 = 3 4 3 Explanation. — Invert the divisor and multiply. i)T 2. 14 ^■m 2J3 3/5 4/2 lU 7) 7 12 11. 15. 1)4 2/5 i)^ 8 l)-2- ». 2^4 12- 1)1 16. 4)1 5. 9. 4^3 ivT 13. 4:)i 17. 5)i To divide an integer or a fraction by a fraction, invert the terms of the divisor and proceed as in multiplication of fractions. To Divide an Integer by a Mixed Number 124. Divide: 1. 12by2|. Peocess 12^21=12^1 2 2. 12 by 31 5. '69 by 30| 8. 100 by 41| Explanation. — mixed number to fraction. Divide. Reduce the an improper 3. 25 by 61 4. 6. 84 by 42| 7. 9. 600 by 84| lo. 49 by 51 76 by 12f 1000 by 961 84 INTERMEDIATE BOOK To Divide a Mixed Number by a Mixed Number 125. Divide: 1. Ill by 81 Process 2 Explanation. — Reduce 2 X ^4 4 ^^^ mixed numbers to improper 11 J -5- 8^ = ^^ = — fractions and divide. yL/ X o o 2. 7f by 2| 3. 41 by 3L 4. 21 by 3?- 5. 4| by 5| 6. 7^^ by 14f 7. 12f by 16f 8. 5| by 251 9. 24| by IQi lo. lOQi by 50i Written Problems 126. 1. How many caps can be purchased with $ 21, if each cap costs $1? 2. If a book costs $ 2, how many books can you buy with $4? If a book costs $f, how many books can you buy with $ 2 ? 3. Mrs. Jones spent $ 18 for ribbon, paying $ |- a yard. How many yards did she buy? 4. How many yards of lace can be bought for $25 at $1 a yard? 5. If it takes f lb. of flour for each loaf of bread, how many loaves can be made from one barrel of flour weighing 195 lb. ? 6. Mary uses f lb. of sugar for a cake. How many cakes will 27 lb. of sugar make ? COMMON FRACTIONS 85 7. In a sclioolroom | of a box of chalk is used each school day. How many days will 9 boxes last? 8. A bootblack uses J^ of a box of blacking for three pairs of shoes. How many pairs can he black with 3 boxes ? How many boxes does he need for 60 pairs of shoes ? 9. If 21 yd. of cloth are needed for a coat, how many coats can be made from 35 yd. ? How many yards are needed for 12 coats? 10. During the month of July a laborer was idle ^ of the time. How many days was he idle ? 11. At $ 2 J per volume, how many books can be bought for $ 18 ? 12. Find the cost of 37 electric globes at $^ apiece. How many globes can be purchased for $ 5 ? 13. A certain postage stamp is -| in. by f in. Give its area. How many stamps of this size will it take to cover completely a page 7 in. by 6 in. ? 14. If 5 bu. of wheat cost $3|-, what is the cost of 1 bu. ? 15. If 6 boys earn $ 1^ in 1 hr., what part of a dollar does each earn ? 16. Charles has $ 2J. How many railroad tickets at $J each can he purchase? 17. What is the cost of 21 books at $^ each? 18. What is the cost of one pencil, if 6 cost 25)^ ? 86 INTERMEDIATE BOOK 19. If one electric globe costs $J, how many dollars will 7 globes cost? 20. How many electric globes, at $ J apiece, can be bought for $2f ? 21. How many pencils, at 3^^ apiece, can be purchased f or 7 ^ ? For 21 ^ ? 22. What is the cost of 8 pencils at 3^^ apiece ? 23. Mary buys 6 notebooks and pays 50^. What is the price of each ? 24. At 2^j^ apiece, what is the cost of 5 lemons ? 25. How many lemons, at 2^^ apiece, can be bought for 20^ ? 26. How many sheets of paper, at ^^ a sheet, can you get for 11^ ? 27. How many sticks, | yd. long, can you saw from a pole 2 yd. in length ? Draw a diagram of the pole and show the points of division. 28. A peddler has 17f pecks of peanuts. How many times can he fill a measure that holds J of a peck ? 29. How many tons of coal, at $ 5 J a ton, can be bought for $57.75? 30. If it takes 20|- yd. of canvas to make a tent, how many yards are needed for 7 tents ? 31. If f of a sack of flour will last a family 1 week, how many weeks will Gf sacks last the family ? REVIEW Written Exercise 127. 1. Multiply each of the following by 1^: 22, 46, 50, 48, 64, 68. 2. Divide each of the following by |: 76, 66, 84, 140, 126, 114. 3. Reduce to mixed numbers : -y^, J-f 1, -y-, 125 493 15 11 ' 12 • 4. Change to improper fractions: 1^, 211, 31^ 131 942. 751 -*-^5? "^^3? *^2- 5. Which is larger, l|l or 421 ? 6. Multiply each of the following by 12 : 211, 31 61, lOf, 25|, 111 7. Divide each of the following by 6 : S^, 8 J, 71, 211 14i, 6|. 8. Perform the following operations: 2^-^5|■, "2 • ^4? -"-8 • 4' 8 • 4 • Find the cost of : 9. 21 lb. of cheese at 15^ a pound. 10. 2^ gal. of molasses at 50^ a gallon. 11. 7:1^ lb. of coffee at 40^ a pound. 12. 4| yd. of ribbon at 20^ a yard. 13. 24 shovels at $ f each. 87 88 INTERMEDIATE BOOK 14. 5 tons of coal at $ S^- a ton. 15. IJ bu. of apples at If a bushel. 16. lOJ- A. of land at $200 an acre. Tell the quantity of goods purchased : 17. $ 1^ worth of vinegar at $ J a gallon. 18. 51 j^ worth of berries at S^^ a quart. 19. 85 j^ worth of milk at 8J^ a quart. 20. Lard at 12^^ a pound, and pay $ 1. 21. Butter at 20^i^ a pound, and pay 63^. 22. Candy at 60^ a pound, and pay 15^. 23. Tomatoes at 16|^ a can, and pay 50^. 24. Oranges at 45j^ a dozen, and pay $1.35. 25. Silk at $ 2J a yard, and pay $ IJ. Problems 128. 1. An envelope is 6 J in. by 3^ in. How much greater is the length than the width ? Find the perimeter. 2. A wagon and its load of coal weigh 3^ tons. The empty wagon weighs |- ton. Find the weight of the coal. 3. A ranchman sells f of his corn crop and then J of it. What part has he left ? 4. Find the perimeter of an envelope 7^ in. long and 5 in. wide. What is its area in square inches ? REVIEW 89 5. John buys ^ lb. of cheese and gives ^ of it to James. What part of a pound does James get ? 6. If a boy sells f and ^ of his marbles, what part has he left ? 7. Draw a figure and show that f of | an inch is f of an inch. 8. A carpenter cuts board, f ft. long, into pieces J ft. long. How many pieces does it make ? 9. If a boy earns $f a day, and spends $f a day, in how many days can he save $ 1 ? 10. Draw a figure and show that f in. divided by J in. gives 2 as the answer. 11. It took Louise f of an hour to embroider 2 leaves. How long did it take her to embroider lleaf? 12. George picked 7 qt. of berries and sold them for $ ^. What did he get for each quart ? 13. Albert earns $ 5^ a week for 6 weeks. How much does he earn? 14. James has If lb. of candy and is allowed to eat I of a pound a day. How many days will the candy last? 15. If 2 tons of coal cost $ lOJ, what will 6 tons cost? 16. What is the cost of 7^ yd. of silk at $ IJ a yard? 90 INTERMEDIATE BOOK 17. How many yards, at $ f each, can you buy for $81? 18. How many times larger is $25 than $5? $123 than $7? 19. What must you multiply 1 J by to obtain 7| ? 20. If James earns $ 21 a day, how many days must he work to earn $30? 21. By what must you divide 10-|- to obtain 7J? 22. A traveler spends $ 16 J in 6 J days. How much does he spend each day ? 23. Find the cost of 20|- lb. of sugar at 5|-j^ a pound. 24. The length of a fourpenny nail is If in., a sixpenny nail 2 in. If 4 nails, 2 of each kind, are placed in a line, end to end, how long a line will they make? Oral Exercise 129. 1. What does the denominator of a frac- tion indicate? The numerator? 2. Make a drawing and show that J = f . 3. Reduce to sixths: * I i I ^ H If- 4. Reduce to eighths: i i i li 2f 3J. REVIEW 91 5. Reduce to ninths: f i 5 14 4f- 6. Reduce to twelfths: lit* 4 li 2|. 7. Reduce to integers or mixed numbers : ¥ ¥ ¥ ¥ 2 3 4 ¥ fl> ¥ ¥ ¥ ¥ ¥ ¥ M- 8. Reduce to improper fractions: 131 121 15| 10 * iif 20f, 501 601 1001 11 i . 12f lOf 9. Reduce to their lowest terms: A 12 12 6 A A ^% ^'o A T6 A 2 16 A .%■ Oral Exercise 130. 1. Multiply each by ^: 4 t f f 1 f iV- 2. Find ^ of each: 1 i i 1 3. 4 i f 3. How much is ^ of each : 2 1 f i 4 3" 14 24? 4. Take |- of each: i 1 1 f f 4 i- 5. Find J of each: 4 i f i t i 1- How much is f of each: i i i 1 H ^ 1? Find ^ of each : f 1 1* 7 4i 13 i- Find f of each: i i ^ -H A t's 1- 92 INTERMEDIATE BOOK 6. 7. 8. Oral Exercise 131. 1. Divide | by each of the following: 1 1 2 3 2 5. 5 3 2 3 ¥ :S" 6 12- 2. Divide f by each of the following: t 3 2 3 3 ^ ^' 3. Divide f by each of the following : 4 i i I li li- 4. Divide f by each of the following: I f * t f il tt- Oral Problems 132. 1. A grocer sold 1^ dozen eggs to one per- son and f dozen to another. How many dozen eggs did he sell? 2. A boy spent -^ oi a, dollar for paper and $ f for an arithmetic. How much did he spend for both? 3. A man goes to market and spends f of his money for goods and -^ for dinner. What part of his money is left? REVIEW 93 4. If a pound of tea costs J a dollar, what will 1^ pounds cost? 5. A man owned f of a farm and sold ^ of his share. What part of the farm did he sell? 6. Of f of an acre of land, bordering on the Mississippi River, ^ was washed away. What part of an acre was washed away? 7. A clerk has IJ months' vacation, | of which is spent in Colorado. How long was he in Colorado? 8. A piece of ribbon 3|- ft. long is divided into strips ^ ft. long. How many strips are there ? 9. How many pieces of paper, each -| in. long, can be cut from a paper 4 in. long ? 10. Letters are mailed at the rate of 2^ an ounce, books at the rate of ^^ an ounce. How much more expensive is letter postage than book postage? 11. How many pounds of tea can be purchased with $ 6 at the rate of $ | a pound ? Written Problems 133. 1. A boy earned $3^ one day, $1 the next day, and $ | the third day. How much did he earn in the 3 days ? 2. A father earns $4| a day; his son earns $ 1|- a day. How much more does the father earn in a day than his son ? 94 INTERMEDIATE BOOK 3. A road is built 1^ mi. along level ground, 2f mi. along rising ground, and 5^ mi. along ground sloping downward. How long is the entire road ? 4. During three days the sun shone 9}, 8|, and 7f hr., respectively. How many hours of sunshine were there in all ? 5. A train travels 12^ mi. faster per hour than a steamboat. How many miles farther than the boat does the train travel in 16 hr. ? 6. A flower bed is 40|- ft. long and 5i ft. wide. How many square feet in its area ? 7. A farmer sold 235f lb. of maple sugar at 17f ^ a pound. How much did he receive ? 8. A room is 15| ft. long and 121 ft. wide. What is the cost of a molding extending entirely around it at 5^^ a foot ? 9. A ship is worth $ 100,000. A man owns -f^ of it. If he sells | of his share, what is the value of what he still owns ? 10. How many yards of cloth can be bought for $140 at $ If a yard? 11. What is the price of hay, when 5| tons are worth $ 69 ? Written Exercise 134. Reduce to the least common denominator : REVIEW 95 Written Exercise 135. Perform the operations indicated : 1. l + ii+l=? 2. 10-2|-3f=? 3. lf+lf-f=? 4. llj-f + f=? 5. 101 + 111— T=? 6. 20| + 7|-6i = ? 7. 2fxitxf 8. 61 X 16 X 2| 9. 144 X 8f X 1 10. Six 31x3^ 11. SOxS^Vxi 12. 1x16^x3 13. 9f^4f 14. If-A 15. 27f^l6f 16. ¥x|-ii 17. 14i^5| 18. 67501^33^ 136. Drill Device — Magic Squares 1 f i i ^ i 4 1 1 i A A i i i f i 7 6 1 i 1 1. In the first magic square, find the sum of the three numbers in each line, each column, and each diagonal. 2. Do the same in the second magic square. 3. In the third square, fill the vacant places, so that the sum of the three digits in each line, column, and diagonal will be If. 96 INTERMEDIATE BOOK Suggestive Questions 137. 1. A man pays a nickel carfare twice a day. How can you find the fare he pays in a month ? 2. If you know a man's salary per month, how can you find his yearly salary ? 3. If you know the number of days in each month, how can you find the number of days in a year? 4. Each member of a class needs a new pencil every month. How can you find the number of months a gross of pencils will last ? 5. If a telegrapher knows the number of words he can send per minute, how can he figure the time it takes him to send a given number of words ? 6. John has a certain amount of money. If he knows the cost of writing tablets, how is be to find how many tablets he can buy with the whole of his money ? 7. A merchant sells shoes at a price that en- ables him to double his money. How can we find what they cost him per dozen pairs of shoes ? 8. A confectioner has a number of pounds of candy which he wishes to put up into boxes, all of the same size, each holding a given fraction of a pound. How is he to find how many boxes to order for the candy? REVIEW 97 9. If you know how fast a train travels, how can you determine the time it takes the train to go a given distance? 10. If you know how much a plant grows in a month, how can you find the average amount of growth per day? 11. If James can read a certain number of pages per hour, how can you find the number of pages he can read in a week, when he reads a fixed num- ber of hours each day ? 12. If you know the number of stories in a building and the height of each story, how can you ascertain the height of the building? Drill Exercise 138. 1. Find the sums of the three fractions in each line, each column, and each diagonal. 2. Add f to each of the nine fractions. Add ^J to each. 3. Subtract each fraction from 1^. From 2. 4. Multiply together the pairs of fractions in the first two columns. The pairs of fractions in the last two columns. 5. Divide each fraction in the first line by the fraction below. 4 i * i f f 1 f ^ 98 INTERMEDIATE BOOK 6. Divide each fraction in the second line by the fraction below. 7. Divide each fraction in the first column by the fraction to its right. 8. Multiply the fractions along each diagonal by |. 9. Subtract each fraction along the diagonals from f . 10. Add each fraction along the diagonals to 1^. QUANTITY AND COST Written Exercise 139. Find the quantity and cost : 1. 2 bbl. of flour, each 196 lb., @ $4.65 per barrel. 2. 7 tubs of butter, each 60 lb., @ 27^^ per pound. 3. 6 bbl. of pork, each 285 lb., @ $8.50 per hundredweight. 4. 241 lb. boxes of apricots, each 12 lb., @ 9^^ per pound. 5. 6 boxes of dates, each 301b., @ $2.25 per box. 6. 75 bbl. of pork, each 200 lb., @ 5|^ a pound. 7. 125 bales of cotton, each 4 tons, @ 8J^ a pound. 8. 30 sacks of grain, each 200 lb., @ 80^ per hundredweight. 9. If 1 of a box of peaches cost 45^, what is the cost of a whole box? 10. If I of a barrel holds 66f gallons of pine tar, how many gallons will the barrel hold? 99 100 INTERMEDIATE BOOK Written Problems 140. 1. If Robert can solve 9 problems in half an hour, how many can he solve in IJ hr., at the same rate? 2. How many miles can James walk in 3 hr., if he walks 7 mi. in 2 hr.? 3. John paces off the length of a fence and finds it to be 36 paces. If 3 of his steps measure 8 ft., how many feet long is the fence? 4. If 5 lb. of apples can be bought for 15j^, how many pounds can be bought for 45^ ? 5. Josephine reads 50 pages of history in 3 hours or pages in 27 hours. 6. If two boys working together, at the same rate, can do a piece of work in 7|- hours, in what time can one do it? 7. If a certain supply of provisions lasts 2 men 18 days, how long will it last 9 men? 8. The Atlantic liner Lusitania went from Queenstown to New York in 4 days and 19.9 hours, at an average speed of 24 miles an hour. How far is it between Queenstown and New York ? 9. The Mauretania, a sister ship, attained an average speed of 25.83 miles an hour. How much farther will she travel in 10 hours than the Lu- sitania? QUANTITY AND COST 101 10. One year the skating record* foi* 100 yd:' ' was 11^ sec. or yd. per second, 6|:' — '-r4'ft! pfe^\i i second. 11. The skating record for 400 yd. was 46^ sec. or yd. per second or ft. per second. 12. The distance of 880 yards in skating was made in 1 minute, 36 seconds. How many yards per second is this? 13. In the year 1903 the United States pro- duced 234,000 tons of cane sugar and 214,825 tons of beet sugar. In 1906 it produced 243,000 tons of cane sugar and 431,796 tons of beet sugar. By how many tons did the increase in the production of beet sugar exceed the increase in the production of cane sugar? 14. Colorado produced 2974 million pounds of beets in one year, from which 334 million pounds of sugar were manufactured. How many pounds of beets did it take to yield one pound of sugar? 15. A merchant buys 10 doz. pairs of shoes for $ 360 and sells them at $ 4 a pair. What is his profit? 16. A grocer buys 20 doz. eggs for $ .25 a dozen and sells them at $ .35 a dozen. How much does he gain? 17. If 6 doz. oranges cost $2.40, what will 1 doz. cost? 5 doz. ? 102 INTERMEDIATE BOOK 18. A dealer buys knives at $ .35 and sells them ;»i $:.50. Y^^hat is his profit on 100 knives? 19. Find the cost of 10 lb. of butter at $i a pound. 20. Find the cost of 5^ lb. of butter at $ J a pound. 21. What is the value of 2^ acres of land at $ 1 8 an acre ? 22. A carpenter earns $.45 in |- of an hour. How much does he earn in an hour ? 23. Only f of a class are present. If 30 are present, how large is the class ? 24. If ^ of a flock of sheep are 100 sheep, how many sheep in the entire flock ? 25. f of the distance between two cities is 27 mi. What is the distance ? 26. The distance between two towns is 55 mi. How much is |- of that distance ? 27. A train runs 35 miles in f of an hour. What is the rate per hour ? 28. James buys 10 tons of soft coal and pays $ 6 2 J. What is the cost per ton ? (Use business fractions.) 29. What is the cost of 16 notebooks at $.12| apiece ? (Use business fractions.) 30. Find the cost of 24 doz. eggs at $.37^^ a dozen. QUANTITY AND COST 103 31. How much must you pay for 40 yd. of cloth at $.371 a yard? 32. A merchant buys 1000 yd. of lining at $ .03 a yard and sells at $ .05 a yard. Find the profit. 33. A real estate man buys a lot for $ 1,200. He divides it up into 3 small lots and sells each lot for $ 600. How much does he make ? 34. Another real estate man buys a lot for $ 2,500, pays $ 100 taxes, and then sells it for $ 2,750. How much does he gain ? 35. At $ .08 a pound for rice and $ .05 a pound for sugar, how much will 10|- lb. of each cost ? 36. A woman buys 3 boxes of starch for $ .50 and a bag of flour for $ .75. She gives the clerk a $ 5 bill. How much change does she receive ? 37. What is the cost of 15 hair brooms at $ 3.25 each and 3 mops at $ 5.60 per doz. ? 38. A business house orders 7 quartered-oak flat-top desks at $ 19.85 each and 7 revolving arm- chairs at $ 9.33 each. What is the total cost of this order ? 39. A student buys 3 books at $ 1.25 each and a student's notebook for 37 cents. He gave in pay- ment a $ 10 bill. How much change should he receive ? 40. Find the cost of 4^ yards of calico at 6 cents a yard; 5 J yards of gingham at 24 cents a yard. 104 INTERMEDIATE BOOK Oral Exercise — Cash Register 141. This picture shows a cash register. If a purchaser pays for goods amounting to $ 3.45, the Sri. A Cash Register clerk presses the keys marked $3,40^, and 5 ^. This prints the amount of the purchase on a shp of paper within, pushes up cards, showing the purchaser the sum paid, and opens the cash drawer. Suppose the purchaser has given the clerk a $ 5 bill. The clerk puts the money into the drawer. QUANTITY AND COST 105 He may take out the following change : a 5-cent piece, a 50-cent piece, and one dollar. He hands the change to the customer, in the same order, and says, " Three fifty, four, five dollars." 1. What keys must be pressed to indicate the payment of each of the following sums: $1.50, $1.46, $2.35, $4.75, $3.05, $2.14? 2. What amount of money is shown by the card register when the following keys are pressed : Dollars Cents Cbnts Dollars Cents Cents 1 20 5 1 60 7 2 30 6 2 90 8 3 80 5 4 10 5 3. If a purchase is $ 1.65 and the purchaser gives the clerk $ 2, what change should he receive ? 4. State what keys the clerk must press, what coins he may take out of the drawer, and what he will say to the purchaser in each of the following cases : AMoitnt Money Given Amount Money Given OF Purchase IN Payment op Pxtrohase IN Payment $1.75 $2.00 $3.85 $4.00 $2.55 $3.00 $1.97 $2.00 $2.95 15.00 $4.05 $5.00 5. Let some of the pupils of the class act as clerks, while other pupils make supposed purchases. Which pupils were able to act as clerks without making mistakes ? 106 INTERMEDIATE BOOK Written Problems 142. Prices of Household Goods Tea Kettle $ .49 Range $42.50 Bread Box .55 Sideboard 15.65 Water Bottle .23 Refrigerator 12.75 Broom .15 Kitchen Table 3.50 Teapot .53 Dining-room Chair 2.75 Towels, per doz. 2.15 Kitchen Chair 1.25 Oilcloth, per yd. .30 Dining Table 24.65 Linoleum, per yd. 1.20 China Closet 19.75 1. Mary bought a tea kettle, a water bottle, two brooms, a dozen towels. She pays her bill with a $ 5 bill. How much change does she receive ? 2. Mr. James ordered a china closet, a dining table, a kitchen table, and a range. How much was his bill ? 3. Mrs. Hathaway has $45.60 in the bank. How much does she lack in order to purchase 10 yd. of linoleum, 2 doz. towels, a dining-room chair, a sideboard, and a china closet ? 4. Mother sends Martha with a $10 bill to buy 2 yd. of oilcloth, a kitchen table, a teapot, a bread box, and a kitchen chair. How much change should she bring home ? 5. Mr. Newman purchases a sideboard, a refrig- erator, a dining table, and a bread box. He has $ 99.15 in the bank, but wishes to keep at least $ 50 on deposit in the bank. How much can he pay on account ? QUANTITY AND COST 107 6. Purchase any six articles you like and find the cost. 7. A man earns $ 100 a month. His monthly expenses are $45.60. Is the balance sufficient to pay for a range, a kitchen table, two dozen towels, a teakettle, and a bread box ? 8. Mrs. Grant purchases a sideboard, 10 yd. of linoleum, 3 brooms, a kitchen table, 2 kitchen chairs, a refrigerator. She pays $ 30 on account. How much did she have charged ? 9. Cash sales were $ 125.35 on Monday, $ 130 on Tuesday, $175.25 on Wednesday, $43.40 on Thursday, $150.10 on Friday, and $247.65 on Saturday. How much were the sales for the week ? Drill Device 143. At the completion of the study of fractions use the drill device. Continue the drill daily for short periods of rapid accurate work. Vary the numbers to be added and subtracted and to be used as multipliers and divisors. Change the numbers on the drill. Speed and Accuracy Speed and accuracy should be developed in the fundamental operations. Practice should be con- tinued on drills of this kind, in fractions, decimals, and percentage until a speed of about 60 correct answers a minute can be obtained. 108 INTERMEDUTE BOOK 1 1 :^ ^ ^ \C0 ^ ^ o (M "«* U5 CD t> + tH rH 1 1 1 o O O o o o o "5 jj 1 1 00 t- ?o lO Tf CO 1 1 ^ P yA 1 1 kJ s ;^ cg\ ;5^ ::^ ■^ CO (N p 00 o a rH tH rH iH I 1 1 ^ SCO ;j^ :^ C^^ ::)^ :;^ 1 (M (N »-H o (M r-\ rH iH rH 1 rH iH tH rH 1 1 1 SI SCO c^\ :^ ^ ^ ^ + 1 1 1 DECIMALS Review 144. Study the three ways of writing decimal fractions. 1. One tenth 2. One hundredth 3. One thousandth 4. Five tenths 5. Six hundredths 6. Seven thousandths 7. Twenty-five hundredths 8. Seven hundred fifty- six thousandths A fraction whose denominator is 10, 100, 1000, or 1 with any number of ciphers annexed, is called a decimal or a decimal fraction. A decimal fraction may be written in three ways: In words, as, one tenth, six hundredths. As a common fraction, as, ^, yo^. By using the decimal point, as, .1, .06, 756. Decimal fractions or decimals are usually written with a decimal point. 109 A .1 TUir .01 1000 .001 A .5 100 .06 iooo .007 ^^ .25 1^ .756 no INTERMEDLA.TE BOOK 145. Written Exercise 1. Make a chart like the illustration. Orders of Whole Numbers Orders of Decimals Millions Thousands Units or Ones Thousandths M C O 1 i ■D C 3 I (A C .2 1 c o h (A C o i tn -a c rt to 3 O x: ■M ■o d) •a c 3 I to ■o c to ■M c a> H- to c to 3 O x: h to ■a c 3 to c to 0} c O o to 'E to i2 to J= ■!-« ■D 0) 1- T3 i I to x: ■M ■a c nJ to 3 O Sd Period 2d Period 1st Period 1st Period 4. 5. 2. Count the orders of integers shown on the chart. 3. Count the orders of decimals. Name the orders of each. Write on the chart the following numbers : Four and five tenths. Seven and six tenths. Five and thirty-six hundredths. Six and seven hundredths. Ten and one hundred twenty-five thousandths. In these numbers and indicates the position of the decimal point. DECIMALS 111 Study Exercise 146. 1. Into how many small squares is the large square di- vided? One of the smaller squares is ^fo of the large square. Write this as a deci- mal fraction. 2. How many of these smaller squares in ^ of the large square ? Then, i 100 = .50 = .5. 3. In I" of the large square there are small squares. Then, i = ro"u-= -^^ ^^ ^^® large square. 4. Shade the smaller squares which together stand for ^ or .07. For ^ or .10 or .1. For M or .13. 5. The entire large square = t — t — s" =" T¥o* 6. Tell from the drawing J^o_ = ^^ _2^ = ^^, ^ 10* 60 107 .75 = h . ? 