^Wi i 1 I I ^^'^ IN MEMORIAM FLORIAN CAJORl INTERMEDIATE ARITHMETIC BY BRUCE M. WATSON SUPERINTENDENT OF SCHOOLS, SPOKANE, WASH. AND CHARLES E. WHITE PRINCIPAL OF FRANKLIN SCHOOL, SYRACUSE, N.Y. D. C. HEATH & CO., PUBLISHERS BOSTON NEW YORK CHICAGO Copyright, 1907, By D. C. Heath & Co. 1a2 INTEODUCTION The Intermediate Arithmetic is divided into two parts, each containing a full year's work. Throughout the book the pupil is led to see, in each new topic, an extension and application of principles previously learned. Fractions are treated as expres- sions of division. The work in decimals is presented as an extension of the decimal scale of notation to numbers smaller than one. Percentage is treated as an application of decimals — a familiar topic under a new name. There is no formal divi- sion of percentage into " cases," but the pupil is led to apply his knowledge of the relation of product and factors in deter- mining the process to be employed in the solution of each indi- vidual problem. All technical commercial terms are reserved for later consideration. The work in denominate numbers is confined to such prob- lems as people of the present generation are likely to meet in their daily vocations. Extended reductions and intricate measurements are not required. Attention has been given to the development of ideas of proportion, the real purpose of the so-called " ratio exercises " found in some courses of study. An effort has been made to shorten the course and simplify the work, to a reasonable degree, by reducing each topic to as few cases as possible, and by employing the simplest and most generally applicable processes. The study of arithmetic should furnish the child a means of interpreting mathematically the world about him. It should iii iv INTRODUCTION enable him to measure and relate the facts of geography, history, and science. It should bear directly upon the vital interests of the home and family. Care has been taken in the selection of problems to meet these requirements, so far as possible, with due regard to the mathematical content of the exercises, which must always be the first consideration. I INTERMEDIATE ARITHMETIC ^ IIs^TERMEDIATE ARITHMETIC PART ONE NOTATION AND NUMERATION 1. That which tells how many is number; e.g. 11, 14 (books), 25 (cents). 2. One is a unit ; e.g, 1 dollar, 1 house, one. Every number is made up of units. Three contains 3 units. Twenty contains 20 units. 3. Expressing numbers in figures or letters is notation ; e.g, 7, 29, VII, XXIX. 4. Expressing numbers by means of figures is Arabic notation ; e.g. 13, 4728, 23806. 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called significant figures, because each figure has a value. The figure 0, called zero, or naught, has no value, but is used to give the significant figures their proper places in a number. 5. Expressing numbers by means of letters is Roman notation : e.g. VIII, CD, XCIV. 3 ARABIC NOTATION ARABIC NOTATION ! h "5 o 3 rt o Jz; 00 s H < ^ o <2 ■C = .- "D £ ^ - O TJ "V 3 -O C = C C O C = 3a>--3a>j=3 J^ X|-CQII-Sl|-HX|-3 46 5, 20 9, 315, 087 This number is read, four hundred sixty-five billion^ two huu" dred 7iine million^ three hundred fifteen thousand^ eighty-seven. A comma (,), sometimes called a separatrix, is used between periods to aid in reading numbers. 7. Oral 1. Name the periods in this number. Name the places. 2. How many periods are there ? How many places ? 3. How many places are there in each period ? 4. How does the name of each period compare with the name of its right-hand place ? 8. Written Express in figures: . 1. Two hundred thousand, sixteen. 2. Eleven thousand, two. NUMERATION 5 3. Four million, six hundred eight thousand, three hundred seventy-five. 4. Twenty-five thousand, seven. 5. Nineteen thousand, seventeen. 6. Twenty-seven million, six hundred fifty. 7. Eighty million, six hundred nine thousand, four hundred twenty-eight. 8. Six hundred twenty million, seventeen thousand, four hundred seventy-seven. 9. Four hundred thirty-six thousand, fifty-one. 10. One hundred fifty-seven million, six hundred eight thou- sand, four hundred seventy-seven. 11. Three billion, fifty-seven million, four hundred seventeen thousand, sixty. 12. Write a number containing five places. 13. Write a number containing three periods. How many places does it contain ? 14. Write a number containing three periods in Avhich the thousands' period has no value. NUMERATION 9. Naming the places of figures and reading numbers is numer- ation. Thus, to numerate 43,008,160, we should say "Units, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions, ten-millions — forty-three million eight thousand one hundred sixty." 10. Numerate the numbers below : 1. 385 4. 315,129 7. 8,460,000 2. 1,421 5. 6,785,342 8. 423,000,501 3. 25,678 6. 35,000,730 9. 8,003,040,631 KOMAN NOTATIOIT ROMAN NOTATION 11. The Roman notation uses seven capital letters to express numbers, as follows : I (1), V (5), X (10), L (50), C (100), D (500), M (1000). 12. These letters are combined according to the following principles of Roman notation: 1. Placing a letter after one of greater value adds its value to that of the greater, 2. Placing a letter before one of greater value subtracts its value from that of the greater. 3. Placing a letter between two letters of greater value subtracts its value from their sum, 4. Repeating a letter repeats its value, 5. Placing a bar over a letter multiplies the value of the letter hy 1000. ILLUSTRATIONS 13. 1. X = 10, V = 5, XV = 10 + 5 =15. Which prin- ciple does this illustrate ? 2. V = 5, I = 1, IV = 5 — 1 = 4. Which principle does this illustrate ? 3. C = 100, L = 50, X = 10, CXL = 100 + 50 - 10 = 140. Which principle does this illustrate ? 4. X = 10, XXX = 10 + 10 + 10 = 30. Which principle does this illustrate ? 5. D = 500. D = 500000. Which principle ? 6. CCCLX = 360. Which principle ? 7. MCM = 1900. Which principle ? 8. MDCLXVI = 1666. Which principle ? 9. Write a number to illustrate each principle. ADDITION 7 10. Express in Roman notation all numbers from 1 to 100. Note. — For many years the Roman notation was the one chiefly used in Europe. The ancient Greeks also had a system of notation that employed the letters of the Greek alphabet. Both of these systems were awkward and were not easily used in making computations. The Arabic system of notation, now employed by all the great nations of the world, was used first in India, and afterward brought to Europe by the Arabs. ADDITION 14. Addition is the process of uniting two or more numbers into I one number ; e.g. 2 + 5=7. 15. The numbers added are addends; e.g, 3 + 10 = 13; 3 and 10 are the addends. 16. The result of addition is the sum ; e.g, 8 books and 7 books are 15 books; 15 is the sum. 17. The addends and the sum are called the terms of addition. 18. Oral Add: 1. 31 5 71 7 63 2' 8." 56 61 6/ 46 8 34 7 26 « 19 9 9 71 Note. — In adding a column, always look for combina- tions of two or three figures whose sum may be taken as an addend. Thus, in finding the sum in Example 1, say 9, 19, 26, 34, 46, 56, 63, 71. Read the column downward, making similar combina- tions. 8 ADDITION Add: 2. 4 3. 9 4. 5 5. 6 6. 8 7. 8 8. 5 9. 8 5 4 9 6 7 6 7 2 6 6 6 7 9 5 6 4 5 2 4 6 2 2 4 6 2 7 3 8 9 3 3 7 3 5 5 2 1 7 2 3 4 8 2 7 6 9 4 8 3 4 4 5 7 1 6 4 9 3 5 6 3 8 9 7 814^8736 10. Add 25 ant? 47. 25 + 40 = m. Say 25, 65, 72. 65 + 7 = 72. Ans, In a similar way find the sums indicated in exercises 11-25 : 11. 28 + 26 16. 62 + 28 21. 62 + 24 12. 42 + 75 17. 57 + 36 22. 65 + 34 13. 63 + 29 18. 72 + 29 23. 53 + 46 14. 27 + 38 ^ 19. 35 + 26 24. 64 + 44 15. 45 + 34 20. 44 + 38 25. 73 + 27 26. A drover bought 8 cows, 5 horses, and 10 sheep. How many animals did he buy ? 27. Fred paid 10 dollars for a goat, and 12 dollars for a cart How much did both cost him ? 28. In a certain class 28 pupils were present, 5 were absent on account of sickness, and 4 were absent for other reasons. How many pupils belong to the class? 29. A farmer sold 25 bushels of apples to one man, 10 to another, and 8 to another. How many bushels did he sell? ADDITIOI!^ 9 30. John had 40 cents in his bank. He added 8 cents on Monday, and 10 cents on Wednesday. How much money had he then in his bank ? 31. A man paid 6 dollars for paint, and 10 dollars for labor. How much did he pay for both? 32. Bought sheep for 50 dollars, turkeys for 12 dollars, and chickens for 8 dollars. How much did they cost? 33. A man in repairing his house paid 65 dollars for lumber, 8 dollars for paint, 1 dollar for nails, and 30 dollars for labor. What was the cost of his repairs? 34. How many fish did Mr. A catch in four days if he caught 12 the first day, 8 the second, 9 the third, and 7 the fourth ? 35. A girl spent 5 cents for car fare, 4 cents for pencils, 8 cents for paper, 10 cents for ribbon, 15 cents for lunch, and had 9 cents left. How much money had she at first? 19. Written Add, and test your work hy adding downward: 1. 28 2. 639 3. 1050 4. 126 5. $115.85 39 874 394 149 ' 327.15 76 596 769 1260 495.27 42 421 564 1004 160.03 89 397 285 986 . 598.09 I§ 269 784 24 784.06 97 7. 857 8. 283 9. $208.40 10. 1356.24 98 943 2075 32.03 35.09 79 268 298 26.07 2.15 68 207 963 18.94 30.05 40 976 859 236.29 5.16 87 ' 888 876 28.15 304.29 10 ) ADDITION 11. 2673 12. 837 13. 628 14. 8063 846 2964 4307 259 1025 418 526 8264 92 3825 8279 1287 837 842 428 428 642 29 4273 3064 4983 561 746 42379 8698 29 394 6507 2789 387 786 93289 15. Three boys went fishing, and caught 16 perch, 19 pickerel, and 8 black bass. How many fish did they catch in all ? 16. Two trains starting from the same place ran two days in opposite directions. One ran 530 miles the first day and 525 miles the second, while the other ran 492 miles the first day and 510 miles the second. How far apart were they at the end of the two days? (Illustrate by a picture.) 17. A man bought coal for $5.60, wood for $3.45, and a stove for $45. What was the whole cost? 18. There are 112 bushels of wheat in one bin, 175 in an- other, and 234 in the third. How many bushels in all ? 19. There are 218 pages in my reader, 245 in my arithmetic, 195 in my geography, and 189 in my language book. How many pages in the four books ? 20. The Bum of all the sides of a figure is its perimeter. 1. Find the perimeter of a figure whose sides are 39 inches, 45 inches, 28 inches, b^ inches, 75 inches, and 17 inches. 2. What is the perimeter of a seven-sided piece of land whose sides are 209 feet, 683 feet, 129 feet, 463 feet, 928 feet, 93 feet, and 290 feet ? (Illustrate.) SUBTRACTION 11 ^^ SUBTRACTION 21. Subtraction is the process of finding the difference between two numbers; e.g. 21— 1 = 14:; 13 cents — 5 cents = 8 cents. 22. The number from which we subtract is the minuend. The number subtracted is the subtrahend. The result of subtraction is the difference or remainder. The difference is always the number that must be added to the subtrahend to obtain the minuend ; e.g. 17 — 9 = 8. 17 is the minuend, 9 is the subtrahend, and 8 is the difference or remainder. 23. 77ie minuend, subtrahend, and remainder are called the terms of subtraction. 24. Oral From 83 take 57. 83 - 50 = 33. Say 83, 33, 26. 33 - 7 = 26. Ans. In a similar way find the differences indicated in exercises 1-15. 1. 38-19 6. 72-26 11. 63-25 2. 27-18 7. 66-37 12.. ,48-19 3. 42-15 8. 92-48 13. 51-27 4. 61-22 9. 87-39 14. 84-47 5. 81-36 10. 42-29 15. 75-39 16. Frank lives 12 blocks from school, and Henry 5 blocks In the same direction. Their homes are how many blocks apart ? How many blocks apart would they be if Henry lived 5 blocks from school in the opposite direction ? (Illustrate.) 17. Mary added two numbers, and the sum was 28. One of the numbers was 16. What was the other ? 18. 20 is how much more than 11? 12 . SUBTRACTION 19. Tom has 20 marbles, and Edward 11. Tom has how many more than Edward ? 20. If you pay 15 cents toward the purchase of a slate cost- ing 20 cents, how much do you still owe ? 21. Nell had 21 chickens, but a dog killed 10 of them. How many were left? 22. Lucy is 20 years old, and her sister is 6 years younger. How old is her sister? 23. John had 1 20. He spent 1 10 for a coat and 1 3 for a hat. How much had he left ? 24. A pole was 19 feet long. I cut off 4 feet at one time and 6 feet at another. How many feet were left ? 25. What number must be subtracted from 43 to leave 25 ? 26. John's heart beat 78 times a minute when he was well, but 130 times a minute during a severe illness. How much faster did the heart beat during illness than in health? 25. Written From 58500 take 26937. 58500 = 50000 + 7000 + 1400 + 90 + 10 26937 = 20000 + 6000+ 900 + 30+ 7 31563 = 30000 + 1000+ 500 + 60+ 3 In finding the difference, we write only this : 58500 26937 31563 and say, 7 from 10 = 3, 3 from 9 = 6, 9 from 14 = 5, 6 from 7 = 1, 2 from 5 = 3. To test the work, add the subtrahend and remainder. If the minu- end is obtained, the work is correct. Do not write the numbers again, but make the test with the numbers as they stand. In what other way may we test subtraction? SUBTRACTION 13 Subtract and test results 1. 2819 674 5. 2763 1289 9. 92874 11392 13. 38264 29842 17. 12.15 1.12 21. 134.28 24.28 2. 10. 14. 82'03 1276 37284 9287 94210 8206 19327 8291 18. $35.28 17.05 22. .21 27.13 3. 7. 4295 597 36801 18463 11. 42840 38706 15. 92593 87246 19. $25.18 1.15 23. $17.80 16.75 7306 1807 8. 18003 921 12. 98301 26942 16. 27075 18092 20. $36.51 16.82 24. $75.00 24.32 25. From seventeen thousand sixteen, take nine thousand four hundred eighty-seven. 26. From seventy-two thousand three hundred eleven, take forty-six thousand nine hundred sixty-one. 27. Take eight thousand four, from thirty thousand! REVIEW AND PRACTICE 26. Oral 1. Read 359,016,007,138 ; $3,894,760.15; 1,010,101. 2. ReadCLIX; DCCXXXVI; MCMXIIL 3. Numerate 3,057,608. 4. Jennie bought a skein of Shetland floss for 10 cents, 3 skeins of embroidery silk for 12 cents, and a pair of knitting needles for 10 cents. How much change should she receive from a 50-cent piece? 14 REVIEW AND PRACTICE 5. The sum of 3 numbers is 100. Two of them are 29 and 37. What is the other? 6. Albert has earned 15 cents, 25 cents, and 17 cents toward a pair of gloves that cost f 1. How much more money must he obtain in order to pay for the gloves ? 7. Helen, Howard, and Henry wanted a canoe that cost $47. They obtained f 19 by renting their row-boat, their mother con- tributed $12, and their father the remainder of the price. How much did their father give ? 8. Edith made some purchases, gave the clerk a dollar, and received in change three cents, one nickel, two dimes, and a quarter. What was the amount of her purchases? 27. Written 1. A farmer having 456 bushels of corn sold 84 bushels to one man and 135 bushels to another. How many bushels did he have left ? 2. A man started to walk 112 miles in three days. He walked 32 miles the first day, and 41 miles the second. How far must he walk the third day to complete the journey ? 3. I bought a cow for $42, another for $48, and a third for $56. For how much should I sell them to gain $28? 4. A lady bought sugar for 65 cents, tea for 55 cents, molas- ses for 72 cents, butter for 84 cents, starch for 25 cents, and gave in payment a five-dollar bill. How much change should she receive ? 5. The distance by rail from Galveston to San Antonio is 572 miles, from San Antonio to Tucson 932 miles, and from Tucson to Los Angeles 501 miles. What is the distance by rail from Galveston to Los Angeles ? REVIEW AND PRACTICE 15 6. Two vessels start from points 850 miles apart and sail toward each other. How far apart are they when one has sailed 246 miles and the other 352 miles? (Illustrate.) 7. A man sold one horse for $145 and another for $182. On the first he gained 1 23, and on the second 1 36. What was the cost of both? 8. A boy bought apples for $A5 and pears for $.62, and sold them all for $1.50. What was his profit? 9. John sold 62 newspapers, Frank 48, and Henry 27 less than both of them. How many did Henry sell ? 10. A grocer sold butter for $45 and cheese for $62. On the butter he lost $6 and on the cheese he gained $14. What was the cost of both? 11. A farmer bought a barrel of flour for $6.35, sugar for $2.15, coffee for $1.46, tea for $1.20, and gave in payment $3.15 worth of butter and the remainder in cash. What did he pay in money? 12. The sum of 52 and 64 is how much greater than the difference between 124 and 69? 13. From a flock of 320 sheep 76 were sold at one time and 112 at another. How many remained ? 14. A man bought 148 bushels of potatoes from A, 216 bushels from B, 183 bushels from C, and afterwards sold all but 137 bushels. How many bushels did he sell? 15. The sum of three numbers is 342. Two of the numbers are 84 and 96. What is the third number? 16. John's father gave him $2.25, and his uncle gave him $1.40. He earned enough besides so that he bought, with the whole, a suit of clothes for $8. How much did he earn? 16 REVIEW AND PRACTICE 17. Claude took 987 steps in coming to school, Francis 865, and Alice 398 less than the number taken by both the boys. How many steps did all three take? 18. A ship loaded with iron sailed from Cleveland to a port 332 miles west of that city. A car loaded with machinery at Cleveland was taken to a city 619 miles east of Cleveland. How far apart were the ship and the car when each had reached the end of its trip? (Illustrate.) 19. The first Thanksgiving was in 1621 and the day has been observed every year since. How many times has the day been observed? 20. A father and his three sons earned $2461 in a year. The first son earned f 676, the second $456, and the father $1080. How much did the third son earn? 21. A train started from Chicago with 324 passengers. On the way to St. Paul 185 passengers left the train, and 149 came aboard. How many passengers were on the train when it reached St. Paul? 22. A retail grocer bought at a wholesale grocery three bar- rels of apples for $4.50, a box of lemons for $2,70, and three barrels of flour for $ 12. 30. He handed the wholesale grocer one gold piece and received 50 cents in change. What was the value of the gold piece ? 23. During one week, a man put into the bank $687, drew out $489, put in $348, drew oat $298, and then had $1386 left in the bank. How much had he in the bank at first? 24. A farmer having 215 acres of land, used 21 acres for corn, 36 for oats, 29 for barley, 18 for potatoes, 52 for meadow, and the rest for pasture. How many acres were used for pasture ? MULTIPLICATION 17 MULTIPLICATION 28. Multiplication is taking one number as many times as there are units in another ; e.g. 6 times 9 are 54. 29. The number multiplied is the multiplicand ; the number by which we multiply is the multiplier ; the result of multiplica- tion is the product; e.g. 12 times 20 are 240. 20 is the multi- plicand^ 12 is the multiplier^ and 240 is the product. 20 and 12 are factors of 240. 30. The multiplier^ multiplicand^ and product are called the terms of multiplication. 31. The multiplier and multiplicand are factors of the product. The product is the same in whatever order the factors are taken; e.g. 6 times 7 are 42, and 7 times 6 are 42; 3x5x4 are 60 and 4x3x5 are 60. 32. ORAL EXERCISES (Reviewing work of primary arithmetic) 3 4 5 2 6 1 8 7 12 10 11 9 7 12 9 11 5 10 4 1 3 6 2 8 1. Multiply every number in the upper line by each number in the lower line. 2. How do we multiply a number by 10 ? By 100 ? By 1000 ? 3. Multiply 7 by 10 ; by 100 ; by 1000. 4. Multiply 34 by 10 ; by 100 ; by 1000. 5. Multiply 11 by 3 ; by 30; by 300. 6. Multiply 12 by 7 ; by 70 ; by 7000. 18 MULTIPLICATION 7. The product is a multiple of the multiplicand. Of what other number is it a multiple ? 8. Of what number is 33 a multiple? 9. Name 5 divisors of 24. 10. Multiply 23 by 12. 11. Multiply 23 by 10, also by 2. Add the two products. How does this result compare with the answer to question 10? 12. Multiply 30 by 5, then by 20. Add the two products. The sum is how many times 30 ? 13. 90 times 48, added to 6 times 48, are how many times 48? 14. 7 times 786, 40 times 786, and 3 times 786, added, make how many times 786? Q-ive the products at sight : 15. 86 16. 307 17. 315 18. 73 19. 32 7 5 10 100 20 20. 14 21. 28 22. 25 23. 16 24. 28 30 200 300 700 1000 25. 75 26. 103 27. 212 28. 31 29. 205 2000 40 50 400 600 30. Edward feeds his horse 11 quarts of grain per day and his chickens 3 quarts. He feeds his horse how much more in 30 days than he feeds his chickens in the same time ? 31. Find the cost of 50 five-cent postage stamps. 32. 11 and 11 are the factors of what number ? 33. Of what number are 3, 7, and 5 the factors ? 34. What must be paid for a dozen junior baseballs at 70 cents apiece ? MULTIPLICATION 19 33. Written Multiply 1283 by 967. 1283 967 8981 7698 11547 1240661 We multiply 1283 by 7, 60, and 900, and then add the results, which are called partial products. The sum of the partial products is the product required. We omit ciphers at the right of the partial products after the first. Read each partial product as if the ciphers were ex- pressed. Multiply $34.79 by 806. $34.79 806 20874 27832 128040.74 1. 324x24 2. 296x39 3. 387x45 4. 263x56 5. 892x63 6. 728x75 7. 398x84 8. 987x98 9. 516x31 10. 798x43 11. 896x79 12. 598x36 13. 287x49 14. 799x99 Observe that the right-hand figure of each partial product is written directly under the figure by which we multiply to obtain it. Cents in either factor give cents in the product. 15. 296x28 16. 694x39 17. 206x54 18. 128.15x28 19. 134.98x27 20. 119.84x46 21. $7.85x124 22. 128.75x15 23. $36.91x45 24. $126.93x87 25. 26. 27. 28. $.17.85x48 $19.63x49 $75.10x97 $16.35x764 29. $280.52x236 30. $356.04x328 31. $987.62x475 32. $396.41x641 33. $806.04x879 34. 238x307 35. 5126x208 36. 934x9000 37. 1027x2005 38. 386x1080 39. 527x2300 40. 4008x7003 41. $29.05x108 42. 4040x8356 20 MULTIPLICATION 34. PROBLEMS 1. a. What is the perimeter of a square farm whose side is 309 rods ? 5. What is its area ? 2. a. If it costs 14.78 a day to support a certain family, how much does it cost for a month of 31 days ? h. How much does it cost for a month of 28 days ? e. How much does it cost for a year ? 3. There are 2000 pounds in one ton. How many pounds are there in 496 tons ? (Solve it in the shortest way.) 4. There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, a. What is the number of seconds in a day ? h. In a week ? 5. a. How many minutes are thef^ in the month of May? h. In the month of February ? c. In the month of September ? 6. Alice hemmed 4 dozen handkerchiefs, each 12 inches square. How many inches of hem did she make ? 7. A manufacturer put up 13 tons of cereal in sample packages, each containing 1 ounce. How many packages did he make ? 8. 25,079 crates of strawberries, each containing 36 quarts, were shipped from the city of Oswego in one season. What were they worth at 8 cents a quart ? 9. There are 12 things in a dozen, 12 dozen in a gross, and 12 gross in a great gross. How many pens in a case containing 307 great gross ? 10. What is the value of 347 barrels of shredded cocoanut, each containing 105 pounds, at 15 cents a pound ? DIVISION 21 DIVISION 35. Division is the process of finding one of two factors when the other factor and the product are given; e.g. 35 is the prod- uct of 5 and 7. When 35 and 5 are given, we divide 35 by 5 to obtain 7 ; when 35 and 7 are given, we divide 35 by 7 to obtain 5. 36. The number divided is the dividend ; the number by which we divide is the divisor, and the result of division is the quotient ; e.g. 42 -^ 7 = 6. 42 is the dividend, 7 is the divisor, and 6 is the quotient. When the divisor is not exactly contained in the dividend, the part of the dividertd that is left is called the remainder ; e.g. 59 -T- 8 = 7 quotient and 3 remainder. 37. The dividend, divisor, and quotient are called the terms of division. 38. Oral 1. 6 oranges at 3 cents apiece cost 18 cents. Which of these numbers is a product ? Which are factors ? When 18 and 6 are given, how can 3 be found ? When 18 and 3 are given, how can 6 be found ? When 3 and 6 are given, by what opera- tion can 18 be found ? 2. The area of a page of Henry's book is 35 square inches. If the length is 7 inches, what must be the width? If the width is 5 inches, what must be the length ? 