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versely as another, when the one increases in the same ratio as the other de

creases.

NOTE 2.

The word ratio, when used alone, means the direct ratio.

319. When the antecedent and consequent of a ratio are equal, the ratio equals 1, and is called that of equality. Thus, the ratio of 6: 6 88 1, and the ratio of 6 X 4:8 X 3

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1, are ratios of equality. But if the antecedent is larger than the consequent, the ratio is that of greater inequality, and if the antecedent is smaller than the consequent, the ratio is that of less inequality. Thus, the ratio of 15 : 5 = 15 = 3, is a ratio of greater inequality; and the ratio of 7: 14 =, is a ratio of less inequality.

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320. The ratio of two fractions having a common numerator is the same as the inverse ratio of their denominators. Thus, the ratio of: § is ? ÷ 3 = 2, which is the inverse ratio of the denominator 4 to the denominator 8.

321. The ratio of two fractions having a common denominator is the same as the ratio of their numerators. Thus, the ratio ofis & ÷ 93/ 2, which is the ratio of the numerator 6 to the numerator 3.

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322. The inverse or reciprocal ratio of two numbers denotes what part or multiple the consequent is of the antecedent. Thus, inquiring what part of 4 is 3, or what part 3 is of 4, is the same as inquiring the inverse or reciprocal ratio of 4: 3. The inverse ratio of 4 : 3 is 2, and 3 is 2 of 4.

323. In order to compare one number with another, by ratio, it is necessary that they should not only be of the same kind, but of the same denomination. Thus, to compare 2 days with 12 hours, it is necessary that the days be reduced to hours, before we can indicate the ratio, which is 48 hours: 12 hours.

324. If the antecedent of a ratio be multiplied, or the consequent divided, the ratio is multiplied. Thus, the ratio of 6 3 is 2, but 6 X 2 : 3 is 4; or 6: 32 is 4.

325. If the antecedent of a ratio be divided, or the consequent multiplied, the ratio is divided. Thus, the ratio of 18: 6 is 3, but 183: 6 is 1; or 18 : 6 × 3 = 1.

326. If both the antecedent and consequent of a ratio he multiplied or divided by the same number, the ratio is not altered. Thus, the ratio of 8: 2 is 4; of 8 X 2 : 2 × 2 is 4; and of 82: 22 is 4.

REDUCTION AND COMPARISON OF RATIOS.

327. Ratios, being of the nature of fractions, may be reduced, compared, and otherwise operated upon like them.

328. Ex. 1.

To reduce a ratio to its lowest terms.

Reduce 18: 9 to its lowest terms.

OPERATION.

18:9= 18 = 2:

= 2: 1.

Ans. 2: 1.

We cancel in the two terms the common factor 9, and obtain = 2: 1, the answer. Hence

Cancel in the given ratio all factors common to its terms.

EXAMPLES.

2. Reduce to its lowest terms 63: 72.

Ans. .

3. Reduce to its lowest terms 66: 24. 4. Reduce to its lowest terms 4 X 6 X 3:8 X 9 X 2.

Ans.

5. What are the lowest terms of 19 × 5 × 2 × 3: 15 X 12 X 38?

329. To reduce a complex or a compound ratio to a simple one.

Ex. 1. Reduce 5 to a simple ratio.

Ans. 22: 3.

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changed to a simple fraction (Art. 242), and reduced to its lowest terms, gives 22 22: 3, the answer required.

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We express the given ratio in the form of a compound fraction.

which, reduced to a simple one (Art. 329), gives

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answer required. Hence, to reduce a complex or a compound ratio

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330. To find the ratio of one number to another.

Ex. 1. Required the direct ratio of 108 to 9.

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Ans. 12.

Since 9 is the unit or standard of comparison, we make it the consequent (Art. 111) and the 108 the 12 Ans.

antecedent of the ratio, and obtain 108

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We divide the consequent 8 by the antecedent 72, or, which is the same thing, find

the reciprocal of the direct ratio of 72: 8 (Art. 318), by inverting its terms, and thus obtain Ans. Hence,

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The direct ratio is found by dividing the antecedent by the consequent, and the inverse ratio by dividing the consequent by the antecedent.