80 __ ? ■5"? TOO" — ■§" f^n — ? _ ? — ? 20 .OU — YolOr — 4 "~ 2' lOTT ? 40 5"' TOTJ" :§"? 112 INTERMEDIATE BOOK 8. Tell from the illustration what decimal frac- tions are equal to the following common fractions : 1 1 1 _i_ _i 1 1 2' 4' 5' 10? 20? 25' 50* FRACTIONS TO BE i=.25 i=.75 MEMORIZED Oral Exercise 147. 1. How many cents in $.5? In $.2? In $.4? Inf.l? 2. How much more is $ .5 than $ .25? 3. How much more is $ .75 than $.5? In writing money $ .5 is written $ .50. 4. Mary buys candy worth $.35. She gives $J to the clerk. How much change does she receive ? 5. A matchbox contains 100 matches. One match is ^-^q of the entire number. 6. Frank takes |- of all the matches, or matches. 7. Had he taken only J of all the matches, he would have taken matches. J is equal to what decimal fraction ? DECIMALS 113 8. Write as a decimal fraction the part of all the matches you would have taken, if you took 9 matches. 11 matches. 37 matches. 9. Albert had 100 old five-cent stamps in an envelope. He loses .25 of all he had. How many does he lose ? What part of all still remains ? 10. Henry has 100 tin soldiers. He gives away ^ of them to George and J of them to James. How many has he left? What part of the whole is this? Oral Exercise 148. Read 97.58. There are two ways of reading 97.58. (1) Ninety-seven and fifty-eight hundredths. (2) Ninety-seven, point, ^ye, eight. The second way is generally preferred because it is shorter. Read 1.4, 100.75, 45.05, 3.275, 005. Read 0.014, 7.09, 18.71, 19.1, 1.004. Write in the form of decimal fractions : 3 3 5 1_1_ A_l_ A 7 f; 5 5 fi 8 7 .^ 10? 100? 100' -^10? ^10? ^100? "100? ^1000* Write in the form of common fractions : .3, .13, .05, .75, 4.65, 3.03, .575, 005, 1.006, 7.01, 8.10, 10.101. Which is larger 9.7 or 9.07? 9.86 or 9.68? 5.8 or 8.5? 3.6 or 3.60? 4.8 or 5.8 ? 4.320 or 4.32 ? 114 INTERMEDIATE BOOK Addition of Decimals .49. Copy, add, and check : . $ 3.75 2. $735.15 3. $1095.64 16.85 176.45 46.75 195.43 714.36 9.49 236.78 895.85 9864.25 900.00 100.25 897.68 736.45 236.70 1098.46 $ $ $ In adding money, the decimal points are put in a column so as not to confuse dollars and cents. Precisely the same thing is done in adding all decimal fractions. The decimal points are kept in a column. Do not forget to write the decimal point in the sum. 101.13 5. 973.1 6. 796.125 709.09 79.175 15.471 1000.15 100.65 210.005 793.75 462.456 75.6 434.14 75.5 400.06 11.25 176.47 750.005 125.17 55.125 66.543 Observe the groups of 10 when such groups occur. DECIMALS 115 7. Copy the numbers in each column. Add vertically and horizontally : 12.34 + 21.22 +916. +567.8 = 56.70 +232.4 +928.3 + 5.432 = 88.90 + 25.26 + 89.76 + 4.678 = 21.26 +272.8 +. 9.476+ 48.65 = 9. + + + = 12.14 + 2.905+ 99.15 + 364. = 13.73 + 3.062+ 88.25 + 98.9 = 16.27 + 31.35 + 77.5 + 897.5 = 17.85 + 33.35 + 66.25 + 6.432 = 18.05 +353.6 + 5.125 + 78.23 = + + + = 8.479 + 698.9 +795.3 + 478.79 = 584.7 + 89.67 + 93.57 +' r879 6 = 49.76 +875.4 + 6.932 + 43.715 = 75.42 +196.8 + 58.75 + 791.45 = 66.53 +789.15 + 4.676 + 89.176 = 78.90 + 67.817 + 798. + 458.9 = 634.5 +819.43 + .95 + 45.61 = 4.571 + 278.9 +638.7 + 79.05 = + + + = 10. Copy and add the answers to 7, 8, and 9. 11. Separate Ex. 9 into two parts. Add and find the sum of the answers. Compare with the answer to 9. 116 INTERMEDIATE BOOK Subtraction of Decimals 150. Copy and subtract : I. $10.75 2. 11.251b. 9.28 7.45 lb . 3. 119.45 yd. 4. 120.6 mi. 73,15 yd. 14.75 mi. Since 6 tenths are equal to 60 hundredths, we may write 120.60 in place of 120.6. Then, 120.60-14.75 = 105.85. 5. $419.1 -$296.7. 6. $786.55 -$654.6. 7. 786.4 T. - 6.98.7 T. 8. 1000 lb.- 736.45 lb. 9. Subtract 1.795 from 12. lo. 2.364 from 5.7. Process Process 12.000 = 12. 5.700 = 5.7 1.795= 1.795 2.364 = 2.364 10.205 = 10.205 3.336 = 3.336 How do you check subtraction ? II. Subtract 69.7895 from 79.123. Multiplication of Decimals Oral Exercise 151. 1. If a boy earns $ 1.25 a day, he earns $ 12.5 or $ 12.50 in 10 days, and $ 125 in 100 days. In multiplying by 10, move the decimal point DECIMALS 117 one place to the right. How do we multiply by 100? By 1000? 2. Multiply the following by 10. Also by 100 : $7.35, 7.75 T., 95.5 lb., 73.47 ft., .1 in., 75 bu., $2,335, 791.9, 78.732 mi., .05 oz. 3. Multiply 35.2 by 200. Multiply by 2. Then multiply the product by 100. 4. Multiply each by 10. Multiply each by 20 : 1.11 3.4 4.33 3.23 10.2 10.24 10.15 2.05 2.44 2.25 .25 .04 5. Multiply each by 100. Multiply each by 300 : $ 1.11 2.22 lb. 30.3 ft. 3.2 yd. 2.2 doz. 10.5 1.05 3.23 2.21 lb. .25 6. Multiply each by 10. Multiply each by 40: $1.21 2.11 11.2 T. 10.2 yd. 4.1 pk. 3.02 1.22 1.05 .25 .5 Oral Problems 152. 1. If the wages of each man in a factory employing 100 men are $ 2.25, what is the amount of the payroll for one day? 2. If a train goes .75 mile a minute, how far will it go in 100 minutes? In 200 minutes? 3. A boy walks 112.5 yards per minute. How many yards in 10 minutes? 118 INTERMEDIATE BOOK To Multiply a Decimal by an Integer or an Integer by a Decimal 153. 1. Multiply .75 by 9. Process Explanation. — If we were multiplying .75 75 by 9, the product would be 675. 9 But we are multiplying -^ by 9. Hence 5 75 the product is |^, or 6.75. 2. Multiply 64 by .8. Process Explanation. — If we were multiplying 64 64 by 8, the product would be 512. .8 But we are multiplying 64 by -^. Hence 51.2 the product is ^^-^-, or 51.2. TO MULTIPLY A DECIMAL BY A WHOLE NUMBER I. Multiply as in whole numbers. II. Point off as many decimal places in the product as there are decimal places in the multiplier. Written Exercise 154. Multiply: 1. .15 by 8 2. .25 by 9 3. .09 by 15 4. 3 by $1.15 5. 13 by $2.25 6. 11 by .95 7. 12 by 10.5 8. 21.75 ft. by 29 9. 2579 lb. by .87 lo. 458 by .009 11. .043 by 798 12. 457987 by .0009 DECIMALS 119 Written Exercise 155. Multiply each number in columns B, C, D, and U, by the corresponding number in column A. A B C D E 1. 19 .62 1.05 10.36 6.45 2. 109 .705 8.25 19.75 14.95 3. 28 .901 9.28 109.6 121.5 4. 307 .025 7.01 203.5 906.7 5. 412 .909 8.75 100.9 976.5 6. 505 .097 1.89 23.45 8.756 7. 620 .202 3.333 89.01 45.67 8. 136 1.012 .963 75.75 UA4: 9. 809 8.97 7.09 45.45 67.5 10. 879 .009 .008 .007 .006 To Multiply a Decimal by a Decimal 156. 1. Multiply .64 by .8. Process and Explanation If we were multiplying 64 by 8, the product would be 512. But we are multiplying -^^ by ^ . Hence the product is yu^, or .512. 2. Multiply 4.28 by 1.15. Process and Explanation If we were multiplying 428 by 115, the product would be 49,220. 120 INTERMEDIATE BOOK But we are multiplying ^f|- by ^^. The product is ^Uih or 4.9220, or 4.922. TO MULTIPLY A DECIMAL BY A DECIMAL I. Multiply as in whole numbers. II. Point off as many decimal places in the product as there are decimal places in both factors. Oral Exercise 157 . Answer at sight 1. .9x.2 2. .8x.3 3. .6x.4 4. .5x.5 5. .4x.7 6. .3x8 7. .2x.8 8. 10 X. 8 9. 10 X. 3 10. .4x10 11. 1.1 X .2 12. 1.2 X. 3 13. .4x2.1 14. .5x1.2 15. 3x3.1 16. .3 X 3.1 17. .5x1.1 18. 5x.ll 19. .llx.l 20. .17x2 21. .2 of .4 22. .4 of .8 23. 1.1 of .9 24. 7 of .12 25. 12x12 26. 1.2x12 27. 1.2x1,2 28. 1.2 X. 12 29. 1.1x12 30. 11x1.2 31. 1.1 X. 12 32. .llx.l2 33. 13x11 34. 13x1.1. 35. 13 X .11 36. 1.3 X. 11 Written Exercise 158. 1. Find the product of 195 and 32. Also of 19.5 and 3.2. DECIMALS 121 2. Which of the three products is the largest : 1.2x1.8, 12x18, .12x1.8? Multiply : 3. 2.5 X 3.2 4. 6.7x7.5 5. 77.7x6.6 6. 6.3x8.5 7. 9.8x1.2 8. 37.5 X. 05 9. 16.5x12.9 10. 3.5x2.06 11. 1.07 X. 35 12. .01 X. 5 13. 19.1 X. 01 14. .05 X .96 15. .7 of 35.6 16. 1.25 X. 7 17. .1x7.5 18. 7x6.8 19. .9 of 10.7 20. 12.5 X. 85 21. .2 of 7.3 22. 7x6.8 23. .6 of 11.3 24. 9.17 X. 65 25. .3 of 9.4 26. 91x87 27. .05 of 9.8 28. 8.06 X. 54 29. .03 of 8.5 30. 9.1x8.7 31. .15 of 125 32. 0.25 X .25 33. .33 of 6.1 34. .64X.75 35. 92.34x81.4 36. 80.43 x 76.4 37. 764.1x17.9 38. 63.01x70.6 39. 19.84x1.25 40. 670.9x15.2 41. 99.9x8.88 Written Problems 159. 1. The wheat yield of a field of 19.6 acres is at the rate of 27.5 bushels per acre. What is its value at $ 1 a bushel ? 2. If 1 ft. of lead piping weighs 1.82 lb., what is the weight of a piece of piping 22.5 ft. long ? 3. If an aeroplane travels at the rate of 67.5 mi. an hour, how far will it travel in 3.4 hr. ? 122 INTERMEDIATE BOOK 4. At 25.3 mi. per hour, how far can you ride in 4.5 hr.? 5. A carpenter earns $ 4.25 a day. How many days will he have to work in order to earn $ 180 ? 6. A farm consists of 120 acres. It is valued at $ 7500.50. What is the value per acre ? 7. To find the diameter of a circle, divide the length of the circle by 3.1416. What is the diam- eter of a circle, if its length is 25 ft. ? 8. What is the diameter of a circle, if its length is 50 ft. ? Division of Decimals To Divide Integers and Decimals by Integers 160. 1. 20 divided by 5 may be written 5)20, 2/-, or 20-^-5. 2. Since 1.56 x 10 = 15.6, we have 10)15.6 = ? 3. Since 1.565 x 100 = 156.5, we have 100)156.5 TO DIVIDE A NUMBER BY 10 Move the decimal point in the dividend one place to the left. In multiplying by 10, the decimal point is moved one place to the right. In dividing by 10, the decimal point is moved one place to the left. DECIMALS 123 4. Tell how to divide a number by 100. By 1000. Written Exercise 161. 1. Divide 222.4 by 200. Divide 222.4 by 100, and the result by 2. 1.112 100)222.4 = 2.224, 2)2.224 2. 10)35.4 100)35.4 3. 100)5.6 20)4.4 4. 100)115.6 100)22.2 5. 10)55.5 50)55.5 10)3.7 100)3.7 20)44 20).44 200)22.2 200)222 30)36.3 40)4.8 6. 90)3600 400)3200 300)7200 30)60.6 Written Exercise 162. 1. Divide 427.2 by 24. Process 17 8 X Explanation. — Divide as in whole 24)427.2 numbers. 24 into 42 tens goes 1 ten and 24 a remainder. Write the 1 above the divi- YS7 dend in the tens' column and so on. Place -1 go the decimal point in the dividend and quo- 1 Qo tient under each other. 192 124 INTERMEDIATE BOOK 2. 13)1599 5. 21)51.03 3. 13)159.9 6. 53)424 8. 62)43.4 9. 64)38.4 11. 131)10.48 12. 139)834 14. 17)212.5 15. 23)632.5 17. 83)207.5 18. 39)48.75 20. 17).85 21. 16)1.28 4. X 23)538.2 7. 58)5.22 10. 127)88.9 13. 151)120.8 16. 97)145.5 19. 76)588.84 22. 19). 57 23. 19).285 24. 16).272 25. 14).182 Written Problems 163. 1. A dozen handkerchiefs sell for $2.88. What is the price of one handkerchief ? 2. Nine boys hire rowboats and pay $2.25. What is each one's share of the expense ? 3. On a football trip the expense of 13 men was $ 14.95. How much did each pay ? 4. If the fare of 35 pupils on an excursion is $15.75, what is the fare of one ? 5. A town in Colorado had 226.3 hr. of sun- shine during a month of 31 da. What was the daily average ? 6. A potato patch contains 67.5 sq. yd. It is in the form of a rectangle 9 ft. long. How wide is it ? DECIMALS 125 7. A white ash, 29.9 in. thick, was 115 yr. old. Find the average yearly growth. 8. A tree grows to a height of 77.9 ft. in 41 yr. What was its average gain in height per year? 9. A train travels 213.6 miles in 6 hours. How many miles is this an hour ? 10. In a long-distance race one man ran 24 miles in 2 hours, 57.6 minutes. He ran 1 mile in minutes. To Divide a Decimal by a Decimal 164. 1. f = ? ■§-§■ = ? Compare the quotients. «9._? 1_8— ? 2_7 — ? 36 _? ^.3 — . "e""- 9~* 12~* 2.5 ? 3.6 ? 1.25 ? 3. .5 5 .6 6 .05 5 2 5 25 . . . Instead of — ^ we may take — , which is easier .5 5 to divide. 2.5 « 3.6 o 4.5 o 1.25 4. .5 ' .6 ' .5 ' .05 * 5. 1.5^.5 .25)775 .9)4.5 .9)6.3 6. .6)4.2 .7)5.6 .2)1.4 .7)2.1 .45 7. -j='' .8)7.2 .9)4.5 .6)7.2 126 INTERMEDIATE BOOK PRINCIPLE TO BE REMEMBERED Multiplying the dividend and divisor by the same number does not change the quotient. Written Exercise 165. Answer the following : 1. .6)91 4. .11)1.21 7. .8)6.4 10. 1.2)1.44 13. 25)75 2. 1.2)7.2 5. .9).81 8. .08).64 11. 12)1.44 14. 2.5)7.5 3. 1.1)1.21 6. .09).81 9. .12)1.44 12. 12)14.4 15. 2.5)75 16. 2.5).75 17. .12)1.32 is. 1.3)910 Written Exercise 166. Divide: 1. 1.872 by .13. Process 14.4 X 13)187.2 Explanation. — The divisor becomes 13 an integer, if we multiply by 100. We ~57 have .18)1.872 equal to 13)187.2. 52 52 52 DECIMALS 127 2. 77 by 2.5. Process 3Q g Explanation. — 770 -^ 25 gives X the quotient 30 and the remainder 25)770.0 20. If the division is carried one 75 step farther, in order to secure 200 greater accuracy, then write the 200 dividend 770.0 instead of 770. The next digit in the quotient is 8. Check : 30.8 There is no remainder. The exact 2-5 quotient is 30.8. Check by multi- 1540 plying the quotient by the divisor. 616 What should the product be ? 77.00 3. 224 by 2.4 4. 1718.64 by .62 5. $775 by $.25 6. 74.256 by .34 7. $ 8955 by 4.5 8. 10548.72 by .39 9. 1718.64 by 9.3 10. 150.696 by 2.8 11. 6715.1756by.085 12. 566.351 by 6.7 Written Problems 167. 1. What number, multiplied by 1.2, gives the product 14.4 ? 2. Multiplying a certain number by 1.3 yields the product 11.7. What is the number? 3. The product is $ 10.35, the multiplier is 2.3 ; what is the multiplicand ? 4. If a man earns $26.25 in 7.5 days, how much does he earn in one day ? 128 INTERMEDIATE BOOK 5. John earns $ 3.60 a week. His father earns $ 18 a week. How many times greater than the son's are the father's wages ? Written Exercise 168. Divide, carrying the quotient to 3 decimal places. 1. 9.2 by 1.3. Process 7.076 X Explanation. — When we divide 92 13)92.000 by 13, there will always be a remainder, 91 however far we carry the division. When 100 it is not necessary to know the fractions 91 of a cent, we write the answer, $7.07'^. 90 The plus sign shows that the quotient >To 7.07 is not exact and that the true answer Y^ is a little larger. 2. 73-^2.9 3. 97^.41 4. 107^5.3 5. 4.55 -^ 9.7 6. 2.55^1.07 7. .75^8.9 8. $1.8^6.4 9. 7.95 in. -^12 10. 272.5-4-2.72 ii. 7.63 lb. -^17 Written Problems 169. 1. The area of a drawing board is 483.5 sq. in., its length is 23.1 in. Compute its width to the tenth part of an inch. DECIMALS 129 2. A place in Arizona had 305.4 hr. of sun- shine during January. What is the daily aver- age? 3. The salary of the President of the United States is $ 75,000. What is his salary for a day in a year of 365 days ? 4. A man travels 1,896 miles in his automobile during the month of July. How many miles a day does he average ? 5. During March, a few years ago, 406.1 tons of dynamite were used for blasting the Panama Canal. On an average, how many tons of dyna- mite were used a day? How many pounds of dynamite were used a day ? To Change a Decimal to a Common Fraction 170. 1. Change .625 to a common fraction in its simplest terms. Process Explanation. — Write in oi)n_ 625 ^^^ form of a common frac- no« ~,\^ . tion. Cancel. What com- 125 — 25 — 5 Ario , ,^ 200-40-8 ^^^*- mon factors were canceled ? 2. .8 3. 25 4. .75 5. .125 6. .64 7. .05 8. .2 9. .08 10. .35 11. .28 12. .95 13. .96 14. .55 15. .175 i6. .16 17. .48 18. .225 19. .175 20. .125 21. .725 130 INTERMEDIATE BOOK To Change a Common Fraction to a Decimal 171. 1. Change -| to a decimal. Process 3.— 4 ^r Explanation 2. Change 2^5 to a decimal. Process 32 Explanation 283 = 8^25 = 8.00^25 = . 32 ^m 25)8.00 3. Change ^ to a decimal. Explanation. — In changing J to a decimal, we find that the division does not come out exact. ^ cannot be exactly ex- Process pressed as a decimal. In such a case carry .SS"*" the answer to the second or third decimal 3)1.00 place by adding as many ciphers to the dividend as there are to be decimal places in the answer. Write plus after the last figure in the quotient to show the omis- sion of the rest. Written Exercise 172. Reduce to decimals : i 2. i 3. ¥ 4. H 5. H 6. M H 8. It 9. « 10. W 11. H 12. ¥ i 14. 5 6 15. 4 T 16. i 17. 7 11 18. 18 13 7. 13. In examples 13-I8 carry the answer out to decimal places. DECIMALS 131 Written Exercise 173. Reduce to mixed numbers 1. -^Z 2. y 3. -W M 6. fl 7. H 8. I 10 3J_9_ n 10_0 T« 81 8 1_5_ Trt 379 n 1 0_0 174. Reduce to a decimal, carrying the answer out four places : 1. ^i". Process 10.4146- 41)427.0000 ^2 Explanation. — Add ciphers in the dividend to make four decimal places. Indicate the first partial divi- dend. Divide : 17 16 4 60 41 Indicate the incomplete answer by 190 the + sign. 164 260 246 2 573 3 648 a 8 2 9 5 9Jl6 REVIEW Decimal Fractions 175. 1. Eead: .27, 3.01, 4.025, .007, .726. 2. In .32, the unit is divided into equal parts, and of these are taken. 3. How can we tell the denominator of a frac- tion when it is written in the decimal form ? 4. Arrange in the ascending order of value: 5.05, 5.51, 5.5, 5.005. Oral Exercise 176. Add at sight : 1. 3.2 2. 7.8 3. 3.7 4. 10.3 4.5 2.4 4.8 20.8 5. 10.7 6. .07 7. .03 8. 6.44 15.01 .14 .145 1.4 9. 2.07 10. 7.01 11. 9.09 12. 7.993 .155 2.10 3.46 1.007 Oral Exercise 177. Subtract at sight : 1. 7.5 2. 7.4 3. 10.6 4. 7.5 2.5 2.5 4.7 1.05 132 REVIEW 133 8.7 6. 9.7 7. .100 8. .200 8.65 7.9 ■ .001 .005 1.9 10. 3.75 11. 7.111 12. 5.000 .05 1.7 3.01 .001 Oral Exercise 178. 1. Change to common fractions and simpli- fy: .4, .8, 1.2, .25, .75, .5, .55 2. Change to decimal fractions : 1 i 1 3. 2. 3. 4 5? 4? 2? 4? 5? 5? 5 3. Express in dollars and cents : $11 $11 $41 $lf, $2f, $3f Compare and determine which is the larger: 71 and 7.2 8.5 and 8f 7.4 and 7.40 4.4 and 4i 3.3 and 31 2| and 2.6 Drill Exercise 179. 1. Add the numbers in each column. 2. Subtract the lesser numbers from the greater in each column. 3. Find the product of the two numbers in each exercise. 4. Divide the first number by the second. A BCD 1.1, 2.2 1. hi ^o>i .hi 2. i-i .5, .25 i,l 134 INTERMEDIATE BOOK A B X> 3. H, H 2^1 2J,i 2i,i 4. 2, .4 .3,3 %f hi 5. hi l.i 100' -5 •25,^ 6. 101 2 %i 3^,7 31 .7 7. 31, .1 5i,f,r 51,11 2*4' 12 8. A>i A>f I,2i hi 9. hi f>f hi i.25 10. .331 .661 .75, .25 .5, 121 .121 .25 11. •161, 1 |,-5 f, .661 .75,1 12. hi .99,2 .49,1 1,1 13. .5, .6 4.5,2 8.6,2 1.2,3 14. 24, .2 36, ^ 3.6, xk 4.7, .1 15. H,H 21,1 1, .01 .9, A 16. .11, .11 .12, .12 .01, .05 1.3, 13 17. H,H hi hi 1 A 4' 5 18. hi 1, .33i 200, 300 250, 350 19. 2i,i hH TT' to h"^ 20. 3^,1 1.1, .1 2.2, .2 3.3, .3 Oral Exercise 180. Find the product : 1. .7x.6 2. 12x1.2 3. Tx.lll 4. 8x1.2 5. 1.2x1.2 6. .8x.l2 7. .Ix.l 8. 1.2x12 9. 8x1.2 10. .25x5 11. .15x5 12. .05x5 REVIEW 135 Oral Exercise 181. Find the quotient : 1. .74-^10 2. 1.4-f-lO 3. 14-^10 4. 62 -f- 100 5. 6.2-^100 6. 620 -^ 100 7. 14-f-.l 8. 1.5^.1 9. 2.03^.1 10. 24-^.2 11. 2.4-^.2 12. .240 -5- .2 Written Exercise 182. Perform the operations indicated. Check the results. 1. 17x18.5=? 2. 6.4x7.5=? 3. 175x1.5=? 4. 12.5 X. 25=? 5. 17.28 -^ 144 6. 95.2 -^ 3.4 7. 28.7 H- 109 8. 67.67-^67 9. 28.71-9.9 10. 151.3^39 ii. 144 h- 7 12. 194^9 13. 226 -^ 5 i4. 742-^11 15. 7.23^.4 16. 17.69^7 i7. 45.6^.13 18. 18.78 -^ .17 19. How much is i of 123 ? 20. How much is |^ of 560 ? 21. How much is |- of 1780 ? 22. Find 1 of 75 23. Of 39.9 24. Of 947 25. Of 9003 26. How much is 81.7 less -^^ ^^ ^^^ ^ 27. From ^ of 964 take J of 435. 28. The quotient is 9.36, the divisor is 7.3. Find the dividend. 136 INTERMEDIATE BOOK 29. Find the divisor when the dividend is 21.42 and the quotient 6.3. 30. Find the quotient to 3 decimal places, when the dividend is .074 and the divisor is .3. Written Problems 183. 1. If a man saves $ 32 a month, how long will it take him to save $432? 2. If a car conductor earns $ 2.25 a day, how long will it take him to earn $776.25? 3. If 9.75 tons of coal cost $ 47.26, what is the price of 1 ton? 4. What is the cost of 12 bales of cotton at $.12 a pound, if each bale weighs 410.8 pounds? 5. Two motor cars start from the same place and travel in opposite directions, one at 15.3 miles an hour, the other at 18.4 miles an hour. How far apart are they at the end of 6.3 hours? 6. The daily wages of each employee in a factory were increased $ .18. The daily total amount paid in wages was thereby increased $ 84.06. Find the number of employees. 7. A torpedo boat has a speed of 38.4 knots an hour. What is its speed in miles, if 1 knot is 1.15 miles ? 8. If 9.8 inches of snow when melted make 1 inch of water, how much snow is necessary to make .55 inch of water? REVIEW 137 9. If a merchant buys 125 suits of clothes at $ 13.25 each and sells them at $ 22.50 each, what is his profit ? 10. Find the product of the sum and the differ- ence of 43.25 and 13.76. 184. FRACTIONS FREQUENTLY USED IN BUSINESS 10^=^$^ 25^ = $i 121^ = $i 50^ = $| 75^ = $!' i6f^=n 20^ = $^ 40^ = $f 331^ = $! Oral Exercise 185. Find the cost. Use fractions. Quantity Kate Quantity Eatb Quantity Bate 1. 32 1b. 12iJ^ 6. 40 oz. 20^ 11. 36doz. ^H^ 2. 18 1b. 161^ 7. 80 bu. 25^ 12. 96 yd. 121 f^ 3. 16 1b. 75 f 8. 32 bu. 11.25 13. 60bbl. 75 j^ 4. 24 1b. 12i^ 9. 24 ft. 12i^ 14. 46 yd. $1.50 5. 15 yd. 66|)* 10. 66 yd. ^1.66f 15. 32 bu. 11.12^ Written Problems 186. 1. How long will it take a person to earn $63.25, if he earns $2.75 a day? 2. A train runs at the rate of 36.5 mi. per hour. How long will it take to run 277.4 mi.? 138 INTERMEDIATE BOOK 3. The area of a floor is 396.8 sq. ft. ; its length is 25.6 ft. Find its width. 4. How long will it take a man to walk 8035 mi., if he walks 3.75 per hour ? 5. A factory employs 305 men who work 8 hr. per day. If the daily payroll is $ 512.40, what is the average wage paid per hour ? 6. A farmer paid three men $ 72 to shock his corn, wages being $ 1.50 a day. How many days did each man work? 7. A steel rail, 30 ft. long, weighs 72 lb. per yard. How many men are needed to carry it, if each man carries 120 lb.? 8. A city lot containing 950 sq. ft., and 190 ft. deep, sells for $ 1560. What was the price per foot of frontage? 9. How many states the size of Delaware, 2050 sq. mi., could be made from Texas, 265,780 sq. mi.? 10. A man takes 6 acres of city land, at $560 per acre, in exchange for 55 acres of farm land. What is the value of the farm land per acre ? DrUl Table 187. Multiply: 1- fxlf 2. if x^ 5. 21x3 6. 31x5 REVIEW 7. 5fx7 8. 5fx9 9. 3fxl2i 10. 4f X 161 11. 2fx6i 12. 5|x2f 13. 32.5x7 14. 45.62 X 29 15. 4.55 X 789 16. 14.15x875 17. 16.5 X. 081 18. 3.7 X. 079 19. .087 X. 097 20. .095 X .008 21. 6.663 X .63 22. 2.34 X. 96 23. .83ix.9i 24. 1.5 X .331 Divide : 25. l-i 26. i-i 27. ff-A 28. f-l 29. A-t 30. 10 -2f 31. 20^31 32. 36-^A 33. 35^31 34. 3f^9 35. 4f-^20 36. 61^12 37. 121^121 38. 91-61 39. 4t-5i 40. 35.5^.75 41. 6.3-^.25 42. .331^.121 43. .661^.161 44. .75^.871 45. .625^.25 46. 42.15 H- .625 47. .625^.871 48. .897^.789 49. 3.75^.0375 50. .0062^.0012 139 DENOMINATE NUMBERS Study Exercise 188. All numbers are either abstract or concrete. A concrete number is one that refers to par- ticular objects, as, 25 sheep, 46 feet, 72 bushels. An abstract number is one that does not refer to particular objects, as, 7, 8, 13, 16. A denominate number is a particular kind of concrete number expressing measure of size or weight, as, 7 feet, 4 pounds. Thus, 25 houses is a number that is concrete but not denominate. 24 days is a number that is both concrete and denominate. A denominate number of one denomination is called a simple denominate number. If it has two or more denominations it is called a compound de- nominate number. 2 ft. is a simple denominate number. 2 ft. 4 in. is a compound denominate number. Oral Exercise 189. 1. Name six abstract numbers. 2. Name six concrete numbers. 3. Name a concrete number that is not de- nominate. Name one that is denominate. 140 DENOMINATE NUMBERS 141 Tables 190. Memorize LINEAR MEASURE 12 inches (in. or ") = 1 foot (ft. or ') 3 feet = 1 yard (yd.) 16.5 feet = 1 rod (rd.) 320 rods = 1 mile (mi.) 1760 yards = 1 mile TIME 60 minutes (min.) = 1 hour (hr.) 24 hours = 1 day (da.) 7 days = 1 week (wk.) 365 days or 12 months (mo.) = 1 year DRY MEASURE 2 pints = 1 quart (qt.) 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.) 32 quarts = 1 bushel WEIGHT 16 ounces (oz.) = 1 pound (lb.) 100 pounds = 1 hundredweight (cwt.) 2000 pounds = 1 ton (T.) Reduction of Denominate Nximbers 191. 1. Reduce 9^ yd. to inches. Explanation Since 1 yd. = 3 ft. 91x3 = 28 ft. =28 ft. 28 X 12 = 336 in. And since 1 ft. = 12 in. 28 ft. = 28 X 12 in. = 336 in. 142 INTERMEDIATE BOOK 2. Reduce 234 in. to yards. Pkocess Explanation 1^2 ^^' Since 12 in. = 1 ft. 12)234 in. 234 in. = 234 -^ 12 ft. 12 =19|ft. 114 61 yd. Since 3 ft. = 1 yd. 108 3)191 ft. 191 ft. = 191 ^3 yd. 6 = ^2- yd. 3. How many inches in 17 ft. ? In f ft. ? In If ft. ? 4. Change to a fraction of a foot : 9 in., 6 in., 2 in., 8 in., 10 in. 5. How many feet in 73 yd. ? In 4^ yd. ? In I yd.? 6. Change to rods : 121 yd., 11 yd., 66 yd. Oral Exercise 192. Reduce to inches : 1. 2 ft., 21 ft., 2 ft. 3 in., 3 ft. 2 in. 2. 