35 is which term in division ? 35 is what of 5 and 7 ? 3. There are 9 square feet in 1 square yard. How many square yards are there in 108 square feet ? 108 is the product of 9 and what other number ? 108 is which term in division ? 108 is what of 9 and 12 ? 22 DIVISION 4. Fred paid 54 cents for some sugar. The number of cents that a pound cost is one factor of 54. What is the other fac- tor ? What, besides 54 cents, must be given in order that we may find the cost of one pound ? What, besides 54 cents, must be given in order that we may find the number of pounds that Fred bought ? 5. What is a. The number of feet in 132 inches ? h. The number of pecks in 72 quarts ? c. The cost of a month's rent at $120 a year ? d. The number of weeks in 77 days ? e. The price of a lawn mower, when 15 lawn mowers cost 1150? 6. The first number in each line below is a factor of every other number in the line. Find the factor not given of each number; 15; 75; 355; 525; 405 84; 217; 280; 763; 497 110; 44; 121; 880; 2211 819; 945; 189; 360; 963 64; 328; 176; 728; 960 7. In the following statements tell which numbers are fac- tors and which are products : a. There are peaches in 7 baskets if each basket con- tains 12 peaches. h. 12 quarts of berries cost 96 cents. c. Jerome's wages for 9 weeks at f a week amounted to 27 dollars. d. 6 fountain pens at $3 apiece cost $ . e. il will pay 20 car fares at apiece. f. 72 cents will buy 12 pounds of sugar at cents a pound. a. 5 h. 7 c. 11 d. 9, e. 8 DIVISION 28 39. Written 1. ' Divide 3125 317)990625 951 396 317 792 634 1585 1585 990625 by 317. 317 is not contained in 9 or 99, but it is contained in 990 three times. This is 3 thousand because 990 is thou- sands. The remainder is 39 thousand. Put 600 with it, and 317 is contained in 396 hundred 1 hundred times, with a remainder of 79 hundred. The remaining figures of the quotient are obtained in a similar manner. When the second figure (from the left) of the divisor is 1, 2, or 3, it is well to use the first figure for a " guide figure " in ob- taining each quotient figure. Thus, in this example, we say, 3 in 9 = ? 3 in 3 = ? 3 in 7 = ? 3 in 15 = ? Note. — Be careful to write the first figure in the quo- tient directly above the right-hand figure of the part of the dividend used in obtaining the first quotient figure. 2. Divide 10192 by 49. 208 49)10192 98 392 892 In finding the second quotient figure, we see that 49 is not contained in 39 ; so we place in tens' place in the quotient and bring down 2 units. 49 is almost 50. We may therefore take 5 for a guide figure. 3. Divide 8531 by 672 12|U Quotient 672)8531 672 1811 1344 In this example the dividend does not contain the divisor an exact number of times, hence there is a remainder. The re- mainder may be written over the divisor as a part of the quotient ; thus, 12|f| quotient. Note. — In dividing by any number not larger than 12, short division should be 467 Remainder used. That is, no work should be written except the dividend, divisor, and quotient. In such examples the quotient may be written either above or below the dividend according to convenience. If the dividend contains cents, and the divisor is a whole number, the quotient also contains cents. 24 DIVISION Solve examples 4-68, and test your work hy multiplying the quotient hy the divisor and adding the remainder^ if there is one, to obtain the dividend : 4. 4503 - 3 26. 28,692 - 9 48. 130,052 - 2 5. 2045 -^ 5 27. 333,333 ^ 11 49. 168,754 - 9 6. 2835 ^ 7 28. 35,621 ^ 7 50. 385,980 - 5 7. 4986 -^ 9 29. 42,963 ^ 6 51. 769,520 -V- 7 8. 2009 -^ 7 30. 50,725 -r- 3 52. 387,052 - 10 9. 3504 ^ 8 31. 82,956 ^ 10 53. 943,769 -r- 12 10. .6 193a. 32. 93,043 ^ 7 54. 748,136 - 4 11. ^_±a 33. 65,407 -f- 5 55. 7,688 - 31 12. iLa840 34. 39,842 - 9 56. 12,978 - 42 13. 12 3 M. 35. 27,392 - 8 57. 31,509 -f- 81 14. 5 8 3 4.5. 36. 63,594 - 9 58. 40,948 -^ 58 15. 11^ 37. 31,493 -^ 6 59. 68,476 -h 68 16. lA&M. 38. 25,324 - 5 60. 168,665 -H 427 17. 2 9 31A 39. 28,764 -f- 4 61. 190,855 -^ 931 18. ^|ij6. 40. 36,099 H- 9 62. 293,004 - 801 19. 30,005 ^ 5 41. 14,412 - 12 63. 129,324 - 756 20. 288,012 - 12 42. 36,930 -f- 11 64. 3,247,654^79 21. 300,010 ^ 10 43. 24,003 ^ 6 65. 294,490 = 98 X? 22. 99,011 -^ 11 44. 30,502 - 8 66. 503x7=637,804 23. 33,264 - 11 45. 29,333 - 11 67. 58,487 has what 24. 29,280 - 12 46. 675,262 - 5 factor besides 143 ? 25. 36,550 -^ 10 47. 349,872 - 8 68. ?x 215 = 66,220 REVIEW AND PRACTICE 26 REVIEW AND PRACTICE 40. Oral 1. A farmer exchanged 12 barrels of apples at |3 a barrel for coal at i 4 a ton. How many tons of coal did he receive ? 2. In what time will a boy earn as much at f 3 a week, as a man earns in 6 weeks at $8 a week ? 3. Nell is 3 years old and Will 5. Their sister's age is twice the sum of their ages. What is the sister's age ? 4. How many gallons of milk will a family use in the month of June if they use 2 quarts a day ? 5. Frank rides 6 miles an hour and Albert 9. a. How far apart will they be in 6 hours if they start at the same time and place and ride in the same direction ? h. If they ride in op- posite directions ? 6. Grace bought 2 dozen lemons. She used | of them for lemonade and gave away 6. How many remained? 7. Helen bought -^ of a yard of cambric. She used half a yard in her dress and wasted ^ of a yard in cutting. How much was left ? 8. A man owed $96. He made 5 payments of 1 12 each. How much did he then owe ? 9. Luther has % 6 and Leon 3 times as much. How much have both? 10. 118 plus 112 is how much less than 4 times $12? 11. The product is 125 and one of the factors is 5. What is the other factor ? 12. The divisor is 11, the quotient 12, and the remainder 9. What is the dividend ? 13. The dividend is 85, the quotient 9, and the remainder 4. What is the divisor ? 26 REVIEW AND PRACTICE 14. How many days will a 12-gallon keg of water last 24 shipwrecked men if each man drinks a pint a day ? 15. If Frances can knit 21 stitches a minute, how many stitches can she knit in half an hour ? 16. The product of 20 and 16 is how much less than 20 times 20? 41. Written 1. How many tons of coal at f 5 a ton will pay for 15 tons of hay at $11 a ton? 2. A man started on a journey of 724 miles. After he had traveled 12 hours at the rate of 32 miles an hour, how far was he from his journey's end? 3. A farmer bought six sacks of flour at §1.25 a sack, 25 pounds of sugar at 6 cents a pound, and two pounds of coffee at 30 cents a pound. He paid for it in butter at 24 cents a pound. How many pounds of butter were there ? 4. A man having 1738.58 in a bank drew out 1132.75 at one time, f 175.50 at another, and $216 at another. How much money then remained in the bank ? 5. If the divisor is 38, the quotient 209, and the remainder 23, what is the dividend? 6. The product of three numbers is 1260 and two of them are 12 and 7. What is the third? 7. A grocer buys 88 gallons of molasses at $.56 a gallon. For what price per gallon must he sell it in order to gain 112.32? 8. The dividend is 1821, the quotient 32, and the remainder 29. What is the divisor? 9. Two trains start at the same time from points 1216 miles apart and travel toward each other, one at the rate of 35 miles REVIEW AND PRACTICE 27 an hour, the other at the rate of 41 miles an hour. In how- many hours will they meet? 10. A lady bought 12 yards of dress goods at $1.75 a yard, 8 yards of silesia at f.25 a yard, 2 pairs of gloves at $1.45 a pair, 6 handkerchiefs at 8.25 apiece, and 3 yards of table linen at 1.95 a yard. She paid 118.75. What did she still owe? 11. How many pounds of cheese at 14 cents a pound will pay for 3 barrels of flour at $4.20 a barrel? 12. At what rate per hour must a train run to go as far in 9 hours as another train running 27 miles an hour can go in 12 hours? 13. A farmer paid $1125 for cows, horses, and farming tools, and 8 times as much for a farm of 125 acres. What was the price per acre? 14. A farmer had 46 acres of alfalfa. He cut three crops a year. The first crop yielded If tons per acre, the second IJ tons, and the third 1 ton. a. How much was it worth at $8 a ton? h. How many pounds of alfalfa did he obtain? 15. At a certain post office there were sold in one year 12,400 twenty-five-cent stamp books ; 4600 forty-nine-cent stamp books ; 2250 ninety-seven-cent stamp books. How much was received for all of them? 16. A merchant owed a debt amounting to $9892. He made four payments of $1980 each. How much did he then owe? 17. A river is 2174 miles long. A steamboat started at the mouth of the river and traveled up stream for 6 days at the rate of 149 miles a day. The boat was then how far from the source of the river? (Illustrate.) 28 REVIEW AND PRACTICE 18. a, A rural free-delivery mail carrier on a certain route is on duty 298 days in a year and rides 24 miles each day. How far does he ride in a year ? 5. He starts at 8.15 a.m. and returns home at 3.15 P.M. every day. How many hours does he spend on the road in a year ? The number of pieces of mail delivered and collected by him in one month was as follows: Registered Letters Common Letters . Postal Cards . . Newspapers . . Circulars . . . Packages . . . Collected 3 877 297 27 26 The total number of pieces delivered was how much greater than the num- ber collected ? d. If there were 189 families on this route, what was the average number of pieces delivered to each family? e. If this carrier's salary is §900 a year, and it costs him 813 a month to keep his horse, how much of the salary is left to pay him for his year's work ? /. The Post-office Department of the United States pays the carrier's salary. He sells stamps to the amount of $31.50 per REVIEW AND PRACTICE 29 1^^^ month, and sends the money to the Post-office Department. The amount received from this mail route is how much less per month than the cost of delivering the mail? 19. a. In the year 1905 the Syracuse post-office received $411,630.95 for stamps, registering letters, writing money orders, and other postal business. The expense of carrying on the post-office was f of this amount. What was the expense of carry- ing on the post-office ? h. How much did the Post-office Department gain on account of this post-office ? c. In this office 722 sacks and pouches of mail were handled in one day, each sack and pouch con- taining an average of 154 pieces of mail. How many pieces of mail were handled ? d. At the same rate, how many pieces were handled in a year ? 20.. a. Willis has 15 hens. They laid at the rate of 120 eggs apiece in one year. How many eggs were laid by all of them? h. The food for the hens cost $21. Willis sold the eggs at an average price of 22 cents a dozen. How much more did he receive for the eggs than he paid for the food for the hens ? c. What was the profit from one hen ? d. What would be the profit from 75 hens at the same rate ? e. Willis has a rectangular hen park 30 ft. long and 15 ft. wide. How many feet of wire netting will inclose it ? 21. From one hundred twenty-two thousand take eighty- seven thousand ninety-four. 30 INDICATED WORK INDICATED WORK 42. In problems requiring several operations, or steps, it is well first to indicate the operations by means of signs; e.g. 5208 -T- 3 X 8 means that we are to divide 5208 by 3 and multi- ply the quotient by 8. The parenthesis ( ) is sometimes used to inclose certain numbers or expressions which are to be taken together as one thing ; e.g. 18 x (15 + 10) means that 18 is to be multiplied by the sum of 15 and 10. Operations indicated within a parenthesis should always be performed first; e.g. 5208 -r- (3 x 8) means that we are to multi- tiply 3 by 8 and divide 5208 by the product. When the parenthesis is not used, indicated multiplication and division should be performed before indicated addition and subtraction; e.g. 125 + 3x18 — 46^23 means that we must multiply 3 by 18, then divide 46 by 23, then add and subtract results as indicated; thus, 125 + 3 x 18 — 46 ^ 23 = 125 + 54 — 2 = 177. 43. Perform the operations indicated: 1. 5208-f-3x8 2. 5208 H- (3 X 8) 3. 203x607015-596034 4. 203 X (607015 - 596034) 5. 487 + 598 + 645- (2030 -1435) 6. 9346 - (6342 + 347 + 89) +2349 7. 9346-6342 + 347 + 89 + 2349 8. 41983 - 87 X 103 + 47 9. (41983 - 87) X (103 + 47) 10. 2310 ^ 10 X 7 + 604 X 35 11. 2310 -i- (10 X 7) + 604 X 35 INDICATED WORK 81 12. 3056 + 9821-7x48-29 13. 3056 + (9821 -- 7) x (48 - 29) 14. (|1.25x6 + 25x|.06 + 2x$.30)-5-24. See example 3, page 26. 15. $738.58 -(1132.75 + 1175.50 + 1216). See example 4, page 26. 16. 1216 -^ (35 + 41). See example 9, page 26. 44. Indicate and find results: 1. 1.75 less the sum of $.32 and 1.18. 2. 1500 less $275, and th.e result added to 1 132. 3. $18 more than the difference between $27 and $425. 4. The product of 1125 and 8, divided by 125. 5. The sum of 18 yards and 41 yards taken away from 4 times 69 yards. 6. The sum of 498 and 747 divided by the difference be- tween 2342 and 2425. 7. George earns 55 cents a day and Harry 79 cents. How much do they both earn in the month of October, allowing for 4 Sundays ? 8. The product of 162 and 39 divided by the difference of 87 and 61. 9. Frank's earnings for the 6 days of the week were $.43, $.59, $.62, $.79, $.38, $.48. How much more must he earn before he can buy a $5 set of books ? 10. The quotient of 12,848 -^ 16 is how much less than the number of hours in 1 year ? 11. A lady bought 10 yd. of silk at $1.10 a yard and 2 yd. of silesia at 25^ a yard. How much change should she receive from a 20-dollar bill ? 32 SPECIAL CASES UST MULTIPLICATION" AND DIVISION SPECIAL CASES IN MULTIPLICATION AND DIVISION PRINCIPLES 45. 1. Each removal of a figure one place to the left increases its value tenfold; e.g. 5 = 5; 50 = 5 x 10; 500 = 50 x 10. 2. Each removal of a figure one place to the right diminishes its value tenfold', e.g. 500-^10 = 50; 50 -j- 10 = 5. 46. Oral 1. What is the shortest way to multiply by 10 ? By 100 ? By 1000? By 10000? By 1 with any number of ciphers annexed ? 2.8x10 = ? 8x100 = ? 8x1000=? 8x10000 = ? 3. Cutting off a cipher from the right of a number is the same as moving all the figures one place to the right. How does it affect the value of the number ? 4. How many ciphers must be cut from the right of a number to divide the number exactly by 100 ? By 1000 ? By 10000? 5. Divide each of these numbers by 10 : 50; 600; 5290; 36000; 4500; 321560. 6. Divide each of these numbers by 100 : 300; 4500; 6000; 78000; 70000; 831000. 7. Divide 9600000 by 10000. 8. Multiply each of these numbers by 10, 100, and 1000 : 7; 61; 20; 310; 402; 910; 653; 1020. 47. Written 287 1. Multiply 287 by 3700. 3700 2009 86l 1061900 Product SPECIAL CASES IN MULTIPLICATION AND DIVISION 3B 2. Divide 435600 by 1800. 242 Q uotient 18j9P)4356p 36 75 72 36 1800 = 18 X 100. Therefore we divide by 100 and then by 18. How do we divide by 100 ? 3. Divide 83,645 by 13,000. Q^%%%\ Quotient 78 6 Multiple/ and test hy dividing 4. 432 by 20,100 5. 69 by 38,000 6. 420 by 80,000 7. 1242 by 3020 8. 5003 by 960 Divide and test results: 14. 257,830 by 590 15. 4410 by 70 16. 34,376 by 100 17. 1,333,800 by 1900 18. 1,308,580 by 260 19. 572,400 by 3600 20. 42,978 by 300 21. 642,359 by 470 When we divide by 1000, there is a remainder of 648. When we divide by 13, there is a remainder of 5 in thousands' place. 5000 + 648 = 5648, whole remainder. the product hy the multiplier: 9. 208 by 6500 10. 320 by 420 11. 86 by 12,000 12. 409 by 30,800 13. 6900 X 413 22. 8,205,900 by 4200 23. 367,298 by 1600 24. 368,700 by 3600 25. 496,789 by 420 26. 805,060 by 3090 27. 367,059 by 7800 28. 8,079,600 by 71,000 29. 4,380,700 by 3210 34 IDEAS OF PROPORTION 48. Oral IDEAS OF PROPORTION Fig. a. Fig. B. 1. Figure B is how many times as large as figure A ? 2. If figure A is 1 inch long, how long is figure B ? 3. If A and B are pieces of cloth, and A is worth f 5, what is B worth ? 4. If A is a piece of land containing 10 acres, what is B ? 5. If A is a piece of land worth $12, what is B worth ? 6. If B is worth |60, what is A worth ? 7. If A and B are blocks of wood and A weighs 9 pounds, what does B weigh ? 8. If B weighs 39 ounces, what does A weigh ? 9. If A and B are fields, and A can be plowed in 4 days, how long will it take to plow B ? <5d ■ 66 66 66 66 10. a, 10 pears are how many times 2 pears ? h. 2 pears are what part of 10 pears ? c. If 2 pears cost 3 cents, 10 pears cost cents. d. Two pears are worth cents, when 10 pears are worth 25 cents. IDEAS OF PROPORTIOl^ 36 11. a. One dollar is how many times one dime ? h. One dime is what part of one dollar ? c. If one dollar will buy 30 pencils, one dime will buy pencils. d. If one dollar will pay for 70 apples, one dime will pay for apples. e. Frank can earn one dollar in hours if he can earn one dime in two hours. /. If one silver dollar weighs one ounce, one silver dime weighs ounce. 12. If John rides 9 miles in 2 hours, in what time can he ride 27 miles at the same rate ? Analysis : 27 miles are 3 times 9 miles. Therefore, if John rides 9 miles in two hours, he can ride 27 miles in 3 times two hours, or hours. Solve and analyze each of the following problems : 13. If a man's wages for 12 hours are 5 dollars, in how many hours will he earn 120 ? 14. If 20 men can do a piece of work in 5 days, how long will it take 10 men to do the same ? 15. When 75^ will buy 6 pineapples, how much should be paid for 2 pineapples ? FACTORS AND MULTIPLES FACTORS AND MULTIPLES 49. One of the numbers that are multiplied to produce a num- ber is a factor of that number ; e.g. 2, 3, and 5 are factors of 30 because 2 x 3 x 5 = 30. 50. A number that exactly contains another number is a mul- tiple of that number ; e.g. 21 is a multiple of 7. It is also a multiple of 3. 51. A number that is composed entirely of whole units is an integer ; e.g. 7, 13, 200. Can you name a number that is not an integer ? 52. A factor that is an integer is called an integral factor ; e.g. 8 is an integral factor of 56. 53. A number that is not the product of integral factors other than itself and 1 is a prime number; e.g. 2, 3, 5, 7, 11, and 13. 54. A number that is the product of integral factors other than itself and 1 is a composite number; e.g. 16, 24, 35, 1000. 55. A factor that is a prime number is a prime factor; e.g, 13 is a prime factor of 26. A number that is exactly divisible by 2 is an even number; e.g. 2, 4, 6, 8, 10. A number that is not exactly divisible by 2 is an odd number ; e.g. 1, 3, 5, 7, 11. Note. — In finding the factors of a number it is customary to consider only integral factors. 56. Oral 1. Give the factors of 21 ; 35 ; 49 ; 77 ; 26 ; 39 ; 34 ; 15 ; 91. 2. Name three factors of 30. 3. Name as many factors of 24 as you can. PRIME FACTORS 37 4. Of what numbers are 7, 2, and 13 the prime factors ? 5. Name four multiples of 9. 6. 132 i^ the product of 12 and what other factor? 7. Name all the prime numbers smaller than 50. 8. 84 is the product of three factors. Two of them are 2 and 6. What is the other ? 9. Of what number are 2, 3, 5, and 7 the prime factors? 10. Give the prime factors of 15; 25; 21; 33; 77; 30; 42; 51. 11. 5, 2, and what other number are the prime factors of 70 ? 12. Give two factors of 30 that are not prime. 13. What even number is prime ? 57. Rule for finding whether a Number is Prime or Composite. 1. If the given number is odd^ divide it by 3. 2. if 3 gives a remainder^ divide the given number by 5. 3. Continue this process, using each prime number in order as a divisor, until an exact divisor is found, or until the divisor equals or exceeds the quotient. If no exact divisor is found until the divisor used equals or exceeds the quotient, the number is prime. Otherwise it is composite. e.g. To find whether 71 is prime or composite, 3 )71 5 )71 7 )71 11)71 23 — 2 rem. 14 — 1 rem. 10 — 1 rem. 6 — 5 rem. Since the divisor 11, is greater than the quotient 6, and no exact divisor has been found, 71 must be prime. Even numbers need not be tested ; for no even number, ex- cept 2, is prime. Why ? 38 PRIME AND COMPOSITE NUMBERS 58. In finding the factors of a number, it is useful to remem- ber that a. A number is divisible by 2 if the figure in units' place is even. l. A number is divisible by 5 if the figure in units' place is or 5. 59. Find whether each of these numbers is prime or composite : 1. 143 5. 211 9. 121 13. 231 17. 437- 2. 123 6. 221 10. 97 14. 161 18. 401 3. 324 7. 119 11. 213 15. 87 19. 593 4. 163 8. 208 12. 215 16. 78 20. 395 60. Written 1. Find the prime factors of 7020. 2 7020 By what kind of numbers do we divide? Why? Which divisors do we use first? What besides the divisors is a prime factor? 3 . 5 . 13 Prime factors, Ans. 2 3510 3 3 1755 585 3 195 5 65 2. 13 2.3.3 2. I the prime factors of: 120 8. 45 14. 3381 20. 169 3. 42 9. 189 15. 667 21. 561 4. 6Q 10. 665 16. 310 22. 1001 5. 110 11. 429 17. 399 23. 1265 6. 105 12. 425 18. 1287 24. 682 7. 462 13. 414 19. 253 25. 729 CANCELLATION 39 CANCELLATION 61. Dividing both dividend and divisor by the same number affects the quotient how ? 462 _ ;2 X ^ X 7 X ^ _ ^ ^ . We might express this work as follows : dividing both divi- dend and divisor by 2, then by 3, then by 11 : 7 77 m ^ = 7 Quotient n 1 Taking out the same factor from both dividend and divisor is cancellation. 62. Solve hy cancellation: 1. Divide 86 X 27 X 49 X 38 X 50 by 70 x 18 x 15. 2. (28x38x48)^(14x19x24x2x2)=? 3. (26 X 5 X 54) - (13 X 5 X 6) = ? 4. What is the quotient of 36 x 48 x 16 divided by 27 x 24 X 8 ? 5. Divide 5 x 45 x 7 x 20 by 49 x 5 x 4 x 9. 6. Divide 5 X 51 X 7 X 9 X 4 by 17 X 20 X 12 X 7 X 2. 7. Divide 25 x 2 x 72 x 14 by 6 x 9 x 120. 8. How many bushels of potatoes at 50 cents a bushel must be given in exchange for 15 pounds of tea at 40 cents a pound ? 40 REVIEW AND PRACTICE «o r\ 7 REVIEW AND PRACTICE 63. Oral 1. Name the letters used in Roman notation and give the value of each. In finding the sums i and difEerei ures first, thus : 36 + 46= ? 36 + 40 = 76 76+ 6 = 82 .4ns. Say 36, 76, 82. 2. Mnd the sums ' 36 + 47 89 + 27 81 + 29 62 + 38 76 + 39 48 + 24 48 + 53 36 + 17 3. Find the differences : 28-19 41-14 31-13 62-28 43-16 97-58 93- -27^ = ? 93 - -20z = 73 73- -7 = 66 Ans. Say 93, 73, 66. 82 + 69 78 + 36 38 + 78 29 + 92 29 + 33 26 + 35 42 + 71 42 + 99 31-14 45-36 75-37 109 - 87 62-19 203 - 174 81-45 76-59 58-29 311-82 4. Crive products at sight: 403 X 10 86 X 200 86 X 100 15 X 40 22 X 10,000 18 X 300 14 X 20 12 X 6000 5. Find results: 27,000 + 13,000 345,000 ^ 100 218 - 38 6250 -f- 10 550x100 435-^10 19 x40 16 x500 200 xl90 403 x8 8324 -f-100 2800 ^400 1635- 4-200 REVIEW AND PRACTICE 41 6. Henry can row a boat 20 rods in a minute, and Eva can row 15 rods in a minute. If Eva is 60 rods ahead of Henry, in how many minutes can he overtake her ? 7. a. How many strokes of a force pump are required to fill -J of a tank that holds 200 gallons of water, if a pint is pumped at each stroke ? h. How long would it take at 20 strokes per minute ? 64. Written A Force Pump Find sums and test your work. Can you do it in four min- I T'V _ J 1 Jl -1 . utes ? Do not copy addends. a, 49 5. 235 c. 8749 d. 1346.25 392 419 3254 29.48 48 786 286 934.29 6759 592 39 98.65 24 839 458 813.78 864 496 3476 92.48 9837 318 239 9.62 481 745 8375 46.78 28 932 468 932.86 938 467 9628 ^8.93 2. Subtract and test ; a, 4352 h. 38290 c, 4001 d. 603040 1987 8199 102 13048 3. Divide and test: a. 153825 by 25. h. 49386 by 78. c. 12634 by 500. d, 983,700 by 1500. e. 863,426 by 19,000. /. 163,801 by 690. 4. 2, 3, 5, 7, 11, 13, and 17 are the prime factors of what number ? 42 LEAST COMMON MULTIPLE 5. What prime factor beside 19 and 11 has 8987? 6. Indicate the work and solve: a. Divide by 37 the result obtained by adding 111 to the product of 148 and 6090. h. A merchant bought 345 pounds of wool of one man, 3067 pounds of another, 468 pounds of another, and 384 pounds of another ; and sold ^ of it at 27 cents a pound. What did he receive for the part sold ? 7. Make and solve a problem that might be indicated thus : 110.00 - (1.35 4-12.20 + 16.19 + 1.18). 8. Solve by cancellation, (48 X 36 X 55 X 26) -^ (12 x 22 x 13). LEAST COMMON MULTIPLE 65. Oral 1. 3 X 4 = ? 12 is what of 3? Of 4 ? 2. 2 X 6 = ? 12 is what of 2? Of 6 ? 3. Name all the numbers of which 12 is a multiple. 4. Define multiple. 