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(d.) Dividing the divisor multiplies the quotient; thus,

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that is, dividing the divisor by 5 multiplies the quotient by 5.

(e.) It follows, from (a) and (b), that the greater the dividend, the greater is the quotient; and the less the dividend, the less the quotient.

(f.) Also, from (c) and (d), that the greater the divisor, the less is the quotient; and the less the divisor, the greater the quotient.

80. From the illustrations in Art. 79 we see that any change in the dividend causes a similar change in the quotient, and that any change in the divisor causes an opposite change in the quotient. Hence,

(a.) Multiplying both dividend and divisor by the same number does not affect the quotient; thus,

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(b.) Dividing both dividend and divisor by the same number does not affect the quotient; thus,

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(c.) It follows from (a) and (b), that the operations of multiplying and dividing by the same number cancel (that is, balance) each other; for example,

If a number is multiplied by any number, and the product divided by the multiplier, the quotient will be the multiplicand; thus,

8 x 756, and 567 8, the multiplicand.

=

Also, if a number is divided by any number, and the quotient multiplied by the divisor, the product will be the dividend; thus,

15 ÷ 3 = 5, and 5 × 3 = 15, the dividend.

81. These general principles may be more briefly stated as follows:

1. Multiplying the dividend multiplies the quotient; and dividing the dividend divides the quotient (Art. 79, a and b).

2. Multiplying the divisor divides the quotient; and dividing the divisor multiplies the quotient (Art. 79, c and d).

3. Multiplying both dividend and divisor by the same number, or dividing both by the same number does not affect the quotient (Art. 80, a and b).

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135. (42367) x 4 + 3 x 4-2166=?

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140. 849 X 43 714 x 4 ÷ 3 - 135 x 4 ÷ 3 = ?

83. Oral Exercises for Review.

1. In a certain orchard there are 43 apple-trees, 10 peachtrees, 7 pear-trees, 8 cherry-trees, and 6 quince-trees; how many trees are there in the orchard?

2. A lady bought a silk dress for 22 dollars, a bonnet for 9 dollars, a shawl for 6 dollars, a pair of shoes for 2 dollars, a pair of gloves for 1 dollar, and had 10 dollars left; how many dollars had she at first?

3. A man paid 75 dollars for a horse, 9 dollars for a saddle, 3 dollars for a bridle, and 1 dollar for a whip; what did they all cost him?

4. A miller bought 54 bushels of wheat, 10 of rye, 8 of corn, and 3 of buckwheat; how many bushels of grain did he buy?

5. David has 37 cents in his purse, 8 in one pocket, 9 in another pocket, and 7 in his hand; how many cents has he?

6. James had 37 cents, and George gave him 7, Charles 6, Samuel 9, John 8, and he bought a top for 10 cents; how many cents had he left?

7. A lady gave 50 dollars for a watch, 9 dollars for a chain, 1 dollar for a key, and 5 dollars for a pin; how many dollars did she give for all?

six;

8. Two men start from the same place and travel in opposite directions, one at the rate of two miles in an hour, and the other how far apart are they in 7 hours? How far in 10 hours? 9. If the interest of one dollar is six cents a year, what is the interest of three dollars for two years?

10. How many oranges, at 4 cents apiece, must be given for 12 lemons, at 3 cents apiece?

11. How many pounds of beef, at 9 cents a pound, will pay for 6 pounds of nuts at 12 cents a pound?

12. If 8 men can do a certain piece of work in 6 days, in how many days can 4 men do the same?

13. If 9 men can do a piece of work in 8 days, how many men can do the same in 12 days?

14. If 4 men can do a piece of work in 6 days, in how many days can 3 men do twice as much work?

15. A farmer sold 2 tons of hay at $20 a ton, and for pay received 8 yards of cloth at $4 a yard, and the rest in money; how much money did he receive?

16. If a man earns $24 in 2 months, what will 2 men earn in 6 months?

17. At 12 cents a pound, how many pounds of beef can be bought for 144 cents? For 96 cents?

18. How many wheelbarrows, at 7 dollars apiece, can be bought for 84 dollars? For 63 dollars?

19. A boy sold 6 doves for 72 cents; how many cents did he receive for 1 dove? For 3 doves? For 4 doves?

20. A boy sold chickens at 25 cents apiece, and received 1 dollar, or 100 cents, for them; how many chickens did he sell? 21. If syrup is worth 20 cents a pint, how many pints may be bought for 1 dollar?

22. What is the price of a shawl, if 9 shawls cost $108?

23. A boy divided 80 chestnuts equally among 5 of his companions. How many did he give to each?

24. If 8 quarts make a peck, how many pecks will 96 quarts make? 84 quarts? 48 quarts?

25. A man divided $84 equally among his 3 sons and 4 daughters; how many dollars did he give to each ?

26. A boy having 7 melons, gave away 2 of them, and sold the rest for $1; what did he receive for each of those he sold?

27. A man having 75 dollars, bought 7 sheep, and had $5 left; what did he pay for each sheep?

28. A boy bought 5 hens at 20 cents each, and paid for them with apples at 10 cents a dozen; how many dozen did it take?

29. A man had 75 sheep, and bought 5 more; he then distributed them equally into 8 pens; how many sheep did he put in each pen?

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30. The area of California is 155980 square miles, and of France 204092 square miles. How many square miles does California lack of being as large as France?

31. The earnings of a man and his three sons are $4475 and their expenses are $2845. If the balance is equally divided among them, what will each have?

32. How many yards of cloth in 5 bales, each bale containing 25 pieces, and each piece containing 175 yards?

33. From a farm containing 1243 acres there were sold at one time 324 acres and at another time 472 acres. How many acres remain ?

34. Paid $55 for a harness, $147 for a carriage, and for a horse as much as for the carriage and harness. What was the cost of all?

35. Bought 230 acres of land at $35 an acre, and built a house that cost $1975, and a barn for $1187.50. What was the cost of all?

36. From two million seven hundred ninety take one thousand two and forty-nine thousandths?

37. A man paid $1775 for a lot, $345.50 for digging the cellar and putting in the foundation for a house, $ 4250 for building the house, $ 1844.75 for furniture, $215.25 for grading and laying out the grounds. If he had $ 12400 when he began, how much had he left?

38. A man bought three farms; the first contained 648 acres, the second 495 acres, the third as many acres as both the others. How many acres in the three farms?

He

39. A student had $200 with which to fit up his room. spent for a carpet $ 40, for a set of furniture $ 42.50, for bedding $18.25, for curtains $6.75, for a mirror $4.75, for a bookcase $23, for an easy-chair $7.50, for a lounge $ 10, for pictures $25. How much did he have left?

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