2^ ft., 4^ ft., 31 ft., lOjL ft- 3. 3f ft., 2| ft., 4| ft., J5 ft. 4. 51 ft., If ft., 7f ft., 6A ft. 5. 4A ft., 2Yj ft., 3^ ft., 4^5_ ft. 6. 5,iv ft., 6^^ ft., J^ ft., 411 ft. 7. Change to ounces : 4 lb., 2 lb., 10 oz , 2^ lb., 31 lb.. If lb. DENOMINATE NUMBERS 143 8. How many hours in : 2 da. ? 21 da. ? If da. ? 21 da. ? f da. ? 9. Reduce to minutes : 1^ hr., 2^ hr., 3^ hr., J^ hr ^^ hr 10. Reduce to seconds: ^V i^i^., -^ mm., -^ min., ^ min. 11. How many days in 21 wk. ? in 3f wk. ? in 31 wk.? 12. In reducing from larger or higher units to smaller or lower units, do you multiply or divide ? 13. How many bushels or parts of bushels in 8 pk. ? in 10 pk. ? in 12 pk. ? in 14 pk. ? 14. Change to pecks or parts of pecks : 8 qt., 16 qt., 12 qt., 20 qt. 15. Reduce to quarts : 8 pt., 9 pt., 10 pt., 11 pt. 16. In reducing from smaller or lower units to larger or higher units, do you multiply or divide ? Written Exercise 193. 1. A grocer buys apples at $ 1.25 a bushel and sells them at 50^ a peck. Find his profit on one bushel. 2. A bushel of peanuts costing $ 1.35 is sold at 5^ a pint. What is the gain ? 3. A coal dealer sells coal by the sack (weighing 100 lb.) at the rate of 25^ each. What is his profit per ton, if he buys the coal at $ 3 a ton ? 144 INTERMEDIATE BOOK 4. A man works 8 hr. daily; at this rate how many hours does he work in 3 weeks (omitting Sundays) ? 5. A man's business office is 420 rd. from his home. If he walks to his office and back once every day, how many miles does he walk in 16 days ? 6. If pears cost $ 1.35 a bushel and are sold at 60^ a peck, what is the profit on 2 bu. ? 7. A dealer sells peanuts at 3j^ a bag. If two bags hold 3 pints, how much does he get in selling 1 pk. of peanuts ? 8. How much cheaper is it to buy a pound of candy at 30^ a pound than at the rate of 10^ for 4 ounces ? Tables 194. Memorize. SQUARE MEASURE 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet (sq. ft.) = 1 square yard (sq. yd.) 640 acres (A.) = 1 square mile (sq. m.) LIQUID MEASURE COUNTING 2 pints (pt.) = 1 quart (qt.) 12 units = 1 dozen (doz.) 4 quarts (qt.) = 1 gallon (gal.) 1 2 dozen = 1 gross Study Exercise 195. 1. Draw a diagram and explain why it is that, while 3 ft. = 1 yd., it takes 9 sq. ft. to make 1 sq. yd. DENOMINATE NUMBERS 145 2. In the same way, explain why it takes 144 sq. in. to make 1 sq. ft. 3. Pace the distance of a little over 200 ft. (208 -^ ft.). Then imagine a square that long and wide. This square covers one acre. How many feet of fence are needed to inclose this square ? Oral Exercise 196. 1. How many square inches in 2 square feet ? 2. How many square feet in 13 square yards ? 3. How many acres in 100 square miles ? 4. How many quarts in 12 gallons, liquid measure ? 5. How many units in one gross ? 6. How many square yards in 27 square feet ? 7. How many square miles in 6400 acres ? 8. How many dozen in 84 units ? Study Exercise 197. Eeduce 435 in. to feet and inches (avoiding fractions). Process 36 ft. Explanation. — The answer is 36 12)435 in. ft. 3 in. Notice that the remainder 3 36 is inches. 75 The dividend and remainder are 72 always the same kind of measure. 3 in. 146 INTERMEDIATE BOOK Written Exercise 198. 1. Reduce 1179 sq. ft. to sq. yd. 2. Reduce 1290 sq. ft. to sq. yd. and sq. ft. (avoiding fractions). 3. If 5280 ft. make a mile, how many square feet make a square mile ? 4. Reduce 37 gallons to pints. 5. A small farm contains 160 acres. What part of a square mile is this? How many such farms can there be on a square mile ? 6. A town lot is 20 yd. wide and 180 ft. deep. How many yards of fence will inclose it ? What is its area in square yards ? 7. A farmer bought 60 acres of timber land in South Texas at $46.75 an acre. Then he paid $ 12.25 an acre for clearing the land. How much was his total outlay ? 8. On 39 acres he plants rice and the yield is worth $45 an acre; 10 acres planted with corn yield 40 bushels per acre, worth 78^ a bushel; 1 acre yields strawberries that sell for $ 150. Find the total value of the crops. 9. Multiply 5|- by 5J. Then explain why it is that SOJ sq. yd. make 1 sq. yd. 10. Reduce 20|- bu. to quarts. To pecks. 11. How many quarts in 7 bu. 3 pk. and 7 qt. ? The liquid quart is less than the dry quart. 4 DENOMINATE NUMBERS 147 liquid quarts make 1 gal., but it takes 8 dry quarts to make 1 pk. 12. If 31.5 gal. make a barrel, and 2 barrels a hogshead, how many gallons are there in 13| hogsheads ? 13. How many bushels in 564 pecks? In 164 dry quarts ? 14. A merchant buys blackberries at 12^ cents a quart and sells them at $.23 a quart. What is his profit on 68 quarts ? 15. A milkman buys 8 gallons of milk at $.15 a gallon and sells it at $.07|- a quart. Find his profit. 16. How many pounds and ounces are there in 450 oz. ? 17. How many hundredweight are there in 7 J tons ? How many pounds ? 18. Which is more, 45 hundredweight or 2^ tons ? How much more ? 19. What is the difference in weight between a gross ton and the ordinary ton ? 20. A man sells 7^ tons of coal at $ 3.25 a ton. Find the selling price. 21. Change 26 lb. 3 oz. to ounces. 22. Change 8000 lb. to hundredweight, also to tons. 23. How many pounds in 5 T. IS^ cwt. ? 148 INTERMEDIATE BOOK Written Problems 199. Study the daily army ration. Daily army rations to a United States soldier in garrison comprise the following articles, measured in ounces : Fresh beef 20 Prunes 1.28 Flour 18 Coffee 1 12 Baking powder .08 Sugar 3.2 Beans 2.4 Evaporated milk .5 Potatoes 20 Salt .64 Black pepper .04 Lard .64 Cinnamon .014 Butter .5 Flavoring extract .014 1. How many soldiers can be rationed for one day on 96 oz. salt? Process Explanation. — . 64 oz. salt 96 -^ .64 = 150 Arts, supply one soldier for one day. 96 oz. salt supply as many soldiers for one day as .64 is contained in 96. 2. How many soldiers can be rationed for one day from each amount named below ? 48 oz. salt 12 oz. black pepper 10-lb. keg of butter 1 cwt. of sugar 100 lb. potatoes 96 lb. beans 180 lb. flour 7 oz. cinnamon 6 lb. lard 3 lb. flavoring extract 70 lb. coffee 64 lb. prunes 16 lb. baking powder 500 lb. beef DENOMINATE NUMBERS 149 3. How many days can 10 soldiers be rationed from each amount named below ? 11 lb. sugar 8.4 oz. baking powder 183| lb. flour 252 lb. 8 oz. beef 13f lb. beans 65| lb. potatoes 77.3 lb. prunes 76.4 lb. coffee 46 lb. 15 oz. milk 120 lb. 13.6 oz. salt 13|- lb. black pepper 23.96 oz. cinnamon " 6.9 lb. lard 212 lb. 3 oz. butter PRACTICAL EXERCISES AND PROBLEMS Written Problems 200. 1. One year there passed through the canals at the " Soo " about 90 million bushels of wheat and 60 million bushels of other grain. How many million bushels of grain were shipped through the " Soo " ? 2. How much more wheat is carried than other grain ? 3. Duluth and Superior have together 27 grain elevators with a joint capacity of 35 million bushels. Buffalo has 28 elevators with a joint capacity of 23 million bushels. How much smaller is the average capacity of a Buffalo elevator ? 4. If it takes 4| bu. of grain for one barrel of flour, how many million barrels of flour are ob- tained from 150 million bushels of grain ? Does the answer exceed 31^ million barrels? 150 INTERMEDIATE BOOK 5. Besides the grain there passed through the " Soo " locks about 71 million barrels of flour. Add this to the 31^ million barrels. 6. If one barrel of flour yields 250 one-pound loaves of bread, how many million loaves can be made from 39 million barrels of flour ? 7. Allowing each individual 1 loaf a day, how many days would this supply New York City, which has a population of 3^ million ? 8. How many common years and days is this ? 9. Allowing each individual 1 loaf a day, how many days would this amount of bread supply the 90 million inhabitants of the United States ? 10. How many months and days would this be, counting 30 days to a month ? 11. 16 million tons of coal were transported in one year on Lake steamers. If this coal were sold in equal amounts to each individual in a city of 500,000 inhabitants, how many tons would each person get ? 12. If a vessel carries a cargo of 10,000 T. of coal, how many vessels would be required to carry 16 million T. ? 13. If such a cargo of coal can be loaded on a ship in 12 hours, how many tons are loaded per hour ? 14. Find the cost of transporting 8 million T. of coal on the Lakes from the East to Duluth, a dis- DENOMINATE NUMBERS 151 tance of about 1000 rni., at $.35 a ton. What is the cost of transportation per mile ? 15. On an average, one large freight vessel passes from Lake Huron through the Detroit River to Lake Erie every 12 minutes, day and night, during the 8 months of navigation. How many vessels pass in 8 months (of 30 days each) ? 16. One year there were 571 steel ships carrying freight on the Great Lakes. What was the average tonnage, if the tonnage of all ships was 2 million ? Written Problems 201. 1. Study and compare the height of tall buildings in New York City. Building Height in Fket No. OF Stories Trinity Church Flatiron 234 286 375.5 382 419 617 700 750 909 20 Pulitzer Park Row 22 26 Times 28 Sinerer . . 42 Metropolitan . Woolworth New Equitable (plan) 46 51 62 2. How much higher will the New Equitable building be than the Trinity Church, Flatiron, and Pulitzer taken together ? Than the Park Row and Times combined ? 152 INTERMEDIATE BOOK 3. Compute to a tenth of a foot the average height of a story in each building. Which has the highest stories ? 4. How many times higher will the New Equi- table be than Trinity Church? Than the Times building ? 5. The flagpole of the New Equitable will extend 150 feet above the top of the building. How far above the street will the flag float ? 6. The Eiffel tower in Paris is 75 feet higher than the New Equitable will be. How much higher than the Eiffel tower will the flag on the New Equitable be ? 7. The New Equitable will have 8 passenger elevators running all the way to the top. What is the combined length of the 8 elevator shafts ? 8. If the New Equitable elevators travel 600 feet a minute, how long will it take an elevator to travel, without stops, from the ground floor to the top? 9. The Metropolitan is built on ground 75 ft. square ; that is, 75 ft. long and 75 ft. wide. The Singer on ground 65 ft. square. By how many square feet does the former ground exceed the latter? 10. The Metropolitan Tower has a clock with a dial 25^ feet in diameter. The circumference of a circle is about 3.14 times the length of the diameter. DENOMINATE NUMBERS 153 How far is it around this dial ? How many feet must the extremity of the gigantic minute hand move every minute ? Gas Meter 202. A gas meter has three dials. By reading these dials we can find the number of cubic feet of ^9^^T^ March 4, 1914 ^-^^^ ^"PT^ \P^f^ April 3, 1914 gas used. The figures on the dial at the right denote hundreds of cubic feet, the figures of the middle dial denote the thousands of cubic feet; the fig- ures on the dial at the left denote ten thousands of cubic feet. While the hand of the right dial makes one revo- lution, the hand of the middle dial moves through one division: while the hand of the middle dial 154 INTERMEDIATE BOOK makes one revolution, the hand of the left dial moves through one division. The dials are read from left to right by taking the figures which the hands have just passed. Thus, the meter in the picture above gives the figures 3, 6, 1. This means 30,000 en, ft. plus 6,000 cu. ft. plus 100 cu. ft. = 36,100 cu. ft. Notice that it is necessary only to write 361 and annex two zeros. 1. The figure shows the meter in Mr. Jackson's house on March 4, 1914, and on April 3, 1914. Read the meter for each date. 2. How many cubic feet of gas were consumed from March 4 to April 3 ? What was the cost of the gas at $ 1 per 1000 cu. ft. ? DENOMINATE NUMBERS 155 3. At $1 per 1000 cu. ft., what is the cost of 4,320 cu. ft. ? Of 800 cu. ft. ? Of 76,800 cu. ft. ? 4. The following is a gas bill sent to Mr. A. Hart : SCo Consolidated Qas Company of New York, JBr« Branch Offiice, 133 EAST 15th STREET, near Irving Place TELEPHONE 4901 STUYVESANT cA&w- yo-ik ^itif For Gas Supplied from j/o/^. /,\oS&(y: f, 191(5" Arrears 191 Present State of Meter 76,8 00 Previous State of Meter 7^,^00 / , ^ 00 cubic feet of Gas at 80c per M This charge is made in con- formity with the opinion and decree of the Supreme Court of the United States rendered in the suit brought on May i, 1906, by the Consolidated Gas Company of New York in the United States Circuit Court. Received payment for the Company. 62 Make a bill for Example i similar to the one shown here. 5. If an ordinary gas burner consumes 6 cu. ft. of gas per hour and a Welsbach burner consumes 4 cu. ft. per hour, how many cubic feet will both consume in Vl\ hr. ? 6. If gas costs $ 1 per 1000 cu. ft., how much is saved in a month of 30 da. by using the Welsbach burner ? 156 INTERMEDIATE BOOK 7. At various times read the gas meter in your home or the home of a friend, and make out bills for gas consumed, using the rate actually charged. 203. Time Tables Condensed Time, Chicago to Omaha Read Down Read Up Limited Daily Express Daily Mi. 9:16 a.m. 10:32 P.M. 10:21 A.M. 11:44 P.M. 40 12:10 P.M. 1:87 A.M. 114 2:00 P.M. 2:26 a.m. 183 8:45 P.M. 8:40 A.M. 358 8:60 P.M. 8:45 A.M. 358 11:06 P.M. 1:10 P.M. 603 Stations Eock Island Lines Limited Daily Express Daily Lv Chicago. ..Ar. Joliet Bureau Davenport . . . Ar. Des Moines Lv. Lv. " " Ar. Ar. . ..Omaha. ..Lv. 4:59 P.M. z 2:10 P.M. 12:25 P.M. 7:36 A.M. 7:20 A.M. 3:00 A.M. 7:25 A.M. 6:20 a.m. 4:25 A.M. 2:35 A.M. 9:42 P.M. 9:30 P.M. 4:40 P.M. z Trains stop to let off passengers from Colorado and Points west. 1. What are the terminal points given in this table ? 2. What other cities are given ? 3. How many trains are given in this table? Which run west ? Which run eastward ? 4. Name the time the former leave Chicago; also the time the latter leave Omaha. 5. How long a stop do the trains make in Des Moines? 6. How far is it from Des Moines to Omaha? From Joliet to Davenport ? DENOMINATE NUMBERS 157 7. In what time does the westbound express run from Bureau to Davenport ? From Des Moines to Omaha? 8. In what time does the morning train run from Omaha to Davenport ? 9. What is the fare from Chicago to Des Moines at 2^ a mile? At 3^ a mile? 10. Can a passenger from Colorado, passing through Omaha in the morning, get off at Joliet ? 11. In what time does each of the four trains travel between Chicago and Des Moines ? Between Des Moines and Omaha ? 12 . What is the distance of each city from Chicago ? Areas 204. 1. Review square measure. Art. 194. 2. A plane figure is a part of a plane bounded by straight or curved lines. 3. Construct with ruler a plane figure bounded by four straight lines. How many sides has the figure ? How many corners or angles has it ? 4. A plane figure bounded by four straight lines is called a quadrilateral. 5. Construct a quadrilateral that has square corners, or right angles. What is this figure called ? 6. A rectangle is a quadrilateral whose angles are right angles. 158 INTERMEDIATE BOOK 7. In the figure how many squares in each row? How many rows are there? If each square is one square inch, how many square inches in the area of the figure ? 8. How is the area of any rectangle found ? 9. The area of a rec- tangle is the base times the altitude. 10. A triangle is a plane figure bounded by 3 straight lines. 11. The line on which a rectangle or a triangle stands is called its base. 12. How does the shaded triangle A compare in area with the rectangle A ? 13. How does the shaded part in B (triangle B) compare in area with the rectangle B ? 14. In A the rectangle and the triangle have the same base and the same height. Is this true of B ? A triangle is exactly half the rectangle of the same base and height. 15. The area of the triangle is the base times the altitude, divided by 2. 16. If rectangle A is 10 sq. in., what is the area of triangle A ? 17. If rectangle B measures 16 sq. in., what is the area of triangle B ? DENOMINATE NUMBERS 159 18. If rectangle B is 6 in. long and 4 in. high, what is its area? What is the area of triangle B? Rectangle A Triangle A (the shaded part) . Rectangle B Triangle B (the shaded part) . 19. The perimeter of a triangle or rectangle is the sum of the lengths of its sides. Written Exercise 205. Find the areas : Rectangles Base Height Base Height 1. 36 ft. 105 ft. 11. 175 ft. 348^ ft. 2. 79 ft. 437 ft. 12. 175 ft. myd. 3. 63 ft. 84 in. 13. 96 ft. 2^ rd. 4. 48 in. 4 ft. 14. 180 rd. 29f rd. 5. 96 in. 5 yd. 15. 274f rd. 48f yd. 6. 100 in. 6rd. 16. 29| yd. 17i ft. 7. 50 yd. 10.5 ft. 17. 75.79 rd. 18.80 rd. 8. 127 yd. 37.75 ft. 18. 96.37 ft. 29.98 ft. 9. 40 rd. 19.66 yd. 19. 48.24 yd. 3.48 rd. 10. 80 rd. 28.8 rd. 20. .75 in. .8 in. 160 INTERMEDIATE BOOK Triangles Babe Height Base Height 1. 4|ft. 5A ft. 11. 1 ft. 4 in. • 3 ft. 7 in. 2. 7§in. 6|in. 12. 5 ft. 8 in. 4 ft. 6 in. 3. 8iyd. 7fft. 13. 25 ft. 11 in. 16 ft. 9 in. 4. 25.5 rd. 12.8 rd. 14. 12 ft. 10 in. 20 ft. 5 in. 5. 36.87 rd. 14.24 rd. 15. 3 yd. 2 ft. 4 yd. 1 ft. 6. 5fyd. 6.8 yd. 16. 5 yd. 1 ft. 1 yd. 2 ft. 7. 28.48 in. 2|yd. 17. 8 rd. 2 yd. 2 rd. 2 yd. 8. 18.5 ft. 18.5 ft. 18. 3 rd. 2 ft. 4 rd. 6 ft. 9. 12|in. 12| in. 19. 5 rd. 1 ft. 4 rd. 1 ft. 10. 10.5 in. 20,5 in. 20. 10 rd. 10 ft. 10 rd. 10 ft. Written Problems 206. 1. The pages of a book are 5^ in. wide and 10 in. high. How many square inches in the area of the page ? 2. If one of these pages is cut in two from corner to corner, what is the area of each part ? 3. An envelope is 4 in. by 5 J in. How many square inches in its area ? 4. What is the area of a postage stamp ^ in. by 5. Which covers the greater surface, a triangle with a base of 8 in. and an altitude of 7 in., or a triangle with a base of 6 in. and an altitude of 9 in. ? 6. The first baseman, second baseman, and catcher are on the corners of a triangle. Call the DENOMINATE NUMBERS 161 line from the home plate to first base the base of the triangle and the line from the first to the second base its height. The base and height are each 90 ft. Find the area of the triangle in square feet. Also in square yards. 7. A mason has 1000 tiles, each ^ ft. square (that is, ^ ft. long and ^ ft. wide). How many square feet of floor can he lay with them ? 8. A cement walk 35 J ft. long and 6 ft. wide costs 10^ a square foot. What is its total cost? 9. A rug is 7 yd. by 6|- yd. How much larger is this than another rug 6 yd. by 7 yd. ? 10. How many square yards of wall can be covered with a roll of paper ^ yd. wide containing 85 yd. ? 11. A public hall is 150 ft. long and 20 yd. wide. Find its floor space in square feet. 12. A rectangular garden is 17^ yd. long and lOi yd. wide. How many yards of fencing would it take to inclose it ? How many square yards of ground in the garden ? 13. Which is more, a piece of ground 25| rd. by 10 rd., or one containing two acres ? 14. How many acres of land are there in I- sq. mi. ? 15. One flower bed is 35 ft. by 20 ft., another is 70 ft. by 10 ft. Find their perimeters. Find their areas. Can two rectangles have equal areas but different perimeters ? 162 INTERMEDIATE BOOK 16. A teacher orders a slate blackboard with a wooden molding around it. If the blackboard is to be 2 1 yd. long and 1| yd. wide, how many square yards of slate must be ordered ? How many yards of molding ? 17. In making boxes, pieces are cut from sheets of pulp board so as to waste as little as possible. How many pieces, each 2'' x 2'', can be cut from a sheet 14'' by W ? Can waste be avoided ? 18. How many pieces, each 3" x 4'', can be cut from a sheet 15'' x 16" ? Can waste be avoided ? 19. How many pieces, each 4'' x 5", can be cut from a sheet 16" x 18"? Can waste be avoided? 20. Make similar problems of your own in which there is waste. In which there is no waste. 21. From one sheet, make a box, without cover, of the dimensions 5" x 3" x 2". How large a sheet is needed? 22. Draw a similar figure showing the sheet needed in making a box 7" x 4" x 3". How much is the waste ? 23. How much is the waste in making from one square sheet a box 4" x 4" x 4" ? How large a sheet is needed ? 24. How much is the waste in making a box 5" X 3" X 2" from a sheet 9" x 8" ? 25. How many covers the size of this book cover can be made from a piece of cloth 1 yd. square ? DENOMINATE NUMBERS Angles 163 Bight Angle Acute Angle Obtuse Angle 207. 1. What' kind of angle do the hands of a clock make at 9 o'clock ? At 3 o'clock ? 2. What kind of angle do the hands of a clock make at 10 o'clock ? At 11 o'clock? 3. When do the hands form the larger angle, at 11 o'clock or at 2 o'clock ? 4. What kind of an angle do the hands of a clock form at 4 o'clock ? At 5 o'clock ? 5. Is the angle formed by the hands larger at 8 o'clock than at 7 o'clock ? At 9 o'clock than at 8 o'clock? 6. A right angle has been divided into 90 equal parts, called degrees. Surveyors and draftsmen usually give the size of angles by telling the number of degrees which they contain. 90 degrees (90°) = 1 right angle How many degrees are there in 2 right angles ? In 3 ? In 1 of a right angle ? In J ? In | ? 164 INTERMEDIATE BOOK Drawing and Construction Exercise 208. 1. Measure the sides of this triangle to find its perimeter. 2. Measure the height of this tri- angle. Find its area in square inches. 3. Find the area of a triangle which is twice as high and has a base twice as long as that of the triangle shown in the figure. 4. Construct 5 triangles. Measure the base and the altitude to find the perimeter and the area of each. 5. Construct a triangle with one angle a right angle. 6. Draw a large triangle similar to the triangle in Example i of this exercise. Cut off the 3 angles of the triangle and set them together on one side of a straight line as illustrated. Then draw a line as indicated. 7. To how many right angles are the three angles of a triangle equal ? 8. If a right angle is equal to 90°, how many degrees in all the angles of a triangle ? DENOMINATE NUMBERS 165 Written Exercise 209. 1. If the three angles of a triangle are equal, what part of a right angle is each ? 2. If two angles of a triangle are each just | of a right angle, how big is the third ? 3. If one angle is f of a right angle, and an- other angle is f of a right angle, how much is the third angle ? 4. If two angles are each f of a right angle, how large is the third angle ? 5. If one angle of a triangle is ^ of a right angle and another is | of a right angle, what is the third angle ? 6. Find in the walls and ceiling of your room two lines forming a right angle. 7. Express in degrees the sum of the angles of a triangle. 8. Two of the angles of a triangle are 65° and 73°. What is the third angle ? 9. How many right angles will exactly fill the space about a point in a plane ? 10. A wheel has 8 spokes. What is the angle between two neighboring spokes ? 11. Through how many degrees does the minute hand turn in an hour ? In half an hour ? In 15 minutes ? In 5 minutes ? 166 INTERMEDIATE BOOK 12. What angle do the hour and minute hands form at 1 o'clock ? At 2 o'clock ? At 5 o'clock ? 13. How many degrees in each angle of an equi- angular triangle ? 14. How many degrees are there in the four angles of a rectangle taken together ? Time Exercises and Problems 210. 1. Compute the number of seconds in one day. 2. Reduce 90,630 seconds to days, hours, min- utes, and seconds (avoiding fractions). 3. Find the number of seconds in ^ of an hour. 4. Reduce 400 seconds to a fraction of an hour. 5. Change 114 hours to days. 6. How many weeks in 192 days ? In 200 days? 7. Which months have 31 days ? Which 30 ? How many has February ? 8. A clock ticks 138 times every minute. How many times in 10 seconds ? In one hour ? 9. A man in walking takes 3 steps every 2 seconds. How many steps does he take in an hour ? 10. If he takes 90 steps per minute, how many does he take per hour ? DENOMINATE NUMBERS 167 11. If he takes 5,400 steps per hour, and each step measures 3 ft., how many feet does he travel per hour ? 12. A man walks at the rate of 4 mi. an hour. At this rate, how many rods will he walk in 20 min.? 13. In a 10-acre nursery of white ash trees there were 63 rows of trees, 630 trees in each row. In 9 weeks gophers killed ^ of the trees by gnawing their roots. How many trees were destroyed in 9 weeks ? In 1 week ? In 1 day ? 14. A gopher is known to have dug an under- ground burrow 480 ft. long in 2 days, throwing up hillocks of loose earth at intervals of about 4 ft. How many hillocks were thrown up at this rate in 2 days? The Difference in Time between Two Dates 211. 