66. A number that exactly contains two or more numbers is a common multiple of those riumbers ; e.g. 12 is a common mul- tiple of 2, 3, 4, and 6. 36 is also a common multiple of 2, 3, 4, and 6. Can you name any other common multiple of 2, 3, 4, and 6? 67. The smallest number that exactly contains two or more num- bers is their least common multiple (L. C. M.); e.g. 18 is the least common multiple of 3, 6, and 9. 36 is a common multi- ple of 3, 6, and 9. Why is it not the least common multiple? LEAST COMMON MULTIPLE 43 68. Oral Find the L. C. M. of. 1. 2 and 3 2. 2, 3, and 4 3. 4 and 6 4. 9 and 6 5. 10 and 6 6. 8 and 6 7. 5, 3, and 2 8. 1, 2, 6, and 4 9. 2, 3, and 9 10. 5, 4, and 2 11. 7, 4, and 2 12. 10, 5, and 4 13. 2, 4, 8, and 12 14. 4, 5, and 12 15. 7 and 8 16. 16 and 32 17. 2, 3, 6, and 5 18. 4, 9, 3, and 12 69. When the least common multiple is a large number, the following direct method is employed in finding it. Let it be required to find the L. C. M. of 12, 15, and 18. 12 = 2 X 2 X 3 15 = 3 X 5 18 = 2 X 3 X 3 What kind of factors have we found ? A number, in order to contain 12, must have what prime factors? What prime factors must it have in order to contain 15 ? 18 ? A number that contains 12, 15, and 18 must have how many factors 2 ? How many factors 3 ? How many factors 5 ? What is the smallest number that has the factors 2, 2, 3, 3, and 5 ? What, then, is the L. C. M. of 12, 15, and 18 ? The prime factors may be easily found in this way: 2 112 15 18 ^ — fi — T^ Q ^^ what kind of numbers do we divide ? 3 5 3 2 X 3 ><; 2 ;x; 5 X 3 = 180 L. C. M. 44 GREATEST COMMON DIVISOR 70. Find the L. C. M.: 1. 18, 27, 30 8. 15, 60, 140, 210 15. 10, 15, 6, 14 2. 9, 12, 18 9. 24, 42, 54, 360 16. 48, 20, 21 3. 16, 48, 60 10. 25, 20, 35, 40 17. 9, 36, 45, 63, 42 4. 21, 27, 36 11. 14, 21, 35, 45 18. 25, 15, 30, 50 5. 36, 40, 48 12. 24, 48, 96, 192 19. 13, 19 6. 18, 24, 36 13. 15, 18, 20, 60 20. 2, 3, 4, 5, 6 7. 15, 30, 21, 28 14. 16, 24, 40 21. 7, 8, 9, 10 GREATEST COMMON DIVISOR 71. A number that will exactly divide two or more numhers is a common divisor of those numbers ; e.g, .5 is a common divisor of 30, 40, and 60. 72. The largest number that will exactly divide two or more numbers is their greatest common divisor (G. C.T),)\ e.g. 10 is the greatest common divisor of 30, 40, and 60. Note. — A common divisor is sometimes called a common factor^ and the greatest common divisor is sometimes called the highest common factor. 73. Numbers that have no common divisor are prime to each other ; e.g. 13 and 15. 74. Oral 1. Find the G-. 0. D. of: a. 6, 9, 12 e. 8, 24, 40 L 30, 45, 60 5. 10, 30, 35 /. 14, 28, 42 /. 18, 27, 36 c. 2, 10, 16 g. 33, 22, 77 h. 12, 24, 36, 48 d. 12, 30, 18 h. 21, 27, 39 I. 24, 32, 48 2. Name two numbers of which 7 is a common divisor. 3. Name three numbers of which 9 is a common divisor. 4. Name two numbers which are prime to each other. GREATEST COMMON DIVISOR 46 5. What is the greatest number that will exactly divide 12, 30, and 36 ? 6. Name two numbers of which 11 is the G. C. D. 7. Tell which of these pairs of numbers are prime to each other: a. 12 and 7 b. 16 and 20 c. 19 and 21 d. 8 and 15 75. Written 1. Find the greatest common divisor of 336, 504, and 924. 336 = ;2x;2x2x2x^x / 504 = ;2x;2x2 x^x3x7 924 = ;2 X ;2 X ? x 7 X 11 2 X 2 x 3 X 7 = 84 G. 0. D. Factoring the numbers and selecting the common prime factors, we find them to be 2, 2, 3, and 7. Since all of them are factors of each of the given numbers, their product, 84, is the greatest common divisor required. The common prime factors may easily be found in this way: 2 336 504 924 2 168 252 462 3 84 126 231 7 28 42 77 4 6 11 2 • 2 • 3 • 7 Common prime factors. Find the a. 0. D. : 2. 63,42 3. 90,105 4. 112, 168 5. 132,156 6. 40, 60, 80 8. 36, 48, 24 14. 63, 126, 189 9. 40, 56, 72 10. 18, 54, 32 11. 45, 60, 90 12. 36, 72, 81 15. 36, 81, 135 16. 91, 143, 156 17. 192, 400, 240 18. 168, 210, 308, 350 7. 64, 144, 560 13. 44, 121, 33 19. 1980, 945 46 REVIEW OF INTEGERS 20. Find the greatest number that will exactly divide 189, 378, and 504. 21. Find all the common prime factors of 360, 540, and 450. 22. Find the product of all the common prime factors of 108, 144 and 360. 23. Find a number that is prime to 210. 24. Name three numbers of which 11 is the greatest common divisor. REVIEW OF INTEGERS 76. Oral 1. (15-4)x(3 + 2) = ? 2. 15-4x3+2=? 3. Numerate 137,640,507,239. 4. Name the periods in the above number. 5. ReadDCXLIV. 6. What is the value of ^^? 7. 35 + 48 = ? (35 + 40 = 75; 75 + 8 = 83. Say 35, 75, 83.) In the same way add : a. 6S and 29; b. 58 and 15; e. 49 and 33; d. 67 and 24. 8. The product of two factors is 45. If one factor is 9, what is the other? If one factor is 15, what is the other? If one factor is 6, what is the other? Which of the factors given in your answer is not an integral factor? 9. The product of three factors is 108. Two of them are 4 and 3. What is the other? REVIEW OF INTEGERS • 47 10. Find tlie difference by subtracting the tens first : a. 64-25; 5.81-32; c. $.76 - $ .28; (^. 11.27 -1.79. 11. 24 X 20 = ? 30,700 -- 100 = ? 1200 -^ 200 = ? 22. 235 X 1000 = ? 208 X 10 = ? 4000 -5- 400 = ? 13. 1,500,000-^1500 = ? 14. Edward earned $3 one week and $6 the next. How much was left after he had spent | of it? 15. The change for 1.24 from 11.00 is |.06 + |.70=$.76. Say 6, 76. Find the change from 11.00 for: a. $.28 c. $.18 g. $.69 ^.$.52 2. $.79 A;. $.72 h. $.10 d. $.42 /. $.37 A. $.39 j. $.83 L $.35 16. a. One day is what part of a week? If a man pays $84 for a week's travelling expenses, they average how much per day ? h. At the same rate, what would they be for 11 days ? 17. What part of $5 is $21? If $5 will buy 20 splint baskets, how many such baskets will $ 2^ buy ? 18. How many times is $20 contained in $400? 19. Whatpart of $400is$20? 20. A cent is what part of a dime ? 7 cents are what part of a dime ? 21. If a dime will pay for 20 steel hooks, how many such hooks will 7 cents buy ? 22. 6 is what part of 12 ? 23. When a hardware merchant makes a profit of $1.44 on 12 window screens, what does he make on 6 of them ? 48 REVIEW OF INTEGERS 77. Written Test and time yourself on the first eight examples. 1. Add : 2. Add: 3. Subtract: 4. Subtract 287 627 807,204 230,007 965 438 99,197 150,008 473 796 287 . 69 342 109 5. 2059x78 218 246 781 627 6. 786 X 205 -f- 210 968 749 280 3578 7. 302 X (4780 - 3874) 421 372 642 7986 . 8. 346793-^5700 568 8144 9. A park in the shape of a rectangle, 135 rods long and 48 rods wide, contains how many square rods of land ? How many acres ? 10. A certain street is 6 rods wide. How long must it be to contain 77 acres of land ? 11. A farmer has 20 cows and feeds each of them two quarts of corn meal a day. How long will 100 bushels of corn meal last them ? (Solve by cancellation.) 12. A man owed his grocer f 135. He paid | of the debt in labor and the rest in cash. a. How much cash did he pay ? h. How many days did he work, if he received $2 a day ? 13. In the year 1905, 1,027,421 immigrants came to this country ; 317,000 of them settled in New York State, 222,300 in Pennsylvania, 20,000 west of the Mississippi River, and the rest FRACTIONS 49 in other parts of the country, a. How many settled in other parts of the country ? h. The entire number was how many times the number that settled west of the Mississippi ? REVIEW OF FRACTIONS (^Studied in the Primary Arithmetic) M, i I I I I I I I I I N. i 78. Oral 1. This figure shows that \ — ^-^. What else does it show ? 2. Make a figure to show that f = J. 3. Make a figure to show that f = ^. 4. Make a figure to show that |^ = f and ^ = f • 5 . What is the value of ^ ' ^1 H ^? f? J^: > 1? ¥? 6 . Express in lowest terms: I; f; A; 1 ' is> ri' A ' 2^' 7. * + i = ? 16. 1 + 1 = ? 25. f2- f=? 8. * + i + i = ? 17. i + i = ? 26. A + J = ? 9. f + j = ? 18. |-i = ? 27. tV + t=? 10. i-i = ? 19. f + f = ? 28. T>2 + i = ? 11. i + i = ? 20. *-J = ? 29. i- i = ? 12. i-i = ? 21. J + f = ? 30. i + 1 = ^ 13. i + i = ? 22. J + i = ? 31. i + i=? 14. i--i = ? 23. A + i = ? 32. 1- tV = ? 15. i + i = ? 24. l%-i = ? 33. f- i = ? 50 FRACTIONS 34. Harold bought a melon. He gave |- of it to Clarence and ^ of it to Howard. What part of the melon did he keep ? 35. A blotter is 1^ inches long and 3| inches wide. What is the sum of the two sides ? Of the two ends ? What is the perimeter ? Draw the blotter, full size. 3. 41Jj 4. 25^ 268|_ _8^ 7. 84| 8. 152f 91^ 801-1 79. Written Add: 1. 12i 13i 2. 58i 25j 28| S. 18J 6. 325| 5| 27f 41i Subtract and test your work: 9. 43f 10. 85V^ 29J 37| 13. 132f 14. 203 24A 161 11. 401f 12. 204-j^ 123J 131 15. 83 16. 401 If HA j5|_ 17. What must be added to 5J to make 16| ? 18. What must be taken from 18^^^ to leave 6|? 19. Laurence bought a pencil 6lf inches long and cut it into 3 pieces, two of which were 3| and 2^1 inches long. How long was the third piece? Prove the correctness of j^our answer by drawing a line 6^| inches long and cutting off pieces 3| and 2^^ inches long. 20. Indicate by signs the work required for example 19. FRACTIONS 61 FRACTIONS 80. One or more of the equal parts of a unit is a fraction; e.g. i' |! f ; *A- 81. A fraction is always an expression of division. For ex- ample, if 1 inch is divided into 8 equal parts, each part is -J of an inch. If a line 7 inches long is divided into 8 equal parts, one part is -J of an inch long. That is, 1 in. -v- 8 = J in., and 7 inches -i- 8 = |- inch. Take your rule and draw a line 1 inch long. Divide it into 4 equal parts. How long is one part? Draw a line 3 inches long. Divide it into 4 equal parts. Measure one of the parts. 3 inches -^ 4 = ? Draw a line 5 inches long. Divide it into 8 equal parts. Measure one of the parts. 5 inches -^-8 = ? 3h-7 = ? 9^11 = ? 82. The number above the line in a fraction is the numerator. It is always a dividend. In the fractions ^, ^, ^, |j, the numerators are 1, 7, 15, and 23. 83. 77ie number below the line in a fraction is the denominator. It is always a divisor. In the fractions J, -|, ^^^ -||, the de- nominators are 3, 9, 5, and 12. 84. The numerator and denominator are the terms of a frac- tion; e.g. the terms of -^j are 7 and 11. 85. The value of a fraction is the quotient obtained by dividing the numerator by the denominator. REDUCTION OF FRACTIONS 86. Changing the form of a number without changing its value 'ds reduction ; e.g. 8 pt. = 4 qt.; $7 = 700 ct. ; 7 ft. = 2^ yd.; ^^=3; i| = f;| = A- 52 KEDUCTION OF FRACTIONS REDUCTION TO LOWEST TERMS 87. A fraction is in its lowest terms when the numerator and denominator are prime to each other ; e.g, |^, f^^ ||^. 88. Oral 1. Dividing both dividend and divisor by the same number affects the quotient how ? 2. 1^ compares how with ^ ? 3. -^j compares how with | ? What did we do with the terms of ^^ to obtain ^ ? 4. Show by these circles that i=h t = |- x\ = h ^ = -1- A=l- iV = t- fo = h 5. How are these fractions reduced to lowest terms ? 6. Reduce to lowest terms : |; |; |; |; |; -^; |; f; -fji h f^; A; t'j; ^r' f^; ^; ^; A; if; if; if; if; if; A; IXL. _8_. 8 12' 12' 1?- 89. Written Reduce || to lowest terms. || = || = J 4^«. We divide both terms by 2 and then by 3. If we use the greatest common divisor (6), we shall need to divide only once, thus f f = |. Note. — We may often save time by remembering that an even number will never exactly divide an odd number. Can you tell why? REDUCTION OF FRACTIONS 63 22. ^1 23. IH 24. 3% 25. T% 26. m 27. m 28. m 90. Reduce to lowest terms : 1- If 8- l¥? 15- ^s¥j 3. M -10. 5^ 17. 11^ 4- M 11- ^J 18. ^ 5. If 12. Ill 19. ^ 6- M 13- iifs 20. iH '• A% "• SWV 21. ^% 29. Express in lowest terms 230 -f- 345. 30. Express in lowest terms 98 divided by 392. 31. Express in lowest terms 437 -4- 2484. 32. Express in lowest terms the quotient of 288 divided by 504. 33. What are the lowest terms of if J ? REDUCTION OF IMPROPER FRACTIONS TO INTEGERS OR MIXED NUMBERS 91. A fraction whose numerator is smaller than its denominator is a proper fraction; e.g. f, ^^p \^, The value of a proper fraction is always less than 1. 92. A fraction ivhose numerator equals or exceeds its denomi- nator is an improper fraction, e.g. |^, |, ^|. The value of an improper fraction compares how with 1 ? f 93. A numher that is composed of an integer and a fraction is a mixed number; e.g. 5|, lOi 201-j\. 54 REDUCTIOlSr OF IMPROPER FRACTIONS 94. Oral 1. A boy has two half dollars. That is the same as how many whole dollars ? Six half dollars equal how many whole dollars ? How do you find it ? 2. Eleven half dollars make how many dollars and how many halves over ? How do you find it ? Write it. 3. How many quarters make a dollar ? 4. How many dollars are there in 8 quarters ? 40 quarters ? 5. Fifteen quarters make how many dollars and how many quarters over ? Write it. What do you do to find it ? 6. I = how many whole ones ? | ? -^ ? | ? 7. 4=? | = ? Jj^ = ? -V^=? 8. A fraction is an expression of what operation? 9. How may we find the value of a fraction ? 10. Define the value of a fraction. Mnd the values of: 11 I 15. -\Q- 19. -2g5- 23. \3. 27. 2L^Sl 31. 1| 12. f 16. I 20. Aj^ 24. If 28. ^| 32. f| 13. f 17. J^ 21. i^ 25. ^9^ • 29. ^f 33. ff 14. I 18. ^^ 22. -^ 26. If^ 30. -J| 34. -L\^ 95. Written 1. 1|1 3. ^^ 5. i^^ 7. If^il 9. %4 2- W 4- -Vf 6. ^ 8. ^ 10. i||il 11. -fj*^ 14. ^f^ 17. S/ji 20. A|^A 12. il4|i 15. -V^a 18. i'jl^ 21. i|-fA 13. -^ 16. ^ 19. 2J^^ 22. -4111 REDUCTION OF INTEGERS AND MIXED NUMBERS 55 REDUCTION OF INTEGERS AND MIXED NUMBERS TO IMPROPER FRACTIONS 96. Oral 1. How many fourths in 1 circle? In 2 circles? In 3 circles ? In 4 circles ? 2. How many fourths in 4| circles ? In 2| circles ? In 3| circles ? 3. How many eighths in 1 circle ? In 3 circles ? In 2 circles ? In 4| circles ? In 2| circles ? 4. How do you reduce an integer or a mixed number to a fraction ? Reduce to improper fractions 5. IJ 9. 3| 13. 8f 17. 8J 6. 41 10. 2| 14. 4| 18. ^A 7. 3f IX. 4f 15. 5f 19. 8t% 8. 5i 12. 21 16. 64 20. 9i 97. TTn^fgw 1. Reduce 38J to a fraction. 38 = 38 X 9 ninths = 342 ninths. 342 ninths plus 7 ninths = 349 ninths. The work may be expressed thus: 38| = ^^ Ans. _9 342 7 349 56 LEAST COMMON DENOMINATOR Reduce to fractions : 2. 9A 9. 49i^ 16. 19t^ 23. 35^ 3. ^^ 10. 25^ 17. 29A 24. 191^ 4. 25f 11. 59A 18. 149f 25. 203 j8,- 5. 16^ 12. 67t^ 19. 1281 26. 98ii 6. 23^ 13. 89M 20. 137^ 27. 8711 7. 40i 14. 131| 21. 238iJ 28. 138j2^ 8. 3T^ 15. 270|| 22. 491^ 29. 351if LEAST COMMON DENOMINATOR 98. Fractions whose denominators are alike have a common denominator ; e.g. 60 is a common denominator of g^^, l|, and |J. 99. Fractions having the smallest possible common denomi- nator have their least common denominator; e.g. ^-^, ^^, ^. 100. Oral 1. We have found that when we add fractions having dif- ferent denominators, we must first change them to fractions having the same denominator. What shall we call that denominator ? 2. Since the common denominator must contain all the given denominators, it must be what of those denominators ? (A number that exactly contains two or more other numbers is what ?) 3. The least common denominator, then, must be which multiple of the given denominators ? 4. Reduce |, |, and J to fractions having the least common denominator. LEAST COMMON DENOMINATOR 67 How many 12ths in 1 ? (12 --- 4 = 3.) How many 12ths in | ? (^^^ = — . ) How many 12ths in 1 ? (12 ^ 6 = 2.) How many 12ths in f ? ('^^^ = —^ ^ ^ V6 X 2 12 ; How many 12ths in J ? (12 -^3 = 4.) How many 12ths in | ? f^^ = -^.^ ^ ^ V3x4 12; Change the following to fractions having the least common denominator : 5- hi 6- hi 12. f, I, f 13 6 1 6' 2 14. J.f'l 23. h h h i 15. f f ' f . T^ 24. h I \h i 16. J' !. f ^ 25. h tV' h J 17. A' ¥' i 26. f T^I- |. Jl 18. f . I'f ' f 27. f i^ h i 19. f.*'* 28. h f . i' ■h 20. f fA 29. h h h h tV 21. ,f ' h H 30. h h h 1^' tV 22. i'l^' A 31. i' ¥' 16' 32' 2 58 REVIEW AND PRACTICE 101. Written Change •:j^, j^^, Jf and ^^ to fractions having the least com- mon denominator. 7 8 16 17 2 10 15 33 30 3 5 15 33 15 5 5 5 11 5 1 1 11 1 2 X 3 X 5 X 11 = 330 L. C. M. 330 •^ 10 = 33 7 x33 231 10 X 33 330 330 -f-15=22 8 x22 176 15 X 22 330 330 -i- 33 = 10 16x10 160 33 X 10 330 330- - 30 = 11 17 X 11 187 == 30 X 11 330 131 116. 110 18X A7)fi 330' 330' 330' 330 ^^^' Change to fractions having the least common denominator: ^- h f' f 6. 1, 1, 1, J 11. i|, ^5_ II 3. I' 1% i 8. 1 |, I, ^ 13. 1|, ^5_, ^ ^' fit 9- I' 14' A 14. M'l^^'A 5. f , ^2 , if 10. f , ^5^, f , I 15. ^, ^^, J3, 38_ 102. Ora? REVIEW AND PRACTICE 1. What change should I receive out of $2 for a purchase of 1.50? 8.75? $.85? i.45? $1.25? 11.79? $.69? 2. Henry bought a top for 3 cents, some candy for 11 cents, and a pencil for 7 cents. What change should he receive from a quarter? 3. $240 will buy how many typewriters at $60 apiece? At $ 80 apiece ? 4. 8 cows at $ 40 a head cost how much ? 5. What is the cost of 2 bushels of potatoes at 20^ a peck ? REVIEW AND PRACTICE 59 6. 800 + 500 + 700 + 1500 = ? 7. Name the prime factors of 90. 8. Tell the value of -J/; 1^^; 2i; 3^^; J_2j)_(i. 9. Change to improper fractions 8| ; 2|; 17| ; 5^J; 241; 10. What is a fraction ? If you change | to eighths, how will its value be affected ? How will the number of parts be changed ? How will the size of the parts be changed ? 11. How does ^ compare with ^ ? Show this by a drawing. 12. Which is larger, 1 of an apple or ^ of an apple ? -^- or J? fori? 13. Which is greater, | or | ? 14. 29 pounds are how many times 5 pounds? Compare $250 with $50; 1 qt. with 1 pt.; 80^ with 20)?^. 15. Compare 2 cents with 50 cents ; 2 gal. with 3 gal. ; 8 lb. with 64 lb. ; $.25 with $1.50. 16. Find the cost of 24 souvenir cards at the rate of 3 for 5 cents. 17. A windmill turned 20 times a minute with a certain wind. The owner oiled the bearings of the mill and then it turned 24 times a minute with the same wind. a. How many turns per hour were gained by oiling the bearings ? h. How many times as much work did the mill do after oiling as before oiling ? c. What part as much work did the mill perform before oiling as after oiling ? 60 REVIEW AND PRACTICE 103. Written Land Surfaces in Square Miles New York . . . 47620 Rhode Island . . , 1080 Texas .... 262290 Pennsylvania . . . 44980 Nebraska . . . . 76840 Connecticut . . 4850 Delaware . . . 1960 Illinois .... 56000 California . . . 155980 Montana . . . 145310 Kentucky . . . 40000 Massachusetts 8040 New Jersey . . . 7450 New Hampshire . 9000 District of Columbia . . 60 Alaska . . (nearly) 570390 Note 1. — Water surfaces are not included in the above figures. Note 2. — While answering questions 1-5, keep your geography before you, open at the map of the United States. By referring to the map, estimate each answer before computing it, and then compare your estimate with the result obtained by computation. 1. a. Texas contains how many times as much land as New York ? h. It contains how many more square miles of land than New York ? 2. Alaska would make how many states the size of New Hampshire ? 3. Compare, by division, the land areas of: a. Alaska and Illinois. b. Illinois and New Hampshire. c. New Jersey and Pennsylvania. d. Rhode Island and Texas. e. Massachusetts and New York. /. Connecticut and California. ff, Montana and Delaware. h. Rhode Island and District of Columbia. 4. Compare, by subtraction, the land areas of: a. Nebraska and Pennsylvania. ADDITION OF FRACTIONS AND MIXED NUMBERS 61 h. Delaware and New Hampshire. c. Kentucky and Rhode Island. d. Illinois and Massachusetts. e. Texas and Alaska. 5. a. Find which of the columns of land surfaces (top of page 60) indicates the greater number of square miles. b. What is the difference ? Make other problems from the above table. 6. Find the prime factors of 1232. 7. Reduce 365J and 66| to improper fractions. 8. Reduce -^^^ to lowest terms. 9. How many 15ths are there in 39 ? 10. Find the value of ^OgO-; %2_4; J^_t 11. How many 40ths are there in 7| ? 12. 7 is equal to what fraction having 7 for a denominator ? 13. Reduce to lowest terms: «• }lf *• iff 0. Ill d. ii, e. Uf /• ^-h 9- HI 14. Reduce to fractions having the least common denominator : "• 9' IB' 10 ^- 1?' 21' 35 ^' 11' 3' 8 ADDITION OF FRACTIONS AND MIXED NUMBERS 104. A number is in its simplest form when it is in the form of an integer, or a proper fraction in its lowest terms, or a mixed number whose fractional part is in its lowest terms ; e .g. 18, | and 5^ are in their simplest forms ; ■^, |^, -*^ and 8| are not in their simplest forms. Why ? Answers should always be expressed in simplest form, unless the question requires a different form. 105. Oral Add: 1- hi 5. ii 2- hi 6. hi 3- h\ 7. hi 4- f.l 8. hi 62 ADDITION OF FRACTIONS AND MIXED NUMBERS 9. f,J,3 13. 1|,2^,J 10. i,|, 11 14. 3i,lf,5 11. 21 41 1 15. f, 8J, 4 12. f j.jj 16. ^,^,^ 17. A man paid I J for a book, f | for an inkstand, and f J for writing paper. How much did he spend ? 18. Mary had 8f ; her mother gave her $8 J. How much had she then? 19. The addends are 7f , 16 J, IQlf. What is the sum ? 20. Mary walked 5| miles on Monday, 4 miles on Tuesday, 5| miles on Wednesday, and as far during the next three days as during these days. How far did she walk in all ? 106. Written 9 8 6 3 4 1 f If = 2^ S^^- 2x2x3x3x4 = 144, L. C. D. Add lOf , 7f , and 6|. o ■" ? 0^ We add the whole numbers and frac- • t = • ^1^ tions separately, and then unite the 6f = sums. 2^^ = 2^^Sum, 17. 6f , 8|, f , i 18. 3, f , h 1 19. f . 4, f, li 20. 1' h iV t\ 21. 6f , 81 5|, 7| 22. 9i,5f9,J| 23. i f . % i 24. 6i, 7f , 91, 45 SUBTRACTION OF FRACTIONS AND MIXED NUMBERS 63 Add: 4. f, i, f 12. f, 1^, I, i 5. JL,-|,| 13. J,2|,|,6 6- h i I' ^ "• i' I' 2i, tV ^' f' A' 2 0' 2 ^^* 4 9' y 3' 2T 25. What is the sum of 14f 9|, IQi and 12lf ? 26. A man travels 25| miles on Monday, 37| miles on Tues- day, and on Wednesday as many miles as on Monday and Tues- day. How many miles does he travel in three days ? 27. A farmer has 27 J bushels of potatoes in one bin, 133| bushels in another, 47^^^ bushels in- another. How many bushels has he ? 28. How many yards of cloth will I have, if I buy 123| yards, 76| yards, and 58| yards ? 29. 2J yards of cloth are required for a coat, IJ yards for trousers, and | of a yard for a vest. How many yards are required for the whole suit ? SUBTRACTION OF FRACTIONS AND MIXED NUMBERS 107. 7. Oral 1. f-i 6. f-f 11. 12 -H 2. «-J 7. T^.-| 12. 22 -l| 3. l-i 8. il-f 13. 4-lJ 4. i-i 9. 7-J 14. ^-^ 5. T\-f 10. 8-1 15. 11 -8f 64 SUBTRACTION OF FRACTIONS AND MIXED NUMBERS 16. 19|-9f 20. 6-i 24. 121-51 "• i-i 21. 3i-i 25. 81 -4J 18. i-i 22. 15-21 26. il-l 19- i-i 23. 10-f 27. 1-1 108. Written From \^ take |. 1 = 1^ How is 45 obtained? ■Jl Difference From 29J take IS-,;^. How do we obtain |4? 15J|^ = 15^ Difference 1. Take 1 from f. 2. From Jf take ^p 3. Find the difference between l|^ and ^. 4. Take 91|- from 1781 5. H-i 13. U-i 21. 13,^-3^ 6. H-H 14. 81-51 22. 481^^-232,^ 7. 42f-33f- 15. 2101 -109f 23. 862|-46f 8. 198f-49f 16. 12i-5i 24. 2301-140^ 9. H-H 17. n-H 25. 891-431 10. 16f-8f 18. 461-271 26. 807-jV-298} 11. l-i 19. 1867f-976| 27. 190| - 28f 12. l-A 20. 321 _ 26f 28. 281^-371 ADDITION AND SUBTRACTION OF FRACTIONS 65 29. A piece of silk contains 18J yd. How many yards will be left after 13 J yd. are used ? 30. Mrs. Brown bought 4| yd. of broadcloth and used all but 1| yd. How much did she use ? 31. Find the difference between 256J and 149J. 32. A man bought a lot at auction for $92^ and sold it the next day for |105|. What did he gain ? ADDITION AND SUBTRACTION OF FRACTIONS 109. Written 1. From a piece of cloth containing 47|^ yd., 22| yd. were sold to one lady and 5| yd. to another. How many yards remained unsold ? 2. A farmer sold a load of hay for f 13^"^ and another for $16|. He was paid $25. How much was still due ? 3. A lady paid $1-^-^ for a pair of gloves, $S^ for an umbrella, and 1292'^Q- for dress materials. How much should she have left from four ten-dollar bills ? 4. What must be added to the sum of |- and 10| to make 20? 5. From the sum of 109| and 87| take their difference. 6. A grocer drew at one time 9|- gallons and at another time 15f gallons from a tank containing 44^g gallons of oil. How many gallons were left ? 