NOTE 1. Ratios expressed by fractions having different denominators must be reduced to a common denominator, in order to be compared; and then they are to each other as their numerators (Art. 323).

NOTE 2. - When a ratio is expressed in terms inconveniently large and prime to each other, we may find the approximate values of the ratio expressed in smaller numbers, as in other fractional expressions (Art. 309).

EXAMPLES.

3. What is the ratio of 39 to 13?

Ans. 3.

4. What is the ratio of 2 yards 2 quarters to 9 yards?

5. What is the ratio of 21 gallons to of a hogshead?

Ans. 1.

6. What is the ratio of of of $ 2 : 1⁄2 of $0.50 ?

Ans. .

7. What is the inverse ratio of 24: 6?

Ans. 4.

Ans..

8. What part of 36 is 4?

9. What part of a farm of 94A. 2R. 16rd. is 11A. 3R.? 10. Which is the greater, the ratio of 17 to 9, or of 39 to

19?

Ans. 39: 19.

11. By how much does the ratio of 36 × 4 ×

3: 12 × 16 Ans. 13.

× 2 exceed that of 60 ÷ (3 × 5) : 20 × 2 ÷ 8? 12. What is the inverse ratio of .02 : 2.503? 13. Which is the greater, the ratio of of: of, or that of 5: 4?

14. The height of Bunker Hill Monument is 220 feet, and that of the great pyramid, Egypt, 500 feet; what is the ratio of the height of the former to that of the latter?

Ans.

15. A certain farm contains 180 acres, and the township of which it forms a part is 36 square miles in extent. What is the ratio of the latter to the former?

16. Find approximate values for the ratio of 4900 to 11283. Ans., 4, 48, 48, 7765, &c.

17. The ratio of the circumference of a circle to its diameter is 3.141592. Required approximate values for this ratio.

333 355

Ans. 3, 22, 138, 115, &c., or 3, 34, 3,1%, 3,163, &c.

106

ANALYSIS BY RATIO.

331. Operations by analysis may often be much abridged by ratio. Thus, frequently, it is more convenient to multiply or divide by the ratio a number bears to a unit of the same kind, than to multiply or divide by the number itself.

This form of analysis is much used by business men; and, like that by aliquot parts (Art. 114), is sometimes called Practice.

EXAMPLES.

1. What cost 14 tons 15cwt. 3qr. 20lb. of iron, at $60 a Ans. $887.85.

ton?

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20cwt. = , 10cwt. will cost ratio of 5cwt. to 10cwt.

$15. 3qr.

20qr.

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2qr.1qr.

1, 2qr. will cost

Since 1 ton costs $60, 14 tons will cost 14 times $60, or $840. 15cwt. = 10cwt. 5cwt. Since the ratio of 10cwt. to 1 ton or as much as 1 ton, or $30; and as the 5cwt. will cost as much as 10cwt. or Since the ratio of 2qr. to 5cwt. or as much as 5cwt., or $1.50; and as the ratio of 1qr. to 2qr. = 1, 1qr. will cost as much as 2qr., or $0.75. 20lb. 15lb. 516. Since the ratio of 15lb. to 3qr. or 75lb. , 15lb. will cost as much as 3qr., or $0.45; and as the ratio of 5lb. to 15lb. ,5lb. will cost as much as 15lb., or $0.15. The cost of the several parts equals the cost of the whole, or $887.85, Ans.

2. What is the value of 17 acres 3 roods 35 rods of land, at $80 per acre? Ans. $1437.50. 3. What cost 16cwt. 3qr. 10lb. of guano, at $2.50 per cwt.? 4. What cost 27cwt. 1qr. 201b. of coffee, at $14 per cwt.? Ans. $384.30. 5. If 1 yard of cloth cost $5.60, what will 7yd. 3qr. 2na. cost? Ans. $44.10. 6. What cost 7 tons 13cwt. 2qr. of hay, at $20 per ton? 7. What cost 99bu. 1pk. 4qt. of wheat, at $1.92 per bushel?

Ans. $190.80.

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