1. Find the exact number of days from June 16 to Aug. 24. In June there are 14 days remaining ; in July there are 31 days ; in August there are 24 days to Aug. 24. Add 14, 31, 24, to obtain the answer. 2. Find the number of months and days from March 4 to July 10. Find the whole number of months and then the number of days remaining. From March 4 to July 4 are 4 months. From July 4 to July 10 are 6 days. The answer is 4 months and 6 days. 168 INTERMEDIATE BOOK 3. How many days are there from July 7 to July 31 ? From Feb. 3 to Feb. 27 ? From March 16 to April 25 ? 4. Find the exact number of days from May 3 to July 22. 5. How many months are there from Jan. 5 to June 5 ? 6. How many even months and how many days over from Feb. 12 to Sept. 25 ? 7. School opens Sept. 12 and closes June 5. How many even months and how many days over are there between these dates ? 8. How many years, months, and days between Sept. 15, 1875 and Nov. 24, 1910 ? From Sept. 15, 1876, to Sept. 15, 1910, are 35 years ; from Sept. 15 to Nov. 15 are 2 months ; from Nov. 15 to Nov. 24 are 9 days. Answer: 35 years, 2 months, 9 days. 9. What was Benjamin Franklin's age at the time of his death, born Jan. 17, 1706, and died April 17, 1790 ? 10. What was George Washington's age at the time of his death, born Feb. 22, 1732, and died Dec. 14, 1799 ? 11. What was Henry W. Longfellow's age at the time of his death, born Feb. 27, 1807, and died March 24, 1882 ? DENOMINATE NUMBERS 169 12. What was Abraham Lincoln's age at the time of his death, born Feb. 12, 1809, and died April 15, 1865 ? 13. What was James R. Lowell's age at the time of his death, born Feb. 22, 1819, and died Aug. 12, 1891? 14. How many years elapsed between the birth of Washington and of Lowell ? 15. How long did Washington live after the death of Franklin ? 16. How old was Lowell when Lincoln died ? 17. How old was Longfellow when Lowell was bom? Review Exercise 212. 1. How many dimes make $ 1 ? Then what part of $ 1 is a dime ? What part are 2 dimes? 3 dimes? 2. How many cents make $ 1 ? Then what part of $ 1 is a cent ? What part are 3 cents ? 7 cents ? 3. In $ 2.35 there are how many whole dollars ? How many dimes ? How many cents besides ? 4. In $ 4:A4: what does each digit stand for ? 5. What part of a dollar is $0.10? S0.60? $0.70? 6. What part of a dollar is $0.05? $0.09? $0.06? 7. What part of a dollar is 5^ ? 15^ ? 65^ ? 170 INTERMEDIATE BOOK 8. 35^ = ^,55^' = ^, $0.45 = ^. 9. What does each digit stand for in $ 15.678 ? 10. How does the value of each digit in $ 6.666 compare with the digit to its right ? Which digit has the least value ? Which the most ? Oral Exercise 213. Tell how many dollars, cents, and mills there are in : 1. $4,765 2. $10,075 3. $0,457 4. $0,043 5. $111,111 6 $10 7. $.01 8. $.001 9. $10.01 10. $100.01 11. $1000.101 12. $1000.01 13. $1001.001 14. $1,001 15. $8,663 Making Change 214. If you owe 85^ and pay the debt with a dollar, the storekeeper gives you 5 cents and 10 cents, saying 85 and 5 is 90 and 10 is a dollar. Make change 1. For $1, when some one pays 70^. 2. For $ 1, when some one pays 65^. 3. For 50^5 when some one pays 35^. 4. For $5, when some one pays $2.75. Make change for $ 1, $ 2, and $ 5 with each of the following numbers : DENOMINATE NUMBERS 171 5. 75f! 55^ 45^ 35^ 85^ 45^ 65^ 6. 77^ 67^ 15^ 31^ 47^ 66^ 18^ 7. sy 62^ 82 «* 78^ 23^ 21^ 56^ 8. 77^ 36<^ 54^ 52 >^ 48^ 34^ 71^ Oral Exercise 215. 1. At 25^ a yard, how many yards can be bought for $ 1 ? For $ 4 ? For $ 17 ? 2. At 33^^ a bushel, how many bushels can be purchased for $ 2 ? For $ 10 ? For $ 15^? 3. How many yards of calico at 20^ a yard can be bought for $ 920 ? 4. $ 10 will buy how many pounds at 12^^ a pound ? 5. $ 30 will buy b ow many dozen at 3 3|^^ a dozen? 6. How many yards of silk costing $ 1.121 per yard can be bought for $ 243 ? 7. How many articles at $ 1.25 each can be bought for $ 75 ? 8. How many articles costing 33^^ each can be bought for $ 20 ? $ 33 ? $ lOf? 9. How many 12 1^ articles can be bought for $2? $7? $15? $7^? 10. How many 25^ articles can be had for $ 75? $10? $10.75? 11. How many 66f ^ articles can be purchased for $10? $14? $76? THE ANALYSIS AND SOLUTION OF PROBLEMS 216. 1. If 2 lamp chimneys cost 12)^, what will 3 cost ? Analysis If 2 chimneys cost 12^, 1 chimney will cost J of 12^, or 6^. 3 chimneys will cost 3x6^, or 18^. 2. What is the cost of 5 drinking glasses, if 2 of them cost 22^? 3. If 3 cakes of soap cost 15^, what is the cost of 2 cakes ? 4. If 2 sugar bowls are sold for 38^, what are 3 sold for? 5. If 3 coffee pots cost $1.20, what will 5 of them cost ? 6. What is the cost of 2 water bottles, if 5 of them cost $1.25? 7. What do we pay for 7 boxes of matches, if 2 boxes sell for 4^? 8. What is the price of 9 brooms, if 2 brooms cost 30^? 172 SOLUTION OF PROBLEMS 173 9. Trout is advertised at 30^ for 2 lb. Find the cost of 5 lb. Of 10 lb. Of 11 lb. 10. If 5 lb. of salmon cost $ 1, what will 9 lb. cost? Written Problems 217. Analyze and solve: 1. If 21 gallons of gasoline cost $ 3.54, what will 49 gallons cost? Solution 3 54 1 gallon costs -j— dollars, zi 49 gallons cost ^' dollars. z J. Cancel factors common to both terms, thus: 1.18 7 7 The process is shorter if the indicated division is sim- plified but not performed. Thus, -^ is ♦ Then 3.54 x49 354 X 49 ^ -, . .. ., — = — — . By so doinff we oiten escape the 21 2100 ^ ^ ^ necessity of dividing. In any case, it is better not to carry out the operations until the last step. 2. What is the cost of 35 lb. of catfish at % 2.67 for 15 lb. ? 174 INTERMEDIATE BOOK 3. If 2 gallons of benzine sell for 25^, what is the cost of 25 gallons? 4. If 20 A. of land sell at $ 310, what will 59 A. sell at? 5. If 2 acres of strawberries yield a crop worth $ 259.65, what will 121 acres yield ? 6. At $ 16.50 per half dozen pairs of gloves, what will 21 dozen pairs cost ? 7. John's salary is $ 700 a year. What is it for 7 months? 8. If f of an acre of garden land sells for $ 375, what will i of an acre sell for ? 1 acre ? 9. At $ 5 a ton of 2000 lb., what will 3000 lb. of coal cost? 3500 lb.? 10. Mutton is quoted at $ 4.60 a hundred pounds. How many pounds can be purchased for $ 230 ? 11. At $ 5.75 a hundred pounds, find the cost of 275 lb. of lamb. 12. Calves sell at $5.10 per hundred pounds. How much will calves weighing 2700 lb. bring? 13. Find the price of cows weighing 3200 lb. at $ 3.25 a hundred pounds. 14. At $ 30 a hundred pounds, find the cost of 1875 lb. of creamery butter. 15. At $ 9.75 a hundred pounds, 7^ tons of timothy hay cost $ . SOLUTION OF PROBLEMS 175 16. If 3 men can do a piece of work in 6 days, how long would it take 1 man to do this work ? 17. How long will it take 10 masons to build a certain foundation for a house, if 3 masons can do it in 20 days ? 18. If 50 ft. of garden hose sell at $ 5.75, what do 125 ft. sell for? 19. When asked his age, a man replied, " f of my age is 22| years." How old was he ? Suggestion 1 J. , . . 45 8 J. ... 45 X 8 m. 8 "* ^'' ^^' '' 273' 8 "^ '' '' Y^- ^^'" cancel. 20. If 8^ yd. of calico cost $.66, what will 57 yd. cost ? 21. If the rent for a house for 9 mo. was $319.50, what is the rent for a year? 22. If a certain sum of money brings $ 198 in 15 months, what will it bring in 20 months ? 23. If 65 % of a certain sum of money, put out at interest, yields $ 390 annually, how much would the entire sum yield at the same rate of interest ? 24. At $14.50 a hundredweight, what will 71 lb. cost ? 171 lb. ? 25. If 5 men can do a piece of work in 12 da., how long will it take 15 men to do the same work? 176 INTERMEDIATE BOOK Solution and Explanation Will 1 man do the work in a longer or a shorter time than 5 men? To find the time it takes 1 man to do the work must you multiply 12 days by 5, or divide 12 days by 5 ? If it takes 1 man 12 x 5 days, must this product be multiplied by 15 or divided by 15, to find the time it takes 15 men to complete the work ? 26. If 12 dredging machines can clear a certain channel in 18 days, how long will it take 16 to do the same work ? 27. If 4 persons eat 5 packages of breakfast food in 15 days, how many persons will eat 6 packages in 8 days ? Solution and Explanation If 5 packages are eaten in 15 da. by 4 persons, then 5 packages are eaten in 1 da. by 4 x 15 per- sons ; 1 package is eaten in 1 da. by — - — persons ; 5 6 packages are eaten in 1 da. by per- 5 sons, and 6 packages are eaten in 8 da. by 6x4x15 c- 36x2x15 o .u — - — - — persons, bmce — - — - — = 9, the an- 5x8 ^ 5x8 ' swer is 9 persons. In this method of solution, the concrete number of the kind required in the answer is put last. SOLUTION OF PROBLEMS 177 In this example, the required number is " persons." We arrange the statement so that " persons " comes last. 28. If 2 launches require 10 gallons of oil to travel 7 hours, how many launches can travel on 15 gallons for 3 hours ? 29. If it takes 2 boys 5 days to build a pigeon house, how long will it take 3 boys working at the same rate ? 30. If 880 bricks are needed for a wall 10 ft. long, 2 ft. wide, and 2 ft. high, how many bricks are needed for a wall 12 ft. long, 1 ft. wide, and 5 ft. high ? 31. A clothier invests $368.55 in boys' coats. How many does he buy, if each coat costs $ 1.89 ? 32. A man fails in business. He owes $7900, and his creditors receive 65% of this. What amount do they receive ? 33. If f of a man's money is $2430, how much has he ? 34. How long is a pole, if ^ of it is 40 ft. ? 35. If 10 yd. of gingham cost $ .75, how much will 87 yd. cost? 36. If 3 teams of horses can plow a field in 1 6 days, how long will it take 4 teams to plow the same field ? 37. If 5 men can do a piece of work in 12 days, how long will it take 7 men to do the same work ? 178 INTERMEDIATE BOOK 38. If 1.8 yd. of silk cost $3.24, find the price of 71 yd. 39. The earth moves in its path around the sun at the rate of 1110 miles a minute. How many times faster does it move than a train which travels 54 miles an hour or miles a minute ? 40. If a train travels 75 miles in If hours, how far will it travel in 7 hours ? 41. Skimming milk by hand, only f of the cream is obtained. In a week a farmer obtained 270 qt. of cream. How much would he have obtained if he had used a separator, which extracts all the cream ? 42. William earns $750 a year, which is f as much as his father earns. How much does his father earn ? 43. If ^j of a certain number is 42, how much is f of that number ? 44. If a stable has enough oats to feed 30 horses 45 days, how long will the oats feed 20 horses ? 45. If a clock gains 1|- minutes in 24 hours, how much time will it gain in 40 hours ? 46. At the rate of 3 miles an hour, I can walk a certain distance in 2 hours 30 minutes. What is my rate when I walk this distance in 3 hours ? 47. If a certain weight of sheet iron, f in. thick, covers 45 sq. ft., how many square feet will the same weight of sheet iron only J in. thick cover? SOLUTION OF PROBLEMS 179 48. Using 4 electric lights, the electric bill is $ 5 a month. What is the bill for 6 months when 3 electric lights are used ? 49. If the interest on a certain sum of money is $ 120 for 9 months, what is the interest on that sum for 25 months ? 50. A farmer raised 1,125 bushels of beets on 3 acres. At this rate, how many bushels could he have raised on 11 acres? Unitary Analysis 218. 1. If 3 packages of rolled oats sell for 24^, what is the cost of 4 packages ? Analysis In problems of this kind it is often easier to find the cost of one unit, then the cost of the required units. In this case find the cost of 1 package, then the cost of 4. Thus, If 3 packages sell for 24^, 1 package sells for ^ of 24^, or 8^. 4 packages sell for 4 x 8^ = 32^. Ans. 2. If 276 is f of a number, what is the number ? Analysis If I of the number = 276, then i of the number = 1 of 276 = ^^=69. f of the number = 5 x 69 = 345. Am, 180 INTERMEDIATE BOOK Oral Exercise 219. Analyze, explaining the process in each case : 1. If 51 is f of a number, what is the number ? 2. If 15 is f of a number, what is the number ? 3. Of what number is 12 the six sevenths part ? 4. 54 is ^Y of what number ? 5. 56 is ^ of what number ? 6. If 48 is f of a number, what is the number ? 7. If 121 is ^- of a number, what is the number ? 8. If 3 men together earn $12 in one day, what will 7 men earn daily at the same rate ? 9. If 14 men pay $56 for board per week, what will 11 men pay at the same rate ? 10. If 4 yd. cost 22^, what will 10 yd. cost? 11. 35 is f of what number ? 12. 27 is f of what number ? 13. 100 is 20 % of what number ? 14. 21 is 1 J times what number ? 15. 40 is .20 of what number ? 16. 24 is ^ less than what number ? 17. 16 is 1^ less than what number? 18. 48 is ^ more than what number ? 19. 36 is I' more than what number ? 20. 75 is 75 % of what number ? APPROXIMATIONS Oral Exercise 220. As a check against absurd results it is desirable that pupils accustom themselves to giving approximate answers. Whenever possible, the ap- proximate results should be found orally. Suppose a pupil wishes to find the cost of 2.8 lb. at $ 1.05 a pound and writes his answer $ 29.40. By a brief mental computation he should detect the error at once. 2.8 lb. is nearly 3 lb., and $ 1.05 is nearly $ 1. 3 lb. at $ 1.00 gives $ 3.00 as an approximate answer. Hence, $29.40 is absurd. The correct answer is $ 2.94. 221. Give the approximate answer. Check with the correct answer. 1. 28 handkerchiefs at $ .24. 2. 22 yards at $.29. 3. 13 wool blankets at $4.95.. 4. 5 gingham skirts at $ .98. 5. 11 suits of clothes at $12.75. 6. 98 yd. of silk at $ .79. 7. Find cost of 1 pair of shoes when 15 pairs cost $ 74.25. 181 182 INTERMEDIATE BOOK 8. Cost of 23 linen suits at $ 3.98 each. 9. Cost of 105 pairs of shoes at $ 2.45 a pair. 10. At 44 1^ a pound, find cost of 2|- lb. 11. Cost of 1 hammock, 31 cost $96.72. 12. The monthly salary of a man who receives $ 2350 a year. 13. Cost of 52 books of fiction at $ .39. 14. Cost of 19 sacks of potatoes at $ 1.25. 15. Suggest problems for testing the ability to estimate answers that are approximately correct. AVERAGES Introduction 222. We have used the term average in our problems and our discussions. The average of 6, 7, 11 is found by adding them and dividing the sum by 3. mi, • 6+7-fll 24 Q Ihe average is = -— = 8. 3 3 In all cases, find the total of the items, then divide the sum by the number of the items. Statistics are, to a large extent, averages. We speak of the average attendance at school, the average temperature, the average number of days of sunshine per month, the average rations for man and animals, the average height of men, the average crop, and so on. What problem can you make requiring that the average be found ? Problems in Averages 223. 1. John is 11 years old, James 12, Harry 10, Wallace 15. Find the average age. 2. A driver earned on successive days $ 3, $ 5, $4, $ 1. What is his average daily earnings ? 183 184 INTERMEDIATE BOOK 3. A merchant's receipts for 3 consecutive days were $ 200, $ 200, $ 500. Find his average daily receipts. 4. At 6 o'clock on 4 mornings the thermometer stood 61°, 59°, 50°, 70°. Find the average tem- perature. 5. A man earns 50 j^ an hour. He works 6 hr. on Monday, 2 on Tuesday, 7 on Wednesday, 10 on Thursday, 8 on Friday and 3 on Saturday. What were his average daily earnings ? Written Problems 224. 1. What is the average weight of 5 bales of cotton weighing 450, 460, 475, 455, 457 lb.? 2. What should ground feed, made from an equal number of bushels of oats @ 28^, barley @ 78^, and corn @ 59^, be sold per bushel, in order to yield a profit of 20 % on the cost ? 3. The weekly salary list of 5 employees in a store is $ 25, $ 20, $ 18, $ 17.75, $ 15.50. What is the average weekly salary ? 4. The cyclometer on an automobile shows that the distances traveled in 4 da. are 130.4 mi., 82.27 mi., 90.01 mi., 207.54 mi. What is the average distance traveled per day ? 5. A pleasure launch uses 5J gal. gasoline the first day, 4|- gal. the second day, 6|- the third day. SOLUTION OF PROBLEMS 185 and 4|- the fourth day. What is the average daily consumption of gasoline ? 225. Table FOR Reference Height Ages Ft. In. 15-24 2^29 30-34 85-39 40-44 45-49 50-54 lb. lb. lb. lb. lb. lb. lb. 5 120 125 128 131 133 134 134 5 2 124 128 131 133 136 138 138 5 4 131 135 138 140 143 144 145 5 6 138 142 145 147 150 151 153 5 8 146 151 154 157 160 161 163 5 10 154 159 164 167 170 171 172 6 00 165 170 175 179 180 183 182 226. Problems Based on the Table 1. The weights given in this table are the averages obtained by weighing 74,162 applicants for life insurance. From this table complete the average weight of men between the ages of 15 and 24, which are not less than 5 ft. nor more than 6 ft. tall. 2. Do the same for each of the six other age periods in the table. 3. Find the average weight of men 5 ft. tall, between the ages of 15 and 54 years. 4. Do the same for each of the other heights given in the table. 5. Make 5 problems based on the table. THE DIRECT METHOD OF SOLUTION 227. It frequently happens that the solution of problems may be performed in a direct method by eliminating useless operations. The results are more speedily obtained in this way and there is less likelihood of error. Oral Problems 1. If butter sells at $ 18.50 per 100 lb., what is the cost of 300 1b.? Solution. — In examples like this, where 300 lb. is exactly divisible by 100 lb., it is easier not to find the cost of 1 lb. Since 300 lb. is 3 times 100 lb., the cost of 300 lb. will be 3 times 118.50, or 8 55.50. This is called the direct method of solution, 2. At $3.50 for 200 lb. of lignite, what is the cost of 6000 1b.? 3. Find the cost of 120 eggs at 23|^ per dozen. 4. Find the cost of 300 bbl. of flour at $ 575 a 100 bbl. 5. If 10 men earn % 175 a week, what will 40 men earn in the same time ? 6. If 10 men can do a piece of work in 36 days, how long will it take 40 men to do the work ? 7. How much will 150 gal. of molasses cost at $12.50 for 50 gal.? 186 THE DIRECT METHOD OF SOLUTION 187 8. Mary bought 10 yd. of cheviot at $3.95. If Lucy bought 30 yd. of cheviot, how much did she pay ? Written Problems 228. Solve, using direct method wherever pos- sible : 1. In a city of 200,000 inhabitants, 93 inhabi- tants out of every 1000 are foreign born. What is the total foreign population ? 2. In the same city there are 2|- grocery stores for every 6 inhabitants, or how many grocery stores in all? 3. If a circle, 16 in. in diameter, has a circum- ference of 50.2 in., what is the circumference of a circle whose diameter is 112 in.? 4. If 56 lb. of rye make a bushel, how many bushels in 1120 1b.? 5. What will 8000 bu. of barley weigh, if 40 bu. weigh 1920 lb. ? 6. If 1 cu. ft. of cast iron weighs 446 lb., what will 144 cu. in. weigh ? 7. What is the cost of floor tiles for a room 35 ft. by 40 ft., at $ 15 per 100 sq. ft. ? 8. A certain mantel tiling costs 40^ per square foot. If a man purchases tiling to the amount of $ 8, how many square feet can he cover ? 188 INTERMEDIATE BOOK Oral Exercise 229. Find the cost : Articles Kate Articles Rate Articles Rate 1. 7 Si0Tl2f 5 2 for 24^ 6 5 cost 55^ 2. 10 4 for 20^ 6 7 cost Mf 3 4 cost 52 ^ 3. 3 2 cost 34^ 2 3 cost 27 ^ 12 7 cost 28 J^ 4. 30 331 cost 100^ 12 12^ cost 100^ 6 16| cost 100^ 5. 6 4 cost 32^ 5 6 cost 72^ 21 30 cost 60 ^ 6. 5 3 cost 36 J? 9 5 cost 75^ 10 4 cost 36 ^ 7. 12 4 cost $44 7 6 cost $36 100 2 cost $3.50 8. 100 3 cost $3.60 100 7 cost $2.10 100 5 cost $5.50 9. 20 13 for $2.60 200 14 for $4.20 30 11 for $1.21 10. 50 6 for 15^ 10 6 for $4.20 4 3 for $3.30 PROBLEMS THAT MAY BE ILLUSTRATED BY SIMPLE GRAPHS Industry 230. 1. Much gold and silver is taken from mines in Colorado, California, and Nevada. Coal is found in many places in the United States. Is the coal taken out of the earth in one year worth as much as the gold and silver ? The lines show the value of several minerals produced in the United States during one year. One inch stands for 120 million dollars. Which mineral repre- sents the greatest value ? Which the least ? How long is the line representing soft coal ? Measure the line to the nearest tenth of an inch and see how nearly correct you are. Soft Coal , Gold & Silver Hard Coal Iron _— ^__--. Copper.. _____-■_ 2. How many million dollars stand for the value of the soft coal for last year ? 3. How long is the line for gold and silver, and how many million dollars does it represent ? 4. How many million dollars of hard coal were produced? 189 190 INTERMEDUTE BOOK 5. How many millions more soft coal than hard coal ? 6. How many millions more of soft coal than gold and silver ? 7. About how much more of gold and silver than hard coal ? 8. How many million dollars of iron were mined? 9. How many million dollars of copper were mined ? 10. The limestone production was valued at 30 million dollars How long a line stands for that value ? 11. Draw a line representing a mineral produc- tion worth 150 million dollars. 12. Construct similar problems of your own. Rainfall 231. 1. The United States government has gauges in different parts of the country by which the exact amount of rain or melted snow is meas- ured. From this diagram, show during which two months San Francisco has the least rain and Santa Fe the most. 2. During which months has San Francisco the most rain and Santa Fe the least ? PROBLEMS THAT MAY BE ILLUSTRATED 191 3. On a strip of paper copy carefully the scale of inches. You can measure easily to the nearest J of an inch or, if you are careful, to the nearest ^ of an Normal Monthly Rainfall m SAN FRANCISCO GAL. 1 in SANTA fE N. MEX. Jj >. g ^ bb m O O 0) "A P inch. If you apply the scale to the line showing the rainfall in San Francisco for January, you find the line 4|^ in. Ions;. c? i * t v *> ^ , Scale of Inches That means that if a tub 12 3 4 5 is left out of doors to catch the rain and snow during January, the rain and melted snow in the tub at the end of the month is, in an average year, 4| in. deep. The tub must not leak. It must be covered during fine weather, to prevent evaporation. 4. Find the number of inches of rainfall for a year in San Francisco. Also in Santa Fe. 192 INTERMEDIATE BOOK 5. In whicli place is the total amount of rain- fall for one year greater ? 6. Which place has a more even distribution of the rainfall throughout the year ? 7. In which place can you more easily dispense with an umbrella during a summer vacation ? 8. During 6 successive years in Santa Fe the number of inches of rainfall per year was 20, 14, 20, 12, 10, 16. Find the average. 9. In some parts of the Panama Canal zone the annual rainfall is as high as 120 in. How many times more is this than 15 in., which is the annual rainfall in Denver ? 10. At Cheyenne, Wyo., the monthly rainfall (in inches), beginning with January, is as follows : 4 5 8 1 5 9, 4 1 5 9, 1 5 9 7 4 3 TO"? TO"' TO"? -'-ro'? ^10"? -*^T0"' ^? -*-T0? 10? TO"? TO? TO* Find the rainfall per year, also the monthly average. Population of the United States 232. 1. The curve AB shows the increase in population in the United States during 110 years. Years are marked off from left to right ; the popu- lation from the bottom line up. One space up stands for 5 million inhabitants. The point A indicates the population in the year 1800. This point is about one space up and stands, in round numbers, for how many inhabitants ? PROBLEMS THAT MAY BE ILLUSTRATED 193 CO o CO 00 o o O y-i 05 05 90 80 s 70 60 2. At the year 1820 the curve is spaces up. This indicates about inhabitants. 3. At the year 1840 the curve is a little more than 3 spaces up, or about 3.4 spaces. The popu- lation was, there- fore, about 3 4 times 5 million, or million. 4. In this way estimate the pop- ulation for 1860, 1880, 1900, and 1910. 5. About how many more in- habitants were there in 1820 than in 1800? In 1900 than in 1880? 6. Estimate the population for the years 1810, 1830, 1850, 1870, and 1890. 7. Make a new graph to show about what the population of the United States is in 1915. What will be the approximate population of the United States in 1920 ? -) 50 '^ 40 30 20 10 A "1 B / / / 1 / / f / / / / / / / / / / / X 1 194 INTERMEDIATE BOOK The Money Value of Training 233. The St. Nicholas (Vol. 31, p. 57) gives statistics of two typical boys. The wages of John, who enters a shop at the age of 16, are compared ^ y y $20 / / >r J /^ yi. express per cent as a decimal. Written Exercise 238. Write as common fractions i in their lowest terms : 1. 