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CANCELLATION.

73. Cancellation is the process of rejecting or cancelling equal factors from both dividend and divisor.

PRINCIPLES.—I. Rejecting a factor from any number divides the number by that factor.

II. Rejecting equal factors from both dividend and divisor does not change the quotient.

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74. 1. Divide 56 × 24 by 48 × 7.

EXPLANATION.-Since 8 will divide both 56 and 48, reject 8 as a factor of 56, retaining the factor 7, and also of 48, retaining the factor 6.

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56 × 24

48 × 7

Again, since 6 will divide both 24 in the dividend and 6 in the divisor, reject 6 as a factor from both, retaining the factor 4 in the dividend. Finally, rejecting the factor 7, common both to the dividend and to the divisor, there remains only the factor 4 in the dividend, which is the required quotient.

2. Divide the product of 25, 18, 4, and 3, by the product of 7, 6, 5, and 3.

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1. The vertical line is much used, and by many is thought to be more convenient, wherever cancellation is applied. It serves the same purpose as the curved line in division, since it separates the dividend or its factors on the right from the divisor or its factors on the left.

2. The horizontal line may be used by those who prefer it.

3. Divide 13 x7×5×3 by 3 × 5 × 7.

4. Divide 42 × 18 × 6 × 4 by 36 × 21 × 6.

84

RULE.-Cancel the factors common to both dividend and divisor, and divide the product of the remaining factors of the dividend by the product of the remaining factors of the divisor; the result will be the required quotient.

5. What is the quotient of 21 × 8 × 60 × 8 × 6 divided by 7 × 12 × 3 × 8x3?

6. Multiply 8 times 32 by 6 times 27, and divide the product by 9 times 96.

7. How many tons of hay, at $18, must be given for 45 cords of wood, at $4 a cord?

8. How many flour barrels, at $.80 each, will pay for 112 bushels of corn, at $.70 a bushel?

9. How many tubs of butter, each containing 56 pounds, at 30 cents a pound, must be given for 7 barrels of sugar, each containing 195 pounds, at 10 cents a pound?

10. How many bushels of apples, at 75 cents a bushel, will pay for 95 pounds of coffee, at 30 cents a pound?

11. A farmer exchanged 45 bushels of potatoes, worth 40 cents a bushel, for 18 pounds of tea. What was the tea worth a pound?

12. From 6 pieces of cloth, each piece containing 24 yards, worth $3 a yard, how many suits can be made, worth $18 a suit?

13. When hay is worth $18 a ton, how many tons must be given for 12 barrels of flour, at $7 a barrel ?

14. If 6 acres of land cost $180, what will 150 acres cost? 15. How many bales of cloth, each containing 40 pieces, and each piece containing 36 yards, worth $3 a yard, must be given for 120 government bonds, worth $108 each?

16. (240 x 56 x 18)+(60 x 28 x 9) = ?

17. (72 × 48 × 28 × 5)÷ (84 x 15 x 7 x 6) = ?

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18. (66 x 18 x 27 x 25)÷(84 x 45 x 7 x 30) = ?

19. (80 × 60 × 50 x 16 x 14)÷(70 × 50 × 24 × 20) = ?

20. How many bushels of oats, at $.60 a bushel, will pay for 12 tons of coal, at $7.20 a ton?

21. How many days' work, at $1.25 a day, will pay for 75 bushels of corn, at $.80 a bushel ?

22. A laborer gave 12 days' work for 48 bushels of potatoes, worth 50 cents a bushel. What were his daily earnings?

REVIEW.

WRITTEN EXERCISES.

75. 1. How many times is 246 contained in 3690 ?
2. How many times can 246 be subtracted from 3690 ?
3. How many times 246 will produce 3690?

4. The factors of a number are 84 + 23 and 427 - 259. What is the number?

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5. The product of two numbers is 14304, and one of the numbers is 240025. What is the other?