7. Mary and Alice live on Bryant Avenue, and their school is on the same street, between their homes. Mary walks 40^2_. rods to school, and Alice 25J| rods. a. How much farther does Mary walk than Alice ? b. How far apart are their homes? 66 ADDITIOI^ AND SUBTRACTION OF FRACTIONS 8. It took Mr. Farmer S^^ hours to plow a field, and 13| hours to plant it. a. How much more time was required for planting than for plowing ? b. How much time was required for both ? 9. 132\-2^3_ + 7|._4i|^ ? Can ^ou find the result in two ways ? 10. a. i| + (li-J) = ? 5. 8i|-(li + J) = ? 11. Roscoe gave ^ of his new writing pad to his sister and ^ to his brother. What part did he keep ? • 12. From 16f + 12| + 51 take 18f + 6|. .; 13. Add |, |, ^^2' '^^^ A ' ^^^®^ subtract IJ from the sum. 14. After reading g^, f, and 1 of a book, what part have you yet to read ? 15. A owns 79^5g acres of land, B 9^^ acres less than A, and C 25^ J acres less than B. a. How many acres has B ? b. How many acres has C ? e. How many acres have all three together ? 16. From 4223^0- take 826|. 17. Take 8^2 ^^^m 17^^. 18. A has I641-. B has $37| less than A. How much money have both ? 19. 19^g yards of twine were cut from a ball containing 69^ yards. The piece that was left was how much longer than the piece cut off ? 20. Add |-, |, and ^J, and subtract the sum from 5. 21. Find the sum of |, ^, -f^^ and ■^\. 22. Find the difference between | and -^^y. 23. A boy walked to his grandfather's in three hours, walk- ing -^^ of the distance the first hour and ^ the second hour. What part of the distance did he walk the third hour ? MULTIPLICATION AND DIVISION COMBINED 67 QUICK TEST 110. 1. 39 cents is how much less than one dollar ? 2. 25x20 = 1000-^? 3. Which is greater, |- or J ? "^ 4. 3400 is the product of 100 and what other number ? 5. Express ||- in simplest form. 6. Express J as 28ths. 7. How far will a motor car run in 12 hours if it runs at the rate of 50 miles in 4 hours ? 8. What is the L. C. M. of 2, 3, 4, and 5 ? 9. What is the G. C. D. of 21, 35, and 49 ? 10. 16 is the sum of 10| and what other number? 111. MULTIPLICATION AND DIVISION COMBINED Written / (20^ 20-1- 4) X (21 - 7) = ? 4 = 5 21-^7 = 3 5x 3 = 15 Ans. or. fxf 20 x21 4x7 = 15 Ans. 20 and 21 are dividends and 4 and 7 are divisors. The result is the same whether we make each division separately and then multiply the quotients, or divide the product of the dividends by the product of the divisoi-s. In many cases the latter way is easier, because we may use cancellation ; e.g. 5 8 I a.(20 + 4)xC21-i-7) = (|x|)=?|^ = 15^«s.; ^7 42 b. (18 + 7) X (28 + 24) X (210 ^ l.S) = ^^-^|i2lM= 42 Ans. ^ 68 MULTIPLICATION OF FRACTIONS Find results : 1. (12-5-ll)x(22--5)x(35-*-6)x(15-f-2) 2. (20 -5- 6) X (55 -^ 10) X (42 -11) 3. (39 -^ 13) X (35 -5- 21) X (12^ 7) X (21 -J- 3) , 42 36 63 *• T^T^li 5. (27 -^ 18) X (35 -5- 75) x (25 -^ 12) x (12 - 7) 6. (68 -I- 7) X (14 - 8) X (35 ^17) 7. (52 -^ 10) X (34 H- 13) X (125 -h 10) 8. (26 -^ 20) X (68 -f- 13) X (125-^35) 9. (70 -f- 17) X (68 -5- 24) X (35 -i- 7) 7510898 49241720 42 ^ 26 ^ 15 ^^* 56 ^ 34 ^ 5 ^ 3 12. Multiply the quotient of 29 divided by 12 by the quo- tient of 84 divided by 29. MULTIPLICATION OF FRACTIONS 112. Any integer may be expressed as a fraction by writing it as a numerator with 1 for a denominator^ e.g. ; 5 is the same as -^ ; 19 is the same as -Ij^ ; |- x 7 x ||- is the same as fx^xM- 113. The word of, between fractions^ means the same as the sign of multiplication ; e.g. |of| = |x|; |of4x^ = fx^X^^^. 114. An indicated multiplication of two or more fractions is called a compound fraction ; e.g. f X | ; ^ X Jf X |f ; f of |. MULTIPLICATION OF FRACTIONS 69 115. Written 1. Find the product of f , f , and -^q. Each of these fractions indicates what operation ? Since all the numerators are dividends and all the denominators are divi- sors, we may find the result by dividing the product of the numerators by the product of the denominators, as in Article 111, using cancellation : 3 8 15 56 ■ Ans. Find the products . : 2. |X| 8. fof^off 14. fxTxif 3- Axf 9. ixi\xl- 15. 1 of 1 x 14 4. ioiJ^ 10. Axfxi 16. Axifx22 5. fofjf 11. AxJ^^i = ^=h r K Fig. 3 K ^ M ^fe W!^/< pJ WMy///M '■Wl/, i- m ¥.:.... „2^ Fig. 4 Fig. 5 Show as many other facts as you can by that figure. To THE Teacher. — Many exercises similar ixD the preceding may be given to interest children and make the topic real to them. We must re- member, however, that these are mere graphic verifications of the rule for multiplication of fractions. They neither prove nor derive the principle. The authority for every operation in fractions is found in the principles of division and the relation of dividend, divisor, and quotientc 72 MULTIPLICATION OF FRACTIONS 117. Oral 1. How much is 1^ of I of an inch ? 2. Illustrate that \oi ^ oi an apple is \ of an apple. 3. Multiply I by i; i by 1; 1 by 1; 1 by 1 4. Howmuch is^of f? foff? ^off? iof^=? 5. A man owned f of a farm and sold ^ of his share. What part of the farm did he sell ? 6. James had i|, and John | as much. How much had both? 7. If a pound of tea costs If, what will \ pound cost? 8. -J of |- of a square yard is what part of a square yard ? Show it by a drawing. 9. Frank gave Harry |- of his apple and Harry gave away J of his piece. What part of the apple did Harry give away ? 10. Mr. Greeley, having an acre of ground, took J of it for a garden. He planted J of the garden to potatoes and |- as much to corn. What part of an acre of corn did he have ? 118. Mixed numbers may he reduced to improper fractions and then multiplied ; tJiua^ I|x8jx^x4 = 2 6 ;I7 5 ^ 50 ,., , gX^x^xf=g- = 16t^«». Written 1. 4fx5J 4. 35|x27| 2. ll,\x7J 5. 2^xl^ 3. 177fx3 ^ 6. Find 3^ of f of 11 of 8^ MULTIPLICATION OF FRACTIONS 73 7. 4ix7| 9. -3-3^x25fx| 8. 5fx8fx^ 10. 3ix9fx6| 11. Multiply 101 by I by f by 6f. 12. Multiply : a. ISJ by 14f . h. 16 by ■^. 13. 5|x2fx20 15. 9|x^x2| 17. ^g x 4 x 51 14. 7ix5|xf 16. 6|x|iXi\ 18. -f^xS0x5^ 19. If a man earns f 2| a day, how much does he earn in 35 days? 20. Multiply f by I by J| by |l by ^\. 21. Find the cost of 16 bushels of oats at 37|^^ a bushel. 22. Mrs. A buys S^ qt. of milk a day. What does she pay for it at 5 J/ a quart? 23. Show by a diagram that 1 of ^ = J. 24. How far can Joe ride in 3| hours if he rides 9J miles an hour ? 25. How many square feet of floor are there in a room 12^ ft. by 71 ft? 26. P^ind the cost of 86 cords of wood at f 4| a cord. 27. Find the value of | of a chest of tea weighing 57 J lb. at' $f per pound. 28. «. I of f of f of f of 1= ? h. -j9^ of f of IJ^ of i\= ? 29. a. T\offfof.3-9^oflfoff = ? h. ioi^^oiiofil^? 30. Mr. Brown earns f 60| a month, and his son | as much. How much does the son earn ? 31. At 1121 a ton, how much will 9^\ tons of hay cost? 32. What will be the cost of 48| yards of cloth at $f a yard? 74 REVIEW AND PRACTICE 33. A man gave 124^5^. acres of land to his two sons, giving I of it to the elder and | to the younger. How many acres did each receive ? 34. If it requires 21| days for a man to dig a ditch, in what time can he dig | of it ? REVIEW AND PRACTICE 119. Oral 1. Read 305,027,503,060. 2. Read XLVI ; CXII ; CCIV ; XCIII. 3. If 2 sheep are worth $7, what are 8 such sheep worth? 4. If 12 books, worth f 8 apiece, will pay for a typewriter, how many books at f 6 apiece would pay for it ? 5. If I make a purchase for $9.15, what change should I receive for a 1 10 bill ? 6. Express in simplest form _6_. lA 10 ' 3 ¥; If; 6^; Y; 4.021 34' ¥ r^ 7. What is the area of the top of this table ? 8. 280^70 = ? 640-^40 = ? 9. Find the product of 32 and 20. 10. ?^|=| i+^ = ? |^? = f 11. Name two numbers that are prime to each other. 120. Written 1. Express in figures one hundred twenty-five million, ten thousand, seven. 2. Find the G. C. D. of 126, 210, and 294. 3. Find the L. C. M. of 720 and 216. DIVISION OF FRACTIONS 75 4. Divide the product of 144, 25, and 56 by the product of 48, 120, and 105, using cancellation. 5. When potatoes are worth 55/ a bushel, how many bushels must be given in exchange for 3 jars of butter, each containing 33 lb. at 25 / a pound ? Indicate the work, and solve by can- cellation. 6. When $150 will buy 189 bushels of wheat, how many bushels will i^50 buy? (|50 is what part of $150?) 7. Express in simplest form; a. |f| b. ^-^ c. |^^ d. 17^|f. 8. Change 18 to ninths. 9. Reduce |J^ to a fraction whose terms are prime to each other. 10. How many 40ths are there in 7|? 11. How many 99ths are there in 89|^? 12. A certain block in our city is \^ of a mile long and J| of a mile wide. What part of a square mile of land does it contain ? 13. Find the area of both sides of a square piece of cardboard whose edge is 15| inches. DIVISION OF FRACTIONS 121. Divide If by f. Since || is a product and | is one of its factors, we may state the question thus: §5^5 ^^^35^^x_? 72 8 ' 72 8 X? In order to find the required factor we must divide the numerator 35 by 5, and the denominator 72 by 8, thus: 35-|-5^7 72-8 9* 76 DIVISION OF FRACTIONS That is exactly what we should do if the question were: 7 72^5 ' 7;2 ^ 9* 9 The latter method is the more convenient, especially when the numerator of the divisor is not exactly contained in the numer- ator of the dividend or the denominator of the divisor in the denominator of the dividend. Therefore, to divide hy a fraction we interchange the terms qf the divisor and multiply. 122. Written 1. Divide 4| by 5|. 2 (How do we treat mixed numbers?) 2. Divide 47 by ^. Solution : 47 -^ GJ = ^ -^ -V- = ^f x ^^ = f | = 7^3 Ans. (How do we treat integers?) 3. fi-sV 9. HHi IS. 8^A 21. ^Hi 4. 1 ^i 10. H^A 16. 10-i-f 22. H^H S. iHf 11. ^+5| 17. 1^14 23. 8J^9f 6. A^l 12. ■^■^H 18. 11^8 24. n-^u 7. if^f 13. il^Si 19. 2i^6J 25. il^^ 8. il^T^ 14. 2+i 20. ^i^H 26. 10|-*-4if 27. By what must f| be multiplied to make f^-? 28. One factor of |^ is J J. What is the other? DIVISION OF FRACTIONS 77 29. How many pieces | of an inch long can be cut from a wire that is 10^ inches long? 30. When 3| lb. of beef steak are worth 57| cents, what is the value of one poUnd? 423. Division of fractions is sometimes indicated hy writing the dividend above and the divisor below a line. Such an expression is called a complex fraction; e.g, A, !_, M M and i^ 8|' 16' 25' 7f ^"^^ If -I are complex fractions.* Read each fraction. A fraction whose terms are integers is a simple fraction ; e.g, W is a simple fraction. 7 1. Reduce r-r to a simple fraction. 81" 3-1- 3~1^26~26* ^^*' _5 2. Reduce 45 to a simple fraction. 6=A^40 = Ax-l = J- Ans 40 17 n^'^jZ) 136* '^^'' 74 3. * Reduce ^^ to its simplest form. 5 2 4. li if «■'? 5. 18| 6 -8 78 DIVISION OF FRACTIONS In examples 4—13 change the given complex fractions to simple fractions by performing the indicated divisions : 8. i^ 10. M 12. i^ 9. i 11. 1! 13. i^L^ 14. If 1^ of an acre of land is worth f 72^, what is the value of an acre at the same rate ? 15. There are 5|^ yards in a rod. How many rods in 70|- yards ? 16. At I 6 J a ton, how many tons of coal can be bought for $73J? EXAMPLES FOR PRACTICE 124. 1. 2| X I -f- IJ = ? 2. Multiply l| by i| and divide the product by 1^. 3. a. 14f -*- 71 = ? 5. 71 -I- 14| = ? 4. Change to a simple fraction - — \c)\' 6. What is one third of one hundred seventy-five and one half? 7. The multiplicand is 1^^ and the product is 2^j. Find the multiplier. 8. Simplify |L±|. AVERAGES * 79 9. How many pounds of sugar at 6^ cents a pound will pay for 12 J dozen eggs at 16 cents a dozen ? 10. When 15 yards of silk cost $ 16J, what is the price per yard ? 11. Divide 75f by 14f . 12. Find the value of ^-li- -^2 16 13. In one month Mr. Finlay earned $ 46|, his wages being $ 2^ a day. How many days did he work ? 14. Divide f of | of 2| by | of f of 7. 15. $ 75 will pay for how much corn at f f a bushel ? 16. Divide the sum of 4| and 5| by their difference. 17. If J of a mile of telephone wire was divided into 14 equal pieces, how long was each piece ? 18. By what must 2^ be multiplied to obtain 2^ ? 19. By what must 3| be divided to obtain l^j ? 20. How many aprons can be made from 10|^ yards of cloth, if 1 J yards are enough for one apron ? 21. Divide 85f by 14f . AVERAGES 125. 1. Jacob weighed six of his chickens, and found their weights to be 68 oz., 40 oz., 63 oz., 47 oz., 55 oz., and 70 oz. What was the average weight of the chickens ? Solution 68 oz. 40 oz. 63 oz. 47 oz. 55 oz. 70 oz. 6)343 oz. Total weight. 57i oz. Average weighty Arts. 80 AVERAGES 2. Harry's marks in spelling for a month were as follows: 1st Week 2d Week 8d Week 4th Week Monday 80 75 78 80 Tuesday 86 90 84 88 Wednesday 83 82 90 92 Thursday 88 80 86 96 Friday 92 76 90 100 a^d. Find Harry's average for each week. e-i. Find his average for all the Mondays, all the Tues- days, etc. j. How much higher was his fourth week's average than his average for all the Mondays ? k. On which day of the week did he spell best ? I. What was Harry's general average for the month ? 3. Here are the standings of seven girls in three examina- tions : Marion Frances Dorothy Helen Jessie Hazel Ruth Arithmetic, 70 93 98 92 70 83 95 Geography, 93 90 76 98 90 84 80 Language, 95 88 95 79 96 85 76 a. Which girl has the highest average ? 5. What is the average of the class in language ? c. In arithmetic ? d. In geography ? e. Find the difference between Marion's average and Hazel's. /. Between Ruth's and Jessie's. g. Find the general average of the class. h. Find the difference between Helen's average and the av- erage of the class. 4. Our outdoor thermometer indicated the following tem- peratures for the mornings of last week : 76°, 82°, 80°, 67°, 60°, IDEAS OF PROPORTION 81 70°, 81°. The week before the record was 60°, 63°, 58°, 57°, 70°, 68°, 72°. For which week was the average temperature higher, and how much higher was it ? IDEAS OF PROPORTION 126. Oral 1. 2 is what part of 6 ? If 6 quarts of beans cost 45 cents, what will 2 quarts cost ? 2. 14 is how many times 2 ? What will 14 pears cost at the rate of 2 for 5 cents ? 3. 18 is how many times 3 ? If a boy is paid 20 cents for 3 hours' work, what should he receive for 18 hours' work ? 4. If a boy works 6 days to earn §4, how long should he worl^ to earn flO ? 5. What should I receive for 5 weeks' work when I earn $16 in 5 days ? 6. One gallon is how many times 1 quart? 10 gallons are how many times 10 quarts ? When 10 quarts of milk cost 60 cents, what should be paid for 10 gallons of milk ? 7. If 14 five-pound jars of butter will last a family a certain time, how many ten-pound jars would last the same time ? 127. Written 1. 3 is what part of 4 ? What should a man pay for three acres of land when 4 acres are worth i 189 ? 2. How many bushels of wheat can be raised on 42 acres of land when 159 bushels are raised on 7 acres ? 3. How many tons of hay can be bought for §3600 when 17 tons cost 1300? 82 REVIEW AND PRACTICE 4. How many 10-gallon cans may be filled from a tank of oil that will fill 155 two-gallon cans ? 5. Five is how many times three ? How far can a man walk in 5 days if he walks at the rate of 67 miles in 3 days ? 6. Find the amount of cloth needed for 36 suits when 17| yd. will make 3 suits. 7. How many 8-quart baskets of peaches would it take to equal in value 5346 bushel baskets of peaches ? REVIEW AND PRACTICE 128. Oral 1. What is the L. C. M. of 8 and 12 ? 2. Name two other common multiples of 8 and 12. 3. Which of these numbers are composite : 15, 13, 29, 36, 71, 83, 87, 91, 97, 99 ? 4. Name two numbers that are prime to each other. 5. Name a number that will exactly contain 13. 6. What is the smallest number that will exactly contain 3, 4, and 6 ? 7. 1 of J = ? 1 and 1 = ? 1 less J = ? ^ times \ = ? 8. From I take |-. 9. A farmer having 60^eep sold \ of them at one time and \ at another. How many had he left ? 10. A field of 3|- acres was planted to corn and potatoes. There were 1| acres of potatoes. How many acres of corn were there ? REVIEW AND PRACTICE 83 11. A lady went shopping with f 10. After spending i3| in one store and $5^ in another, how much money had she ? 12. Change ^| to a fraction whose terms are prime to each other. 13. Change 4J to an improper fraction. 14. Find the value of -2^-. 15. My mother paid 8 cents for one melon, 7 cents for another, and 10 cents for another. What was the average cost? 16. Show by means of a circle that l of ^ = 1. 129. Written 1. Draw a clock face, using Roman numerals. Let the hands indicate a quarter past 9. 2. Find the wages of 8 men for 5| days at f 3J a day. 3. 15 sheep at |2| apiece will pay for how many yards of cloth at 1 1 per yard? 4. A watch gains 1 J seconds every day. How many min- utes does it gain in the months of June and July? 5. There are 609 pupils in a school, j- of whom are girls. How many boys are there ? 6. Divide ^ by M. 7. A man spent | of his money and had $60 left. How much had he at first? 8. a. 15fxl6J = ? h. 16f-v-33i=? c. 5f-^li = ? 9. (2j4-f + 3)^| = ? 10. If 6^ bushels of rye cost $5|, what is the cost of 1 bushel? 84 REVIEW AiNTD PRACTlClii 11. Change to a simple fraction ^ ^ , 6| 12. |78| will buy how many barrels of flour at f 4| a barrel? 13. Find the value of 2l. 14. The area of a wall map is 976J square inches. Its length is 42|^ inches. Find its width, 15. An alley between two houses is | of a rod wide and 7| rods long. How many square rods of land does it contain ? 16. a. The bunches of bananas hanging in this fruit stand contain respectively 98, 124, 62, and 140 bananas. What is the average number of bananas per bunch ? b. The two smaller bunches are red, and sell at the rate of 2 for 5 cents. What are they all worth ? e. What will the others bring at 15 cents a dozen ? d. If the four bunches were bought at 11.09 per bunch, what will be the entire profit on the sales ? REVIEW AND PRACTICE 85 17. Mr. Scotese, the fruit dealer, bought 5 bushels of apples for $3.25. They average 136 apples per bushel. He sells them all at the rate of 4 for 5 cents. a. What is the cost per bushel ? h. What is received for a bushel ? c. What is the profit on 5 bushels ? 18. a. He sold a dollar's worth of pears at the rate of 3 for 5 cents. How many pears did he sell ? h. If he should put the pears in baskets, 12 in each basket, and offer them to you at 14 cents a basket, or at the rate of 4 for 5 cents, which offer would you take ? 19. a. He bought peaches at $1 a basket and sold them at 15 cents a quart. There were 10 quarts in a basket. What was his profit on 5 baskets of peaches ? h. If 9 of these peaches would make a quart, and they were sold at 2 cents apiece, what would be the profit on a basket of peaches ? 20. What was his profit on 50 baskets of Delaware grapes bought at 12|^ a basket and sold at 18^ a basket? 21. He bought chestnuts at $4.00 a bushel and sold them at 20 cents a quart. What did he gain on a quart ? 22. Spanish shelled peanuts cost him $9.80 a sack, each sack containing 140 pounds. He sold them at 12 cents a pound. What was the profit on a sack ? 23. He bought 50 watermelons at 23 cents apiece and sold them at 40 cents each. What was his profit ? 24. He paid f 8 a hundred for cantaloupes and sold them for 12 cents apiece. How much did he gain on a hundred? 25. John earned $ 6| one week, 1 8f the second, and $ 7^ the third. What were his average earnings per week? 26. How many pounds of sugar, at b^ cents a pound, are equal in value to 6J dozen eggs, at 15 cents a dozen ? 86 ALIQUOT PARTS 27. Divide If of -^^ by 3f times l^j. 28. Divide f of f of ^ x 3^ by f of if of f 29. Solve in the easiest way : a. If 6 acres of land cost $ 438, what will 42 acres cost ? h. How many loads of earth can be bought for $80 when $400 will buy 1135 loads? 30. -^ is how many times ^^2 ALIQUOT PARTS 130. One of the equal parts of a number is an aliquot part of that number; e.g. 8 oz. is an aliquot part of 16 oz. because 8 oz. is ^ of 16 oz. ; 16 J cents is an aliquot part of 100 cents because 16| cents = J of 100 cents. Find the number of cents in If; I J; I J; ||; $1; ||; The answers you have given are all what kind of parts of a dollar? Prove the correctness of the following table : PARTS OF A DOLLAR 5 cents = 1 2V ^H cents = ||- 61 cents = $ Jg . 37^ cents = $| 8J cents = lyV ^^ cents = |l 10 cents = $^^ 62J cents = if 12 J^ cents = -I J 66| cents = If 16f cents = i^ 75 cents = $f 25 cents = $ J 87^ cents = $f Which column in the table gives aliquot parts ? This table should be committed to memory like the multiplication table, because its use will shorten many problems, e.g. 33 books, at $ .16| each, will cost 33 x $^ = $ 5^. i ALIQUOT PARTS 87 131. Oral Multiply : 1. 121- cents by 16 , 7. 37J cents by 8 2. 16f cents by 12 8. 50 cents by 15 3. 25 cents by 20 9. 62^ cents by 8 4. 33^ cents by 27 10. 66f cents by 9 5. 6i cents by 16 11. 75 cents by 4 6. 81 cents by 24 12. 87J cents by 8 13. What is the cost of : 16 pounds of bacon at 12| ^ a pound ? 16 balls at 50^ each? 36 yards of ribbon at 331^ a yard ? 36 pounds of candy at 25 j^ a pound? 8 pounds of tea at 62| ^ a pound ? 14. When 4 geographies cost $ 3, what is the cost of one ? Of 9? Of 11? Of 15? (There is an easier way to find the cost of 20 geographies. What is that way?) 15. At $.75 apiece, what must be paid for 3 chairs ? 4 chairs? 6 chairs ? 16 chairs ? 40 chairs ? 132. Written What is the cost of : 166 pounds of pork at 121 cents a pound ? 348 pounds of veal at 16| cents a pound ? 265 boxes of strawberries at 25 cents a box ? 1215 yards of flannel at 33|- cents a yard ? 3580 pounds of honey at 20 cents a pound ? 748 pounds of tea at 50 cents a pound ? 88 DECIMALS Oral 1. At 25^ a pound, how many pounds of butter can be bought for ^8 ? (How many pounds can be bought forfl? For $8?) Divide : 2. $3 by 331/ 5. $9 by 121^ 8. $6 by 331/ 3. 15 by 25/ 6. II by 6^ 9. |4 by 121/ 4. 12 by 81/ 7. 110 by 50/ lo. $2 by 25/ 11. $3 divided by 81/ =? 12. At 25 cents apiece, how many hats can be bought forf 6? 13. At 25 cents a pound, how many pounds of cheese can be bought for 15? 14. At 16| cents a dozen, how many dozen eggs can be bought for 14? 15. How many pounds of beef can be bought for $4 at 16|/ a pound ? 16. At 33 J / a yard, how many yards of linen can be bought for 1 10? 17. How many penknives can be bought for 16 at 33|^ cents iece ? 18. 24 X 121/ = ? 19. f 24 ^ 121/ = ? apiece ? DECIMALS 133. Each removal of a figure one place to the right affects its value how ? We have used this principle thus far in dealing with in- tegers only ; but it holds true also for numbers smaller than one. Thus, DECIMALS 5000. .■ 500. = 5000. J- 10 50. = 500. -^10 Moving 5 to the right . 5. = 50. .5 = 5, hlO -10 = ^ \: .05 = .5 - -10=ifo .005 = .05 H -l^ = ToV .0005= .005 H -10 = Tol( 734. = 7340. -- 10 Moving all the figures 73.4 = 734. -^10 = 73^^ to the right 7.34 = 73.4 -f- 10 =7^3^ .734= 7.34-: L0 = 3^,V 89 Notice that we place the decimal point (.) at the right of units' place. This shows where the integer ends and the frac- tion begins. The places at the right of the decimal point are called decimal places and are named much like those at the left, thus : •♦-• w (A 4>' Vt JZ T3 C c« w 3 o 3 o M C "O ■o C ■o f rt ? (A a> rt a> ■!-• vt I- JZ 1. (O ■C 1- 3 ■a (A (O ■D 3 +< T5 o O c C +i c C O c C := J= 3 3 — H X f- 3 h I h \- X S 4638-754966 .7 is read seven tenths. .75 is read seventh-five hundredths. .754 is read seven hundred fifty -four thousandths. . 7549 is read seven thousand five hundred forty-nine ten-thou- sandths. 90 DECIMALS Oral 1. Read .8; .49; .786; .4923; .56249; .38T654. 2. Read 500; 50; 5; .5; .05; .005; .0005; .00005. 3. Read .25; .36; .39; .47; .365; .3; .7; .403; .07; .009. 4. How many cents make one dollar ? What part of a dollar is one cent ? 7 cents ? 19 cents ? 41 cents ? 97 cents ? 5. We write 25 cents, $.25, because it is 25 hundredths of a dollar. One dime is 1 tenth of a dollar. How would you write it ? Write on the blackboard ; 5 dimes ; 7 dimes ; 8 dimes ; 3 dimes. 6. 4321.648 is read 4321 and 648 thousandths. Notice that and is used between the integer and the fraction. Read: 6.42; 17.5; 4.23; 583.97; .640; 7.640; 439.018; 9341.215; 68.43; 70.9; 893.047; 903.03; 642.008; 95.249. Read: 3.4; .0034; .987; 2048.017; .00315; 200.02. Write in words on the blackboard : 35.08 ; 6.5 ; .084 ; .6082; 235.235; 64.105; 308.02; 56.081; 30.130. 