10%. Process tV PERCENTAGE 199 2. 6% 3. 4% 4. 12% 5. 16% 6. 18% 7. 20% 8. 25% 9. 50% 10. 40% ii. 75% 12. 48% 13. 64% 14. 24% 15. 72% is. 96% Drill Exercise 239. Memorize these relations : 50% = J 100% = 1 331%=!=. 33^ 25% = J 75% =1 66|%=f = .66| 20% =1 10% = ^ 12i%=i=.12i 2%=^V 62J% = | 37J% = |=.37i iro=jh 80% = f 16|%=i=.16f Written Exercise 240. Write as common fractions in their lowest terms : 1. 121 2. 331% Process 121% = .121 = 1 3. 66t 371% 871% Comparison 241. 1. How many fifths of AisBI (7? D? Ul 2. How many hundredths of^is^? Z)? (7? B1 3. What per cent of A is (7? ^? i)? ^? 4. What per cent of Z) is ^? A B D E 200 INTERMEDIATE BOOK 5. What per cent oi Cis JEI Z)? 6. What per cent of ^ is J5' ? i)? (7? 7. What part is 50 % of J. ? 20 % of J. ? 8. What part is 50 % of ^ ? 25 % of ^ ? . 9. What part is 331% of CI 66f % of (7? 10. What part is 50 % of i) ? 100 % of i) ? Oral Exercise 242. 1. How many hundredths of an inch is | in.? fin.? fin.? fin.? fin.? fin.? fin.? fin.? 2. What per cent of 1 in. is f in. ? f in. ? fin.? fin.? fin.? fin.? fin.? fin.? 3. What per cent of 2 ft. are 6 in. ? 3 in. ? 2 in. ? 1 in. ? 4 in. ? 5 in. ? 9 in. ? 4. What part of 1 inch is 50 % of it ? 25 % of 1 inch ? 5. What part of J an inch is 25 % of it ? 50 % of! inch? Study Exercise 243. Consider the exercise 5 % of 400 is 20. In this exercise we call 400, the base; 5%, the rate ; and 20, the percentage. The number upon which the percentage is found is called the base. The number of hundredths to be taken is called the rate or rate per cent. PERCENTAGE 201 The result obtained by finding a certain per cent of the base is called percentage. When you have the product of two numbers, how do you check your result ? If you divide the percentage by the base, what is the quotient ? Or, if you divide the percentage by the rate (expressed decimally), what is the quotient ? We have, therefore, the following principles : The percentage is equal to the base multiplied by the rate. The rate is equal to th*e percentage divided by the base. The base is equal to the percentage divided by the rate. 1. Given the base $ 2300, the rate 8 %, compute the percentage. 2. Given the rate 11%, the base 3587 lb., find the percentage. 3. When you know the base and the rate, how do you find the percentage ? If we write h for base, r for rate, percentage, we can indicate these more briefly by means of equations oi thus, and p for principles • formulas. p = hxr, h r 202 INTERMEDIATE BOOK Find the Percentage Written Exercise 244. 1. Find 5 % of 100 bu. Explanation. — Apply the formula Process p = h x r. Change 5 % of 100 bu. = Yo"^ of 100 bu. the per cent into a = 100x-^- decimal fraction. _ g ^^^ The exercise then becomes an exercise in fractions. Solve by fractions. In the exercise what is the base ? What is the rate ? What is the percentage ? Perform the operation by fractions when it is possible to do so to advantage. 2. Find 5% of 600 1b. 3. 6 % of 50 yd. 4. 20 % of 60 oz. 5. 30 % of 400 bu. 6. 12% of 900doz. 7. 3 % of 800 mi. 8. 5% of 80^. 9. 16% of 20T. 10. 50^ 7 of 40 men. Written Exercise 245. Apply the formula. Solve and explain. Find 6 % of : 100 400 150 250 300 200 1000 900 500 700 PERCENTAGE 203 Find 8 % of each of the following : 200 700 900 150 250 450 500 550 600 750 What is 10% of: 350 A. 130 T. 145 bu. 700 men 870 bu. 750 bricks 930 ft. 670 lambs 360 da. 750 ft. Written Problems 246. Read the problems carefully. Select the formula to be used. Decide whether to use frac- tions or decimals in the solution. Then solve the problem. 1. A farmer has 200 sheep, of which 5 out of every hundred, or 5 %, are black. How many sheep are black ? In this problem it is required to find a per cent of a number. What is the base ? What is the rate ? What is the formula to be used ? Process 2. John buys a $2.50 hat at 10% off. How much does he save by this reduction in price ? 204 INTERMEDIATE BOOK Process by Common Fractions 10% of $2.50 = J^ of $2.50 = $21x33^ 2 i^ 2 4 = $.25 Explanation. — Change the per cent to a common fraction. Change the number of dollars to an improper fraction. Multiply, using can- celation. Express the fraction of a dol- lar in cents. Process by Decimals 10% of $2.50 = .10 of $2.50 = $2.50x.l0 = $.25 Explanation. — Change the per cent to a decimal fraction. Multiply as in deci- mals. Written Problems 247. 1. A suit of clothes, marked $15.50, is sold at 10 % off. How much in dollars and cents was saved by the buyer ? 2. A merchant offers a reduction of 15 % for cash purchases. Mary's mother buys goods to the amount of $ 150. How much does she save by paying cash ? How much does she actually pay ? 3. A school has 360 pupils enrolled. Of these 5 % are absent. How many pupils are absent ? How many are present ? PERCENTAGE 205 4. A man sold a bicycle that cost him $ 60, and lost 15 % of the cost. What was the selling price ? 5. Last year a man earned $ 1500. This year he earned 8 % more. How much more does he earn this year than last ? 6. An agent sells a lot for $1250 and receives 2 % of this sum for selling it. How much does he receive ? 7. A real estate man bought a house for $ 5750 and then sold it at a profit of 6 % . What was the selling price ? 8. A farmer buys 32 cows at $55 each. For what sum must he sell his stock to realize 10 % on the sum paid ? 9. A boy bought a bicycle for $ 25 and sold it at a loss of 40 % . How much did he lose ? 10. A ranchman lost 5% of his herd of 4560 sheep. How many sheep were left ? 11. A farm costing $ 6000 was sold at a gain of 7 %. What was the amount gained ? 12. An orchard has 8 rows of trees with 10 trees in each row. Five per cent of the trees are dead. How many trees are alive ? 13. At a city election there were cast 5600 votes, of which the successful candidate received 59 %. How many votes did he receive ? 206 INTERMEDIATE BOOK Written Exercise 248. 1. Find 36% of $32.50. Check the answer. Process 32.50 .36 19500 97500 Explanation.— 36% = .36. Hence, 36% of $32.50 is the same as .36 off 32.50, or .36 x 132.50 = $11.70, the answer. 11.7000 Check: 1170 -^ 36 = 32.50 2. 15% of $279.60 3. 75 % of $ 988 4. 161% of $960 5. 12% of $304 6. 7 % of $ 1234 7. 371% of $875.50 8. 24|% of $1758.97 9. 871% of $565.50 10. 3i-% of $75.60 Find the Rate Written Exercise 249. 1. If the base is $ 375 and the percentage $52.50, find the rate. Process .14 375)52.50 37 5 15 00 15 00 .14 = ^1^=14% Am. Explanation. — Apply the formula r =p -i-b. Write the quotient as a per cent. PERCENTAGE 207 Find the rate : 2. Base $83.40, percentage $4.17. 3. Base $66, percentage $4.62. 4. Base $ 37.50, percentage $ 13.50. 5. Percentage 540 lb., base 4500 lb. 6. Percentage 5.88 bu., base 36.75 bu. 7. Percentage 559.5 mi., base 746 mi. 8. Percentage $ 882, base $ 840. Written Problems 250. Use pencil only when necessary. 1. Herbert writes 200 words, but misspells 20 of them. What per cent does he miss ? The problem is to find what per cent one num- ber is of another. What is the base ? What is the percentage ? What formula to be used ? Process 2. A fruit raiser planted 200 orange trees, 7 % of which died. How many trees died ? Which is the base ? Which the rate ? Which the percent- age? 3. A fruit raiser planted 200 orange trees, 14 of which died. What per cent died ? 4. Of 300 children enrolled in a school, 5 % are absent. How many are absent ? 208 INTERMEDUTE BOOK 5. From Problem 3 make up an example in which the base and the percentage are given, and the rate is to be found. 6. With the numbers $ 500 and 5 %, make up a problem, to find the percentage ; then a problem to find the rate. 7. A boy had 200 stamps, but lost 6. What per cent did he lose ? In other words, 6 is what per cent of 200 ? 8. The loss in weight of 800 lb. of wheat in drying was 16 lb. What was the rate of shrink- age? 9. A small army of 600 men has 66 officers. What per cent of the army are officers ? 10. A tank filled with 2500 lb. of salt water taken from Great Salt Lake contains 475 lb. of salt. What per cent of salt is there in the lake water ? 11. 5000 lb. of water from the Atlantic Ocean contains 180 lb. of salt. What per cent of salt is there in the water ? 12. A man at the seashore allows 6600 lb. of salt water to evaporate and he finds that 264 lb. of salt remain behind. What per cent of salt is there in the water ? 13. A man with a yearly salary of $ 1500 spends $112.50 on clothes. What rate per cent of his salary is thus spent ? PERCENTAGE 209 Find the Base 251. 1. If the percentage is $ 122.45 and the rate 16 %, what is the base ? Process $ 765.31+ J.71S. ,16)$ 122.45.00 112 104 96 85 80 50 48 20 16 Check the answer thus : $ 765.31 X. 16 = ? Explanation. — Apply the formula h = p-T-r. Change 16% to .16 and multiply both dividend and divisor by 100 in order to remove the decimal point from the divisor. This multiplication may easily be performed by moving the point two places to the right. Why ? Check. Find the base : 2. Percentage 80.75 T., rate 17 %. 3. Rate 6^ %, percentage 32.5. 4. Rate 5|- %, percentage 41.47. 5. Rate 23 %, percentage 107.18 lb. 6. Rate .06, percentage $ 30. 7. Percentage 11.4, rate .12. 8. Percentage 1139.45, rate 3^. 210 INTERMEDIATE BOOK Written Problems 252. 1. Mr. Jones borrows $ 400 and pays 6 % interest a year. How much interest does he pay yearly ? 2. Mr. Jones pays $24. interest a year, which is 6 % of the money he borrowed. How much did he borrow ? Suggestion $ 24 -f- .06 = $400 3. How many pupils in a school of 200 pupils are absent if 3 % are absent ? 4. From Problem 3 make an example in which the percentage and the rate are given, and the base is to be found. 5. A man paid $ 18 interest a year, which was 6 % on the money borrowed. How much did he borrow ? 6. If 6 % of a number is 24, what is the number ? 7. A merchant sold a motorcycle for 80 % of its cost and received $ 160. How much did the machine cost ? 8. Mary spends 15^ at a fair, which is 10 % of what she had. How much did she have ? 9. In a battle an army lost 1815 men, which was 3 % of the number of men engaged. How many men took part in the battle ? PERCENTAGE 211 10. An orchardist has 75 orange trees, which is 60 % of his number of lemon trees. How many lemon trees has he ? 11. A merchant saves $875 a year, which is 35 % of his earnings. Find his earnings. 12. An agent collected money for me, and I paid him $ 14.40 for his services. This was 6 % of what he collected. How much did he collect ? Oral Problems 253. 1. Albert has 48^ and spends 25 % of it for writing paper and 50 % of it for firecrackers. How much does he spend on writing paper? How much on firecrackers ? 2. William has 96^ and spends 50 % of it for entertainment. How many cents has he left ? Had he spent only 25 %, how much would be left ? 3. A farmer bought a house for $ 200 and sold it at a loss of 7^ % . What did he receive for it ? 4. A ranchman bought 400 sheep from one man, and 75 %" as many sheep from another. How many did he buy all together ? 5. A merchant buys suits at $ 20 each and wants to sell them so as to make 40 % on the cost. How high must he mark each ? 6. A boy gains $ 5 by selHng a bicycle. This is a gain of 10% of the cost. What is his per cent of profit ? 212 INTERMEDIATE BOOK 7. A real estate dealer buys a lot for $ 2000 and sells it for $ 2100. What is his per cent profit ? 8. If I sell my watch at a gain of $ 6, I gain 25 % . Find the cost and the selling price of the watch. 9. A boy buys a dozen stamps for 25^ and sells them for 30^. What is his per cent of profit ? 10. A girl had a number of roses. She gave away 10 of them. This was 50% of the whole number. How many roses had she in all ? Written Problems 254. 1. In a ward 12^ % of the registered voters did not vote at the last election. There were 70 who did not vote. What was the total registration ? 2. Of the votes cast, 55% were for one party and 45% for the other. The winning party had 56 more votes. How many voted ? 3. Of the 5800 registered voters in a city, 899 fail to vote. What per cent fail to vote ? 4. 65 % of the blossoms on a small apple tree failed to develop into fruit. The tree bore 77 apples. How many blossoms did it have ? 5. A poultry raiser set 150 eggs. 12 % failed to hatch. How many eggs did hatch ? 6. A farmer raised 921 bu. of wheat. He sold 33|^% of them at $ 1.05 a bushel and the rest for 95^. What did he get for the whole crop ? PERCENTAGE 213 7. If unseasoned lumber is 18% water, what will 125 T. of grain lumber weigh after it is seasoned ? 8. An acre of land produces 12,750 lb. of sugar beets. If the beets are 12^% sugar, how many pounds of sugar were obtained from these beets ? 9. If a beef weighing 1200 lb. contains 192 lb. of tallow, what per cent of the whole weight is tallow? 10. On an experimental farm 200 seeds were planted to test them. Of these only 105 sprouted. What per cent of the seed was good ? 11. If an ounce of flower-seed costs 30^, and 50% of the seed is good, what is the price per ounce of the good seed ? 12. A fruit dealer buys a crate of oranges for $2.50 and sells them at 2^ each, making a profit of 40 %. How many oranges are there in the crate ? APPLICATION OF PERCENTAGE Oral Problems 255. 1. Sugar costing a merchant 5^ a pound is sold by him for 6 ^ a pound. How much is his profit on 37 lb. ? What per cent of the cost is his profit ? 2. If a merchant buys sugar at 4 ^ a pound and sells it at 5^, what per cent of the cost is his profit? 3. If an automobile is bought for $1200 and sold later at a loss of 5 %, what is the selling price? 4. A dealer in stationery buys pencils at 40 ^ a dozen and sells them at 5 j^ apiece. What is his per cent of profit ? Notice that the per cent of gain or loss is always figured on the cost of the goods or on the sum invested. 5. A grocer buys grapefruit at 7 ^ each and sells them at 10 ^ each. What per cent of profit does he make ? 6. He buys lemons at 25 ^ a dozen and sells them at 35 ^. What per cent is his profit ? Exercises and problems of this character are sometimes classified under the heading Gain or Loss. 214 APPLICATION OF PERCENTAGE 215 Written Problems 256. 1. A city lot was bought for $600, and sold at a loss of 15 %. What was the loss ? What was the selling price ? 2. A dealer paid $ 450 for a pair of horses and sold them at a profit of 25 %. Find the selling price. 3. A merchant sold a piano and gained $ 50. If it cost him $ 400, what was his per cent of gain ? 4. A merchant has $ 15,000 invested in a store. His profit this year is $3000. What is his per cent of profit ? 5. What is the per cent of profit, when you buy at $ 600 and sell at $ 700 ? 6. If I buy table water at $ 1.10 a dozen bottles and sell it at 2 bottles for 25 ^, what is my per cent of profit ? 7. If I buy pickles at $25.92 for a gross of bottles, and sell at 25^ a bottle, what is my per cent of profit ? 8. Clothes that cost $2000 were damaged by fire and sold at a loss of 17 %. How much was lost ? 9. What is the gain on bank stock bought at 88 and sold at 96 ? 216 INTERMEDIATE BOOK Written Exercise 257. Find the selling price, the per cent gain or loss, or the cost: 1. The cost is $ 40, the loss is 5 %. 2. Loss 7 %, cost $ 1000. 3. The cost is $16, the gain 25 %. 4. Gain 121%, cost $250. 5. The selling price $ 80, the cost $ 75. 6. The selling price $ 120, the cost $ 100. 7. The selling price $ 225, the cost $ 180. 8. The selling price $ 240, the cost $ 250. 9. The selling price $ 850, cost $ 100. 10. The selling price $ 1275, the cost $ 1500. Make problems to illustrate these relations. Discount 258. Merchants, manufacturers, and business houses frequently make a reduction or a discount from the catalogue price or the list price to those who buy goods in large quantities and to those who pay cash for their goods. Discounts are often offered by merchants in order to increase trade. 1. A merchant offered a reduction of 5 % on purchases amounting to $25 or more. One cus- tomer bought $ 30 worth of goods. What reduc- tion was made ? APPLICATION OF PERCENTAGE 217 Any reduction made from a fixed price is called a discount. Discounts are usually reckoned as so many per cent of the fixed or list price. The price after the discount is taken off is often called the net price. 2. A suit of clothes, marked at $35, is offered at 10 % off. How much is the discount and how much is the selling price? 3. Goods damaged by fire were sold at the following discounts : Marked price :$ 15 $39 $60 $120 $24 $77 Discount: 20 % 10 % 5% 1% 121% 50% Find the reduced prices. Discount is simply an application of percentage. It involves no new principles. The marked price or list price corresponds to the base. The rate of discount is the rate per cent. The discount expressed as a sum is the per- centage. Oral Exercise 259. Find the discount and the reduced selling price when the fixed or list price and the rate of discount are : 1. 30^ 10% 2. $44 25% 3. 75^ 50% 4. 40^ 15% 5. $40 75% 6. 48^ 25% 7. 56^ 121% 8. $50 6% 9. $1 2% 218 INTERMEDIATE BOOK Written Problems 260. 1. What is the net price on a set of books, listed at $ 25, when a discount of 20 % is allowed ? When a discount of 5 % is allowed ? 2. A shipment of bananas was slightly damaged in transit. It had been valued at $ 120, but the buyer agreed to take it at a discount of 25 %. How much was the discount in dollars? How much the selling price ? What loss did the shipper sustain, if the bananas cost him $ 95? 3. A lawn mower, listed at $ 5.40, is sold at 10 % off. Find the selling price. 4. The marked price on a piano being $ 500 and the rate of discount 20 %, how much is the discount in dollars ? If you were given the rate of discount and the amount of the discount, how would you find the list price ? If you were given the list price and the discount, how would you compute the rate of discount? 5. An automobile is listed at $1750. From this price there is a discount of 10 % and still an- other discount of 6%. What is the net price of the automobile? Written Problems 261. 1. If a set of books, listed at $ 36, sells for $ 27, find the rate of discount. APPLICATION OF PERCENTAGE 219 Find the first discount and price. Then find the second or net price. Process $S6 — $27 = $9 Explanation. — The discount is '25 $36-127 = 89. 36)9.00 9 -r- 36 = ^ = 25 %, the rate of dis- rj2 count. JgO" Solve by fractions when possible. 180 Thus: 9-36 = /^=i=25%. 2. Find the net price on goods listed at $ 3360, when a discount of SS^% is allowed. 3. A merchant offers cloth at $ 1.50 a yard, sub- ject to a discount of 16|%. What is the net price a yard? How man}'^ yards can be bought for $45? What part of $lare 16f^? 4. Find the net price of a table listed at $36, discount 16f %. 5. A store sells desks at 30 % off. What is the selling price of one marked $25.75? 6. A hardware merchant gives an order for 350 lb. of nails at 24^ a pound, discount 35%. How much was his bill? • 7. A man buys 50 ft. of garden hose at 9|)^ a foot, discount 15 %. How much does he pay? 8. A merchant advertises 25 % discount on cash purchases. A lawn mower, listed at $ 7.65, can be bought for what sum? 220 INTERMEDIATE BOOK 9. An article, marked $2.50, sells for $2.25. What is the rate of discount? 10. If an article, listed at $ 3.80, sells at $ 3.42, find the rate of discount. 262. Find the numbers that belong in the blank spaces : List Prick Ratb of Discount Net Price List Price Rate of Discount Net Price 1. $4.50 16f% 11. $ 525.75 1 515.23 2. 3300 33r/o 12. 278.40 4% 3. 4760 12|% 13. 979.97 5% 4. 860 $731 14. 275.50 259.07 5. 950 16% 15. 478.88 431.00 6. 200 150 16. 9648.72 8% 7. 540 351 17. 4887.75 10% 8. 1750 4% 18. 8425.25 16|% 9. 1800 90 19. 9637.75 7510.20 10. 2175 5% 20. 7475.40 25% Commission 263. When a person is engaged as an agent to transact business for another, he usually is paid a certain per cent for his services. The amount thus paid is called commission. 1. An agent sells a lot for me at $ 800 and I pay him 5 % of this sum for his services. How much was his commission? 2. John Smith sells for John Brown 10 A. of land at $ 50 an acre at a commission of 3 % . How APPLICATION OF PERCENTAGE 221 much was Smith's commission? After taking out his commission, how much more does he turn over to Brown? Commission is one of the applications of per- centage. The sum collected, the value of the goods or property bought or sold, the sum men- tioned in the contract, corresponds to the base. The rate of commission is the rate per cent. The commission expressed as a sum is the per- centage. Oral Problems 264. 1. If an agent charges 8 % of one month's rent for his services, what is his commission for finding tenants for a house that rents for $60 a month? For a house that rents for $ 75 a month? 2. A lawyer collects a debt of $ 500 and receives 10% of it for his services. How much does he send his employer after deducting his commission? 3. The contract for building a house is $ 6000. The architect is allowed 5 % for supervising the construction. What is his commission? 4. An agent sells books on commission. He sells a $40 set and receives a commission of 25%. What is his commission ? 5. A doctor gave his accounts to a collector. If the collector receives 20 % for collections, what will he receive on a $25 account? 222 INTERMEDIATE BOOK Written Problems 265. 1. I bought through an agent 10 bags of coffee, each containing 120 lb., at 15^ a pound. The agent charged 4 % commission. How much did I pay the agent? How much did the coffee cost me, including the commission? 2. Acting as an agent, I buy for W. Burgess & Co., 75 barrels of flour at $ 4.50 a barrel. I charge a commission of 3 % . How much commission do I receive? 3. A real estate agent receives for his services 5% of the rents collected on three houses, one renting for $ 40 a month, another for $ 75 a month, and the third for $87 a month. How much does he receive a year for his services? 4. A merchant gave bills, aggregating $ 360, to an agent for collection. The latter succeeded in collecting only $240 and reported the remainder uncollectible. How much does the merchant lose if he pays the agent 10% on the amount collected ? What per cent of the $ 360 did the merchant lose? 5. An agent for farm machinery sells to a* farmer a mowing machine for $250. After de- ducting his commission of 12|-% he sends the balance to the manufacturers. How much does he send ? APPLICATION OF PERCENTAGE 223 6. An auctioneer received $23.40 for selling $ 588 worth of goods. What was his rate of com- mission ? 7. I, as agent, sold a carload of fruit for $ 476. If I retained 7 % for my services, how large a sum did I transmit to my employers? 8. A commission merchant sold 60 boxes of oranges at $ 2 a box. Find his commission at 5 % and the amount sent to his employer. 9. An agent collects house rent for the owner of the house and charges 5 % of the rent of his services. If the rent is $50 a month, how much does the agent get a year? 10. Find the commission on the sale of $ 700 worth of goods at the rate of 7 % on the selling price. Drill Exercise 266. Use drill device for Percentage, Commission, and Discount. Change the rate. Use 5, 10, 50, 75, etc. Change the numbers. 500 400 300 200 100 550 450 150 160 175 80 75 50 X .20 INTEREST Introduction 267. If I borrow money from John Smith, not only must I return the money to him after a cer- tain period of time mutually agreed upon but I must pay him for the use of his money. This extra sum that I pay him is called interest. Interest is the money paid for the use of money. The sum borrowed is called the principal. The principal and the interest, added together, give the amount. When we say that the rate of interest is 4 % or 6 %, we mean that the rate per annum (by the year) is 4 % or 6 % . 1. What is the interest on $300 for 1 yr. at 6 % ? at 3 % ? 2. What is the interest on $500 for 1 yr. at 5 % ? at 4 % ? 3. What is the interest on $ 600 for 6 mo. at 5 % ? In finding the interest for 6 mo. you first find the interest for 1 yr. and then divide by 12. 224 INTEREST 225 Oral Exercise 268. Find the interest for 1 mo. : 1. $ 500 at 7 % 2. $ 60 at 6 % 3. $1000 at 6% 4. $7000 at 4% 5. $ 700 at 8 % 6. $ 70 at 4 % 7. $800 at 5% 8. $900 at 7% Find the interest on each of these sums for 6 mo. Tell how to find the interest on $ 100 for 6 mo. at 6%. Tell how to find the interest on a sum of money for 3 mo. Oral Exercise 269. Find the interest for 6 mo. ; for 3 mo. : 1. $200 at 6% 2. $800 at 5% 3. $ 300 at 8 % 4. $ 900 at 4 % 5. $500 at 5% 6. $1000 at 6% 7. $ 700 at 8 % 8. $ 800 at 9 % 9. $400 at 7% 10. $2000 at 3% Tell how to find the interest on a sum of money for 3 mo. ; for 1 mo. ; for any number of months. Written Exercise 270. 