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326. If both the antecedent and consequent of a ratio be 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 X 2 is 4; and of 8 ; 2:2 = 2 is 4.

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OPERATION,

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

328. To reduce a ratio to its lowest terms.
Ex. 1. Reduce 18:9 to its lowest terms. Ans. 2:1.

We cancel in the two terms the 18:9= = * = 2:1. common factor 9, and obtain 4 =

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. š. 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 X 5 X 2 X 3 : 15 X 12 X 38 ?

329. To reduce a complex or a compound ratio to a simple one. Ex. 1. Reduce 51 : 4 to a simple ratio. Ans. 22 : 3.

We express the 54 53 : = 44 = = = 22 : 3 Ans. given ratio in the

form of

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

8 5 2. Reduce to a simple ratio.

Ans. 7 : 15.

OPERATION.

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7:

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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 15 7:15, the answer required. Hence, to reduce a complex or a compound ratio to a simple one,

Proceed as in like operations with fractions.

EXAMPLES

Ans. 35 : 24.

Ans. 1:2.
Ans. 2 :1.

Ans. 5 : 8.

* 3. Reduce 5 : 4 to a simple ratio.
x 4. Reduce 131 : 27 to a simple ratio.
5. Reduce 6.25 : 3.125 to a simple ratio.

4:16 6. Reduce

25 : 10)

to a simple ratio.

6 > 7. Reduce 9:27 Sto a simple ratio.

108 : 12 8. Reduce 121 : 61

to a simple ratio.

3 :

Ans. 3 : 2.

Ans. 6:1.

OPERATION.

Ans. š.

OPERATION.

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Ans.

330. To find the ratio of one number to another. Ex. 1. Required the direct ratio of 108 to 9. Ans. 12.

Since 9 is the unit or standard of 108 : 9 = 198 = 12 Ans.

comparison, we make it the conse

quent (Art. 111) and the 108 the antecedent of the ratio, and obtain 108 12 Ans. 2. Required the inverse ratio of 72 to 8.

We divide the consequent 72 : 8 inverted नी

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 a

| Ans. Hence, 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.

Ans. 3. 4. What is the ratio of 2 yards 2 quarters to 9 yards ?

3. What is the ratio of 39 to 13 ?

Ans. 1

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

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

Ans. 7. What is the inverse ratio of 24 : 6? 8. What part of 36 is 4 ?

Ans. s. 49. 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 X 4 X 3 : 12 x 16 X 2 exceed that of 60 - (3 X 5): 20 X 2=8? Ans. 18.

12. What is the inverse ratio of .02 : 2.503 ?

13. Which is the greater, the ratio of 1 of } : f of ļ, 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. 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. 1, 4, 3, 2, 165, &c. 17. The ratio of the circumference of a circle to its diameter is 3.141592. Required approximate values for this ratio.

Ans. 3, 47, 138, iis, &c., or 3, 34, 3166, 3195, &c.

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. 201b. of iron, at $ 60 a ton ?

Ans. $ 887.85.

OPERATION

$ 60.00 = cost of 1 ton.

14

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$ 840.00

14 tons. (10cwt.: 1 ton=*); } of $ 60= 30.00 10cwt. (5cwt.: 10cwt.= }); } of $30 =

15.00 = 5cwt. (2qr. : 5cwt. 1u); It of $15 1.50

2qr. (1qr. : 2cwt. = }); of $ 1.50 = 0.75 =

1qr. (15lb. : 3qr. = }); } of $ 2.25 = 0.45 = 15lb. (51b. : 151b. = }); of $ 0.45 = 0.15 = 5lb.

Ans. $ 887.85 = 15 T. 3qr. 201b. 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 20cwt. = 1, 10cwt. will cost ļ as much as 1 ton, or $ 30; and as the ratio of 5cwt. to 10cwt. = 1, 5cwt. will cost as much as 10cwt. or $ 15. 3qr.

2qr. + Iqr. Since the ratio of 2qr. to 5cwt. or 20qr. 1o, 2qr. will cost to as much as 5cwt., or $ 1.50; and as the ratio of lqr. to 2qr. 1, lqr. 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 f 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 ?

Ans. $ 1437.50. 13. What cost 16cwt. 3qr. 10lb. of guano, at $ 2.50 per cwt. ?

4. What cost 27cwt. lqr. 20lb. 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. lpk. 4qt. of wheat, at $1.92 per bushel ?

Ans. $ 190.80.

per acre

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The quantity being nearly 100 bushels, we find the cost of 100 bushels by annexing two ciphers to $ 1.92, the cost of 1 bushel, and obtain $ 192, from which we subtract the cost of 2pk. 4qt., the difference of quantity between that given and 100 bushels; the cost of 2pk. $ 0.96 ; and that of 4qt. $ 0.24 ; $ 192 $ 0.96 + $ 0.24 $ 190,80 Ans. 8. What cost 19yd. 3qr. 2na. of cloth, at $ 4.40 per yard?

Ans. $ 87.45. $ 9. How much must be paid for 24A. 3R. 20p. of land, at $ 32 per acre ?

X10. How much must be paid for 1991b. 12oz. of butter, at $ 0.30 per lb. ?

Ans. $ 59.925. X 11. What cost 714 yards of broadcloth, at 15s. 6d. per yard ?

Ans. 553£. 7s. \ 12. How much must be paid for the services of a man 2y. 9mo. 15da., at $ 450 yer year?

Ans. $ 1256.25. 1 13. If 1 acre of land cost $ 80.50, what will 25 acres 2 roods 35 rods cost?

Ans. $ 2070.35+. 14. What cost 498lb. of tea, at 2s. 6d.

per

lb. ? *15. If lcwt. 2qr. 12lb. of alum can be purchased for $ 4.05, how much can be purchased for $28.35?

Ans. 11cwt. lqr. 91b.

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OPERATION.

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$ 28.35; $ 4.05 = 1cwt. 2qr. 121b. X 7 = 1lcwt. lqr. 91b. Ans. Since the ratio of $ 28.35 to $ 4.05 = 7, $ 28.35 will purchase 7 times as much as $4.05. By multiplying what the latter will purchase by the ratio, we have the answer required.

* 16. If ilgal. 3qt. lpt. of molasses cost $5.83, what will 35gal. 2qt. lpt. cost?

Ans. $ 17.514. 17. If 24yd. 3qr. of cloth cost $ 49.50, what will 12yd. lqr. 2na. cost?

18. If 17bu. 2pk. 4qt. of oats be paid for 14bu. 3pk. of salt, nat quantity of oats must be paid for 73bu. 3pk. of salt?

Ans. 88bu. Opk. 4qt. * 19. If $ 9.75 will purchase 1T. 2cwt. 2qr. 15lb. of coal, how much will $ 3.25 purchase ? * 20. If a train of cars move at the average velocity of 27m. 3fur. 20rd. per 1h: 20m., how far will it move in 4h. ?

Ans. 82m. 2fur. 20rd.

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