134. The product of equal factors is a power : e.g. 4 is a power of 2 because 2x2 =4 8 is a power of 2 because 2x2x2 =8 81 is a power of 3 because 3x3x3x3 = 81 100 is a power of 10 because 10 x 10 = 100 Name 3 other powers of 10. 135. A fraction whose denominator is 10 or a power of 10 is a decimal fraction ; e.g. -f^, -^^-q, .25, .3421. Only decimal fractions can be expressed by use of the decimal point as in the DECIMAL FRACTIONS 91 last exercise. When a decimal fraction is thus expressed, how may we tell what the denominator is ? Name some decimal fractions not given here. 136. A number that is composed of an integer and a decimal fraction is called a mixed decimal ; e.g. 2.5 ; 130.35 ; 21.007. 137. A fraction that is expressed hy writing the numerator above ayid the denominator below a line is a common fraction; e.g. h I' 9' if' I'V- 138. Change to the decimal form: 1- i¥(y 6. fi^^ 11. ^^ 16. A 2- t¥o '• m 12. ^JjV "• iV 3- A% 8- h 13. ,^ 18. 5^^ *• A 9- lb "• T^% 19. 8J^ 5- 'Uo ' 10- 1^7 15- 1% 20. 64^Vo^ Change to common fractions and read: 21. .36 26. .485 31. 5.6 22. .7 27. .016 32. 5.06 23. .125 28. .16 33. 5.006 24. 12.2 29. .06 34. 5.600 25. 6.25 30. .6 35. 5.060 Write^ first as common fractions^ or mixed numbers, then as decimals : 36. Four tenths. 37. Seventy-five hundredths. 38. One hundred twenty-five thousandths. 92 . WRITING DECIMALS 39. Sixteen, and forty-eight hundredths. 40. Twelve, and four tenths. 41. Six tenths. 42. Six hundredths. 43. Six thousandths. 44. How many decimal figures are required to express thousandths ? Hundredths ? Tenths ? 45. Read the numerators only in examples 36 to 43. Write the following as decimals^ and read the numerator and denominator of each : 46. Two hundred eighty-two thousandths. 47. Fifty-six hundredths. 48. Seven tenths. 49. Six hundred thousandths. 139. Oral 1. What part of 10 units is 1 unit ? 2. What part of 1 ten is 1 unit ? 3. What part of 2 hundreds is 2 tens ? 4 In the number 555, what is the value of the first 5 at the right ? The second 5 ? The third 5 ? 5. Upon what does the value of any figure depend ? 6. In the number 555, the value of tho first 5 is what part of the value of the second 5 ? 7. -f^ is what part of 2 units ? 8. In the number 5.5, the value of the right-hand 5 is what part of the value of the left-hand 5 ? READING AND WRITING DECIMALS 98 9. In the decimal .555, what is the value of the first 5 at the right ? The second 5 ? The third 5 ? 10. Name .225; .3478; the denominator of .6 ; .17 .06; .049; .207; .3007. ; .105; .006; .05; 11. Read the numbers in question 10. 12. Read: a. .368 h. .894 j. 37.005 Tc. 25.2036 s, .421 ^^. .691 c. .5328 I. 38.000006 u. .81 d. .2053 m. .498869 i;. .637f e. 25.623 n, 4.9836 «^. .4378^ /. 7.0063 0, 49.836 X. .809Jj- g. 28.3005 h. .28962 i. 15.605 j9. 498.36 g. .000400 r. .0004 y. .430^ z. .6842J TTn'^e deciinally: 1. Eight tenths. 2. 29 hundredths. 3. Sixteen, and 284 thousandths. 4. 4584 ten-thousandths. 5. Twenty -five hundredths. 6. Twenty-five thousandths. 7. Twenty-five ten-thousandths. 8. Twenty-five hundred-thousandths. 9. Twenty-five million ths. 10. 1650, and 464 thousandths. 94 WRITING DECIMALS 11. One thousand one, and 36 hundred- thousandths. 12. Sixteen, and six thousandths. 13. Seven hundred eighty-four millionths. 14. Twelve hundred-thousandths. 15. Seventy-five ten-thousandths. 16. Seven hundred five thousandths. 17. Seven hundred, and five thousandths. 18. Four thousand three ten-thousandths. 19. Four thousand, and three ten-thousandths. 20. Twenty-four, and five hundred-thousandths. 21. Seventy-one, and seven hundred-thousandths. 22. Four hundred thirty-five, and four thousandths. 23. Eight thousand three hundred forty-one ten-thousandths. 24. Ninety-nine, and eighty-six thousandths. 25. Seventy-eight, and four thousandths. 26. Nine thousand seven, and two hundred seven" ten-thou- sandths. 27. One, and one hundred thousandths. 28. One thousand one, and one hundred one thousandths. 29. Ten, and ten ten-thousandths. 30. One hundred, and one hundred ten thousandths. 31. One hundred one, and one hundred ten-thousandths. 32. One thousand one ten-thousandths. 33. One thousand, and one ten-thousandth. 34. Two hundred seven thousand, and two hundred seven thousandths. 35. Six, and six thousand ten-thousandths. ADDITION AND SUBTRACTION OF DECIMALS 95 ADDITION AND SUBTRACTION OF DECIMALS 140. Since decimal figures increase in value from right to left, like the figures in whole numbers, we may add and subtract decimals as we add and subtract whole numbers, taking care to write them so that the decimal points are all in a column, thus : 4.375 391.42 .35 165.70316 28.3065 225.71684 Difference 351.294 384.3255 Sum The vacant places in the addends and in the minuend are treated as if they were occupied by ciphers. Add: i 1. 2.2 2. 3.25 3. 4.5 4. .004 34.5 7.163 .168 4.1 79.89 15.0032 2.12 16.1563 5. .175+1.75 + 17.5 + 175. 6. 145 + 14.5 + 1.45 + .145 + .0145. 7. 3.2 + 14.0063 + .006 + 25.384 + .1. 8. .8 + .446 + 59.3 + 2.575 + 1.0056 + .3. 9. 1.45 + 2.365 + 96 + .96 + 15.863 + 4.3 + .0004. 10. 446 + 44. 6 + 37562 + 9 + .8 + . 321 + .16. 11. 21.0005 + . 3842 + . 1 + .005 + 3.6 + .158. 12. 1.0006 + 2001.1 + .003 + ^.b + 11.1111. 13. 205.07 + 301.2 + 687.9124 + 83.045 + 200. 14. .308 + 308. + 8.09 + 9.0786 + .859. 15. 2378. + 23.50 + .890 + .089 + 1.0886. 96 ADDITION AND SUBTRACTION OF DECIMALS .41 . 1. Subtract : a. 24.3 h. 2.86 C. 4. d. 2.46 4.5 1.325 1.15 .005 2. 7 -.15 7. 29.325-15.14 3. 1-.004 8. 3.852 -.125 4. 13-2.1 9. 1.1111 -.0011 5. 3.256-1.05 10. 500- .05 6. 256.1-1.256 11. 25.3894-15.005 - • 12. From twenty-eight, and twenty-five thousandths take fourteen, and twenty-five hundredths. 13. From one tenth take one thousandth. 14. a. Which is the greater, fiJty thousandths or five hundredths ? h. Three tenths or three hundred thousandths ? 15. Take one thousandth from one thousand. 16. From 5 hundred take 5 hundredths. 142. Find results : 1. 175 - 30.23. 2. .015 + 1.05 + .57 + 5.7 + 1.04 + .0045 + 75.36. 3. 50.4 - .504. 4. 25.006 + 200.00008 + 6.00005 -f 49.005 + 300.059. 5. 2.005 4- 5.5 4- 25.010 - 3.2045. 6. Find the sum of two hundred forty, and four hundred fifty thousandths ; thirty-four, and three hundredths ; six hun- dred four, and six hundred four ten-thousandths ; fifty, and five tenths. 7. A boy had two balls of kite string. One contained 145.3025 yards and the other 84.3502 yards. He made a kite string 200.02 yards long. How much string had he left ? DIVIDING BY MtlLTIl^LES Oi" t:EN 97 8. Subtract: a. 32.854 c. 86.2 e. 21.101 9.378 43.948 7. b. 19.042 d, 36.015 /. 28.78 16.854 . 24.008 21.987 9. There are four villages on the same road. From the first to the second is 8.46 miles; from the second to the third, 10.5 miles; from the first to the fourth, 25 miles. Make a picture of the road and find the distance from the third to the fourth village. MULTIPLYING AND DIVIDING BY MULTIPLES OF TEN 143. Oral 1. How can we multiply decimals by 10 ? By 100 ? By 1000 ? 2. Multiple/ by 10 : 5.25; .06; 3.7; 593.207; 6.800; 9.16; 82; 420; .035; .0061. 3. Multiply by 100: 61.843; 3.215; 75.16; 3.18; .65; 2.3; 5; 520. 4. Multiply .0612 by 10 ; by 100 ; by 1000 ; by 10,000 ; by 100,000 ; by 1,000,000. 5. By what must .0503 be multiplied to obtain .503? 503. ? 50.3? 5030.? 6. Moving the decimal point one place to the left is the same as moving all the figures of the number one place to the right. For example, moving the decimal point one place to the left in the number 42.3, it becomes 4.23. How does this affect the value of the number ? 98 MULTIPLYING BY MULTIPLES OF TEN" 7. What, then, is the easiest way to divide a decimal by 10 ? 8. Divide by 10 : 35.; 247.; 385.; 16.; 24.3; 2.59; 347.69; 8.137; 42.69; 394.68; .725; .042. 9. How may we divide decimals by 100 ? By 1000 ? By 10,000 ? 10. Divide hy 100 : 3567.8; 937.; 635.25; 42304.; 687.96; 485.03. 11. Divide hy 1000 : 986; 5321; 63,485; 983.7; 4284.25. 12. Divide hy 10,000 : 389,076; 42,831; 68,379.5; 425. 13. By what must 8193 be divided to obtain 819.3 ? 8.193 ? 81.93? .8193? .08193? .008193? 14. How must the decimal point be moved to change tenths to hundredths ? Thousandths to tenths ? Thousandths to millionths ? 15: Divide as follows : 64.2 by 10; 83.75 by 10; 63.59 by 100; 4251 by 10,000; 33 by 1000; 5 by 100. 16. What is the effect of moving the decimal point three places to the left ? One place ? Two places ? Four places ? Six places ? 17. Find results : 3.5x10; 6.5-^10; 83-^1000; .987-^10,000; .8432 x 100,000. 18. What is the effect of moving the decimal point to the left one place ? To the right two places ? To the left four places ? To the right six places ? 19. How may we multiply a decimal by 10,000 ? Divide a decimal by 1000 ? Multiply a decimal by 100,000? MULTIPLICATIO^r OF DECIMALS MULTIPLICATION OF DECIMALS 144. Multiply 6.41 by 3.2. 6.41 = 641 -- 100 3.2 = 32-5-10 6.41 X 3.2 = 641 X 32 + 100 -*- 10 641 6.41 32 3.2 1282 1282 — -_^ 1923 1923 \4 20512 = 641 X 32 20.512 = 641_x 32 -^ 100 -$- 10 How did we divide by 100 and by 10 ? From this we see that to multiply decimals we multiply the factors as whole numbers and point off in the product as many decimal places as there are in both factors ; e,g, 2.8 1.25 .005 25 8 .6 m jM 22.4 .750 .00015 1.60 Find the products : 1. .18 X .15 8. 13.3 X 1.3 2. 1.0005 X .2 9. 100 X .01 3. 2.5 X .06 10: 100.56 x .0005 4. m X .005 11. 25.32 X 1.05 5. .005 X 1.6 12. 2.84 X .25 6. 25.05 X 1.15 13. 3.28 x 1.125 7. 2.863 X 100 14. 1.111 X 1000 100 DIVISION OF DECIMALS 145. Oral 1. One of the factors has two decimal places and the other has five. How many decimal places has the product ? 2. When there are five decimal places in one factor and one in the other, how many are there in the product ? 3. When there are eight decimal places in the product and five in one factor, how many are there in the other factor ? When there are six in the product and four in one factor? Five in the product and three in one factor ? DIVISION OF DECIMALS 146. Divide 10.96516 by 4.67. 2.348 Quotient 4.67)10.96-516 934 We first divide as in whole numbers. Since the -— — • dividend is a product and the divisor one of its fac- tors, the other factor, or quotient, contains as many 1401 decimal places as the number of decimal places in 2241 ^l*® dividend, less the number of decimal places in the divisor. Mistakes may be avoided by observing the fol- lowing 1868 3736 3736 RULE FOK PLACING THE DECIMAL POINT When the divisor is an integer^ place the decimal point in the quotient directly over the decimal point in the dividend (or under in short division^. When the divisor contains decimal places^ make a dot on a line with the tops of the figures as many places at the right of the deci- mal point in the dividend as there are decimal places in the divisor. Place the decimal point in the quotient directly above this dot (or below in short division). MULTIPLICATION AND DIVISION OF DECIMALS 101 Note 1. If there is a remainder after all the figures of the dividend have been used, we annex ciphers to the dividend, and continue the division until there is no remainder, or until a sufficient number of decimal places have been obtained in the quotient. Note 2. When the dividend contains fewer decimal places than the divisor, we annex ciphers to the dividend until it has as many decimal places as the divisor. Written Find the quotients and test 1. 60.8-^-1.6 2. .00075^.05 3. 25. 50 -f-. 34 4. 15.2 -3.04 5. .27560-^265 6. 90.978-4-3.54 7. 38.4444-4-177 8. 14.4 -4-. 0018 9. 1.127-4-4.9 10. .9156-4-12 11. 315.432 -.48 12. 1.5906 -4- .6 13. 375 -4- .125 . 14. 125 -4- .125 15. 1000^.001 16. .001-4-1000 17. 25-^.25 18. .25-4-25 hi/ multiplication : 19. 18.65-4-100 20. 266.4^.036 21. 2.107 -4-. 35 22. .100854-4-3.879 23. 125874.^.486 24. 9801. -4-. 99 25. 2976.^4.96 26. 164. 32 --.208 27. 347.76^.368 28. .0006478 -4-. 079 29. 2.5826-5-69.8 30. 98.07^.210 31. 20.852 -4- .52 32. .0023322 - .0026 33. 676.8 -4- .08 34. 1273.998 -4-. 199 35. 357.6-4-2.98 36. .75897-4-810 102 MULTIPLICATION AND DIVISION OF DECIMALS 147. Find the products and test hy division : 1. 3.2x3.6 16. 7.0001 X. 0603 2. 86 X .09 17. 43.55 x .06 3. 9.8 X. 005 18. 2354 X. 008 4. .039x57 19. 39.04x2.08 5. .00356x6.8 20. 6.80x86732 6. 6394. X .029 21. 2400. x .387 7. 864. X .278 22. .0406 x 3080 8. .00967x240 23. 920x63.7 9. 839.42 X .0015 24. .992 x 3001 10. 208.7x30.9 25. 2460 X. 0039 11. 930x6.80 26. 1234. x .56 12. 4203 X .0076 27. 9.924 x .0106 13. 69 X. 00035 28. .0204x20.40 14. 406 X. 000039 29. 78.08 x .025 15. 7.92x1.002 30. .8060x300 148. Oral .7 = -j"^, or 7 divided by 10. .305 = %%, or 305 divided by 1000. .581 =^ , or 58 J divided by 100. In like manner tell the meanings of the following decimals : 1. .8 6. .89 J 11. .029^ 16. .05 J 2. .416 7. .6f 12. .007^ 17. .034 3. .21 8. .39f 13. .103f 18. .0165 4. .3879 9. .48^ 14. .2134f 19. .00017J 5. .200 10. .873^ 15. .4070^ ao. .OOOf CHANGING DECIMALS TO FRACTIONS 103 CHANGING DECIMALS TO COMMON FRACTIONS OR MIXED NUMBERS 149. .072 = 10 00 — ilr - 18.25 = 18^^0 = = 18i Reduce to common fractions or mixed numbers in simplest form : .1. .8 8. .125 15. 16.75 2. .25 9. .875 16. .00125 3. .35 10. .375 17. .054 4. .75 11. .455 18. .0250 5. .64 12. .025 19. .01375 6. .52 13. .561 20. .342 7. .38 14. .368 12 Solution : .342 = 34f ^ 100 = 2^ X yi- =i? Ans. 21. .12J 31. 2.331 5 41. .166f 22. .621 32. 'H 42. .19^ 23. .06} 33. 42.621 43. .5621 24. .18i 34. 97.087J 44. 400.40f- 25. .03| 35. 56.131 45. 361.41-1 26. .25f 36. 158.06} 46. 2042. If 27. .871 37. 409. 6| 47. 79.00f 28. .66f 38. •OTA 48. 308.00|| 29. .361 39. .261 49. 2890.901 30. 16.25 40. .012f 50. 98.000|f 104 KEDUCTION OF FRACTIONS TO DECIMALS . 150. Oral Using aliquot parts^ reduce the following to mixed numbers: 1. 158.50 6. 16.05 11. 15.081 2. 139.331 7. I32.12J 12. 422.331 3. |5.16| 8. $19.10 13. 100.14|- 4. 17.25 9. II8.I42 14. 603.16| 5. 119.20 10. 16.02 15. 99.02 151. At sights reduce the following to common fractions or mixed numbers : 1. .621 4. 12.875 7. 29.871 10. 43.4 2. .66| 5. 13.02 8. 3.6 11. 12.371 3. .75 6. 430.625 9. 7.8 12. 54.375 REDUCTION OF COMMON FRACTIONS AND MIXED NUMBERS TO DECIMALS 152. Written Change -{^ to a decimal. .3125 16)5.0000 48 "20 16 ^g= 5 -5-16 = . 3125 Ans, . 40 Write 9j-^g as a mixed decimal. 32 "so 80 REDUCTION OF FRACTIONS TO DECIMALS 105 Ret iuce ifo decimals .* 1. t 11. H 21. *l 31. 19,^ 2. 1 12. A 22. If 32. 72T 3. A 13. i 23. M 33. llr 4. i 14. li 24. H 34. 12^fT 5. A 15. 3iV 25. i¥ff 35. 14f| 6. 1 16. f 26. i¥ff 36. StIt 7. 1 17. ^ 27. 12A 37. ISiV 8. 3\ 18. ^r . 28. H 38. 2AV 9. iV 19. II 29. If 39. 5A 10. If 20. 2#j 30. ^2V 40. 3^ 153. ^ fraction in -lowest terms whose denomiyiator contains other prime factors than 2 arid 5 cannot be reduced to an exact entire decimal; e.g. |, f, if, j\, If, ^\. Such a fraction may be reduced to a decimal of nearly the same value by carrying the division to a certain number of deci- mal places, thus : Reduce ^| to a decimal of four places. ,U01^^Ans. 26.)19.0000 182 — 18=_9_ •'^307 is almost equalto ||. ^g 2 6 13 rp^g gxact value of if is .TSOT^V 200 182 "Is The result may be expressed, .7307-1- 106 REDUCTION* OF FRACTIONS TO DECIMALS 154. Written Reduce to decimals of three places: 1- 1 7. if 13. 8A 19. 42ii 2. ^ 8. A 14. 231 20. til 3. ^\ 9. if 15. 681t\ 21. 2¥^ *• f 10. W 16. Hy 22. 43A 5. f 11. 2^^ 17. II 23. 16A 6- A 12. 151 18. M 24. in A COMMON FRACTION AT THE END OF A DECIMAL 155. .21 = .2+ a ofi^, or 2^. or , .05). .2 + .05 = .25 In a similar manner, we may show that, .271 = .275, .3841 = .3845, etc. Also, that .21 = .225, .341 = .3425, etc. Also, that .8| = .875, .06f = .0675, etc. Also, that ,^ = .9125, .07| = .07375, etc. Oral Express as entire decimals: 1. a. '^ J. $.17J c. .3601 d. 71 e. .0041 2. a. 'H 5. 3.7^ c, 19. 20 J (^. i.39} 6. .145^ 3. a. .05f h. l.lf c. $21.46f d, .033f e. .090f 4. a. •7i 6. 6.81 c. 80. 3 J (^. 12f e. 1.9i 5. a. ^2^ h, 3. 97 J c. 150f (?. 24.01 e. 29.0| 6. a. 1.30f J. 2.45| j = 4% PERCENTAGE 151 Answer the following questions, using common fractions instead of decimals when it is easier to do so. 6. Find: k. 6 J % of 16 days I. 60 % of 50 ft. m. 371% of 64 n. 87| % of 96 0. 83 J % of 18 trees p. 121% of 600 q, 331% of 60 years r. 4% of 150 «. 81 % of 3600 people t. 25 % of 836 miles a. 331% of 12 5. 10% of 200 c. 16f% of 30 da. d. . 81 % of 144 sq. in. e, 75 % of 28 gal. /. 66|%of 27cu. ft. g, 20% off h. 40% of 1100 z. 90% of $200 y. 62 J % of 24)2^ 7. 9 is 25 % of what number ? 8. 32 is 50 % of what number ? 9. 75 is 10 % of what number ? 10. 4 is what per cent of 16 ? (^ = what per cent ?) 11. 6 is what per cent of 9 ? (f = | == what per cent ?) 12. 1 is what per cent of 16 ? 13. 3 months are what per cent of 18 months ? 14. 3 books are what per cent of 15 books ? 15. 17 are what per cent of 1 84 ? 16. 9 men are 75 % of how many men ? 17. 10 min. are 16|% of what? 18. 2 is what per cent of 40 ? 19. 80% of 35^ is what? 20. 20 % of my money is 16 ^. How much money have I ? 21. 8 cents are 66| % of what ? 152 PERCENTAGE 22. Frank is 15 years old. Julia's age is 16| % of Frank's. How old is Julia? 23. Will is 8 years old and Ethel is 7. Elhel's age is what per cent of Will's? 24. I gained f 7 in selling my watch. If that was 12 J % of the cost, what did it cost ? 25. Out of 25 words, Charlie missed one. What per cent of the words did Charlie miss ? 26. Edith had 9 examples right. If there were 10 in the lesson, what per cent of them did she have right ? 27. — ■ = how many hundredths ? 100 100 ^ 28. 100 % — 90 % = what per cent ? 29. If I spend 90 % of my money, what per cent do I have left ? 30. Having spent 90 % of my money, I had % 2 left. How much money had I at first ? 31. A grocer sold 90 % of a barrel of sugar and had 35 lb. left. How many pounds did the barrel contain at first ? 32. Hubert gave away 75 % of his apples and had 6 left. How many had he at first ? 33. Harry took a silver dollar to the store and bought 2 lb. of cheese at 18 ^ a pound. On the way home he lost 12J % of the change. How much change did he lose ? 34. A boy sold 66f % of his chickens and kept 20. What per cent of them did he keep ? How many had he at first ? How many did he sell ? 36. Sarah answered correctly 80 % of the questions that, came to her and missed one. How many questions came to her ? How many did she answer correctly ? PERCENTAGE 153 36. Alfred attended school 98 % of the days of the term. If he was absent 2 days, how many days were there in the term ?. 37. A boy was sick and stayed out of school 5 days in one month. If there were 20 days of school in that month what per cent of the time did he attend school ? 38. 62 1 % of a cask of vinegar leaked out. If there were 15 gallons left, how many gallons did the cask hold ? 39. .20 + .10 = how many hundredths? 40. 20 % + 10 % = how many per cent ? 41. A butcher bought 200 lb. of beef. He sold 20 % of it to one man and 10 % of it to another. What per cent of the beef did he sell ? What per cent was left ? How many pounds were left ? 42. What per cent of this oblong is -4.? J5? O? m If A is 50 sq. in., what is Bl (7? i>? El What per cent of the oblong are A^ jB, and Q together ? 43. In a shipwreck, \ of the crew were lost. What per cent were saved ? If 80 men were saved, how many were lost ? 44. Alice, having read 75 % of a book, has 50 pages yet to read. How many pages does the book contain ? 45. A man has traveled 60 % of the distance from New York to Chicago and has 400 miles yet to travel. What is the whole distance ? 46. When 25 days of the month of November are past, what per cent of the month is yet to come ? 47. If your schoolroom is 40 feet long and 30 feet wide, its width is what per cent of its length ? Its length is what per cent of its width ? A B C D E 154 PERCEKTAGE 186. Written 1. An army of 19,000 men went to the front. 12| % of them were killed in battle, and 25 % of them died of wounds and sickness. How many were left ? 2. A collector for a newspaper started out with bills amounting to ^840. He collected $156. a. What per cent of the bills did he collect ? b. What per cent did he fail to collect ? 3. 35 % of the apples in an orchard were unfit for market and could not be sold. If 1300 bushels were sold, what was the entire yield ? 4. 39 % of the 4700 blossoms on a cherry tree were blasted and the rest became fruit. How many cherries did the tree bear ? 5. 98 % of the men in a certain city can read and write. If there are 1398 men who cannot read and write, how many men are there in the city ? 6. 23 % of the men in a certain city work in factories. If there are 9200 men who work in factories, how many men are there in the city ? 7. 45 % of a jeweler's goods were stolen, a. If he had i 16,500 worth of goods left, what was the entire stock of goods worth ? h. How many dollars' worth were stolen ? 8. In an orchard of 3600 trees, 25 % were pear trees, 15 % peach trees, 10 % plum trees, and the rest apple trees. How many apple trees were there ? 9. My gas bill for one month was f 1.80. Five per cent of it was deducted for prompt payment. What was saved by pay- ing promptly ? 10. A merchant bought a piece of cloth for f 65 and sold it for 130 % of its cost. What did he receive for it ? PERCENTAGE 155 11. This load of hay weighs 2200 lb., the wagon 1200 lb., and the team 2600 lb. a. The weight of the hay is what per cent of the entire weight ? h. The weight of the wagon is what per cent of the entire weight? c. The weight of the team is what per cent of the entire weight ? d. How many tons must the bridge support ? 12. a. In 1905 the Chicago baseball team won 92 games and lost 60. What per cent of all the games played did they win ? h. The Boston team won 78 and lost 74. What per cent of all the games did they lose ? c. The New York team played 149 games and won 47^9^"^ % of them. How many games did they lose ? 13. By selling paper at 150 % of its cost, a stationer receives 90 cents a package for it. What is the cost of a package of this paper ? 14. Twelve pounds of seed for a lawn contained 21 lb. of white clover seed. What per cent of the mixture was white clover seed ? 156 A FRACTION IN THE MULTIPLICAND 15. a. If a pine plank weighs 45 lb. and an oak plank of the same size weighs 72 lb., the weight of the pine is what per cent of the weight of the oak? h. The weight of the oak wood is what per cent of the weight of the pine ? 16. A grocer bought 100 lb. of soda for $3.50 and sold it for 142|^ % of its cost. What did he receive a pound for it ? A FRACTION IN THE MULTIPLICAND 187. In multiplying a large mixed number by an integer, time may often be saved by multiplying the whole number and the fraction separately, then adding the products, thus : 314| X 5 = ? 3141 5 3| = I X 5 1570 = 314 X 5 1573f = 3141 X 5 Multiply: 1. 248f by 5 6. m^-^ by 8 li. 224J^ by 9 2. 39jl- by 3 7. 35f by 7 12. 42^^ by 12 3. 42^ by 6 8. 49^^ by 13 13. 65|-| by 16 4. 8501 by 15 9. 207fby3 14. 201ff by 16 5. 292 by 11 10. 38^gbyl6 15. 431|J by 72 188. In division, if either dividend or divisor contains a common fraction that cannot be easily reduced to a decimal, it is sometimes helpful to multiply both dividend and divisor by the denominator of the fraction, thus making both dividend and divisor integers, or simple decimals ; e.g. : A FRACTION IN THE DIVISOR 167 .05f 148.74 Multiplying both dividend and divisor by 7, 2814. Quotient .37 1041.18* (Multiplying the dividend and 74 divisor by the same number 301 affects the quotient how ?) 296 51 37 148 148 >. 1. 2.295 -^ .05| 6. .3125 -*- .02^ 2. 10.44 -J- .04| 7. 787.2 ^ .931- 3. 36 -^ .13J 8. 1.024 -^ .005 J 4. 96.9 ^ .15| 9. 218.24 -^ 1.13| 5. .0256-1- .071 10. 385.35 -h 5.2f 11. Find the number, of which : a. 72 is 5{ % /• 701.4 is 4| % h. 10.5 is 11^% 9- 284.4 is 1051 % c. 24.64 is 391% h. 5.775 is 116f % d, 12.834 is 14f% i. .3155 is 901 cj^ e. 1263 is 171% h .833 is 108y\ % 12. Mr. Fitch rained 14.60 in se illinof a waffon. This was 6| % of its cost. What was the cost of the wagon ? 13. The average attendance in a certain school was 640 pupils. If this was 91 f % of the number registered in the school, how many were registered ? 158 REVIEW AND PRACTICE REVIEW AND PRACTICE 189. Oral 1. What number is composed of 5 units, 7 tens, and 3 thousands ? 2. Read XLIV; CCLXII; DCXCI; MC IVIII; CDLIV. 