1. Find the interest on $350 at 6% for 6 mo. ; for 4 mo. ; for 3 mo. Q 226 INTERMEDIATE BOOK Process 7 ^ 2 Explanation. — Multiply the sum at interest by the rate of interest expressed as a decimal fraction and by the number of months expressed as part of a year. Use cancelation when possible. 2. What is the interest on $475 at 5% for 6 mo. ? for 3 mo. ? for 4 mo. ? 3. Reckon the interest on $465.50 for 2 yr. at 6%. 4. Compute the interest on $505.50 for 3 yr. at 5 % . 5. Find the interest on $ 675.25 for 5 yr. at 3%. 6. Find the interest on $ 325.40 at 4 % for 1 yr. and 6 mo. 7. By how much does the yearly interest on $ 375 at 5% exceed the yearly interest on $400 at 4%? 8. How much are the principal and interest together on $575 at 5% at the end of one year? What is the amount ? 9. Find the amount at the end of two years of $565.50 at 4%. 10. Find the amount of $ 1000 for 3 yr. at 6 %. INTEREST 227 Oral Exercise 271. 1. What is the interest on $ 120 at 5 % for 1 yr. ? 1 mo. ? 2. Find the uiterest on $ 240 at 10 % for 1 yr. 1 mo. 3. Compute the interest on $ 200 at 6 % for 1 yr. ; for 1 mo. 4. How much interest must you pay on $ 300 at 8 % for 1 yr. 1 mo. ? 5. What is the interest on $300 at 6% for 2 mo. ? 2 mo. are what part of 12 mo. ? 6. What is the amount of $200 at 6% for I yr. and 1 mo. ? 7. Find the amount of $ 600 at 4 % for 1 yr. and 1 mo. 8. Find the interest on $ 120 at 5 % for 1 mo. ; for 7 mo.; for 11 mo. 9. What is the interest on $ 240 at 10 % for II mo. ? 10. What is the interest on $ 500 for 2 yr. 8 mo. at 3 % ? Written Exercise 272. Find the interest : Pkin. Rate Time Prin. Rate Time 1. $300, 6%, 2 mo. 2. $200, 3%, 2 mo. 3. $ 100, 5 %, 3 mo. 4. $ 50, 6 %, 4 mo. 228 INTERMEDIATE BOOK Pbin. Eate Time Pein. Kate Time 5. $150, 6%, 6 mo. 6. $500, 6%, 4 mo. 7. $600, 4%, 5 mo. 8. $400, 4%, 3 mo. 9. $430, 5%, 6 mo. lo. $650, 6%, 9 mo. 11. $ 1000, 4 %, 8 mo. 12. $ 1560, 5 %, 8 mo. 13. $2575, 6%, 10 mo. i4. $8975, 7%, 7 mo. 15. $6875, 8%, 8 mo. Study Exercise 273. Find the interest and the amount : 1. $ 525.75 for 2 yr. and 5 mo. at 3 % . Process Int. for 1 yr. Int for 2 yr, $525.75 X.03 $15.77 Int. for 1 mo. $1.31 12)$ 15.77 $15.77 X.02 $31.54 Int. for 5 mo. $1.31 X.5 $6.55 $31.54 Int. for 2yr. 6.55 Int. for 5 mo. $38.09 Int. for 2 yr. 5 mo. $ 525.75 Principal 38.09 Interest $563.84 Amount Explanation. — Find the interest on the sum at the re- quired rate for one year. Find the in- terest for the num- ber of years. Find the interest for one month, -^^ the in- terest for 1 year. Find the interest for the number of months. Add the interest for the years and the months. Add the principal to the in- terest to get the amount. INTEREST 229 Written Exercise 274. Find the interest and the amount : 1. $307.25 at 6% for 5 yr. 2. $906 at 5% for 1^ yr. 3. $175.40 at 4% for li yr. 4. $ 263 at 7 % for 1 yr. 3 mo. 5. $ 650 at 5 % for 1 yr. 1 mo. 6. $ 780 at 4 % for 2 yr. 9 mo. 7. $ 690 at 7 % for 3 yr. 8 mo. 8. $ 360 at 5 % for 4 yr. 7 mo. 9. $ 1200 at 9 % for 5 yr. 5 mo. 10. $ 94.60 at 6 % for 1 yr. 6 mo. Written Exercise 275. Find the interest and the amount : 1. $ 785.45 for 1 yr. 2 mo. at 41%. $785.45 .045 392725 314180 Process $5.89 (2 mo.) 6)35:35 $35.34 5.89 Explanation 41% = .045 2 mo. = Joflyr. Use 135.35 as $ 35.34525 (1 yr.) $ 41.23 Ans. interest for 1 yr. 2. $375.35 for 3 yr. 4 mo. at 5i%. 3. $ 750.45 for 2 yr. 6 mo. at 4|%. 4. $ 4850.75 for 5 yr. 4 mo. at 5^%. 230 INTERMEDIATE BOOK 5. $ 7525.25 for 1 yr. 8 mo. at 2| %! 6. $ 8759.40 for 4 yr. 9 mo. at 2|% . 7. $2002.60 for 1 yr. 10 mo. at 4f %. 8. $5000 for 1 yr. 1 mo. at 61%. Written Problems 276. 1. I borrow $375 at 7% for 1 yr. and 3 mo. How much must I pay back at the end of that time? 2. I borrow $2000 at 6% for 2 yr. With this money I buy a small lot and build upon it a cottage. At the end of two years, I sell the cottage and lot for $2500. Do I gain or lose by this transaction? How much? 3. A real estate agent buys a house for $ 1500. He must pay $ 50 every year for taxes, insurance, and repairs. For how much per month must he rent the house to make 6 % on the cost of the house? 4. I borrow $ 60 from Mr. Brown for 9 months at 6%. How much interest must I pay him? 5. Mr. James borrows $75 from a friend and promises to pay it back in 4 months at 6 % . How much is the interest for that time? Find the amount that Mr. James must pay at the end of the 4 months. INTEREST 231 Interest Accounts 277. Any person ten years of age or over may deposit money at interest at the post office, with the security of the United States government for repayment. Money may also be deposited in banks, savings banks, and trust companies where it will draw interest according to the rules of the institution. In postal savings banks, interest is allowed at the rate of 2 % for each full year that the money remains on deposit. The interest is computed from the first day of the month following the day on which the money is deposited. In savings banks and trust companies the in- terest rate varies. It is usually 3 % or 3 J %. The interest is usually computed from the first day of the month following the day on which the money is deposited. In these institutions interest is usually paid quarterly or semiannually. Written Exercise 278. Compute the interest on deposits in postal savings banks as follows : 1. $ 50, deposited May 6, 1914, withdrawn June 5, 1915. 2. $70, deposited Jan. 4, 1914, withdrawn Dec. 17, 1914. 232 INTERMEDIATE BOOK 3. $95, deposited April 29, 1915, withdrawn May 1, 1917. 4. $ 75, deposited June 15, 1914, withdrawn June 1, 1916. 5. $ 64, deposited Sept. 3, 1913, withdrawn Oct. 1, 1915. Written Exercises 279. 1. Find out how often interest is paid on accounts in the local savings bank. 2. What rate of interest is paid by the local bank on savings accounts ? 3. Find out how interest is computed on savings bank accounts. 4. Make an account of savings deposited in a savings bank. Can you compute the interest on the account at the end of one year ? 5. How is interest computed using the interest table ? BILLS AND CHECKS Introductory 280- 1. Mrs. James Downs makes several pur- chases at the store of Milton Bros. & Co. The articles are shown in the following bill : San Francisco, Sept. 15. Mrs. James Downs, Bought of MILTON BROS. & CO., 165 Market St. Phone 204 2^ lb. steak 1 lb. tea 4 lb. butter 10 lb. beet sugar 28 j^ 45 j^ 35^ 70 45 40 50 05 What are the articles purchased ? What is the price of each article ? How much is the entire bill ? Where and when were the articles sold ? Who is the buyer ? The seller ? After the bill was paid by Mrs. Downs, the clerk wrote the w^ords " Received payment, Milton Bros. & Co. per J. F." The words show that the bill is paid. The bill is now called a " receipted bill." " J. F." are the initials of the clerk's name. 233 234 INTERMEDIATE BOOK The receipted bill should be kept by the buyer to show that the bill has been paid. 2. Mr. J. W. Jones goes to the bookstore of Burgess & Co. and buys books and stationery. He receives a bill and pays it. What does the clerk in the store write on the bill to show that it has been paid ? The receipted bill is as follows : Madison, Wis., Dec. 1. Mr. J, W. Jones, Bought of BURGESS & CO., 105 State St. Phone 714 Nov. 12 1 arithmetic 1 geography 3 lead pencils Received payment Burgess & Co. per B. C. 45 85 15 45 3, Mrs. A. Smith orders from Gerry & Co. : 4 lb. coffee at 30^ a pound 18 lb. sugar at 5^^ a pound 5 gal. molasses at 60.^ a gallon Make out the bill which Gerry & Co. send Mrs. Smith. How much does Mrs. Smith owe ? Indi- cate what must be written on the bill to show that it has been paid. BILLS AND CHECKS 235 4. Make out bills of the following items, using your father's name as buyer and the name of your local merchant as seller : 2 bags salt @ $ .10 15 yd. silk @ $1.50 3 bu. potatoes @ .75 12 yd. muslin @ .09 10 lb. prunes @ .121 9 yd. lace @ .65 5. Make out 4 bills for goods purchased of the local dealers. 6. If a person has money deposited in a bank, he may pay a bill by check, thereby ordering the bank to pay the sum specified. Form of a Check JVo, 89 ISanJt of aEtsconsin Madison, Wis., fwUf /^. 1^/6. Pay to the order of famv&o. W^oa^.- ..f 36.^6 ^klvtl^ jiA}-& a.'yvcL //^-r^rvr-i^or-r^^^rrrc^r-^- >c^ ^.Dollars ^o-yva u ^v&c^. Name the sum paid. Who ordered it paid ? To whom is the amount paid ? In what bank does Donald Gregg have his money deposited ? How much does this check reduce the amount of Donald Gregg's deposit in the bank ? When James Ward presents the above check at 236 INTERMEDIATE BOOK the bank, he writes his name on the back of the check and either receives the $ 35.45 from the bank in cash or has it added to his own deposit in the bank. The writing of his name across the back is called indorsing the check and is evidence that the sum has been received by him. The bank returns the indorsed check to Donald Gregg, who keeps it as evidence that the sum has been paid. 7. John Ward has his money deposited in the First National Bank of Lawrence, Kansas. He wished to pay Richard March of that city the sum of $ 105.25. Write the check. 8. Write out checks of your own on the sup- position that you had money deposited in the Second National Bank of Liliputia. PROBLEMS ON INDUSTRY 281. Table for Reference in Agricultural Problems Kind of Cboh AvEBAGB Amount OF Febtilizees PEK ACKE Average Amount OF Seed PER Acre Average No. of Weeks to Maturity Corn 10 T. 10 qt. 15i Fall wheat 18 T. 2bu. 42 Oats 7iT. 5ipk. 13 Potatoes 171 T. 14 bu. 16 Turnips 10 T. 2 1b. 10 Problems Based on the Table 282. 1. How much more fertilizing material is used per acre for fall wheat than for corn ? What per cent of 10 T. is this excess ? Comparing pota- toes and com, what per cent of 10 T. is the excess ? How much less fertilizing material per acre 2. is used for oats than corn ? 10 T. is this difference ? What per cent of 3. What per cent of the fertilizers used for corn is that used for wheat ? 4. How much fertilizing material is needed for SOJ A. of potatoes ? 7 A. of oats ? 5. How many acres of land for wheat can be fertilized from 50 loads of material, f T. per load ? 237 238 INTERMEDIATE BOOK 6. How many quarts more of wheat than of corn are needed on 8 A. ? 7. How many pecks less of oats than of wheat are seeded on 10 A. ? 8. How many more acres can be seeded with 154 bu. of potatoes than with 20 lb. of turnips ? Written Problems 283. Refer to the table on page 237 for facts necessary to the solution of the problem. 1. When may we expect corn planted on May 15th to mature? Process Mav 16 da Explanation. — Count off the weeks J orj -j * on a calendar, or proceed thus : 15^ wk. T ^Ql^^* =^^^^ ^^•' ^"^ ^^P*- ^ ^""^ ^^^ ^^• July 31 da. Therefore the required date is Sept. 1 Au g. 31 da. or 2. 108 da. 2. How many more acres can be seeded with 493 bu. of potatoes than with 63 bu. of turnips ? 3. About what date may we expect wheat seeded Sept. 25th to mature? 4. How many bushels and pecks less of oats are seeded on 74^ A. than of wheat over an equal area ? 5. How much fertilizing material is needed for 210J A. of potatoes and 2T|- A. of oats ? PROBLEMS ON INDUSTRY 239 6. What per cent of the fertihzer per acre on a wheat field are the fertilizers commonly put on a 1-acre potato patch ? 7. What per cent of the fertilizers for 4 A. of oats are the fertilizers needed for 2 A. of wheat ? 8. Mr. Jones agrees to cover 12|- A. of corn land with fertilizers at $ .75 a load, and to allow 5% discount for payment within 10 days. How much will he be paid, if he allows the discount ? 9. A ranchman buys potatoes at $ .60 a bushel for planting 50^ acres and is allowed a discount of 7 %. How much does he pay ? 10. In seeding tobacco, 1 oz. of seed is applied to 6 sq. rd. How many ounces is this for 1 sq. rd. ? For 160 sq. rd. or 1 A.? 11. What fraction of an ounce of seed is that for Isq. yd.? 12. In case of tomatoes, 6^ oz. of seed usually suffice per acre. What part of an ounce is needed for 1 sq. rd.? 13. Tomatoes seeded in the Southern states ordinarily ripen in 16 weeks, but this year they take 5 days longer than the usual time. On what date should they be picked if they were seeded Feb. 15 ? 14. A farmer plans to harvest his oats on July 25. What would seem to be the most favor- able time for putting in the seed ? 240 INTERMEDIATE BOOK 15. In planting sweet potatoes 11^ bu. are usually needed per acre. How many bushels must be procured for 17.6 A., if 5% more than the amount named are to be planted per acre? Written Problems 284. 1. In the spring and summer the salmon leave the sea and proceed up the rivers to places where the river water is cool. There they deposit their eggs and then die. Young fish develop from the eggs and float downstream into the ocean. Salmon starting up the Columbia River early in March move at first only about 2^ miles a day. How many rods a day is this ? How many yards a day ? How many feet a day ? 2. If they travel 13,200 ft. a day, how far do they go in an hour on an average ? 3. Later the salmon move faster, reaching, per- haps, a speed of 6 mi. a day. How far can they go at that rate in the month of May ? What per cent of 6 mi. is the former rate of 2^ mi. ? 4. Some salmon go up the Columbia River 1000 mi. to tributaries fed by melting snow in Idaho. If a fish left the ocean March 1, traveled 1000 mi., and reached the cold waters in Idaho on Oct. 1, how many days was it traveling ? How many miles did it average a day. If it made 4.7 mi. a day, how many feet a day did it travel ? PROBLEMS ON INDUSTRY 241 5. When salmon come to a low waterfall, they leap atop of it. A salmon jumps atop of one 12 ft. high. How many times higher is this than Harry's running-high-jiimp record of 4 ft. 6 in.? 6. While ascending streams the salmon eat nothing and consequently lose weight. If a salmon weighed 20 lb. at the beginning of its trip and 16 lb. at the end of it, what per cent of its original weight did it lose ? 7. Another salmon weighed at first 30 lb. and then lost 16 %. What was its final weight ? 8. The young fish float down the river, tail foremost. If it takes them 11 months to descend 1000 miles to the ocean, how many miles do they make a month ? 9. If a young salmon is 4 in. long when it first reaches the ocean, and is 2^ ft. long when later it ascends the river, how many times has it increased its length while in salt water ? 10. No other fish is canned so extensively as the salmon. The Columbia River yielded during the 6 yr. ending 1880 about 150,000,000 lb. of salmon. If the fish averaged 30 lb., how many fish were killed ? 11. In one year 31,500,000 pound cans of salmon were shipped from the Oregon coast, valued at $ 3,300,000. Compute to cents and mills the value per can. 242 INTERMEDIATE BOOK 12. The value of the entire salmon catch on onr west coast, including Alaska, exceeds S 13,000,000 annually. If 65% of this is from canning the species known as chinook salmon, how many dollars come from other species of salmon ? 13. A firm desire to expend $ 75,000 upon a salmon-canning factory, but have only $ 36,000 in cash. They borrow the balance at 3^ %. How much interest must they pay annually ? 14. An agent sells 50,000 cans of salmon for this firm, at $ .09 a can, and charges 2 % com- mission. How much money does he send to the firm after deducting his commission ? 15. It is feared that the salmon will be ex- terminated before many years. One year Oregon yielded 39,500,000 lb. of salmon. If the year fol- lowing 10 % less was canned, how many pounds were canned? 16. Salmon are known to have reached the weight of 100 lb. What per cent is this of the average weight of 25 lb. ? 17. In Monterey Bay salmon are caught by trolling, the hook being baited with sardine; 25 fish by one line is a big day's catch. If the fish caught average 19 lb. and sell at $ .05 a pound, what is the value of a day's catch ? 18. If the speed in trolling is 4 mi. an hour, and a man trolls 7f hr., how far will he have gone ? PROBLEMS ON INDUSTRY 243 19. If 5 gallons of oil are used in moving the boat during 7 hours, oil costing $ .16 a gallon, what is the cost of oil for a 21-hour cruise ? 20. One observer found the average weight of salmon in the Columbia River to be 22 lb. and in the Sacramento River 16 lb. What per cent was the latter of the former ? 21. In trolling in Monterey Bay, fishermen let out about 150 ft. of line. Through what distance in yards must the line be drawn in while catching 24 fish ? Written Problems 285. This dia- gram shows the standard method of cutting meat. Sup- pose the weights and pieces of the different kinds to be as follows : Table for Reference Neck, 27 lb. at $ .13 Porterhouse, 95 lb. at $ .30 Chuck, 139 lb. at $ .15 Sirloin, 41 lb. at $ .25 Ribs, 74 lb. at $ .18 Flank, 49 lb. at $.15 Shin, 63 lb. at $ .06 Rump, 41 lb. at $ .15 Plate, 120 lb. at $.13 Round, 123 lb. at $ .20 Shank, 27 lb. at $ .06 244 INTERMEDIATE BOOK Problems Based on the Table 286. 1. Find the cost of 11 lb. of each kind of meat. 2. Find the cost of 12 lb. of each kind of meat. 3. What is the cost of lOJ lb. of neck? of round ? of chuck ? 4. What must you pay for 2 J lb. of ribs ? of shin ? of flank ? If the answer has ^ cent, figure 1 cent in place of ^. 5. What is the cost of 2 lb. of porterhouse and 3 lb. of plate? 6. Find the cost of 5 lb. of rump and 3 lb. of chuck. 7. A purchaser gets 3 lb. of porterhouse and pays the bill with a dollar bill. How much change should he get ? 8. If you order 3 lb. of sirloin and hand over $ .75, how much change will you receive ? 9. How much more will 30 lb. of ribs cost than 35 lb. of chuck ? 10. Mr. Jones buys a piece of flank weighing 30^ lb. and a piece of shin weighing 20^ lb. How much does he pay for both pieces ? 11. How much more or less than the piece in- dicated above does a man pay for all the porter- house and all the sirloin of that one animal, if he buys at the rate of $ .22 a pound ? PROBLEMS ON INDUSTRY 245 Written Problems 287. 1. In 1888 the steamship PhiladelpJda was built, 560 ft. long and 63.3 ft. broad. How many times the breadth was the length ? What per cent of the length was the breadth ? 2. The Colwnbia, built in 1901, was 503 ft. long and 56 ft. broad. Was this broader than the Philadelphia, in proportion to its length, or less ? 3. If you divide the length by the breadth, you find what is often called the " ratio " between the length and the breadth. Find this ratio for each of the following steamers of the Atlantic Transport Line : Mesaba, length 482.1 ft., breadth 52.2 ft., depth 31.6 ft. Minnehaha, length 600.7 ft., breadth 65.5 ft., depth 43.3 ft. Minnewask, length 616 ft., breadth 66 ft., depth 44 ft. 4. In each case the ratio was found to be not far from . It is found that steamers built on this ratio make the best travelers. 5. Find the ratio between the length and depth of the steamers in Exercise 3. Is this as fixed as the other ratio ? 6. The Lusitania, built in 1906, was 790 ft. long, 88 ft. broad, and 60.6 ft. deep. How many feet longer is this than the Minnewask^ What per cent of the length of the Minneioask is this excess? 7. The Olympic is 882 ft. long. What per cent is this of the length of the Mesaba f 246 INTERMEDIATE BOOK 8. The Lusitania has a horse power of 70,000, the Minnewask of 1616. What is the ratio of the former to the latter ? 9. The largest locomotive for moving trains has 4000 horse power. What per cent is this of the Lusitania! s horse power ? 10. A knot used in indicating distances on the ocean is equal to 6086 ft. How far is this in miles ? 11. In 1856 the Persia made the trip .between New York and Queenstown, England, in 9 da. 15 hr. 45 min. In 1908 the Lusitania made this trip in 4 da. 15 hr. Could the Lusitania have gone to Queenstown and back to New York in the time it took the Persia to go one way ? What is the difference in time of the two steamers for the trip one way ? 288. Written Problems Table for Reference Rate Rate Places Places Day Message Night Message. Day Message Night Message Alabama 60-4 50-3 Louisiana 60-4 50-^ California 1.00-7 1.00-7 New Hampshire 35-2 25-1 Colorado 75-5 60-4 New York City 20-1 20-1 Connecticut 25-2 25-1 Tennessee 50-3 40-3 Dist. of Columbia 30-2 26-1 Wisconsin 50-3 40-3 Kentucky 50-3 40-3 Yukon, Dawson 4.25-29 4.25-29 PROBLEMS ON INDUSTRY 247 Explanation. — These rates are between New York City and the places named. A day rate, " 60-4," means 60 <^ for 10 words and 4 ^ for each additional word. A night rate, 50-3, means 50 (^ for 10 words and 3 ^ for each additional word. Address and signa- ture are free. A Night Letter is different from a night message. The standard day rate for 10 words is charged for the transmission of 50 words or less, and \ of such standard day rate for 10 words is charged for each additional 10 words or less. A Day Letter of 50 words or less is transmitted at one and one-half times the standard night letter rate. Problems 289. 1. Find the cost of sending a 15-word day message from New York City to Montgomery, Ala. How much cheaper is a night message ? 2. What is the cost of a 50-word night letter from New York City to Montgomery? What is the cost, if sent as a day letter ? 3. What is the cost of a 12-word message from New York City to Dawson ? 4. A man in New York City sends a 20-word telegram to a friend in a distant part of that city. How much does he pay for it ? Does he save any- thing by sending it as a night message ? 248 INTERMEDIATE BOOK 5. Find the cost of wiring the following mes- sage from New Haven, Conn., to a home in New York City : "Yale-Harvard football game a tie." 6. What is the cost of a 60-word night letter sent from New York City to Denver, Col. ? 7. Find the cost of sending a 15-word telegram from New York City to New Orleans, La., at day rates. How much cheaper is a night message ? What per cent of the cost of a day message is this saving ? 8. A 25-word day message sent from New York City to Madison, Wis., costs how much more than a night message ? What per cent of the cost of the day message is saved by sending a night message ? THE WESTERN UNION TELEGRAPH COMPANY 1.45 P.M. Chicago, Feb. 15, 1911. To The Macmillan Company, 64-66 Fifth Avenue, New York City. Send by express ten copies Tarr's New Physical Geography, twenty copies Cole- man's The People's Health. John Lake. 9. If the rates for Illinois are the same as for Wisconsin, what is the cost of this telegram, night rate? PROBLEMS ON INDUSTRY 249 10. Write telegrams to some of your acquaint- ances, and compute the cost of sending each. 11. What per cent of the rate to Louisiana is the rate to New Hampshire for 10 words? For 20 words ? 12. In 1870 the Western Union Telegraph Co. operated 54,000 miles of telegraphic lines ; in 1908 this had increased to 209,000 miles. What per cent of the miles in 1870 is the increase in miles shown in 1908? On an average, what was the per cent of increase per year ? 13. The number of messages sent in 1870 was 9 million; that sent in 1908 was 63 million or sevenfold the former number. W^hich increased at a more rapid rate, the number of messages or the number of miles of line ? 14. Write messages and letters, then ascertain the cost of sending them from your town to neigh- boring towns. REVIEW Miscellaneous Problems 290. 1. One year the Bureau of Engraving in Washington printed over 35,000 postage stamps a minute. If these stamps are ^ in. long and are placed end to end, how many inches long will the line thus formed be ? How many feet long will it be ? Yards ? 2. If these stamps are -^ in. long and f in. wide, how many square inches of area can be covered with them ? How many square feet ? How many square yards ? 3. It is found that, owing to the wear of the stream upon the rock, Niagara Falls recedes at the rate of 4 ft. a year. How far back will it move, at this rate, in 100 yr. ? How far has it moved during your lifetime ? 4. If 1 cu. ft. of anthracite coal weighs 93.5 lb., how many cubic feet will weigh 3 T.? 5. Pressed brick weighs 140 lb. per cubic foot. Find the weight in tons of a brick wall 30 ft. long, 12 ft. high, and 1^ ft. thick. 6. A cubic foot .of water weighs 62.4 lb., a cubic foot of ice weighs 57.4 lb. What is the 250 REVIEW 251 difference in weight of a cubic yard of ice and an equal volume of water ? 