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Aliquot Parts.

ART. 396.–An Aliquot Part of a number is any number, either integral or mixed, which will exactly divide it. The aliquot parts of a dollar are : $.50

$.124 = $1. $.33

$.10 = $10. $.25 $1.

$.08} = $1. $.20 $1.

$.064 = $16. $.163 = $4.

$.05 = $.

ORAL EXERCISES.

1. What is the cost of 28 grammars at 50 cents apiece ?

Solution. Since one grammar costs $.50, equal to $1, the cost of 28 grammars is 28 times $k, or $22, equal to $14.

2. If you pay 20 cents apiece for 35 chickens, how much do they all cost ?

3. At 163 cents a yard, what cost 48 yards of gingham ? 4. At 64 cents a quart, what cost 84 quarts of milk?

5. How much must I pay for 24 yards of muslin at 8} cents a yard ?

6. A farmer sold 64 pumpkins at 12 cents apiece : how much did he receive ?

7. A stationer bought 120 bottles of ink at 25 cents a bottle : how much did he pay ?

8. A grocer sold 16 quarts of sirup at 33 cents a quart : how much did he receive ?

9. A farmer's son sold 38 quarts of chestnuts at 121 cents a quart: how much did he receive ? 10. What is the cost of 124 fowls at 50¢ each ?

11. What are 36 quarts of nuts worth at 124 cents & quart ?

12. How much must I pay for 81 yards of sheeting at 33} cents a yard ?

13. How many bottles of ink at 67 cents each can be bought for $1 ?

WRITTEN EXERCISES 1. A dealer sold 273 pairs of rubber shoes at 33} cents a pair : how much did he receive ?

2. If a man saves 163 cents a day, how much will he save in 18 days ?

3. What cost 840 histories at 25 cents each ?

4. What must be paid for 624 bushels of potatoes at 50 cents a bushel ?

5. What is the cost of 768 copy-books at 67 cents each ? 6. What will 340 posts cost at 33} cents each ?

7. What is the cost of 18 pieces of calico, each containing 24 yards at 64 cents a yard ?

8. Find the cost of 48 yds. of carpet at $1.25 per yard. 9. What is the cost of 120 hats at $1.50 each ? 10. What cost 45 shovels at 75 cents each ?

11. What cost 8 bales of Sea Island cotton, each containing 300 pounds, at 163 cents a pound ?

12. A farmer purchased 12 yards of calico at 83 cents a yard, 24 pounds of sugar at 64 cents a pound, and 6 yards of cloth at $1.163, a yard: what was his bill ?

13. A dealer sold 15 tons of coal at $5.20 a ton : how much did he receive ?

14. A grocer bought 144 gallons of kerosene at $.127 a gallon : how much did he pay ?

15. I sold 64 cords of wood at $4.064 à cord : how much did I receive ?

16. What must be paid for 40 barrels of flour at $8.05 per barrel ?

17. What is the cost of 448 books at $1.124 each ?

Ratio and Proportion.

ART. 399.-Ratio is the relation that two similar numbers bear to each other : thus, the ratio of 6 to 3 is 2 ; of 9 to 3 is 3 ; of 12 to 2 is 6.

ART. 400.—Ratio can exist between like things only.

ART. 401.—The Sign of ratio is the colon (:); thus, 18 : 6 means the ratio of 18 to 6. Ratio is sometimes expressed in the form of a fraction. Thus, f expresses the ratio of 6 to 2.

ART. 402.—The Terms of a ratio are the two numbers compared; thus, in the ratio 24 : 8, the terms are 24 and 8.

ART. 403.—The Antecedent is the first term.

ART. 404.-The Consequent is the second term.

ART. 405.—The antecedent and the consequent together form a Couplet.

ART. 406.—A Ratio is found by dividing the antecedent by the consequent.

ART. 407.-A Simple Ratio is the ratio of two numbers; as 4:2, and 10 quarts : 4 quarts.

ART. 408.—A Compound Ratio is a ratio whose terms are the products of the corresponding terms of two or more

16:18 simple ratios. It is generally expressed thus

15:20 S This is 30 : 360, or reduced to its simplest terms, 1 : 12.

ORAL EXERCISES.

Find the ratio of

1. 16 to 4; 14 to 17. 4. 24 to 6; 6 to 24. 2. 18 to 6; 6 to 18. 5. 7 to 21 ; 21 to 7. 3. 16 to 8; 8 to 16. 6. 27 to 9; 9 to 27. 7. 2 to 20; 10 to 2; 12 to 4; 3 to 21.

8. If the antecedent is 9, the consequent 36, what is the ratio ?

9. If the consequent is 7, the antecedent 28, what is the ratio ?

10. The consequent is 16, the antecedent 32: what is the ratio ?

11. The consequent is 40, the ratio 5 : what is the antecedent?

12. The antecedent is 9, the ratio : what is the consequent ?

13. The consequent is 33, the antecedent 11 : what is the ratio ?

14. The antecedent is 15, the consequent 75: what is the ratio ?

15. The antecedent is 13, the ratio 5 : what is the consequent ?

16. The antecedent is ž, the ratio 3 : what is the consequent ?

17. The consequent is f, the antecedent i'v : what is the ratio ?

18. The consequent is 3, the ratio }: what is the antecedent?

19. What is the value of the ratio 14.3: 42.9 ? 16.25 : 3.25 ?

NOTE.-Since the antecedent represents the dividend and the consequent the divisor, factors common to both terms may be canceled.

Simple Proportion.
ART. 409.-A Proportion is an equality of ratios.

The equality of two ratios may be expressed in the usual way, by the sign of equality, but the use of the double colon (: :) for this purpose is more common. Thus, 6: 30 ::8:40 means that 6 : 30

8 : 40, that is, the ratio of 6 to 30 equals the ratio of 8 to 40, each being equal to 1.

In reading such a proportion we say, “The ratio of 6 to 30 equals the ratio of 8 to 40,” or, “6 is to 30 as 8 is to 40."

ART. 410.—The Terms of a proportion are the numbers compared.

ART. 411.–The first and fourth terms are called the Extremes ; the second and third terms are called the Means.

ART. 412.—The Couplets are the equal ratios, hence, the first couplet must be the first and second terms, and the second couplet the third and fourth terms.

ART. 413.-Simple Proportion is an equality of simple ratios.

ART. 414.—Compound Proportion is an equality of ratios of which one or both are compound.

When the first three terms of a proportion are given, the fourth can readily be found.

To show this, let take the simple proportion 3: 9::4:12. We see that the first term, or antecedent, is multiplied by 3 to make the second term. It follows, therefore, that since 4 bears the same ratio to the last term that 3 does to the second term, we have only to multiply the third term by 3 to obtain the last term.

By similar reasoning, the first term can be ascertained when the others are given. In general, when any three terms are known, the remaining one may be found.

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