3. G-ive results rapidly^ adding or subtracting the tens* figures first: 36 + 45; 29 + 32; 57 + 76; 93+28; 93 -27; 84-45; 72 + 39. 4. Grive quickly the number of: a. Quarts in 98 pt. j. Feet in 2 rd. b. Pecks in 28 bu. k. Dollars in 36,000 cents. c. Hours in a week. I. Gills in a gallon. d. Seconds in 1 hour. m. Days in two common years. e. Inches in 2 yd. n. Tons in 1600 lb. /. Square inches in 2 sq. ft. o. Square rods in 10 A. g. Square yards in 450 sq. ft. p. Yards in 10 rd. h. Cubic feet in 2 cu. yd. q. Days in 14 wk. i. Dimes in |15. r. Days in a summer. «. Cubic inches in a box 5 in. by 2 in. by 1 in. 5. A half dollar, a quarter, 2 dimes, and a nickel are how many cents ? 6. i + i + A = ? 10. 15-J = ? 7. 3x8x? = 48 * 11. 18- If = ? 8. 8x9 = 6 X? 12. 5| + 13f=? 9. 88h-? = 8 13. 7|-f=:? 14. When 36 men can earn a sum of money in 15 da., how long will it take 12 men at the same wages to earn the same amount ? 9 men ? 6 men ? 72 men ? REVIEW AND PRACTICE 159 15. If 6 men earn 8 dollars in a certain time, how many men can earn $ 16 in the same time at the same wages ? $ 32 ? 164? 14? 16. If 10 men earn $ 200 in 8 days, how many dollars will twice as many men earn in that time at the same rate ? ' 17. fof8bu.= ? 21. 36 is y9^ of what? 18. I of? = 14. 22. iof^f = ? 19. 27 is -fj of what ? 23. f of what= 15 qt. ? 20. What part of 18 is 15? 24. | of what = ^y? 25. What are the prime factors of 84? 26. Name two numbers that are prime to 12. 27. How many 42ds are there in |? 28. What is the least number that exactly contains 6, 15, and 20? 29. Name three numbers of which 7 is an exact divisor. 30. Give two composite numbers that are prime to each other. 31. How may we tell whether a number is prime or not ? 32. What is the greatest number that will exactly divide 26 and 39 ? 33. What is the smallest number that 6, 8, 12, and 16 will divide ? 34. Crive results at sight : a. 362 -- 10 /. 14 x 200 k. .06 of 500 h. 4900-1000 ^. 99-f-.l I. lof = 7| e. 29 X .01 h. .224 T = lb. m. 5% of 25 cZ. 834-10,000 z. 23,400-200 n. 121% of = 7 e, 29x1000 y. .12x50 o. ^V = % 35. What per cent of a ton is 400 lb. ? 36. Compare .84 x 25 with 84 x .25. 160 REVIEW AND PRACTICE 37. (16 + 4) X (43 -23) = ? 38. What must we do with |, |, and | to find out which is greatest ? 39. Numerate 23516.00562. 40. Compare the values of |, |^|, and i^. 41. Find the cost of: a. 64 lb. of pork at 12 J ^ h, 600 boxes of berries at 8| ^ c. 96 gal. of molasses at 50^ d. 16 doz. oranges at 37 J ^ e. 54 yd. of matting at 33 J ^ /. 48 1b. butter at 25^ g. 16 knives at 62 J ^ h. 2 doz. sleds at 87|^ apiece ^. 80 lb. rice at 6^^ 42. How many bushels of beets will §12 buy at 33^^ a bushel? 43. f 16 will rent a boat for how many hours at 16^ an hour ? 44. What change will be left from a f 20 bill after paying for 12 hours' labor at 37^ ^ an hour ? 45. At 16|^ a dozen, how many ears of corn will $2 buy ? 46. ^ of ^ lb. = oz. 47. What is the area of a rectangle 5^ inches by 4 inches ? 48. What is the cost of a dozen tomato plants at the rate of 4 for 5 cents? 49. At 10^ a dozen, how many sheets of sandpaper will 5^ buy? REVIEW AND PRACTICE 161 50. A man spent 20 % of his salary for rent, 10 % for cloth- ing, 5 % for fuel and light, 25 % for food, and 20 % for other things. What per cent of his money did he spend ? What per cent did he save ? If he saved |400 a year, what was his salary ? 51. What is 120% of 5 miles ? 190. Written 1. Write in figures forty-two thousand, and two hundred five ten-thousandths. 2. 209 X 87,000 0,6%^). 3. 235,404 -^ 468 (fesQ- 4. 23,945 -^ 160 {te%t^, 5. 302,050 - 92,059. 6. 17 hr. 35 min. = how many minutes ? 7. 639,800-- 700 (te%t), 8. 4320 square yards = how many square rods ? 9. When 60 bu. of oats grow on an acre of ground, how many bushels grow on a square rod ? 10. The highest ten batting averages in the National Base- ball League in a certain year were .377, .363, .356, .328, .317, .316, .315, .311, .308, .304. What was the average of all these ? 11. Using cancellation, divide 48 x 54 x 200 by 18 x 108 x 25. 12. How many pieces of sheeting, 39 yd. in a piece, worth 12^ a yard, would pay for 52 hours' work for 36 men at 32 ^ an hour ? 162 REVIEW AND PRACTICE 13. Find the L. C. M. of 23, 37, 32, 36, and 56. 14. Find the G. C. D. of 48, 60, and 78. 15. Change to lowest terms: 16. Reduce to simplest form : a. Iff. h. 82fA. c. ^i-. d. ^^. e. ^. /. ^^. ff- H^- 17. Change to improper fractions : a. 399|J. b. 48|J. c. 18^. d. 5&^\. 18. Change : a. 7| to 24ths. b. U|| to a fraction whose terms are prime to each other, c. |, |, and -^^ to fractions having the least common denominator. 19. A wagon which cost $52 J was sold for f 46|. a. What was the loss ? h. What per cent of the cost was lost ? 20. A salesman cut 19| yd. of cloth from a piece containing 38|^ yd. How many yards remained ? 21. The remainder is 632|^ and the minuend 965|. What is the subtrahend ? 22. a. After John had spent J of his money for a book, ^ of it for a knife, and ^ of it for oranges, what per cent of it was left ? b. If he had $.36 left, how much had he at first ? 23. What can 6 men earn in 4J wk. at $12 J a week ? 24. ^x8xifx6|x|f=? 25. A man sold a horse for $160 and a cow for | as much. What did he receive for both ? 26. Find the area and the perimeter of a rectangle 12^ inches by 8|^ inches. 27. Find I of f of -^-q. REVIEW AND PRACTICE 163 28. A man sold | of his farm and had 90 acres left. How many acres did the farm contain ? 29. Simplify^. 3 30. A man put in the bank f 252, which was f of what he received for a wood lot. What was the selling price of the wood lot ? • 31. A merchant lost | of his money and has §123.50 left. How much had he at first ? 32. Simplify ¥^. i of 2i 33. The sum of two fractions is -^||. One of them is -Jf . What is the other ? 34. Add ten and five thousandths, three and seven tenths, forty-seven millionths, five hundred five thousandths. 35. Find the sum of eight and thirty-five thousandths, seventeen and fifty-three thousandths, fifty and fifty-four millionths, five hundred two and nine ten-thousandths. 36. Find the sum of 6.06; 70.50; 6.0765; .00365; 101.09; 28.56741; 50.005. 37. Express in figures and add : 25 thousandths, 12 hun- dredths, 26 ten-thousandths, 8 hundred-thousandths, 7 mil- lionths, 2375 hundred-thousandths. 38. Add seventeen thousandths, eighteen ten-thousandths, sixty-four millionths, fifteen ten-millionths, five hundred two hundred-thousandths, and from the sum subtract eighty-four hundred-thousandths. 39. a. .0375 is how much greater than ^^0^? 6. 12.5-9.0025 = ? 164 REVIEW AND PRACTICE 40. Reduce to common fractions or mixed numbers: a. .00125. h. .0875. c, 3.625. d. 4.032. e. .83J-. 41. Multiply and test your work : a. .00375 by 400 /. 34.05 by | 5. 5.275 by 5000 g. .000568 by 1.07 c, 5.64 by .006 h, 4.32 by .15 d, 35.005 by .008 i, '.0316 by .58 e, 350.5 by 8.04 j. .375 by 2.05 h. Four and twenty thousandths by twenty-six and nine tenths. 42. How much a ton do I pay for coal when .375 of a ton costs me 81.875? 43. At what rate per hour is a launch running when it goes 132.3 miles in 13.5 hours ? . 44. A farmer buys groceries and sells farm produce to the grocer as follows : Groceries Farm Produce 10 gal. oil at 2^f 25 lb. cheese at 18 f 50 lb. sugar at 5|^ 40 bu. potatoes at 58/ 2 boxes soap at |3,25 J T. hay at $12 3 dozen oranges at 40 >^ 20 doz. eggs at 25/ 5 gal. molasses at 40/ 7 bu. pears at $1.50 20 lb. coffee at 35 / 72 lb. smoked ham at f .13 In whose favor is the balance of this account, and how much is due him ? 45. 10% of a man's income was paid for rent. If his rent was il5 a month, what was his income per year? 46. After using 70 % of his month's wages, Jerry had 114.40 left. What were his month's wages ? REVIEW AND PRACTICE 165 47. a. A schoolroom 30 ft. square and 12 ft. high contains how many cubic feet of air ? h. If there are 30 pupils in the room, how many cubic feet of air are there for each pupil ? 48. a. Supposing a cubic foot of ice to weigh 62 J lb., what is the weight of a pile of ice 12' by 10' by 8' ? 5. How many three-ton loads would it make ? c. If 17| % of the ice melts in handling, how many pounds can customers receive from this pile of ice ? 49. Amos sells vegetables for Mr. Robbins, the gardener, and is allowed to keep 12| % of all the money he takes in. a. If he earned 13.41 in a week, what was the amount of his sales for that week? h. How much did Mr. Robbins receive ? 50. I bought 75% of a carload of sugar and sold |^ of my share. What per cent of the carload did I sell ? 51. 97% of the pupils of a certain school are present. If 21 are absent, how many pupils belong to the school ? 52. Three days are what per cent of a week ? 53. A piano was sold for 1700. This was 140% of what it cost the dealer, a. How much did the dealer pay? h. How much did he gain ? 54. H. J. Howe, jewelry merchant, sold to Mrs. James R. Hazzard \ doz. silver table-spoons at $30 a dozen, one dozen silver table-forks at $25 a dozen, one tea urn $15.50, one kitchen clock, $2.00. Who is the debtor? The creditor ? Make out the bill and receipt it. 55. What is the amount of my bill for 4J lb. of mutton steak at 16^, 1^ lb. of tea at 48^, 5 lb. coffee at 34^, and two lb. raisins at 15^? 166 DENOMINATE NUMBERS DENOMINATE NUMBERS 191. A number that is composed of units of weight or measure is a denominate number; e,g, 10 doz., 215 cu. in., 2 gal. 3 qt. 1 pt. 192. The name of a unit of weight or measure is a denomina- tion ; e.g, ounce, square foot, minute. 193. A denominate number that is expressed in two or more denominations is a compound number; e.g. 1 yd. 2 ft. 7 in.; 2 lb. 14 oz. 194. TABLE OF LIQUID MEASURE 4 gills (gi.) = 1 pint (pt.). 2 pints = 1 quart (qt.). 4 quarts = 1 gallon (gal.). Oil, vinegar, molasses, and other liquids are shipped in barrels or casks of various sizes. But for the purpose of indicating the capacities of vats, tanks, reservoirs, etc., 31 J gallons are called a, barrel (bbl.) and 63 gallons a hogshead (hhd.). 195. Oral 1. 5 gal. = pt. 2. 1 hhd. = bbl. 3. What will 48 pt. of cream cost at f 1.20 per gallon? 4. 1 bbl. is what per cent of 1 hhd.? 5» How many pints in 10 gal.? 6. At 4 ^ a pint, what is the cost of 6 qt. of milk? 7. 4 gal. 2 qt. 1 pt. = pt. DRY MEASURE 167 8. A tank contains 10 bbl. of oil. How many gallon cans will it fill? 9. 63 qt. is what part of a hogshead ? 10. A gallon contains 231 cu. in. How many cubic inches are there in J of a gallon ? In ^ of a gallon ? In -jl^ of a gallon? In 1 qt.? 11. How many gallons and quarts are there in 50 quarts ? 12. A cistern that holds 10 hhd. of water holds how many barrels? How many gallons? 13. One pint is what per cent of one gallon? 14. 10 % of a barrel is how many gallons ? 15. If 1 qt. of sirup can be made from 20 oz. of maple sugar, how many ounces will make a gallon of sirup ? 16. 33i % of a hogshead is how many gallons ? 196. TABLE OF DRY MEASURE 2 pints (pt.) = 1 quart (qt.). 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.). 197. Oral 1. 64 qt. = pk. 2. 1 bu. = qt. 3. If 2 qt. of cherries fill a jar, how many jars will 2 bu. fill ? 4. 1 pk. is what per cent of a bushel? 5. 1 qt. is what per cent of a peck? 6. Elsie, Nina, and Robert gathered 4 bu. of chestnuts and sold them for 10^ a quart. How much did they receive ? 7. 10 bu. = pt. 168 AVOIRDUPOIS WEIGHT 8. ^hu. +^ pk. = qt. 9. 8 qt. = what part of 2 bu. ? 10. What is gained on a bushel of hickory nuts bought for $2 and sold at 10^ a quart ? 11. A barrel of potatoes containing 2| bushels will sell for how much at 20^ a peck? 12. 3 bu. and 3 pk. of apples at $1 a bushel cost how much? 13. If a bushel of oats weighs 32 lb., what is the weight of 3} pk. ? 14. How many bushels of apples at 25 ^ a peck can be bought for 120? 15. A bushel of corn and a peck of wheat are ground to- gether. What per cent of the mixture is corn? What per cent is wheat ? 16. 37 J 9^ of a bushel is how many quarts ? 198, TABLE OF AVOIRDUPOIS WEIGHT 16 ounces (oz.) = 1 pound (lb.). 2000 pounds = 1 ton (T.). 2240 pounds = 1 long ton. 100 pounds =1 hundredweight (cwt.). The term hundredweight is used less than formerly, although its value (100 lb.) is still taken as a unit in quoting freight rates and prices of various articles, when the quantity used makes this a convenient unit of weight. The long ton is used in wholesaling certain mining products. The ton of 2000 lb. is sometimes called a short ton. LINEAR MEASURE 169 199. Oral 1. How many ounces are there in 1 ton ? 2. At 48^ a pound, what must be paid for 4 oz. of tea ? 3. 1 % of a ton is how many pounds ? 4. 1 cwt. is what per cent of a ton ? 5. What is the cost of a ton of corn meal at f 1.25 per hun- dredweight ? 6. How many short tons equal 1 long ton ? 7. What is the cost of 500 lb. of hay at $12 a ton ? 8. How many pounds of coal at $6 a short ton can be bought for 11.50 ? y 9. A car was loaded at the mines with 10 long tons of coal. How many pounds of coal did it carry ? 10. One ounce is what per cent of a pound ? 11. 5 lb. of candy will make how many 4-ounce packages ? 200. TABLE OF LINEAR MEASURE 12 inches (in.) = 1 foot (ft.). 3 feet = 1 yard (yd.). 5J yards or =1 rod (rd.). lejfeet 320 rods = 1 mile (mL). 201. Oral 1. How many inches in 10 yd. ? 2. One foot is what per cent of a yard ? 3. One inch is what per cent of one foot ? 170 LINEAR MEASURE 4. How many rods are there in 2 mi.? In 10 mi.? In 100 mi.? 5. 12 yd. 2 ft. = ft. 6. 33|^ % of 2 rd. = how many feet? 7. 10 rods are how many feet ? 8. 12J % of a mile is how many rods ? 9. 8 J % of a foot is how many inches ? 25 % of a foot ? 50%? 16f%? 331%? 10. How many rods are there in the perimeter of a lawn that is 33 feet square ? 11. Draw on the blackboard a line 1 yd. long, using no measure. Measure and correct it. On your paper draw a line 3 1 in. long, using no measure. Measure and correct it. 12. Estimate the length and breadth of your schoolroom. Test your estimates by measuring. 13. How many feet high do yo\i think your schoolroom is ? Can you find a way to measure it without climbing ? Measure it, and see how nearly correct your estimate is. 14. This oblong represents a field. ^ inch stands for 1 rod. Measure the sides, and tell how many rods long and wide the field is. 15. Draw on the blackboard a plan of your schoolroom floor, using ^ inch for 1 foot ; that is, draw the floor to the scale of 1' to J". SUKFACE MEASURE 171 16. This square represents a square mile. Can you tell what the scale is ? Each small square is what part of a square mile? How many rods of fence would be needed to inclose a square field con- taining ^ of a square mile ? 17. 320 rd. The width of this floor is 16 ft. Can you find the scale of this plan ? What is the length of one long side? What is the greatest length of the room? 18. If you go 30 inches at a step, how many steps will you take in going 30 feet? 19. If a row of corn contains five hills to a rod, how many hills are there in a row a quarter of a mile long? 202. TABLE OF SURFACE MEASURE 144 square inches (sq.in.) = 1 square foot (sq. ft.). 9 square feet = 1 square yard (sq. yd.). 30 J square yards = 1 square rod (sq. rd.). 160 square rods = 1 acre (A.). 640 acres =1 square mile (sq. mi.). 172 SURFACE MEASURE 203. Oral 1. Without a measure draw a square inch. Measure and correct it. 2. Without a measure draw on the blackboard a square foot and a square yard. Measure and correct them. Divide the square yard into square feet. Divide the square foot into square inches. 3. How many square inches are there in two square feet ? 4. Estimate the number of square yards in the floor of your schoolroom. Measure it, and see how nearly right your esti- mate is. Make an estimate of the area of each wall, and test it by measuring. Measure your school lot, and find what part of an acre it contains. 5. How many square inches are there in -|- sq. ft. ? 6. One square yard is ^ of how many square feet ? 7. A 5-inch square contains how many square inches ? Draw it. 8. A rectangle 6'' by 12" is what part of a square foot ? 9. How many tiles 6'' square will cover a floor 10 ft. by 5 ft.? 10. 8 sq. in. are what part of an 8-inch square ? 11. How many square yards are there in 4 sq. rd. ? 12. 40 sq. rd. are what per cent of an acre ? 13. A room is 15 ft. by 12 ft. and 9 ft. high. How many square yards are there in the floor ? Draw a plan of it to the scale of y = 1'. How many square yards are there in one long wall ? In one short wall ? Draw a plan of each wall. VOLUME MEASURE 173 How many square yards of plastering are needed for the ceiling ? 14. How many acres are there in a farm 160 rd. long and 100 rd. wide ? 15. How wide must a field be to contain 10 A. if it is 40 rd. long ? 16. A 10-acre field is 20 rd. wide. How long is it ? B A 17. The scale of these plans is 16' to 1''. Find the perimeter and area of the surface represented by each. 204. TABLE OF VOLUME MEASURE 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.). 27 cubic feet = 1 cubic yard (cu. yd.). 205. Oral 1. Define a cube. 2. How many edges has a cube ? How do they compare ? 3. How many cubic inches are there in a 6-inch cube ? 174 TABLE OF TIME 4. Prove that a tight tin box 7 in. by 11 in. by 3 in. will hold one gallon. 5. A block of wood 4 in. square must be how long to con- tain 96 cu. in. ? 6. 18 cu. ft. are what part of a cubic yard ? 7. A box 12 ft. long and 3 ft. wide must be how deep to hold 4 cu. yd. of sand ? 8. 8 J % of a cubic foot is how many cubic inches ? 9. A 2-inch cube is equal to how many 1-inch cubes ? 10. How many 2-inch cubes would make a 4-inch cube ? 206. TABLE OF TIME 60 seconds (sec.) = 1 minute (rain.). 60 minutes = 1 hour (hr.). 24 hours = 1 day (da.). 7 days = 1 week (wk.). 365 days = 1 common year (yr.). 366 days = 1 leap year. Ten years are called a decade^ and one hundred years make a century^ but these terms are not used in arithmetical cal- culations. The four thirty-day months may be remembered easily by the following old rhyme : " Thirty days hath September, April, June, and November." February has 28 days, with 29 in leap year. The other months have 31 days. TABLE OF COUNTING 175 207. Oral 1. How many minutes are there in a working day of 8 hr. ? 2. A man who works for 30 ^ an hour receives how much a minute ? 3. A train that is running at the rate of 2 miles in 3 minutes goes how many miles in an hour? In 10 hours? In 24 hours? 4. A boy who is idle 15 minutes in every hour wastes what per cent of his time ? 5. How many hours are there in a week ? In the month of June ? 6. How many hours have we for work in a morning session of school if it begins at 9 o'clock and closes at 11.45, allowing a quarter of an hour for recess ? 7. How many days are there in the fall months? 8. Close your book. Recite the table of time and the names of the months, giving the number of days in each month. 208. TABLE OF COUNTING 12 =1 dozen (doz.). 12 doz. = 1 gross. 20 =1 score. 209. Oral 1. How much apiece do oranges cost at 40^ a dozen? 2. One dozen is what per cent of 1 score ? Of 1 gross? 3. How many pens are there in a gross ? 4. If I buy pens at 72 ^ a gross and sell them at 1 f^ apiece, how much do I make on a gross ? On a dozen ? On a pen ? I gain what per cent of the cost ? 176 PAPER MEASURE 5. "Fourscore and seven years ago" was how many years Ago? 6. A merchant bought fiber pails at f 3 a dozen. How much apiece did he pay ? If he sold them at 35 ^ apiece, what did he gain on one ? On a gross ? What per cent of the cost did he gain ? 7. A merchant buys shoe brushes at 11.20 a dozen and sells them at 15 ^ apiece. How much does he gain on one ? What per cent of the cost does he gain ? 8. What is the cost of 6 cans of Alaska salmon at 98 ^ a dozen cans ? 9. What is the cost of a gross of pencils at 40^ a dozen? 10. A man's age is threescore and ten years. How many years old is he ? 11. Bars of soap at 1 9 a gross are how much a dozen ? 210. TABLE OF PAPER MEASURE 24 sheets = 1 quire. 20 quires = 1 ream. The terms bundle (2 reams) and hale (5 bundles) are seldom used. The denomination quire is used mostly in measuring the finer grades of writing paper. Wrapping paper is sold by the pound or by the thousand sheets. Many kinds of paper are sold in packages of five hundred or one thousand sheets. Pack- ages of five hundred sheets are sometimes called reams, 211. Oral 1. How many sheets of paper are there in 2 quires ? In 4 quires ? In ^ of a ream ? In 3 quires ? In |^ ream ? In 10 reams ? ARC AND ANGLE MEASURE 177 2. One quire is what per cent of 1 ream ? Of ^ ream ? Of •| ream ? 3. What is the profit on 10 quires of paper bought at 14 cents a quire and sold at a cent a sheet ? 4. A package of 500 sheets of paper contains how much more than twenty quires ? 5. A stationer sold 10 quires out of a package of 1000 sheets of paper. How many sheets were left ? What per cent of the package was sold ? What per cent was left ? 6. A stationer made a dozen tablets, each containing 72 sheets of paper. How many quires were used for each tablet? The paper cost 40^ a ream. What was the cost per quire ? What was the cost of the paper in one tablet ? What was the cost of the paper for a dozen tablets ? If the backs and labor cost 28^ for a dozen tablets, what was the entire cost of a dozen tablets ? If they were sold for 10^ a piece, what was the gain on a dozen tablets ? The gain was what per cent of the cost ? What was the gain on a gross of tablets ? 7. One quire of paper will make how many leaves if each sheet is folded into 8 leaves ? 8. If 12 sheets of a certain kind of paper weigh one pound, how many pounds will 5 quires weigh ? 9. 960 pages in a book would require how many leaves ? If one sheet makes 4 leaves, how many sheets are required ? How many quires ? 212. TABLE OF ARC AND ANGLE MEASURE 60 seconds (") =1 minute (')• 60 minutes = 1 degree (°). An arc of 360° = 1 circumference. 178 ARC AND ANGLE MEASURE 213. The difference in direction of two angle ; e.g. that meet is an 214. The lines that meet to form an angle are the sides of the angle. Lines are read by means of letters placed at their extremities. Angles are read by means of letters placed at the extremities of their sides. In the angle ABQ the lines AB and BO are the sides. 215. The sum of all the angles that can he formed around a point in a plane is 360°, 120 120 In figure 1 there are three angles about a point. Add the numbers of degrees. Fig. 1 In figure 2 there are five angles about a point. Add the numbers of degrees. 90° 90' 90' 90' Fia. 3 Fig. 2 In figure 3 there are four angles about a point. Add the numbers of degrees. Draw eight equal angles about a point. How many degrees are there in each angle ? Make other questions about these angles. ARC AND ANGLE MEASURE 179 216. Oral 1. How many angles are there in Fig. 3, page 178 ? How do they compare ? Each of these angles is a right angle. How many degrees are there in a right angle ? 2. When the hour hand of the clock is at 12 and the minute hand is at 9, they form what kind of an angle ? At what other number could the minute hand point to make a right angle with the hour hand at 12 ? An angle of 90° is a right angle. An angle that is greater than a right angle is an , obtuse angle. An angle that is less than a right angle is an acute angle. 3. Draw a right angle. How many degrees are there in it? Divide it in the middle by a line. How many degrees are there in each of the angles thus formed ? 4. Make a drawing of a wagon wheel with 6 spokes. The spokes form angles of how many degrees ? Put in twice as many spokes. How many degrees are there in the angles ? Double the number again, and tell the size of the angles. What kind of angles are these ? 5. Draw an angle that you think is about an angle of one degree. 6. One minute is what part of a degree ? Can you think of something that is like an angle of one minute ? 7. The minute hand of a clock passes through how many degrees in 12 hours ? In 1 hour ? 8. What kind of angle (right, obtuse, or acute) is formed by the hour and minute hands of a clock at two o'clock ? At five o'clock ? At eleven o'clock ? 180 ARC AND ANGLE MEASURE 9. 10° = how many minutes ? 20° ? 40° ? i° ? J/ ? 10. 1° = how many seconds ? 