7. A building for a poultry show is 200 ft. by 152 ft. How many feet is it around the building ? How much floor space is there in the building? If there are over 100 exhibitors and half of the entire floor space is given over to them, how much floor space may be assigned to each ? 8. A trough is 6 ft. long, 2 ft. wide, and 1 ft. deep. How many cubic feet of water does it hold ? 9. A trunk is 3 ft. long, 2 ft. wide, and 1.5 ft. deep. Find its capacity in cubic feet. 10. A box is 6 in. long, 3 in. wide, and 2 in. deep. How many blocks, of the shape and size of cubic inches, may be packed into this box ? 11. George rode his motor cycle for 3 hr. at an average speed of 27.75 mi. an hour. How far did he go? 12. A schoolroom is 9.75 yd. long and 8 yd. wide. How many square feet in its floor area ? ' 13. Mr. Jones has 500 chickens on a plot of ground 150 ft. square. How many feet of fence has he around the plot ? How many square feet of area are there in the plot ? How many square feet is this for every chicken ? 14. One season a baseball team made 18 trips between Chicago and Milwaukee. The railroad 252 INTERMEDIATE BOOK fare amounted in all to $ 275.40. What was the fare for the team for each trip ? What was the fare per individual, if there were 9 men on the team? 15. If the fare is $2.15 to a place 86 mi. dis- tant, what is the fare per mile ? 16. Martha'spent $20.15 during the month of May. How much was this a day ? POSITIVE AND NEGATIVE NUMBERS Introduction 291. The thermometer commonly used in Amer- ica is the Fahrenheit (F.), in which the freezing point of water is marked 32°, and the boiling point 212°. 1. How many degrees are there between the freezing and the boil- ing points of water ? 2. How many degrees are there between the freezing point of water and the temperature of the blood ? 3. At how many degrees above blood tempera- ture does water boil ? Water boils Blood Temc Water freezes Fahrenheit Problems in Temperature 292. 1. How much must the blood temperature be reduced to reach the freezing point ? 2. If at 2 o'clock P.M. the thermometer reads 91° and is steadily falling at the rate of 4^° an hour, at what time will the temperature be 64° ? 3. Between 8 a.m. and 1 p.m. the mercury rose from 65° to 88°. At what average rate per horn- did it rise ? 253 254 INTERMEDIATE BOOK 4. Temperatures below zero are designated by — (minus). For six mornings in succession the temperatures were at 6 o'clock : — 7°, 6°, — 1°, — 10°, 4°, — 3°. What was the average tempera- ture for these six mornings ? 5. In freezing ice cream the mixture of ice and salt reduces the temperature from 60° F. to 28° F. in 12 minutes. What was the average drop of temperature per minute ? 6. An animal was killed and the carcass put in cold storage, where its temperature was reduced 69| %. What is the temperature of cold storage? 7. At Mobile, Ala., the average temperature one year for the month of January was 50°, and for the month of July 80°. The hottest day in the same year reached 102°, the coldest — 1°. Find the difference between the mean temperatures for January and July. Find the difference between the extreme temperatures for the year. 8. Read the thermometer each hour of the school day. Record and tabulate the readings. Find the average temperature. Temperature Chart 293. This chart shows the temperatures of a patient, taken mornings and evenings for 8 succes- sive days. Such records often furnish important information to the physician. This chart shows POSITIVE AND NEGATIVE NUMBERS 255 not only how much the temperature changed, but also how rapidly. Exercise Based on the Chart 294. 1. Read the temperature for each day; estimate the fractions of degrees. 2. How many hours intervene between two suc- cessive readings of temperature ? 3. On what day do you see the greatest differ- ence in temperature ? 4. On what day the least ? 5. When the temperature changes rapidly dur- ing 12 hr., is the line connecting the two tempera- tures steep or not ? 6. The line connecting the two temperatures of the second day is level. What does this show ? 7. Which line is steeper, the one for the first day or the one for the third? What does this show ? 256 INTERMEDIATE BOOK Moonlight Chart 295. This chart shows the hours of moonlight on cloudless nights in Boston, one year during June. One small space to the right stands for one day ; Days 5 10 15 20 Moonlight Chart for June 25 one small space up stands for one hour. Spaces from the bottom line up to the heavy top line show the hours of moonlight. There are about 9 spaces in the heavy top line, showing that the nights were 9 hr. long. On June 5 there were nearly 5 spaces up to the curved line, showing that there were nearly 5 hr. of moonlight during the night, if cloudless. Problems Based on the Chart 296. 1. During what part of the night follow- ing June 5 did the moon shine ? 2. Find the number of hours of moonshine dur- ing the nights following June 10, June 15, June 20, June 25, June 30. POSITIVE AND NEGATIVE NUMBERS 257 3. Express by a common fraction the part of the night that was lighted by the moon on each of these dates. 4. For each of these dates express in per cent the part of the time of the night during which the moon was shining. 5. On what date was nearly the whole night illuminated by the moon ? 6. On what date was there practically no moon? 7. On what dates did the moon shine half the night ? 8. An electric light company agrees to supply street lights in a Boston suburb during the hours of the night when the moon did not shine, as shown by our chart. How many hours were the streets lighted on the 5th, 10th, 15th, 20th, 25th, of June? 9. If a certain district was lighted at the rate of $ 1.25 an hour, what was the cost of lighting for each date named ? 10. If a rival company offered to supply the same kind of light at 15 % less, what would have been the charge for June 5 ? 11. On what night was there no electric light? On what night was the electric light on all the time? 258 INTERMEDIATE BOOK ,12. How much was saved on the night of June 20 by turning off the electric light during moon- light ? 13. By looking at the chart, do you judge that the expense of lighting is reduced to about l, by following the moonlight schedule instead of an all- night schedule ? DOMESTIC POSTAGE 297. To all parts of the United States, including Hawaii, Porto Rico, and the Philippine Islands, also to Canada, Mexico, and the Republic of Pan- ama, — First-class matter : Letters or sealed matter 2j^ an ounce or fraction thereof; this rate holds also for letters to and from England and Germany, postal cards 1 ^ each ; with paid reply card 2 ^ each. Second-class matter: Newspapers and other periodical publications; when sent by publishers or news agents, 1 ^ per pound or fraction thereof ; when sent by others, 1^ for each 4 ounces or frac- tional part thereof. Third-class matter : Books, circulars, pamphlets, proof sheets, or other printed matter, li^ for each 2 ounces or fractional part thereof, sent to a single address. Registered matter : lOj^ in addition to the regular postage. Special delivery: V)^ in addition to the regular postage of first-class matter. Written Problems Based on Rates of Postage 298. 1. How much postage is required for do- mestic letters weighing, respectively, |- oz., 5-|- oz., If oz., 2 oz. ? 269 260 INTERMEDIATE BOOK 2. A newspaper with wrapper weighs 3 oz. How much postage must the publishers pay on 8000 copies ? 3. John Smith remails one of these papers. How many cents postage must be put on ? 4. George sends to a friend a book weighing, with wrapper, 35 oz. How much is the postage ? 5. How many cents postage are needed in all for mailing to friends in Canada 5 letters weighing, respectively, ^ oz., f oz., f oz., 1 oz,, ^ oz. ? 6. The Sunday edition of a paper weighs 7 oz. How much must the publishers pay for an edition of 18,000 ? 7. Mrs. Brown sends a sealed package, weighing 10| oz., by special delivery. She pays — — postage. 8. Mary sends a registered letter, weighing If oz., to a friend in Hawaii. How much does she pay? 9 John mails at Christmas a book weighing 21|- oz., a registered letter weighing ^ oz., and a package of newspapers weighing 9 oz. How much must he pay in all ? 10. A bicycle agent sends out 1000 circulars ad- vertising his machines. Each circular weighs 2|- oz. How much postage must he pay altogether ? 11. What is the postage on a letter weighing 3 oz., sent by special delivery as registered mail ? DOMESTIC POSTAGE 261 The Parcel Post 299. 1. Is Springfield, Illinois, nearer to Chicago than to Indianapolis ? 2. Do these distances seem more than 200 miles each ? Measure the distances as nearly as you can on the map, using the scale of miles indicated on the map. 3. Is Frankfort nearer to Nashville than to Columbus (Ohio) ? 4. Estimate the following straight-line distances. Then measure to see how close you come. Chicago to Nashville Louisville to Wheeling Chicago to Cincinnati Toledo to Indianapolis 5. Make a list of the important articles sold in the place in which you live. Describe the methods by which these articles are distributed to neighbor- ing and distant places. To what extent is parcel post used in this distribution ? Merchandise, and, in general, all matter not clas- sified in the United States postal service as first, second, or third-class matter may be sent by United States parcel post in parcels not greater in size than 72 in. in length and girth combined, nor exceeding in weight 50 lb. for the first and second zones, and 20 lb. for the other zones. The postage on parcels varies with the distance sent, as is shown by the following table : 262 INTERMEDIATE BOOK Table of Rates First Zone Sec- ond Zone Eate Third Zone Eate Fourth Zone Eate Fifth Zone Eate Sixth Zone Eate Sev- enth Zone Eate Eighth Weight Local Kate Zone Rate Zone Eate 1 pound $0.05 $0.05 $0.05 $0.06 $0.07 $0.08 $0.09 $0.11 $0.12 2 pounds .06 .06 .06 .08 .11 .14 .17 .21 .24 3 pounds .06 .07 .07 .10 .15 .20 .25 .31 .36 4 pounds .07 .08 .08 .12 .19 .26 .33 .41 .48 5 pounds .07 .09 .09 .14 .23 .32 .41 .51 .60 6 pounds .08 .10 .10 .16 .27 .38 .49 .61 .72 7 pounds .08 .11 .11 .18 .31 .44 .57 .71 .84 8 pounds .09 .12 .12 .20 .35 .50 .65 .81 .96 9 pounds .09 .13 .13 .22 .39 .56 .73 .91 1.08 10 pounds .10 .14 .14 .24 .43 .62 .81 1.01 1.20 11 pounds .10 .15 .15 .26 .47 .68 .89 1.11 1.32 12 pounds .11 .16 .16 .28 .51 .74 .97 1.21 1.44 13 pounds .11 .17 .17 .30 .55 .80 1.05 1.31 1.56 14 pounds .12 .18 .18 .32 .59 .86 1.13 1.41 1.68 15 pounds .12 .19 .19 .34 .63 .92 1.21 1.61 1.80 16 pounds .13 .20 .20 .36 .67 .98 1.29 1.61 1.92 17 pounds .13 .21 .21 .38 .71 1.04 1.37 1.71 2.04 18 pounds .14 .22 .22 .40 .75 1.10 1.45 1.81 2.16 19 pounds .14 .23 .23 .42 .79 1.16 1.63 1.91 2.28 20 pounds .15 .24 .24 .44 .83 1.22 1.61 2.01 2.40 25 pounds .17 .29 .29 30 pounds .20 .34 .34 40 pounds .26 .44 .44 60 pounds .30 .64 .64 The local rate is applicable to parcels intended for delivery at the office of mailing or on a rural route starting therefrom. The circles drawn on our map have Chicago as their centers and indicate the first, second, third, and fourth zones of distances from Chicago to other DOMESTIC POSTAGE 263 Scale of Miles places. Similar circles, drawn upon the map of the entire United States, showing all together 8 zones, are used by clerks in the Chicago post office to determine quickly the distances from Chicago to other post offices. The post-office clerk in Nashville has the same map of the United States, but the circles are drawn with their centers at Nashville, so as to in- dicate distances from Nashville to other places. Exercise Based on the Rules 300. Which of these parcels are mailable by parcel post ? 1. 10-lb. box, 20'' by 18'' by 10". 2. 9-lb. box, 18" by 15" by 10". 3. 12-lb. box, 25" by 10" by 8". 4. 11-lb. box, 30" by 12" by 9". 5. 81-lb. box, 14" by 14" by 14". Exercise Based on the Table 301. Compute the rate : 1. 8 lb. from Chicago to Columbus (Ohio). 2. 11 lb. from Chicago to Louisville. 264 INTERMEDIATE BOOK 3. 3 lb. from Nashville to Bowling Green. 4. Get a rate card or table for the local city. 5. Make 5 exercises based on that table. Written Problems 302. 1. What is the postage on a parcel weighing 8 lb., sent from Chicago to Nashville ? 2. What is the charge for sending 10 lb. from Chicago to Cincinnati ? To Aurora ? 3. 9 lb. from any post office or delivery on a rural route starting from that office ? 4. How much must a boy pay in all for the fol- lowing parcels which he mails in Chicago : 8 lb. to be sent to Pittsburgh, Pa. 4 lb. to be sent to Nashville, Tenn. 10 lb. to be sent to Springfield, 111. 5. What is the combined length and girth of a box 20 in. long, 15 in. wide, 10 in. deep ? Is it mailable as a parcel, if it does not exceed the limit of weight ? 6. A parcel is 30 in. long and has a girth of 35 in. Is it mailable ? 7. Make problems based on the rates and the size of packages received by the school. 8. Make 5 problems based on parcel post deliv- eries for local industries. VOLUME OR CONTENT Introduction 303. 1. The illustration represents a cube. It is 1 inch long, 1 inch wide, and 1 inch high. Its contents or volume is 1 cubic inch. It has 6 faces. Is each face a square? How large a square? 2. Two such cubes put together form a prism, 2 in. long, 1 in. wide, and 1 in. high. Its volume is 2 cubic inches. 3. If three such cubes are put together in one row, they make a prism 2 in. long, 1 in. wide, and 1 in. high. Its volume is 2 cubic inches. 4. If 2 such prisms, each 3 in. long, are put side by side, they form together a new prism, 3 in. long, 2 in. wide, and 1 in. high. How many cubic inches in that new prism? 5. The edge stands for 1 yard. How many feet does ^ of that line stand for? 6. This drawing is a cube 1 yd. or 3 ft. long, wide, and high. How many square feet are there in 265 266 INTERMEDIATE BOOK one face of this cube? The cube is divided into smaller cubes, each 1 ft. on a side and each a cubic foot. How many cubic feet do you see in the front layer of the large cube? 7. If there are 9 cubic feet in the front layer, how many are in the layer just back of it ? How many in the last layer? 8. How many cubic feet in the entire large cube? How many cubic feet in a cubic yard? 9. You see that 27 is the product of 3, 3, and 3. If you had a prism 3 in. long, 2 in. wide, and 3 in. high, how many cubic inches would there be in it? 10. How many 1-inch cubes can be packed in a box, 4 in. long, 3 in. wide, and 2 in. deep ? 11. A merchant has boxes 1 ft. each way. How many such boxes can be packed in a trunk 3 ft. long, 2 ft. wide, and 2 ft. deep? 12. The draw- ing, as a whole. 12 in. VOLUME OR CONTENT 267 stands for a cubic foot. How many inches long, wide, and high is a cubic foot? Imagine it made up of 1-inch cubes. How many cubic inches in the bottom layer? 13. The other layers are not drawn. Imagine them drawn. How many layers are there, includ- ing the bottom layer? 14. If there are 12 layers, and each layer con- tains 144 cubic inches, how many cubic inches are there in a cubic foot? Cubic Measure 304. Memorize the table. TABLE OF CUBIC MEASURE 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) In this table these relations are established: Length Width Height Volume in. X in. x in. = cu. in. ft. X ft, X ft. = cu. ft. yd. X yd. x yd. = cu. yd. Oral Problems 305. 1. How many 1-inch cubes can be packed into a box 12 in. on an edge? 2. The contents of a tank is 1 cu. yd. A bucket can hold exactly 1 cu. ft. of water. How many bucketfuls will hll the tank? 268 INTERMEDIATE BOOK 3. A bin is 2 ft. by 3 ft. by 4 ft. Does it hold more than a cubic yard, or less ? 4. A box is 20 in. by 10 in. by 9 in. Does it hold more than 1 cu. ft., or less? 5. How many cubic yards of air can a school- room 10 yd. long, 10 yd. wide, and 3 yd. high hold? 6. A man starts to dig a trench 5 ft. by 3 ft. by 2 ft. How many cubic feet of earth must he re- move? 7. Take a candy box and estimate how, long, wide, and deep it is. How many cubic inches in its contents ? How many square inches in the bottom ? The sides ? The ends ? 8. Make a paper box 2 in. long, 1 in. wide, and 1 in. deep. How many 1-inch cubes will it hold ? 9. A coal bin is 10 ft. by 4 ft. by 3 ft. How many cubic feet of coal will it hold ? How many square feet of boards are there in each of the 2 sides of the bin ? In each of the 2 ends of the bin ? In both sides and both ends taken together ? 10. Make an oral problem based on the table. Written Exercise 306. Find the volume of a prism : 1. 2. 3. 4. 5. Length 12 in. 15 ft. 20 in. 17 m. 10 ft. Breadth 7 in. 5 ft. 7 in. 12 in. 9 ft, Thickness 6 in. 31 ft. 5fin. 4|in. 5|ft. VOLUME OR CONTENT 269 Written Problems 307. 1. A bin is 7 ft. by 4 ft. by 3 ft. How much more will it hold than 2 cu. yd. ? 2. A box is 12 in. long, 9 in. wide, and 5 in. deep. How much less than 1 cu. ft. does it hold ? 3. How many cubic feet of earth were removed in digging a cellar 15 ft. long, 13 ft. wide, and 7 ft. deep? 4. How much larger is a box 10 ft. by 5 ft. by 3 ft. than a box 6 ft. by 6 ft. by 4 ft. ? 5. Which holds more, a box 6 in. by 4 in. by 2 in. or one 5 in. by 4 in. by 3 in. ? 6. How many 1-foot cubes are equal to a 2-foot cube (a cube 2 ft. on an edge) ? To a 3-foot cube ? To an 11-foot cube? 7. How many eggs are there in the upper layer of the top box? 8. Each box has two such layers of eggs. How many eggs in each box ? How many in all three boxes ? 9. How many dozen eggs in one box ? In all three ? 10. What is each box worth when eggs are 22^^ a dozen ? 270 INTERMEDIATE BOOK 11. A merchant received 13 such boxes and sold them at 28^^ a dozen eggs. How much money did he take in ? 12. A brick is 8 in. long, 4 in. wide, and 2 in. thick. How many cubic inches in it ? 13. How many such bricks does it take for a cubic foot ? 14. A brick wall is 5 ft. by 3 ft. by 1 ft. How many bricks are there in it ? 15. How long a wall 4 ft. high and li ft. thick can be built with 972 bricks ? 16. A tinsmith is making a water tank, without cover. The tank is 6 ft. long, 4 ft. wide, and 5 ft. deep. How many square feet of tin does he use ? 17. How many cubic feet will it hold ? Allow- ing 62J lb. to a cubic foot of water, how many pounds of water will the tank hold? How much more is it than 3 T. ? 18. Find the number of square inches in the entire surface of a 2-inch cube. Cut thin cardboard into the shape of the drawing shown here. Make a 2-inch cube. Paste together the free edges by using stickers for mounting stamps. 2 m. 2 in. 2 in. VOLUME OR CONTENT 271 19. Find the number of cubic inches in this 2-inch cube. 20. Find the area of the surface of a box and cover 15 in. long, 5|^ in. wide, and 2^ in. high. Find also its volume. 21. A strawberry box is 6'' long, 4'' wide, and 4'' deep. How many such boxes can be packed in a case 2^' long, 1|^' wide, and |' deep ? 22. A rectangular water trough is 4 ft. long, 1 ft. wide, and ^ ft. deep. How many gallons of water will it hold, if there are 231 cu. in. in a gallon ? 23. One cubic yard of earth makes one load. How many loads are necessary to fill up an old cellar 15 ft. by 10 ft. by 6 ft. ? 24. A wagon box is 9 ft. long, 3 ft. wide, and 1 ft. deep. How many times will this be filled with earth in digging a cellar 21 ft. by 9 ft. and 7 ft. deep ? 25. A boy split enough kindling to make a pile 6 ft. long, 1 ft. wide, and 3 ft. high. What part of a cord did he split ? 26. A water trough holds 15 gal. of water. It is 14 in. wide and 11 in. deep. Find its length. 27. If 36 cu. ft. of coal weigh a ton, what is the weight of a wagon load 9 ft. long, 3 J ft. wide and 3 ft. deep? REVIEW Oral and Written Problems 308. Study each problem. Write an answer that is approximately the answer to that problem. Solve the problems and notice what answers are approxi- mately correct. 1. If it takes 7 men 35f da. to do a piece of work, how long will it take 1 man ? 2. Add 15|, lOf, 25i, 30.6, 20.2. 3. How many feet in 120.56 yd. ? 4. Express as a decimal fraction 67|^%. 5. Express as a common fraction 175%. 6. Find percentage on $2609 at 19%. 7. A fruit raiser planted 260 trees. 13 died. What per cent died ? 8. Of 360 pupils, 18 are absent. What per cent are absent ? 9. 1000 lb. of sea water contains 36 lb. of salt. What per cent is salt ? 10. Mr. Hunt borrows money at 6 % and pays $ 25 interest annually. How much did he borrow ? 11. If 5 per cent of a number is 8.3, what is the number ? 272 REVIEW 273 12. What per cent of a pound are 9 ounces ? 13. A man spends 10|-% of his earnings for board. He pays $ 260 a year for board. How much does he earn ? 14. A merchant sold a piano for $ 450 that cost him $ 400. What per cent of the cost did he gain ? 15. 93.08 is 26 % of . 16. 1753 is what per cent of 3506 ? 17. What per cent of a square foot are 80 sq. in. ? 18. What per cent of a mile are 81 rods ? Written Problems 309. 1. How long and wide is one side of the roof of this barn ? 24 2. Snow 2 ft. deep has a weight of about 12 lb. per square foot. How much pressure will 2 ft. of snow exert on the two sides of the roof of this barn? 274 INTERMEDIATE BOOK 3. It takes about 1000 shingles to make 100 sq. ft. of roof. If a bunch of shingles contains 250 shingles, how many bunches are needed for this roof? For a fraction of a bunch, take a whole bunch. 4. If a bunch of shingles sells at 97^, find the cost of the shingles for this roof. 5. If more of each shingle is exposed to the weather, so that only 800 shingles are needed per 100 sq. ft., what is the cost of the shingles for this roof? 6. When a roof is to be covered with slates 6 in. by 12 in., builders use about 500 slates per 100 sq. ft. How many slates will be needed for this roof ? 7. The triangular part of each end of the build- ing, close to the roof, called the gable, is 24' along the base and 9' high. Find its area. 8. If a very strong wind blows squarely against the end of the barn and exerts a pressure of 14 lb. per square foot, what is the entire pressure against the end of the building ? 9. Find the cost of painting the sides of the barn at 23^ per square yard. 10. What is the cost of painting the roof at 19^ per square yard ? 11. Would the bill have been more or less, if the painting of the whole barn had been done at 21|^^ a square yard ? REVIEW 275 12. If the building had been erected of brick, and 14 bricks had been used for each square foot of outside surface, what would have been the bill for brick, at $ 6 per thousand brick ? 13. How much greater would the expense have been, if the brick wall had been built thicker, so as to require 22 bricks, instead of 14, for a square foot of outside surface ? In what ratio is the new cost to the old ? 14. How many cubic yards of space are there in this barn from the ground floor up to the gables ? 15. If well-settled timothy hay runs about 360 cu. ft. to the ton, how many tons of hay will the barn hold when filled up to the gables ? 16. If loose timothy hay runs about 450 cu. ft. to the ton, how many tons of it will the barn hold when filled up to the gables ? Written Problems 310. 1. The cor- ner stone of the Cap- itol at Washington was laid Sept. 18, 1788. Exactly how long ago was tliat ? 2. The old dome was removed in 1856 and the present 276 INTERMEDIATE BOOK dome completed in 1865. Find the age of the new dome. 3. The capitol stands on a plateau 88 ft. above the level of the Potomac River. The height of the dome above the ground is 287 ft. 5 in. The statue of Freedom is 19 ft. 6 in. tall. How high above the ground is the head of the statue ? How far above the Potomac is it ? 4. The Metropolitan Life Insurance Building in New York City is 700 ft. 3 in. high. How much higher is it than the National Capitol ? 5. The Metropolitan Life is 275 ft. 3 in. long and 123 ft. 5 in. wide. The area covered by the United States capitol building is 153,112 sq. ft. How many square feet larger is the area of the latter ? How many times larger is it ? 6. The capitol is 751 ft. 4 in. long, its maximum width is 350 ft. If this width were uniform, how many square feet would the area exceed its present actual area of 153,112 sq. ft.? 7. The dome is of iron and weighs 8,009,200 pounds. Reduce this to tons. 8. The bronze statue of Freedom weighs 14,985 pounds. If bronze is 8.45 times heavier than wood, what would be the weight of a wooden statue of the same dimensions ? REVIEW 277 9. The Senate chamber is 113 ft. long, 80 ft. wide, and 36 ft. high. Find the number of cubic yards of space in it. 10. The Representatives' Hall is 139 ft. long, 93 ft. wide, and 36 ft. high. How many cubic yards of space in it ? 