6. If 2, 2, and 7 are three factors of 364, what is the other factor?

7. What is the difference between 106 × 127, and 15341÷29? 8. The product of three numbers is 107100; one of the numbers is 42, and another 34. What is the third number ? 9. The quotient of one number divided by another is 56, the divisor is 375, and the remainder 124. What is the dividend?

10. A coal dealer bought 250 tons of coal, receiving 2240 pounds for a ton; he sold it at 2000 pounds for a ton. How many tons did he gain?

11. A farmer sold 300 bushels of wheat, at $2 a bushel, and corn and oats to the amount of $750; with the proceeds he bought 120 head of sheep at $3 a head, one pair of oxen for $90, and 25 acres of land for the remainder. How much did the land cost him an acre?

12. How many pounds of sugar worth 13 cents a pound, can be bought for 126 pounds of butter worth 28 cts. a pound? 13. If a man earns $60 a month, and spends $45 in the same time, in what time will he save $900 ?

14. A grocer bought two kinds of a gallon, and the other for 62 cents.

cost a gallon?

syrup; one for 54 cents What was the average

SOLUTION. (54 cents + 62 cents) ÷ 2 = 58 cents.

The average of two numbers is one half their sum, the average of three numbers is one third their sum, etc.

15. A merchant bought three kinds of tea, some at 60 cents, some at 78 cents, and some at 90 cents a pound. What was the average cost a pound?

16. A keeper of a toll bridge received $104 toll on Monday, $97 on Tuesday, $128 on Wednesday, and $99 on Thursday. What was the average daily receipts?

17. Sold three city lots for $1500, $2976, and $1895, respectively. What was the average price of each?

18. If a young man receive a salary of $25 a week, and he pays $8.75 for his board, and $4.65 for other expenses, in how many weeks can he pay a debt of $487.20?

19. A man having $4578, paid out all but $1642 in 8 weeks. What was the average amount paid out each week?

20. Bought 140 acres of land for $7560, and sold 86 acres of it at $75 an acre, and the remainder at cost. How much was gained?

21. The sales of a merchant are $75000 a year. What is the average per week? Per month?

22. How many pounds of sugar, at 20 cents a pound, must be given for 5 pieces of cotton cloth, containing 44 yards each, at 42 cents a yard?

23. Sold a quantity of wood for $2492, that cost $1424, thus gaining $3 a cord. How many cords were there, and what was the cost per cord?

24. Bought 150 acres of land for $7650, and sold it for $63 an acre. How much was the whole gain?

25. If 100 barrels of flour cost $600, what will 350 barrels cost at the same rate?

26. How many pounds of butter, at 40 cents a pound, are worth as much as 1600 bushels of oats, at 75 cents a bushel? 27. If a man gain $638.75 by selling 365 barrels of flour at $9.25 a barrel, at what price did he buy it?

28. A grocer bought 12 barrels of sugar at $16 per barrel, and 17 barrels at $13 per barrel; how much would he gain by selling the whole at $18 per barrel ?

29. A farmer sold 16 cords of wood at $3 a cord, and 40 bushels of wheat at $2 a bushel. He received 15 yards of cloth at $4 a yard, and the remainder in money; how much money did he receive?

30. If a mechanic receive $1500 a year for his labor, and his expenses are $968, in what time can he save enough to buy 28 acres of land at $133 an acre?

31. A planter sold 40 bales of cotton at $68 a bale. With the proceeds he bought 8 horses at $180 each, and with the remainder bought cows at $32 a head. How many cows did he purchase?

32. Bought 288 barrels of flour for $1482, and sold the same for $2058. What was the gain on each barrel?

Find the second member in the following equations :

33. (6300÷15)-(372-3)=?

34. (1246010) x (475-377) = ?

35. 450+(12 × 5)÷+125

?

36. (6070-1200)+(4680÷15) = ?

37. 384+4621-872+(3176÷8×12) = ?

38. 25 × 370-62 × 8+ (335+206) × 45 = ?

39. 16 x 3125-127x0+(972-122)÷50 = ?

40. (125 × 30)÷(25 × 25) × (324—216)+264—450÷5 = ?

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