11. Stand facing north. Turn 90° to the left. In what di- rection are you facing ? Turn 90° farther. In what direction are you facing? Turn 180° farther. In what direction are you facing ? How many degrees have you turned in all ? 217. A plane figure hounded hy a curved line^ every point of which is equally distant from a point within, called the center, is a circle. 218. The boundary line of a circle is its circumference. 219. Any part of a circumference is an arc. The number of degrees in an arc is always the same as the number of degrees in the angle at the center whose sides meet the extremities of the arc, thus : The angle AOB is -J the sum of all the angles at the center, or 90°. The arc AB is \ of the circumference, or 90°. Can you tell the number of degrees in the arc BQl In the angle BOQl Note. — The number of degrees in any angle may be measured by means of a protractor, an instrument with the degrees marked and num- bered. A Protractor UNITED STATES MONEY 181 220. TABLE OF UNITED STATES MONEY 10 mills = 1 cent. 10 cents = 1 dime. 10 dimes = 1 dollar. The gold coins of the United States are the 85, f 10, and $20 pieces, once called the half eagle, eagle, and double eagle. Gold dollars are not in general circulation, although a few of them have been coined. The silver coins are the dollar, half dollar, quarter dollar, and dime. Silver half-dimes are no longer coined. Most five- cent pieces are made of nickel. Most 1-cent pieces are made of bronze, though some nickel and copper cents are in circulation. The mill is not coined. 221. Oral 1. One dime is what part of a dollar ? What per cent ? 2. One cent is what per cent of a dollar ? 3. One mill is what part of a dollar ? What per cent ? 4. 79 cents is what decimal of a dollar? 7 mills is what decimal of a dollar ? 19 cents and 7 mills ? 5. Express as decimals of a dollar : 85 cents 6 mills; 10 cents 8 mills ; 4 cents 7 mills; 8 cents ; 29 cents 1 mill ; 3 mills. 6. 5 mills are what part of a cent ? 4 mills ? 8 mills ? 7. The value of a $5 gold piece is what per cent of the value of a 110 gold piece ? Of a §20 gold piece ? 8. Make other problems about dollars, mills, and cents. 182 TROY AND APOTHECARIES' WEIGHTS 222. TABLE OF TROY WEIGHT 24 grains (gr.) = 1 pennyweight (pwt.). 20 pennyweights = 1 ounce (oz.). 12 ounces = 1 pound (lb.). These weights are used in weighing gold, silver, and some jewels. To get an idea of the weight of a grain, think of the weight of a grain of wheat or rice. 223. Oral 1. How many grains are there in 1 Troy ounce ? 2. A silver dollar weighs about 412| grains. This is how much less than a Troy ounce ? 3. A gold dollar contains 23.2 grains of pure gold, but enough harder metal is put with the gold to make it weigh 25.8 grains. This weight is how much more than 1 pwt. ? 4. Calling the weight of a gold dollar 1 pwt., what does a $20 gold piece weigh? How many dollars in gold would weigh a pound ? 5. How many Troy ounces would $1000 in gold weigh? How many Troy pounds ? 6. What must I pay for a watch chain weighing 240 grains at $1 a pennyweight ? 224. TABLE OF APOTHECARIES' WEIGHT 20 grains (gr.) = 1 scruple (sc. or 3). 3 scruples = 1 dram (dr. or 3). 8 drams = 1 ounce (oz. or 5). This table is used by druggists and physicians in compound- ing medicines ; but medicines are bought and sold by avoir- dupois weight, except in quantities smaller than one ounce. REDUCTION OF DENOMINATE NUMBERS 183 Druggists also use a term fluid ounce^ which is not a measure of weight, but of capacity, and is equal to -^^ of a pint. Thus, a 2-ounce bottle is a bottle that holds ^ of a pint of any liquid regardless of its weight. 225. Oral 1. A druggist buys a pound (avoirdupois) of quinine con- taining 7000 grains. How many 2-grain tablets can be made from it ? 2. How many 2-grain tablets can be made from 1 3 ? 3. How many 3-grain tablets can be made from 1 3 ? 4. A patient takes 5 gr. of a certain medicine every day. How long will 1 3 of it last him ? 5. A druggist made 1000 powders, each containing 2 gr. of ipecac, 2 gr. of muriate of ammonia, and 10 gr. of extract of licorice. There are 7000 grains in 1 lb. avoirdupois. How many avoirdupois pounds did all the powders weigh ? REDUCTION OF DENOMINATE NUMBERS 226. Changing numbers to larger denominations is reduction ascending. 227. Changing numbers to smaller denominations is reduction descending. 228. Oral 1. How many gallons are there in 72 pints ? What kind of reduction is this ? 2. How many minutes are there in 10 ° ? What kind of re- duction is this ? 184 REDUCTION OF DENOMINATE NUMBERS 3. Reduce : a. 2 bu. to quarts. I. h. 64 pt. to pecks. m. c. 17 T. to pounds. n. d. 96 oz. to pounds. 0, e. 11 yd. to rods. P- /. 33 ft. to rods. ?• 9- 5 A. to square rods. r. h. 288 sq. in. to square feet. s. i. 1728 cu. in. to cubic feet. t. h 10 cu. yd. to cubic feet. u. k. 20 wk. to days. V. ^ da. to hours. ^ yr. to days. J niin. to seconds. 20 da. to hours. 240 sec. to minutes. 96 doz. to gross. 12 score to units. 100 quires to reams. 50 reams to quires. 7' to seconds. 720" to minutes. 4. Reduce : a, 51 qt. to gallons and quarts. h. 7 gal. 2 qt. to quarts ; to pints. c, 35 qt. to bushels and quarts. d, 1 bu. 3 pk. to pecks ; to quarts ; to pints. e, 1 T. 370 lb. to pounds. /. 40 oz. to pounds and ounces. g. 15 cwt. 50 lb. to pounds. h, 1 A. 40 sq. rd. to square rods. i, 4 sq. yd. to square feet. j, 100 sq. ft. to square yards and square feet, k. 64 fluid oz. to pints. I, 130 min. to hours and minutes. REDUCTION DESCENDING 229. Compound numbers are seldom expressed in more than two denominations. In measures of time and arcs three denom- inations are sometimes used. Long and difficult reductions are seldom necessary. KEDUCTION OF DENOMINATE NUMBERS 185 Reduce 17 da. 10 hr. 40 min. to minutes. 17 da. 24 number of hours in 1 da. 68 34 408 number of hours in 17 da. 10 418 number of hours in 17 da. 10 hr. 60 number of minutes in 1 hr. 25080 number of minutes in 418 hr. 40 25120 number of minutes in 418 hr. 40 min., or 17 da. 10 hr. 40 min. Reduce 41 A. 20 sq. rd. to square feet. 41 A. 160 number of square rods in 1 A. 2460 41 6560 number of square rods in 41 A. 20 6580 number of square rods in 41 A. 20 sq. rd. 30J number of square yards in 1 sq. rd. 1645 (6580x1) 197400 (6580 x 30) 199045 number of square yards in 6580 sq. rd. 9 number of square feet in 1 sq. yd. 1791405 number of square feet in 41 A. 20 sq. rd. 186 REDUCTION OF DENOMINATE NUMBERS 230. Written Reduce : 1. 49 da. 7 hr. to hours. 2. 79 A. 50 sq. rd. to square rods. 3. 16 yr. 7 mo. 20 da. to days (allow 30 da. for 1 mo.). 4. 9 sq. yd. 6 sq. ft. to square inches. 5. 78 T. 16 cwt. to hundredweight. 6. 59 cwt. 23 lb. to pounds. 7. 48 lb. 15 oz. Troy to ounces. 8. 12 cu. ft. 384 cu. in. to cubic inches. 9. 78 cu. yd. 19 cu. ft. to cubic feet. 10. 48 gal. 2 qt. to pints. 11. 15 bu. 3 pk. to pints. 12. 25° 30' 15^' to seconds. 13. The degrees in 1 right angle to seconds. 14. 158 gross 5 doz. to dozen. 15. 3 mi. to feet. 16. 1 mi. to inches. 17. 1 sq. mi. to square rods. 18. 21 T. 362 lb. to pounds. 19. 4 cu. yd. to cubic inches. 20. 121 yr. 11 mo. to months. 21. 8 mo. 28 da. to days. 22. 42 wk. 3 da. to hours. 23. 17 cwt. to ounces. 24. 12 yd. 19 in. to inches. 25. 2 yr. 4 mo. 3 da. to days. REDUCTION OF DENOMINATE NUMBERS 187 26. 3 yr. 6 mo. 15 da. to days. 27. 17° 3' to minutes. 28. 7 yr. 14 da. to days. 29. 15 mo. 29 da. to days. 30. 42° 12' to seconds. REDUCTION ASCENDING 231. Reduce 3876 sec. to hours, minutes, and seconds. How many seconds = 1 min. ? 387p sec. 3876 sec. = how many minutes and 6^ min. + 36 sec. seconds ? 1 hr. + 4 min. How many minutes = 1 hr.? 1 hr. 4 min. 36 sec. Ans. ^^ "^^^- = ^^^ ^^^^ ^^^^ ^"^ minutes? 6JZ) Written Reduce : 1. 42,876 sec. to hours, minutes, and seconds. 2. 16,307^' to degrees, minutes, and seconds. 3. 8370 da. to years and days. 4. 983 pk. to bushels and pecks. 5. 5834 lb. to tons and pounds. 6. 4376 oz. Troy to pounds and ounces. 7. 892 pt. to gallons and quarts. 8. 508 pt. to gallons and quarts. 9. 8376' to degrees and minutes. 10. 45,360" to degrees and minutes. 11. 4416 sheets to quires. 12. 685 sq. ft. to square yards and square feet. 13. 28,347 cu. ft. to cubic yards and cubic feet. 188 REDUCTION OF DENOMINATE NUMBERS 14. 38,627 sec. to hours, minutes, and seconds. 15. 497' to degrees and minutes. 16. 89,764 lb. to tons and pounds. 17. 49,763 ft. to miles and feet. 18. 42,374 da. to years and days. 19. 94,276 min. to days, hours, and minutes. 20. 13,794 ft. to rods. VARIOUS FORMS OF REDUCTION 232. Written 1. How many inches are there in |- rd. ? (Indicate the work thus : f x-^^x-^^-; then cancel.) 2. Find the number of: a. Inches in f mi. /. Coat-hooks in ||^ gross. 6, Pints in ^g bu. 9- Sheets in l| ream. c. Pints in f bbl. h. Cubic inches in g|^g cu. yd, d. Seconds in l\'' . . i. Minutes in -^^^ wk. e. Ounces in -^^ T. J- Square feet in -^^ A. 3. What part of 5 gal. is 1 gal. 1 pt. ? Note. — Find the number of pints in 5 gal. ; then in 1 gal. 1 pt. Statement of Relation : of 40 pt. = 9 pt. 4. What part: a. Of 1 T. is 324 lb.? /. Of 7 cu. yd. is 5 cu. yd. 9 h. Of 2 T. is 7 cwt. 40 lb. ? cu. ft. ? c. Of 3 gal. is 2 qt. 1 pt. ? 9- Of 2 yr. is 1 yr. 3 mo. ? d. Of 5 da. is 12 hr. 30 h. Of a square mile is 200 min. ? A. 40 sq. rd. ? e. Of a circumference is i. Of 20 gal. is 5 gal. 2 qt. ? 36° 45'? J- Of a week is 18 hr. 8 min . ? REDUCTION OF DENOMINATE NUMBERS 189 5. 3 bu. 2 pk. of potatoes are sold out of a load of 28 bu. a. What part of the load is left? 5. What per cent of the load is sold ? 6. 2 bbl. of cranberries, each containing 3 bu., are worth how much at 12 J ^ per quart ? 7. The speed limit for automobiles in a certain town is 8 miles per hour. That is how many rods per minute ? 8. What is the cost of 5 T. 500 lb. of coal at 15.40 per ton? 9. \ T. is equal to how many ounces ? 10. How long will 16 bu. of corn last Fred's chickens if he feeds them 1 qt. a day ? 11. a. What is the profit on a bushel of chestnuts bought for 82^ and sold at 10 cents a pint ? h. The gain is what per cent of the cost ? 12. How many minutes are there in April, May, and June ? 13. What is the cost of the milk supply for September of a housekeeper who buys \ gal. per day and pays 6|^^ per quart ? 14. What are the yearly wages of a man who earns a cent in 2 minutes and works 8 hours a day and 26 days in a month throughout the year ? 15. If a horse eats 12 lb. of hay per day, how many tons and pounds will he eat in a year ? 16. A box 6'' X 4'' X 2'' contains what fraction of a cubic foot ? 17. I of an acre of land contains how many square feet ? 18. 16 sq. rd. 11 sq. yd. are what part of an acre ? Lb. Oz. 7 8 15 14 23 15 47 5 Add: 190 ADDITION OF COMPOUND NUMBERS ADDITION AND SUBTRACTION OF COMPOUND NUMBERS 233. Written Add 7 lb. 8 oz., 15 lb. 14 oz., 23 lb. 15 oz. 15 oz. + 14 oz. + 8 oz. = 37 oz. = 2 lb. 5 oz. 2 lb. + 23 lb. + 15 lb. + 7 lb. = 47 lb. 47 lb. 5 oz. Ans. 1. 19 ft. 6 in., 17 ft. 10 in., 9 ft. 6 in. 2. 13 A. 17 sq. rd., 19 A. 153 sq. rd. 3. 2 hr. 5 min. 30 sec, 8 hr. 53 min. 47 sec. 4. 8 gal. 3 qt., 15 gal. 1 qt., 16 gal. 2 qt. 5. 81° 19' 35'', 2° 50' 29", 3° 4' 50". 6. 6 T. 480 lb., 7 T. 730 lb., 19 T. 900 lb. 7. 5 yd. 2 ft., 16 yd. 1 ft., 18 yd. 2 ft. 8. 6 pk. 7 qt., 3 pk. 5 qt., 2 pk. 6 qt. 9. 7 cu. yd. 18 cu. ft., 12 cu. yd. 19 cu. ft. 10. 6 yr. 7 mo. 3 da., 7 yr. 8 mo. 29 da. 11. 8 lb. 7 oz., 16 lb. 14 oz., 19 lb. 10 oz. 12. 21 bu. 3 pk., 9 bu. 2 pk., 35 bu. 1 pk. 13. 2 wk. 3 da., 19 wk. 1 da., 20 wk. 6 da. 14. 7 hr. 38 min. 21 sec, 5 hr. 47 min. 29 sec 15. 5 yr- 200 da., 7 yr. 321 da., 8 yr. 179 da. SUBTRACTION OF COMPOUND NUMBERS 191 234. Written From 18 yr. 7 mo. 14 da. take 6 yr. 8 mo. 26 da. 11 yr. 10 mo. 18 da. Difference 7 mo. 14 da. = 6 mo. 44 da. „^- , , , , . « 1Q a 1H. 1Q (Why do we make these reductions?) 18 yr. 6 mo. = 17 yr. 18 mo. \ j j 18 yr. 7 mo. 14 da. = 17 yr. 18 mo. 44 da. 17 yr. 18 mo. 44 da. - 6 yr. 8 mo. 26 da. = 11 yr. 10 mo. 18 da. Suhtraet: 1. 12 yr. 3 mo. 15 da. 5. 4 yr. 2 mo. 18 da. 10 yr. 1 mo. 19 da. 1 yr. 7 mo. 12 da. 2. 9yr. 3yr. 2 mo. 5 mo. Ida. 29 da. 3. 9 yr. 4 yr. 2 mo. 8 mo. 5 da. 5 da. 4. 12 yr. 8 yr. 2 mo. 16 da. 12 da. 6. 7yr. 3 yr. 5 mo. 6 mo. 18 da. 7 da. 7. 9 yr. 8yr. 6 mo. 5 mo. 13 da. 15 da. 8. 1 yr. 3 mo. 7 mo. 15 da. • 9. How many years, months, and days are there from May 30, 1907, to Dec. 5, 1909 ? 1909 yr. 12 mo. 5 da. Note. — December is the twelfth month 1907 vr. 5 mo. 30 da. ^^^ ^^^ *^® fifth. Count 30 da. for a ' month. Find the time from : 10. July 19, 1827, to Mar. 26, 1878, 11. Sept. 20, 1831, to Nov, 15, 1909. 192 SUBTRACTION OF COMPOUND NUMBERS 12. Jan. 7, 1840, to Feb. 8, 1896. 13. May 4, 1850, to Jan. 12, 1861. 14. Oct. 19, 1760, to Aug. 20, 1860, 15. Dec. 12, 1880, to June 5, 1903. 16. July 10, 1809, to Oct. 2, 1893. 17. May 8, 1899, to Feb. 12, 1908. 18. Feb. 12, 1901, to Jan. 30, 1906. Subtract : 19. 5 hr. 54 min. 30 sec. 1 hr. 50 min. 50 sec. 20. 122° 31' 15'' 60° 20' 45" 21. 23 hr. 54 min. 36 sec. 20 hr. 24 min. 48 sec. 22. 7 ft. 6 in. 3 ft. 11 in. 23. 27 gal. 18 gal. 2 qt. 3 qt. 24. 19° 31' 6° 41' 25. 18 bu. Ipk. 17 bu. 3pk. 26. 10 A. 56 sq. rd. 4 A. 106 sq. rd. 27. 16 1b. 4 oz. 51b. 12 oz. 28. 48 ft. 3 in. 27 ft. 9 in. 29. 42' 13" 35' 5^ l^ 30. 5 min. 47 sec. 2 min. . 48 sec. 31. Find the time between Dec. 21, 1620, and July 4, 1776. 32. How much time elapsed from the beginning of the Civil War, April 14, 1861, to the close of the war, April 9, 1865 ? 33. Washington was born Feb. 22, 1732, and died Dec. 14, 1799. How long did he live ? 34. How much time has elapsed from Oct. 12, 1492, to the present time ? SUBTRACTION OF COMPOUND NUMBERS 193 EXACT DIFFERENCES BETWEEN DATES 235. Written I. What is the exact number of days between Dec. 16, 1895, and March 12, 1896 ? Dec. 15 Jan 31 There are 15 days in December after ■J-, ■■ oq the 16th. January has 31 days, February 29 (leap year), and March 12, making 87 March L£ days. Always count the last day. 87 days. Ans. Note. — Every year whose number is divisible by 4 is a leap year, except a centennial year, which is a leap year only when its number is divisible by 400 ; e.g. the year 1896 was a leap year but the year 1900 was not. 2. Mr. Griffith bought a house, Feb. 25, 1896, and paid for it, July 12, 1896. Find the exact number of days between the buying and paying for the house. 3. Find the exact number of days between June 25, 1900, and Aug. 24, 1900. Mnd the exact time between : 4. Sept. 6, 1896, and April 7, 1897. 5. Nov. 11, 1898, and Dec. 4, 1898, 6. Aug. 16, 1907, and Dec. 21, 1907. 7. July 4, 1896, and Aug. 10, 1896. 8. Feb. 23, 1897, and June 4, 1897. 9. Oct. 9, 1899, and Feb. 6, 1900. 10. Nov. 8, 1905, and Oct. 6, 1906. II. A gardener planted an acre of sweet corn on the tenth day of May. It was ready for market on the third day of August. How many days were required for the corn to grow? 194 COMPOUND NUMBERS MULTIPLICATION AND DIVISION OF COMPOUND NUMBERS To THE Teacher. — Little time should be spent upon multiplication and division of compound numbers. In solving problems other than those in longitude and time, it is generally better to reduce the compound num- ber to one denomination before multiplying or dividing. 236. Written 1. Multiply 8 lb. 3 oz. by 9. 8 lb. 3 oz 9x3 oz. = 27 oz. = 1 lb. 11 oz. 9 9 X 8 lb. = 72 lb. 73 lb. 11 oz. Product 72 lb. + lib. = 73 lb. Multiply : 2. 7 lb. 9 oz. by 5. 7. 3. 15 gal. 1 qt. by 10. 8. 4. 14 ft. 11 in. by 3. 9. 5. 17 A. 40 sq." rd. by 5. 10. 6. 19 ft. 7 in. by 4. ii. 12. Divide 32° 15' 30'' by 15. 1 hr. 20 min. 20 sec. by 15. 2 hr. 40 min. 30 sec. by 15. 3 hr. min. 30 sec. by 15. 4 hr. 19 min. 30 sec. by 15. hr. 58 min. 47 sec. by 15. 15 )32° 16' 30'^ 2° 9' 6" Quotient 82° 4-15 = 2° and 2° Remainder. 2° = 120'. 120' + 16' = 136'. 136' -T- 15 = 9' and 1' Remainder. r=60". 60" + 30" = 90". 90" -4-15=6". Divide by 15 and test your work 13. 30° 16' 15" 14. 61° 1'45" 15. 17° 6' 0" 16. 2° 2' 15" 17. 46° 10' 0" 18. 20° 0' 30" 19. 20° 5' 45" 20. 60° 35' 15" 21. 70° 30' 45" 22. 100° 1' 30" REVIEW AND PRACTICE 195 REVIEW AND PRACTICE 237. Oral 1. At 8)^ a quart, what will a bushel of berries cost? 2. How many oranges will 50 cents buy at the rate of 3 foi 10 cents? 3. f of 18 is what part of 72? What per cent of 72? 4. From 4|- bu. take 3 pk. 5. What will 15 bu. of potatoes cost if 7|- bu. cost 1 6 J? 6. At 1 15 a dozen, what will 84 chairs cost? 7. A cube 5 inches long contains what part of a cubic foot ? 8. 9 mo. is what part of a year? 9. I sq. ft. = sq. in. 10. Bought a peck of chestnuts for 80 cents and sold them at 5 cents a half pint. How much did I gain? What per cent of the cost did I gain ? 11. 25 is what part of 75? What per cent of 75? 12. What part of f is |? 13. 15 gal. 3 qt. = pt. 14. Multiply 7.635 by 10; by 1000; by 100. 15. Divide 5.8 by 10; by 100; by 1000. 16. Divide 16.54 by 1654. 17. 36 lb. of coffee for i9 is at the rate of 12 lb. for how much? 18. A wagon was sold for $60, which was -J of the cost. What was the cost ? How much was lost ? What per cent of the cost was lost? 19. How many square rods are there in -^^ of an acre? 20. A ten-acre lot is 20 rods wide. How long is it? 196 REVIEW AND PRACTICE 21. f of 16 is I of what? 22. Andrew spent -| of his money and had 10 cents left. How much had he at first? 23. What is the entire weight of three chickens if their aver- age weight is 2 lb. 11 oz.? 24. What part of |^ is J? 25. ^ is what part of ^? 26. Find 20% of 300; 331% of 72; 16f % of 60; 62J% of 16; 90% of 10. 27. Sugar at $100 a ton is how much a pound? 28. A house was damaged fSOO by fire. This was 10% of the cost. What was the cost ? 29. 87| % of the cost of Glen's bicycle was 1 28. What was the whole cost? 30. If 9 eggs cost 27 cents, what will 6 doz. cost ? 238. Written 1. The month of January is how many minutes long ? 2. If school closes for the long vacation on June 24 and opens on Sept. 6, how many vacation days are there ? 3. If it is now 8.30 a.m., what time will it be in 20 J hours ? 4. Find the exact time from March 15, 1906, to Aug. 2, 1906. 5. How many weeks, days, and hours are there in 9785 hours ? 6. How many rods of fence are needed to inclose a square field 396 ft. long ? 7. What will it cost to paint a ceiling 24 ft. long and 16J ft. wide, at 25 cents a square yard ? 8. What is the length of one side of a square garden whose perimeter is 632 feet ? REVIEW AND PRACTICE 197 9. How many square feet of land are there in a field contain- ing 21 A. ? 10. What will it cost at 65 cents a square yard to cover with matting a floor 21 ft. long and 15 ft. wide ? 11. How many feet of fence will inclose a field 46 rd. by 31 rd. ? 12. Find the perimeter of a room that is 15' 6" by 13' 8". 13. How many rods are there in 2448 inches ? 14. a. If steel rails are 30 ft. long, how many are needed for 8 miles of street railroad track ? b. How many tons do they weigh if each foot weighs 90 lb. ? 15. At the rate of 660 feet a minute, how many miles will a street car run in two hours ? (Indicate and cancel.) 16. What is the cost, at $1.80 per rod, of a fence inclosing a rectangular field 50 rd. by 561 ft. ? 17. Draw a line to represent 1| miles, letting the scale be 80 rd. to an inch. 18. Draw a rectangular building lot 4 rd. by 8 rd., and put on the lot a house 44 ft. by 33 ft., and a barn 22 ft. square, using as a scale 22 ft. to an inch. 19. Three barrels of extract of witch hazel are put up in pint bottles and sold for 20^ a bottle. What is received for all of it ? 20. A grocer bought 40 bbl. of cider at S 2.50 a barrel, made it into vinegar, and sold it at 5 ^ a quart, a. How much did he gain ? b. What per cent of the cost did he gain ? 21. Divide in the shortest way the product of 36, 40, 144, 8, and 160 by the product of 18, 272, and 6. 22. Find the least number that will exactly contain 27, 60, and 24. 198 REVIEW AND PRACTICE 23. Find the greatest number that will exactly divide 231 and 385. 24. In the corner stone of a church were cut these two dates, MDCCCXXI — MCMVII. How many years apart were they? 25. Mr. C. J. Hogan has laid for Mr. Henry Sumner a piece of Portland cement walk 50 ft. long and 5 ft. wide at 12J ^ a square foot. Make out the bill and receipt it. 26. Divide: a. 8.8 by .0008; h. 20.005 by .005. 27. Divide the sum of two thousand and two thousandths by two hundredths. 28. How many times will a bicycle wheel rotate in going a mile if the wheel is 7.2 ft. in circumference ? 29. The product of two numbers is 12.46, and one of them is 24.92. Find the other number. 30. The product of three numbers is .1456. Two of them are 2.6 and .14. What is the third ? 31. Reduce to common fractions in lowest terms : a. .025 d. .06 J g. .625 /. .21 h. .0125 e, .871 h. .0875 k. .0495 c. .371 /. .375 i, .012i h 4.25 32. Express as decimals : d. i^ A. ff Z. 14|^ p. 15| 33. Find the value of: a. 2i + 32-f.ioff + f of J 5. 314^x5 KEVlEW AKD PRACTiC:e l99 e. 8^x21x^x8 /. t^xl8| 34. A hardware merchant bought a bill of goods for $ 560, and marked them to be sold at a price which was 140 % of the cost. a. At what price did he mark the goods ? b. He sold the goods for 90 (fo of the marked price. For what did he sell them ? 35. A jobber bought 2000 wheelbarrows at f 1.25 apiece, marked them at 115 % of their cost, and sold them for 95 % of the marked price. What did he receive for them ? 36. During a storm 1000 bbl. of sugar, or S^ % of the cargo of a ship, were thrown overboard. How many barrels formed the cargo ? 37. Make and solve a problem about $ 420 and 35 % . 38. Mr. Fish paid city taxes to the amount of f 280, which was 1^ % of the value of his property. How much was his property worth ? 39. A load of boards weighing 3500 lb. were put into a kiln and dried. When taken out they weighed 1^ tons. What per cent of the weight of the boards at first was water ? 40. a. Write a problem in percentage in which factors are given to find a product, h. Write another in which one of the factors is to be found. 41. If a pint of cream is used in making two gallons of ice cream, what per cent of the ice cream consists of other things than cream ? 42. I of my money is $ 1260. What is ^ of my money ? 200 REVIEW AN^D PRACTICE 43. This walk is made of sawed Ohio sandstone, cost at 21^^ per square foot. Find its 7Falk\ Walk 44. A man owned f of a mine and sold I of his in- terest for 11710. What was the whole mine worth at the same rate? Walk Scale 5 to ]/i 45. A teamster mixed 600 lb. of wheat bran with a ton of corn meal with which to feed his horses. a. What per cent of the mixture was bran ? b. What per cent was corn meal ? 46. The population of a city to-day is 130,000, which is 125 % of what it was 10 years ago. What can you find ? Find it. 47. Mr. Hunter's ranch in Wyoming contains 3200 acres of land, worth $25 an acre. a. How many square miles of land has he? b. What is its value ? e. One fourth of the land can be irrigated and made worth $75 an acre. Would it pay to irrigate the land at a cost of 120,000? Why? 48. a. Mr. Hunter has 6000 sheep. They are worth $5.25 apiece in the fall and $5.75 apiece in the spring. What is the increase in the value of the entire herd during the winter? h. The increase is what per cent of the value in the fall? Can you tell why they are worth more in the spring ? REVIEW AND PRACTICE 201 The Ranch 49. a. The sheep are separated into two equal flocks and are cared for by two herders, two dogs, and a camp tender. Each herder receives $40 a month and the camp tender 850. The supplies cost 5^ 52 per month. What is the cost of tending the sheep for a year ? ! h. On the twenty-ninth day of June the men started with their flocks for the "summer range" in the mountains 117 miles from the ranch. They traveled an average distance of 6J miles per day. On what day of the year did they reach the summer range ? e. In the spring the sheep were fed 250 tons of alfalfa, worth |5 a ton. A ton of salt lasts them 6 months and costs I 60. Add these items to the cost of caring for the sheep, and find the entire cost of keeping these two flocks for a year. Do you know how much a ton of coarse salt costs where you live ? Why does it cost more at this ranch ? On the Move 202 REVIEW AND PRACTICE 50. a. In early summer 5 men sheared the 6000 sheep in 15 days, receiving 8 cents per fleece for their work. What were the average daily wages of the shearers? h. The wool was bought by a commission man for a jobber in the East at 22| cents per pound. The fleeces weighed 8 lb. apiece, on the average. What did Mr. Hunter receive for the wool ? c. The commission man charged the jobber 10J% of this sum for buying the wool. What did the wool cost the jobber ? A Wool Freighter d. The wool was shipped in sacks 8 ft. long and 4 ft. wide, holding 400 lb. apiece. It was taken to the railroad station in loads of 22 sacks each, drawn by ten-horse teams. How many sacks were left after 3 loads had been drawn ? 