11. The capitol covers an area of 153,112 sq. ft. How many acres is this in round numbers ? 12. The building of the Library of Congress covers 3|- acres. Is this more than the area of the capitol? Find the difference in square feet. 13. The site of the library building is 10 acres. What per cent of it is taken up by the building itself? Find the ratio of the part of the site occu- pied by the building to the whole site. 14. The floor space in all parts of the building, taken together, is 326,195 sq. ft. How many square feet less than 8 A. is this ? 15. The book stacks contain 56 mi. of shelving. How many shelves, each 4 ft. long, furnish this amount of shelving ? 16. How long would it take you to travel 56 miles on a bicycle at the rate of 9 miles an hour? 17. The service of the library consists of 236 employees in the library proper, 70 for copyright, 25 for distribution of cards, 5 for law indexing. 278 INTERMEDIATE BOOK 127 for care of building and grounds. How much, less is this than the total number of the members of Congress, there being 96 senators and 391 representatives ? Written Exercises and Problems 311. 1. How many cubic inches in a peck, if there are 2150.4 cu. in. in a bushel ? 2. How many cubic inches in a pint, if there are 231 cu. in. in a gallon? 3. A liquid quart is less than a dry quart, con- taining only 57f cu. in., while the dry quart con- tains 67|- cu. in. Find the difference between the two. 4. Find the area of a table 5 ft. 7 in. long and 4 ft. 5 in. wide. 5. The area of a drawing board is 7 sq. ft. 72 sq. in. Its length is 3 ft. 4 in. Find its width. 312. Find the missing part in each of the follow- ing surfaces: Figure Base Height Area 1. Rectangle 2 ft. 3 in. 1 ft. 4 in. 2. Rectangle 10 ft. 2 in. 1 ft. 6 in. 3. Rectangle 5 ft. 6 in. 2 ft. 3 in. 4. Rectangle 2 ft. 6 in. 3 sq. ft. 18 sq. in. 5. Triangle 3 ft. 6 in. 2 ft. 4 in. 6. Triangle 4 ft. 7 in. 3 ft. 2 in. 7. Rectangle 3 ft. 6 in. 11 sq. ft. 96 sq. in. REVIEW 279 Written Problems 313. 1. Find the volume in cubic feet of a box 3 ft. 2 in, long, 2 ft. 1 in. wide, and 1 ft. 6 in. high. 2. Find the number of cubic feet of space in a trunk 2 ft. 6 in. long, 1 ft. 6 in. wide, and 1 ft. 8 in. deep. 3. Which trunk holds more, one that is 2 ft. 8 in. long, 1 ft. 8 in. wide, and 1 ft. 7 in. deep, or one that is 2 ft. 6 in. long, 1 ft. 11 in. wide, and 1 ft. 4 in. deep? What is the difference in volumes ? 4. If a ton of coal occupies 36 cu. ft., how many tons of coal will a bin 8' x 4' x 6' hold ? 5. Coal is bought by the long ton (2240 lb.) and sold by the short ton (2000 lb.). How many long tons does it take to gain one short ton ? Table for Reference 314. Weight and Cost of Railroad Cars Type of Cab Weight IN Lb. Capacity Length Width Height Cost Wood Box 37,000 80,000 lb. 36 ft. 8 ft. 6 in. 8 ft. $1,100 steel Coal 42,000 100,000 lb. 31ft. 9 ft. 4 in. 7 ft. 6 in. 1,200 Flat 32,000 80,000 lb. 41ft. 9 ft. 2 in. 950 Day Coach 85,000 68 passengers 60 ft, 8 ft. 10 in. 9 ft. 9,000 Parlor Car 105,000 34 passengers 70 ft. 8 ft. 6 in. 9 ft. 6 in. 15,500 Sleeping Car 115,000 27 berths 72 ft. 6 in. 8 ft. 6 in. 9 ft. 6 in. 19,000 280 INTERMEDIATE BOOK Problems Based on the Table 315. 1. Nine empty wood box cars are discon- nected from a freight train. In their place 8 empty steel coal cars are added on. How much heavier is the train now ? 2. Passengers weigh 150 lb. on an average. Which is heavier, a day coach full of passengers or a parlor car full of passengers ? What is the difference in weight ? 3. If two trains travel on the same route and with the same speed, which needs a more powerful locomotive, the train with 4 flat cars, 5 steel cars, and 10 wood box cars, or the train with 11 flat cars, 4 steel coal cars, and 7 wood box cars, each car in both trains being loaded to its full capacity ? 4. Find the floor area of a wood box car. (Instead of reducing feet to inches and then finding the number of square inches, it is easier in this case to take 8 ft. 6 in. (8.5 ft.) and compute the number of square feet.) 5. How much less is the floor area of a coal car than of a box car ? 6. How much greater is the floor area of a parlor car than the floor area of a day coach ? 7. Which has a larger capacity, a coal car or a box car ? (Take 8 ft. 6 in.= 8.5 ft. ; 7 ft. 6 in.= 7.5 ft.; 9 ft. 6 in. = 9.5 ft.) REVIEW 281 8. Find the number of cubic feet of space for each passenger in a day coach. 9. Find the dimensions of your schoolroom, also the number of pupils in it. See whether the space allowance for each pupil exceeds that of a passenger in a day coach. What is the difference? 10. Why does a passenger in a parlor car pay more than one in a day coach ? How many cubic feet of space are allowed for each in a parlor car ? 11. How many cubic feet are allowed for each berth in a sleeping car ? 12. Compute the cost of the car and locomotive equipment of a train made up of 8 day coaches, 2 parlor cars, and 5 sleepers, and pulled by a Pacific type locomotive costing $ 18,700. 13. If a Mogul locomotive costs 28 % less than one of the Pacific type, find the cost of the former, when the latter is $ 18,700. 14. What is the ratio of the cost of a coal car to that of a box car ? 15. What is the ratio of the cost of a day coach to the cost of a sleeping car ? Since 9000 : 19000 = 9 : 19, we may say that the ratio required is as 9 is to 19. 16. Show that the cost of a parlor car is to the cost of a flat car as 310 is to 19, or, approximately, as 16 is to 1. 282 INTERMEDIATE BOOK 17. An electric locomotive used on one road weighs 160,000 lb. ; that used on another road weighs 180,000 lb. Find the ratio of the first weight to the second. Written Exercise 316. Write in the Arabic notation : 1. lY, Y, VI, VII, VIII, IX, XI, XII. 2. XIII, XVI, XIV, XXI, XXV, XXIV, XXIX. 3. LX, LXV, LXIV, MC, MCC, MCCC. Write in the Roman notation : 4. 13, 14, 17, 18, 19, 20, 30. 5. 36, 47, 54, 69, 99, 95, 96. 6. 94, 200, 500, 600, 400, 1100. GENERAL REVIEW Addition Combinations 317. Add quickly. Practice until the additions can be made perfectly without hesitation. 1. Group I. Group II. Group III. Group IV. 2 4 9 2 5 2 9 3 3 4 3 6 8 6 9 6 2 2 3 4 2. 4 2 8 7 5 4 4 6 3 8 5 8 5 7 3 2 7 5 6 5 3. 2 6 9 8 9 7 6 7 4 9 3 5 9 7 9 4 7 17 2 4. 6 5 8 18 9 6 8 7 9 5 12 8 7 9 8 13 1 When these combinations are perfectly mas- tered, the addition of larger numbers may be made familiar by adding 10 to each number in the lower line. Then 20, 30, and so to 100 may be added. Thus in the second group the lower row of figures will be 14, 14, 16, etc. Oral Exercise 318. Add columns of numbers from 3 to 10 and higher. These columns may be extended to cor- respond with the usual length of columns in business. 284 INTERMEDIATE BOOK 1. 2. 3. 4. 5. 6. 7. 8. 9. 6 5 4 3 2 2 6 3 7 4 3 9 3 8 6 9 3 4 3 4 5 8 3 9 7 5 6 2 2 3 8 3 5 4 2 7 1 4 6 2 4 2 6 9 4 8 8 4 5 7 9 3 2 5 3 5 3 8 2 3 6 3 4 9 2 4 3 6 4 2 8 5 3 When this exercise has been drilled sufficiently, begin with 12, 14, 23, 26, . . . 98, 103, etc., in place of the 2, 4, 3, 6, etc., the first line in each column, to train in the addition of larger numbers. Another step is necessary to secure the practical skill required in daily transactions. 319. Since members of two or more figures are met in everyday business, the ability to carry in addition is indispensable. Teachers may form such columns of numbers of two, three, four, and more figures, and train pupils to add them with accuracy at optimum rate of speed. Later it is well to follow usual method of add- ing (and subtracting) in making change in money transactions. It is not too much to expect pupils to add numbers of two and even three figures, in REVIEW 285 money terms, as high as $100.00. Thus pupils will readily learn to add: 13 + 12 + 25 + 10 + 15 + 25 as 13, 25, 50, 60, 75, 100. They do this in money and should readily carry the power over to related exercises. Exercises should be formed to complete as far as possible pupils' skill in such addi- tion. The following columns may be suggestive. 10. 11. 12. 13. 42 14. 18 16 47 26 65 22 28 32 43 44 13 35 24 72 51 74 52 46 25 36 26 68 24 46 72 18 27 35 87 39 15. 16. 17. 18. 27 46 507 213 52 73 351 625 24 32 112 143 62 45 214 432 27 81 315 107 43 36 421 634 56 65 137 210 286 INTERMEDIATE BOOK 320. Exercises may be brought into close agree- ment with actual practice in money transactions by making problems involving numbers as follows : 1. 2. 3. 4. 5. $2.00 .25 $1.00 .50 $10.00 2.00 .25 2.00 .25 .50 .25 .15 .50 .10 .25 .25 .03 .25 .15 .25 .50 .32 1.25 6. 7. 8. 9. .50 .25 $5.00 $15.00 $25.00 .05 1.00 .25 12.50 .01 .25 2.75 .05 2.19 3.75 3.00 .20 Addition Exercise 321. Add the columns beginning at the bottom, test by adding down, time both methods : 1. 2. 3. 4. 25 96 93 48 87 81 33 84 42 25 62 53 67 37 41 79 66 44 67 67 89 66 88 75 23 75 72 56 34 89 12 94 REVIEW 5. 6. 7. 8. 21 57 13 69 46 22 42 32 59 45 86 45 60 73 47 67 83 19 23 33 49 64 55 71 27 38 63 90 68 43 92 64 45 37 77 287 Make new exercises by rearranging the addends. Addition Exercise 322. Add by columns, then by rows across the page. Find the total sum of the numbers by each method. 1. 2. 3. $3.25 .75 .23 .76 1.50 1.48 1.19 2.20 2.25 .29 .65 .55 .98 9.35 1.75 .85 6.89 6.80 7.63 2.19 .49 .84 3.20 1.98 1.25 1.15 8.75 288 INTERMEDIATE BOOK 3tice addition across the page. It is a form ccurs often in business. 4. 5. 6. 12.65 75.15 31.15 23.40 6.27 16.25 69.25 8.95 6.15 46.37 6.45 2.50 6.55 45.60 50.67 14.75 16.27 25.18 42.20 24.15 7.75 6.49 6.95 1.50 7.25 35.45 62.35- Addition Exercise 323. Add. Try to shorten time by practice : 1. 2. 3. 4. 11 65 26 73 22 32 55 84 64 14 14 23 71 23 27 46 19 57 64 17 36 91 86 29 49 38 95 31 53 45 37 72 27 63 84 18 82 72 25 65 14 49 60 94 15 77 42 73 REVIEW 5. 6. 7. 8. 18 132 346 456 27 640 220 634 33 309 614 581 48 246 325 790 62 125 675 688 12 670 806 857 19 87 642 435 21 75 614 506 67 42 126 783 85 369 350 658 94 401 623 743 33 625 847 387 72 72 637 961 24 155 949 203 289 Dictation Exercise 324- Write from dictation and add. Time your- self in addition. Test accuracy and speed by adding from top to bottom. 1. 2. 3. 4. 423 231 186 301 576 147 724 810 962 376 476 256 128 312 352 100 634 732 471 403 246 673 524 719 107 132 289 322 290 INTERMEDIATE BOOK 5. 6. 7. 8. 523 2376 8361 34,562 879 2837 5496 18,649 391 1549 8067 43,786 418 3412 2952 83,245 573 5835 1694 52,837 726 8043 7209 29,004 607 1979 3573 37,561 9. 10. 11. S 5640.71 $1831 $45| 3763.89 376 125i 2163.25 214| 212^ 7189.66 8651 320 3548.75 4611 404| 2762.57 329 5161 1836.12 105i ' 75 Addition Exercise 325. Add. Test! by adding down columns. 12. 13. 14. 96,423 47,485 76,423 8,402 3,689 89,845 6,075 62,549 . 6,505 8,941 8,632 92,861 6,243 4,815 14,650 8,655 6,842 19,425 75,242 95,586 6,307 68,115 3,704 25,642 REVIEW 291 15. 16. 17. 96,500 98,424 156,742 6,270 16,821 68,045 8,964 42,461 784,924 53,825 53,119 61,075 6,403 55,427 42,740 8,947 43,862 312,625 13,425 31,465 61,842 6,854 55,408 37,896 1,521 87,623 896,420 680,425 62,576 7,256,748 36.017 119,214 847,223 62,422 35,560 67,895 119,627 125,225 967,425 869,421 67,580 5,564,261 Subtraction Exercise 326. Subtract : 1. 2. 3. 4. $156.47 75.89 97.14 76.15 3.29 15.62 38.25 6.27 8. 1914 1775 5. 6. 7. 86.05 69.11 1915 14.20 15.27 1492 9. 10. 11. 1912 1492 1900 1366 850 1399 12. 1912 1792 292 INTERMEDIATE BOOK Subtraction Exercise 327. Subtract the number at bottom of each column, from each of the numbers of the column. Time yourself for each column. Repeat until you can do them unhesitatingly and accurately. How much time can you gain from practice ? 1. 2. 3. 487,205 125,786 6.14 67,481 78,543 115. 695,560 86,491 69.75 72,897 196,325 4.689 846,423 477,584 32.12 952,843 673,962 6.23 65,896 425,759 82.04 712,872 153,861 3.006 49,753 75,708 2.375 4. 5. 6. 9.2 72.3 175.00 14.03 150.605 87.82 6.152 53.715 325.45 35.005 29.45 4575.00 8.141 43.007 500.00 6.007 21.4 50.00 25.340 115.008 375.00 15.261 62.5 400.40 155.01 314.1 100.20 6.3 27.8 1000.10 4.75 18.075 25.50 REVIEW 293 Multiplication Exercise 328. Multiply: 1. 347 by 900 2. 568 by 36 3. 425 by 17 4. 456 by 305 5. 46,420 by 37 6. $4.75 by 9 7. $0.85 by 24 a $.026 by 136 9. $6,483 by 320 lo. 39.5 by 42 11. .036 by 500 12. 8.072 by 4000 13. 5.672 by 8.2 i4. .0375 by .05 15. .0076 by .082 329. Multiply: Multiplication Exercise 1. 2. 3. 4. $25.65 $69.75 $172.50 $67.58 14 213 250 182 5. 6. 7. 8. $96.47 $84.55 67 881 319 86 125 2036 9. 10. 11. 12. 475 9423 6987 527 86 515 484 8643 Division Exercise 330. Divide : 1. 38,642^8 2. 67,479 -^ 9 3. 25,800^60 4. 750,000-^-25,000 5. 182,000^40,000 6. 90,750-^250 294 INTERMEDIATE BOOK 7. 42)365 8. 9. 870 920 10. 6486 11. 1452 12. 38)6758 13. 4237 14. 60086 15. 42118 16. 79)55,869 17. 125,000 18. 640,050 19. 98,425 20. 152)5650 21. 8240 22. 36,472 23. 805,590 24. .25)125.60 25. $48.75 26. $360.10 27. $450 28. 68,429-^.025 29. 1326.14-^3.45 30. 80.742^375 31. 8.60^48 32. $84.00^112 33. 365.702-^36 34. .8386 -^ 46 35. 72,862-^5000 36. 306,125-^2500 37. 3.649^28 38. 90.362 -^ .345 39. 6724.06-^3.32 40. 32,624.6-^.516 41. 8885^.85 42. .36429^.035 43. 6305.5 -5- .007 44. 3460^.025 Exercises in Fractions 331. Add: 1. h^, hi- 2. l>A> hi 3. h T5"? 12? 1- 4. h is> A.i 5. 16f, 25A 671 81|, 12^. 6. 381, 67f, 72|, 96|, 46i, 13|. REVIEW 295 Subtract : '• l-l- 8. i-A- 9- l-A- 10. f-f 11. A- ■1- 12. ^r-f- 13. 31 -2f. 14. n- ■2|. 15. 61-1^. 16. 8t-lf. 17. H- If 18. 93-V-3I. 19. 181-6^. 20. 27f -11 21. 671-83^. 22. 16|-1|. 23. 25| -6f. 24. 469f-67|. 25. BW^-Q^%. 26. 46f -^. 27. 56^3^- 14|. Find the value : 28. f of 275. 29. 1 of 416. 30. 11 of 4800. 31. f of 618. 32. 320 x|. 33. 4800 XjV 34. 2800 xf 35. 36,400x1*3-. 36. 24.45 x|. 37. 9.375 xf. 38. 360x21 39. 250 X If. 40. 450 x2f. 41. 3401x12. 42. 731x36. 43. 2.48 X If. 44. 84.5 x2f. 45. 16|x.8. Find the value: 46. 25 -^f. 47. 16 ^f. 48. i|^12- 49. 25-^21 50. 16| Ki- 5x. 331^4. ^52. 12f^60. 53. 40- + 8f 54. 8^4-5. 55. ,^v-f 56. 7i H-f. 57. 83J+6J 58. 3.75^1 59. .084 -)-f 60. 64.8^-1-. 296 INTERMEDIATE BOOK Combined Fundamental Processes 332. Perform the operations indicated : 1. 2x6 + 8-4= 2. 18 + 6x4-1-3 = 3. 24-j-3x6 + 8= 4. 11 + 6x2 + 8^4 = 5. 27^3x4 + 6-7 = 6. 2xl43-^6 + 8-17 = 7. 78^13 + 26x4 = 8. 65-9x4 + 16x2 = 9. (14x2)-16 + 2x4-^3 = 10. 3 + 7x5^5x3 = 11. (2x8-4)^3 + (30-8x3)-18-f-5 = 12. (235 - 78 -h 6) - 14 X 11 -^ (186 -^31x2 + 5) -4 = 13. (2^ X y) + "s^ 6"~ 14. 25^fx5i-|= 15. 161^6 + 21-3^5x1 = 16. 10xf^(fxfxi)-3A = DENOMINATE NUMBERS 297 DENOMINATE NUMBERS 333. Tables Linear 12 inches (in. or ") 3 feet 16.5 feet 320 rods 1760 yards 5280 feet 6086 feet = 1 foot (ft. or ') = 1 yard (1 yd.) = 1 rod (rd.) = 1 mile (1 mi.) = 1 mile = 1 mile = 1 knot Liquid 2 pints (pt.) = 1 quart (qt.) 4 quarts = 1 gallon (gal.) Dry = 1 quart (qt.) = 1 peck (pk.) = 1 bushel (bu.) = 1 bushel 2 pints (pt.) 8 quarts 4 pecks 32 quarts 10 mills 10 cents 10 dimes 100 cents Value = 1 cent (f ) = 1 dime = 1 dollar ($) = 1 dollar Time 60 minutes (min.) = 1 hour (hr.) 24 hours = 1 day (da.) 7 days =1 week 320 days or (wk.) 12 months (mo.) =1 year (yr.) Counting 12 units = 1 dozen (doz.) 12 dozen = 1 gross 30 units = 1 score Weight 16 ounces (oz.) = 1 pound (lb.) 100 pounds = 1 hundred- weight 2000 pounds 2240 pounds 7000 grains = 1 ton (T.) = 1 gross ton = 1 pound Angle 90° = 1 right angle (rt. Z) 180° = 1 straight angle (st. Z) Square 144 square inches (sq. iu) = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd.) 30 J square yards = 1 square rod (sq. rd.) '^'^ square rods = 1 acre (A.) acres = 1 square mile (sq. mi.) Cubic 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) 128 cubic feet = 1 cord 160 640 298 INTERMEDIATE BOOK TESTS OF MATHEMATICAL ABILITY FOR PUPILS OF THE FIFTH AND SIXTH GRADES Written tests like the ones proposed below, but with changes in the numbers and also slight changes in the wording, should be given, with- out previous notice, limiting the time to 15 or 20 minutes and requiring pupils to work the tests in the given order. Assign more questions than pupils are able to answer in the given time. One examination may be on Part I, another on Part II. As has been suggested by other writers, either or both of two systems of marking may be adopted, (1) marking each example 1 or 0, according as it is right or wrong, or (2) marking each example on the basis of the total number of steps involved. PART I 1. Add 37.5 2. Subtract 980651.19 405.7 786438.255 90.4 3gQ 5 3. Multiply 71.98 by 23.5. 986.1 4. Divide 17.89 by 3.5. Carry answer to 3 decimal places. 5. 21 + 1+^-1 = ? 6. Multiply 5i by I and 7. Divide I by f. simplify the answer. 8. Divide lOf by 4f . 9. 15 % of 48760. 10. 2.5% of 178.65. PART II 1. What is the cost of eggs a dozen, if 7 dozen cost $ 2.73 ? 2. A locomotive has been run 57 times between Chicago and New York. If the distance between TESTS OF MATHEMATICAL ABILITY 299 these cities is 908 miles, how many miles has this locomotive traveled ? 3. Which of the following numbers are exactly divisible by 6 : 46872, 176509, 37932 ? 4. A loaded truck weighs 6J T. The load con- sists of two parts, one weighing 2^ T., the other weighing 3|- T. Find the weight of the truck. 5. The weekly wage list of 5 employees is $ 27, $ 30, $ 32, $ 29, $ 40. What is the average wage per week ? 6. A merchant saves $ 475 a year, which is 35 % of his earnings. Find his earnings. 7. A man buys 40 ft. of garden hose at 9^i^ a foot, discount 15%. How much does he pay? 8. Find the interest on $ 375 at 6 % for 1 yr. and 6 mo. 9. Find the number of cubic feet of space 2 ft. 6 in. long, 1 ft. 6 in. wide, and 1 ft. 5 in. deep. 10. If 3 tons of coal cost $ 14.25, what will 4 tons cost? Printed in the United States of America. T HE following pages contain advertisements of a few of the Macmillan books on kindred subjects The Health Series of Physiology and Hygiene By M. V. O'SHEA Professor of Education, University of Wisconsin ; Author of " Dynamic Factors in Education," etc., and J. H. KELLOGG Superintendent of the Battle Creek Sanitarium ; Author of " Man, the Masterpiece," etc. The Health Series of Physiology and Hygiene presents a complete course in health instruction for elementary schools. It is organized con- veniently in four books that may be used together advantageously and effectively in the series. Each book is, however, complete in itself, and may be used by itself in courses of instruction in physiology and hygiene. Health Habits. The purpose here is to establish the child in the phys- ical habits and forms of conduct that make for bodily health. It says to the child, "These things are desirable. Can you do them in this way? " Health and Cleanliness. The purpose of this book is to interest chil- dren in social service in health; to show the dependence of health and well being upon protection, and especially against infections through germs, and to teach children what to do for themselves and others in case of an emergency. The Body in Health. The human body is here presented as the most remarkable thing in nature, in the variety and delicacy of its action and in the marvelous adaptation of its parts and functions. It presents knowl- edge with sympathy and it leads to an appreciative understanding. Making the Most of Life. This book directs attention to the chief factors in modern life which reduce the vitality and the health of people. It is a forceftU and constructive treatment of health. THE MACMILLAN COMPANY BOSTON NEW YORK DALLAS CHICAGO SAN FRANCISCO ATLANTA The New Sloan Readers By Mrs. KATHERINE E. SLOAN Author of Primary Readers Primer. Cloth, ismo, colored illustrations, vi and 122 pages. $0.30 The First Reader. Cloth, i2mo, illustrated. $0.80 The Second Reader. Cloth, i2mo, illustrated. $0.35 In the New Sloan Readers the author plans to give in three books a basal series of readers that attract and interest the child through content and illustration, and that give to the child in the most direct way and in the shortest time the independent power to read. The material for the lessons in these Readers has been chosen from the best sources of child literature, yet so taken that the les- sons are of primary interest to the child and closely connected with his daily life and experience. The technical drill necessary in the teaching of reading is provided in charming lessons of story, rhyme, and play and does not in any way detract from the interest or re- duce the reading value of the lessons. The method employed in the New Sloan Readers is based upon the thorough presentation of the simple phonetic elements. The lessons have been so prepared, arranged, and grouped that an ade- quate preparation, presentation, and drill on each new element is provided. The phonetic exercises are not added as separate exer- cises nor presented incidentally, but are woven in a simple, natural manner into every sentence. The frequent and formal reviews bring all the elements to the at- tention of the child at opportune times, and so definite progress is made in all the work. THE MACMILLAN COMPANY BOSTON NEW YORK DALLAS CHICAGO SAN FRANCISCO ATLANTA Muscular Movement Penmanship By C. C. lister Director of Penmanship, Brooklyn Training School for Teachers Elementary Book. $0.16 Advanced Book. $0.20 Teacher's Manual {Preparing) The purpose of the series is to furnish a definite plan by which practical writing may be taught in public and private schools with the greatest possible economy of time. In the Manual are directions for the teaching and supervision of writ- ing that inexperienced teachers may need in order to make the work interesting and productive of good results. The chief features of the treatment are correct posture, movement, and good form. Throughout the series, m connec- tion with each lesson, an effort is made to establish the child in habits of correct posture. Correct positions of the body, the arms, hands, feet, pen- holder, and paper, are fully illustrated and the aim has been to treat the matter of posture concisely, yet so clearly that there could be no doubt as to what is meant by correct writing habits. The treatment of the matter of posture is based upon facts and conditions in writing that govern the health of the whole body and the special organs that are immediately concerned in writing. The lessons in this series have been so worked out that a complete system of exercises for the development of muscular movement writing is provided. THE MACMILLAN COMPANY BOSTON NEW YORK DALLAS CHICAGO SAN FRANCISCO ATLANTA English, Spoken and Written By henry p. EMERSON Superintendent of Education, Buffalo, New York, and IDA C. BENDER Supervisor of Primary Grades, Buffalo, New York Book One. Lessons in Language for Primary Grades. Clothy i2mo, illustrated, vii and 217 pages. $0.40x Book Two. Lessons in Language, Literature, and Composition. Cloth, j2mo, illustrated, xv and 27g pages. $0.50x Book Three. Practical Lessons in English, Grammar, and Com- position. Cloth, 1 2mo, illustrated, xiv and ^yb pages. $0.60x The title of this series is significant of its purpose and method. It constantly reminds teacher and pupil that the object sought is to give power to use and appreciate English in oral speech and written form rather than to limit the results of study to a knowledge of grammar. The authors have aimed to make the series practical in the best sense of making it a real help to pupils in the oral and written use of the mother tongue. Instead of relying upon technical grammar to mold the daily speech of children, emphasis is laid upon practice in speaking, reading, interpreting, and writing under the guidance of the teacher. Throughout the book, the preeminent importance of oral practice is recognized. The ear is too often a neglected factor in language teaching. Selections have been introduced which the teacher is to read to the pupils to train them to a perception of nice language val- ues. Pupils are directed to criticise their own language as regards not only the interest of the thought expressed in it, but the quality of its sound also. THE MACMILLAN COMPANY BOSTON NEW YORK DALLAS CHICAGO SAN FRANCISCO ATLANTA THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL PINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. SEP 25 mi\ '7 t94? StP8 1979 I m -\'.t . LD 21-100m-7,'40(6936s) U. C. BERKELEY LIBRARIES THE UNIVERSITY OF CALIFORNIA LIBRARY