51. It may be found from the foregoing problems that Mr. Hunter received §7426 more than he expended on account of his sheep business this year. At the same rate, what would be a man's yearly profit from a herd of 48,000 sheep ? 52. If a herder kills ten coyotes in a year and receives from the government a bounty of $3 for each one, how much will he add thereby to his monthly income? ARTICLES SOLD BY THOUSAND AND HUNDRED 203 ARTICLES SOLD BY THE THOUSAND, HUNDRED, OR HUNDREDWEIGHT 239. Written 1. What is the cost of 8975 bricks at 1 7 per M. ? (M. stands for 1000.) 8975 = 8.975 M. Since 1 M. costs $7, 8.975 M. cost 8.975 x $ 7, or $ .. Ans. 2. What must be paid for 980 soapstone pencils at f .30 per C. ? (C. stands for 100.) 980 = 9.80 C. Since 1 C. costs $.30, 9.80 C. will cost 9.80 x $.30, or | . Ans. 3. Find the cost of 1550 lb. of new buckwheat flour at 12.50 per cwt. 1550 lb. = 15.50 cwt. Since 1 cwt. costs $ 2.50, 15.50 cwt. cost 15.50 x $2.50, or $ . Ans. Note. — In final results, a fraction of a cent, equal to or greater than | cent, is counted a whole cent. A fraction which is less than ^ cent is dropped. 4. Find the cost of each of the following items and the total cost of all of them : a. 27,325 bricks at f 5.15 per M. b. 4900 cu. ft. of gas at $1.20 per M. c. 583 lb. sugar at 14.75 per C. d. 4900 tomato plants at $1.50 per M. e. 1000 laths at 40 ^ per C. /. 3125 cu. ft. of city water at $.14 per C. g. 1 T. fiber paper at $2.50 per C. h. 5600 paper bags at $2.90 per M. i. 16 boxes of envelopes, 250 in a box, at $1 per M. 204 MEASUREMENTS y. 850 Japanese napkins at i.l8 per C. k. 10 bbl. American lump salsoda, 375 lb. in a barrel, at 1.80 per cwt. I. 6500 No. 3 butter trays at 11.60 per M. m. 8950 7-inch picnic plates at $1.75 per M. n. 13,500 cedar shingles at |4.10 per M. 0, 675 lb. light manila bread paper at 13.75 per cwt. MEASUREMENTS » AREAS OF PARALLELOGRAMS '240. A plane figure hounded hy four straight lines is a quadri- lateral; e.g. 241. Lines that are the same distance apart throughout their whole length are parallel lines ; e.g. 242. A quadrilateral whose opposite sides are parallel is a parallelogram. Which of the above figures are parallelograms ? 243. A parallelogram that has four right angles is a rectangle. Which of the above figures are rectangles ? 244. Two lines that meet to form a right angle are perpendicular to each other. 245. The side on which a figure is supposed to rest is its base. AREAS OF PARALLELOGRAMS 205 246. The perpendicular distance from the highest point of a figure to the base, or to the base extended, is its altitude ; e.g. '^ 247. Figures are read by means of letters placed at their angles. Thus, Fig. 1 is read, ''Oblong ABOD.'' Fig. 2 is read, "Tri- angle ABO.'' Read Figs. 3 and 4. The base of the triangle in Fig. 2 is AC. The altitude of Fig. 1 is DQ ov AB, 248. Oral 1. In Fig. 5 the part ^compares how with the part Jf ? 2. The area of the par- allelogram ABCB com- pares how with the area of the parallelogram EFOD ? 3. What is the base of each of these parallelo- grams ? 4. What is the altitude ? What is the area ? 5. Cut from paper different shaped parallelograms. 6. Change them to rectangular parallelograms by cutting off a part similar to ^and placing it like M. 7. How is the area of a rectangle found ? 8. If the base of a rectangle is its length, the altitude is what ? A E B F \k 1, \ V M i \ !< \ 1' \ I'* N' Fig. 5 206 AREAS OF PARALLELOGRAMS 9. If we know the base and altitude of a rectangle, how may we find the area ? 10. Since any parallelogram may be made into a rectangle of the same base and altitude, how may we find the area of a par- allelogram ? 249. The area of a parallelogram is equal to the product of its base and altitude. 250. Oral The following examples relate to parallelograms. Fill in the missing number : Base Alt. Area Base Alt. Area 1. 8 ft. 6 ft. 5. 3J in. 7 sq. in. 2. 71 in. 8 in. 6. 2 ft. 3. 5 mi. 4 sq. mi. 7. 3| ft. 4. 6 yd. 96 sq. yd. 8. 1 yd. 15 sq. ft. Alt. 3J in. 6 in. 8 ft. 1yd. 251. The following parallelograms are drawn to the scale of 1' to ^". Find their areas in square feet. AREAS OF TRIANGLES 207 AREAS OF TRIANGLES 252. A 'plane figure hounded hy three straight lines is a triangle ; e.g. 253. Oral Fig. 1 Fig. 2 Fig. 3 Fig. 4 1. Figures 1, 2, 3, and 4 are what kind of figures ? What -kind of figures are A and B 2 2. In each of the above figures, how does A compare with B? 3. In each of the above figures, how do the base of the triangle and the base of the parallelogram compare ? How do the altitude of the parallelogram and of the triangle compare ? 4. Fold a paper once. By cutting through both leaves of the paper at the same time, you may cut out two triangles. How do they compare ? Put them together so as to make a parallelogram. 5. Repeat this process, cutting triangles, each pair of which differ in shape from the other pairs. Each triangle is what part of the parallelogram having the same base and altitude? 6. How is the area of the parallelogram found? Of the triangle? ^ 208 AREAS OF TRIANGLES The area of a triangle is equal to one-half the product of its base and altitude. 7. The following triangles are drawn to a scale. The bases and altitudes of the triangles which they represent are indicated in the drawings. Find their areas. 17 nrd. 254. Written 1. Find the areas of triangles having the following dimen- sions, using cancellation where possible. When the base and altitude are of different denominations, make them similar before multiplying. Base Alt. Basb Alt. a. 51 ft. 42 ft. y. 15 yd. 15 in. b, 27 in. 15 in. h. 2 mi. 160 rd. c. 6fft. 7Jft. L 11 mi. 1201 rd. d. 3 ft. 7 in. 2 ft. 9 in. m. 4 rd. 3J rd. e. 9fin. 16 in. 71. 2 yd. 50 in. /. 18 yd. 27 in. 0. 3^rd. 12 ft. g, 12 ft. Hyd. p, 101 yd. 5 ft. 4 in. h, 14 rd. 7 yd. q. 7 yd. 7 ft. i. 1 mi. 80 rd. r. 17fin. A^d. MEASUREMENTS: THE CORD 209 2. The above figures are drawn to the scale of 10 ft. to J in. Find their bases, altitudes, and areas. THE CORD 255. The pile of wood in the center of this picture is 8 ft. long, 4 ft. wide, and 4 ft. high. How many cubic feet does it contain ? 128 cubic feet = 1 cord. In the forest, fuel wood for market is generally cut in 4-foot lengths like that in the picture, so that a pile 4 ft. high and 210 MEASUREMENTS: THE CORD 8 ft. long contains a cord of 128 cu. ft. The term cord^ how- ever, is often used to mean any pile of wood that is 8 ft. long and 4 ft. high, whatever may be the length of the sticks. 256. Oral 1. Hold your hand above the floor high enough to show the height of the pile of wood in the picture. Stand as many feet from the side of the room as the pile is long. Show with your hands the length of the sticks. 2. How many cubic feet are there in a pile of wood 8 ft. long and 4 ft. high, if the sticks are 1 ft. long ? Show with your hands the height of this pile. Show its width. Walk far enough to show its length. 3. How many cubic feet would there be in the above pile if the sticks were 2 ft. long ? Show the width of the pile with your hands. 4. How many cubic feet would there be in the pile if the sticks were 3 ft. long ? 1^ ft. long ? Show these lengths with your hands. 257. Written 1. Using cancellation, find the number of cords in a pile ot 4-foot wood : a. 20 ft. long and 8 ft. high. h. 64 ft. long and 4 ft. high. c. 12 ft. long and 6 ft. high. d. 100 ft. long and 7 ft. high. e. 26 ft. long and 5 ft. high. /. 40 ft. long and 4 ft. high. g. 72 ft. long and 7 ft. high. h. 18 ft. long and 2 ft. high. MEASUREMENTS: BUILDING WALLS 211 2. If a pile of 4-foot wood is 48 ft. long, how high must it be to contain 9 cd. ? 3. What must be paid for enough 4-foot wood to fill a shed 26 ft. long, 16 ft. wide, and 12 ft. high at 14.50 a cord ? 4. In the yard of a certain tannery there is a pile of bark 100 ft. long, 24 ft. wide, and 10 ft. high. How many cords of bark are there in the pile ? BUILDING WALLS 258. There are no universal rules for the measurement of masonry. Some masons measure around the outside of a cellar wall to determine its dimensions, while others make allowance for the corners. The method of measurement should be specified in the contract in every case. Quantities of uncut stone are bought by the cord, and usually 99 cu. ft. are taken for a cord. From 21 to 23 bricks 8^' X 4'^ x 2'' are estimated to make a cubic foot of brick wall. 259. Written 7' 6" \_1 \\ \ \ \ \X vX'xx ^\< v<:\<;\^-^^ff'«' \\\ 1 1 { 1 1 1 1 1 I 1 1 \\X\ \V x\ s 1. a. How many cubic feet does this wall contain ? h. If 21 bricks make a cubic foot of wall, how many bricks are used in this wall ? 212 MEASUREMENTS: FLOOR COVERING c. What do they cost at 1 6.30 per M.? d. At the same rate, what would be the cost of brick for a partition wall 50 ft. long, 11 ft. high, and 12 in. thick ? e. When 22 bricks 2'^ x 4'' x 8'' make a cubic foot of wall, how many cubic inches of mortar are used ? 2. A house built in the form of a rectangle 36 ft. by 21 ft. has a cellar 8|- ft. deep. The cellar wall is 1 ft. 6 in. thick. a. Draw a plan of the cellar wall. h. Find the number of cubic feet to be paid for in the cellar wall, measuring around the outside, and making no allowance for the corners. e. If the cellar wall extends 3 ft. above the grolind, how many cubic yards of earth were removed to make the cellar ? d. What was the cost of the brick at $6.50 per M. for a partition wall 12 in. thick, from end to end through the middle of the cellar, allowing 22 bricks for a cubic foot ? e. What was the cost of a Portland cement floor in one half of this cellar at 10 ^ per square foot ? 3. Find the cost of the stone for a wall 297 ft. long, 8 ft. high, and IJ ft. thick at 16.50 a cord (99 cu. ft.). FLOOR COVERING 260. The exact number of yards of carpet, matting, or other covering to be purchased for any given floor is diflicult to deter- mine because of the waste in fitting and in matching figures. We may obtain a fairly correct estimate by dividing the number of square yards of surface to be covered, by the width of the strips, in yards, and then adding a certain amount for waste. A yard of carpet, matting, etc., is a yard of the length of the roll, regardless of its width. MEASUREMENTS: FLOOR COVERING 213 261. Oral 1. A piece of carpet 1 yd. long and | yd. wide will cover how Qyd. much surface? Draw it full size on the blackboard. 2. What is the area of this floor ? 3. If a yard of carpet | yd. wide will cover | of a square yard of floor, how many yards of such carpet will cover 18 sq. yd. of floor ? 4. How many yards of matting 1 yd. wide will cover the same floor ? 262. Written 1. If this floor is covered with carpet | yd. wide, how many strips, running lengthwise, must be purchased ? Note. — When a part of the width of a strip is needed, a whple strip must generally be purchased. 2. How many yards of carpet must be purchased for this floor ? 3. Draw a diagram of each of the floors whose dimensions are given below, and compute the number of yards of material to be i purchased to cover it, running the strips the longer way of the floor : Dimensions Width of Material a. 81 yd. X 5 yd. 1yd. b. 15 ft. X 3 yd. 2 ft. 3 in. c. 101 ft. X 18 ft. fyd. d. 18 ft. X 24 ft. 6 in. 1yd. e. 17 ft. X 27 ft. . 36 in. 214 MEASUREMENTS: FLOOR COVERING /. 9 ft. X 28 ft. |yd. 9' 13 ft. 3 in. X 15 ft. fyd. h. 12 ft. X 16 ft. 1yd. i. 19^ ft. X 29 ft. fyd. h 15 ft. 9 in. X 19 ft. 2 ft. 3 in. k. 11 ft. 3 in. X 14 ft. 27 in. I. 20 ft. X 10 yd. 1yd. m. 16 yd. X 6 yd. 54 in. n. 20 ft. X 38 ft. fyd. 0. 16 ft. X 22 ft. 1yd. V' 15 ft. X 18 ft. 3 in. 27 in. . Written 1. The scale of this drawing is -^q '^ = 1'. Make a drawing like this, using ^" for 1'. Cut it out, and fold it on the dotted lines, making the model of a room. 12' End Wall 16' Side Walt 12' End Walt 16' Ceiling 16' Side Wall a. What is the entire length of the end and side walls ? What is the hei^ ht ? b. How many square feet are there in all the walls ? c. How many square feet are there in the ceiling ? d. How many square feet are there in the walls and ceiling together ? e. How many square yards are there in all ? /. What will it cost to lath and plaster this room at 85 cents a square yard, taking out 5 J square yards for openings ? 2. a. What will it cost to lath and plaster a room 15' by 18' and 9' high at f .30 a square yard, allowing 10 sq. yd. for doors and windows? h. What will it cost at |.28 a square yard, making no allowances ? MEASUREMENTS: WALL COVERINGS 217 3. Measure your schoolroom to the nearest half, third, or fourth of a foot, and estimate the cost of lathing and plastering it at 40)^ a square yard. Allow 3 sq. yd. for each door and window. 4. Estimate the cost of plastering other rooms in your school building in the same way. 5-6. Measure two rooms in your house. Find the cost of plastering the walls at 30 cents a square yard, allowing 2 sq. yd. for each door and each window. Bring your work to school. 7. Find the cost of lathing and plastering the walls and ceil- ings of a room 27 ft. by 18 ft. and 9| ft. high, allowing for 4 windows, each 3 ft. by 6 ft, and 3 doors, each 3 ft. by 8 ft., at 32 cents per square yard. WALL COVERINGS 265. A roll of figured wall paper is usually 8 yards long and J yard wide. How many square yards of paper does it contain ? Ingrain paper is 30 inches wide. Paper hangers generally estimate that a roll of paper will cover from 30 to 34 square feet, after allowing for waste. Woven wall coverings are sold by the square yard. 266. Written 1. Fanny's mother wished to decorate a room in her house, and Fanny estimated the cost. The dimensions were 16 ft. by 21 ft., and 9 ft. 9 in. high. There were 2 doors and 3 windows, each estimated at 2 sq. yd. a. The paper was estimated to cover 30 sq. ft. per roll. How many rolls were needed for the walls ? (You cannot buy a part of a roll.) b. What would the paper cost at 25^ a roU ? 218 MEASUREMENTS: WALL COVERINGS c. The molding to extend all around the room at the top of the walls was sold only in 12-foot lengths, and cost 4J^ a foot. What would it cost ? c?. In preparation for tinting, the ceiling was to be lined with paper at 10^ a roll of 30 sq. ft. What would this paper cost? e. They expected to use two packages of tinting material costing 30^ a package; the putty, glue, flour, etc., were esti- mated at Q^^\ and the labor at two days' work for two men at $3.50 per day for each man. What should have been the total of Fanny's estimate ? /. If they decide to add to the ceiling some relief work which costs f 3.81, and the men can put it on in ^ of a day, how much must Fanny add to her estimate ? 2. Before being tinted, a ceiling 12' x 17' was covered with sheeting 2 yd. wide. What did the sheeting cost at 25 ^ a lineal yard ? 3. a. How much money is needed to buy, at 40^ a square 3^ard, enough crash to cover the four walls of a room 18' x 32' and 10 ft. high, allowing 190 sq. ft. for baseboard and open- ings, and not purchasing a fraction of a square yard ? h. What would it cost to paint the walls and ceiling of this room at f .23 a square yard ? 4. Estimate the cost of painting the walls and ceiling of your schoolroom at f .25 a square yard. 5. Select a room in your own home; measure it. Find the cost of decorating it as your mother would like to have it done. Ask her to tell you what she would like to have put on the walls ; then you make the measurements, compute the amount of material and labor, and the cost. Make and solve other problems in papering. LUMBER MEASURE 219 LUMBER MEASURE 267. A piece of wood 1 ft. long, 1 ft. wide, and 1 in. thick is a board foot (bd. ft.). To THE Teacher. — As material for this lesson, a real board foot — a piece of board exactly 1 ft. long, 1 ft. wide, and 1 in. thick — should be pro- vided. Refer to it in obtaining answers to the oral questions below and whenever pupils seem to answer wide of the mark in this subject. This is very important. 268. Oral 1. How many inches long is a board foot ? How many- inches wide ? How many cubic inches does a board foot contain ? 2. How many board feet piled one upon another would make a cubic foot ? Show with your hands how wide, long, and high this pile would be. 3. A board 1 in. thick, 1 ft. wide, and 6 ft. long contains how many board feet ? Draw it full size on the blackboard, and mark off the board feet. 4. If the board in question 3 were twice as thick, how many board feet would it contain ? How many inches thick would it be ? 5o If it were five times as thick, how many board feet would it contain? 6. How many board feet are there in a piece of board 1 ftc wide, 16 ft. long, and 1 in. thick? 7. If this piece of lumber were 2 in. thick, how many board feet would it contain ? 3 in. ? 4 in. ? 5 in. ? 6 in. ? 8. A piece of inch board 3 ft. long must be how wide to con- tain 1 bd. ftt 220 LUMBER MEASURE 9. A cubic foot of wood could be sawed into how many board feet if there were no waste in sawing? The number of board feet in any piece of lumber is how many times as great as the number of cubic feet ? 269. We may find the number of board feet in a piece of lumber by multiplying the number of cubic feet by 12. The rule commonly used by dealers and mechanics gives the same result, and is stated as follows : To find the number of hoard feet in any piece of lumber^ mul- tiply together its three dimensions, two of them expressed in feet and the other in inches. Lumber that is less than 1 in. thick is counted as 1 in. thick in measuring. 270. Oral 1. About how many feet of lumber (board feet) are there in the top of your desk ? The teacher's desk ? One end of the bookcase ? The cupboard door ? All the shelves in the bookcase ? 2. How much lumber is used in making this box with cover ? (Take outside measurements.) 3. Estimate the amount of lum- ber in a cubical box, including the cover, made of |^-inch lumber, the length of the box being three feet. 4. The door in this hayloft is 4|'x4'. Two battens across the inside are 4' x 6'^ How many feet of lumber are used in the door, the lumber being 1'' thick ? LUMBER MEASURE 221 271. Written 1. How many board feet are there in a piece of timber 18' X le^'x 3^'? (18'xi|'x 3'^= bd. ft.) 2. The floor of a tent 12 ft. by 16 ft. is made of boards 1 in. thick laid close together, a. How many feet of lumber are used ? h. How much is it worth at $ 30 per M. (thousand feet) ? a. Find the number of feet of lumber in this stick. Walk as far on the floor as the length of this piece of timber. Show with your hands how high it is. Show how wide it is. How many feet of lumber does it contain ? h. What are twenty such sticks worth at f 26 per M. ? a. Find the amount of lumber in this plank. 5. Find the cost of ten such planks at | 30 per M. 5. The floor of your schoolroom is |^ in. thick, a. If no allowance is made for sawing and matching, how many feet of lumber are there in the floor? h. If \ of the lumber was wasted in sawing and matching, the floor contains only ^ of the lumber that was bought. How much lumber was bought ? c. What did it cost at ^ 42 per M.? 6. A fence like this, 6 ft. high, extends around two sides and one end of a rectangular garden 40 ft. by t)^ ft. Draw a diagram of the garden, a. How many feet of boards were used ? h. How much did they cost at $ 27 per M. ? 222 LUMBER MEASURE 7. The floor of this bridge is 14 ft. by 8 ft., and made of oak planks 3 in. thick. What did they cost at 1 45 per M. ? 8. Is there a board fence at the rear or side of your school ground ? If so, find the number of feet of boards in it, as a part of to-morrow's lesson. 9. Find the cost of each of the following quantities of lumber : Number of Pieces Dimensions Price per M a. 21 3^^x 12'' X 18' 124 h. 10 ^" X 6" X 20' . $26 c. 75 2"x 4" X 20' $26 d. 6 10^' X 14" X 30' $ 32" e. 3 11'' X 11" X 10' $35 /. 49 l''x 5" X 16' $30 9- 60 2''x 10" X 14' $24 h. 72 2''x 8" X 14' $24 i. m 2''x 6" X 14' $25 h 121 rx 3" X 12' $28 h. 2 6''x 10" X 22' $25 L . 4 6"x 10" X 16' $25 m. 4 2''x 8" X 18' $26 n. 8 S^x 8" X 14' $27 0. 16 y'x 10" X 12' $40 P- 7 2J''x 8" X 16' $45 ?• 600 rx 51" X 10' $42 REVIEW AND PRACTICE 223 REVIEW AND PRACTICE 272. Oral 1. Read: 1,003,050 ; XCIX; MCM ; CDIV; CLI ; 2,050,205. 2. What is the greatest common divisor of 12, 28, and 56? 3. What is the smallest number that exactly contains 4, 5, and 8 ? 4. Give five multiples of 11. 5. Name all the composite numbers smaller than 30. 6. Name the prime factors of 42. 7. What common fraction in lowest terms equals .33J? .871? .625? .37-1? .20? .80? .60? 8. What decimal is equal to f ? J ? | ? f ? 9. Give results: .8x7; .12-J-6; 1.5x.6; 1.03x5; I 7.01 X. 6. 10. Give results : 2000 ^ 20 ; 3.528 x 100 ; 53 -r- 1000 ; 950-50; 2.3x1000. 11. 10 is how many times ^ ? 12. What is the cost of 24 oranges, when 8 oranges cost 25 cents ? 13. What is the difference between^ of 10 and ^ oi 10? 14. If 21 yd. of cloth cost 1 7 J, what will 5 yd. cost ? 15. f of 20 are how many times 4 ? 16. J is contained in 7 how many times? 17. Give results: J-i; ^-^1 1-i; -i + J; l + l; | + 1; A X i' 18. What is the cost of three bananas at 20 cents a dozen ? 224 REVIEW AND PRACTICE 19. What will 4 doz. steel screws cost at 15 i a gross ? 20. If 3 boys can shovel a walk in 15 minutes, how long should it take 1 boy ? 21. When 3 eggs cost 5 cents, what is the cost per dozen ? 22. A rectangle 18 in. by 2 in. contains what fraction of a square foot of surface? 23. \ of 25 is what part of 30? 24. Give results : 41 - f ; | of f ; 4 - If 25. A man had a sum of money. He earned \ as much and then had |9. How much had he at first ? 26. \ inch is what part of a foot ? 27. 40 rd. are what part of a mile ? 28. 40 sq. rd. are what part of an acre ? 29. How many boys are there in a class of 45 pupils if \ of the pupils are girls ? 30. How many cubic feet of stone are there in a stone wall 8' X 4' X 2' ? 31. Ethel cut 20 roses one morning, of which 40 % were red. How many red roses did she cut? 32. One dozen is what per cent of one gross ? 33. A peddler sold 36 pencils at the rate of 2 for 5 cents. What did he receive ? 34. 48^3 + 36-50 = ? 35. 28 days are called a lunar month. One week is what per cent of a lunar month ? 36. How many days were there in February, 1906 ? In Feb- ruary, 1493 ? REVIEW AND PRACTICE 225 273. Written 1. See how quickly you can obtain correct answers and test them: a. The addends are 4,798, 6,430, 895, 49,785, 231, 8,942, 700, 98,346, 209, 98,020, 49, 816. Find the sum. h. The product is 910,386, one factor is 4,398. Find the other factor. e. 7,695 and 493 are the factors that make what number ? d, 2,983, 4,978, 9,399, and 16,897 are the parts that make what number ? e. 83,469 added to what number will make 103,497 ? 2. Find the least number that will exactly contain each of these numbers: 28, 35, 20,42. 3. What is the number whose prime factors are 2, 3, 5, 7, 11 and 17? 4. How many times is the G. C. D. of 48, 36, and 72 con- tained in their L. C. M. ? 5. Reduce to common fractions in simplest form: a. .025 ; h. .375; c. .3125; d. .8375; e. .1275; /. .30; g. .625. 6. Reduce -^^ to a decimal, and write the answer in words. 7. The product of three numbers is l^^. Two of the num- bers are 2^ and 1^^. Find the other number. 8. Which is larger, and how much, | of 64 or |^ of 45 ? 7-^ 9. Change -^ to its simplest form. 8 10. I of 72 is if of what number ? 11. Mrs. Hill's new curtains cost $85.40. This was f as much as her carpets cost. What was the cost of both curtains and carpets ? 226 REVIEW AND PRACTICE 12. In a certain year 200,000 typewriting machines costing $12,500,000 were made in the United States by 10,000 men. a. What was the average cost of the machines ? h. At the average rate, how many men were needed to conduct a factory that turned out 20,000 machines in a year ?