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XII. RATIO AND PROPORTION

321. Ratio. The quotient of one number divided by another number of the same kind is their ratio. The former number is the antecedent and the latter is the consequent.

The ratio is usually written in the form of a fraction and its terms bear the same relation to each other as the numerator and the denominator of a fraction.

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represents the ratio of $10 to $5. The value of this ratio

is, or 2. represents the ratio of a to b. It is usually read, the ratio b

of a to b or a divided by b. The above ratios are also sometimes written $ 10: $5, and a : b. The colon is used here as a sign of division.

The value of a ratio is always an abstract number. (Why?)

ORAL EXERCISE

322. Read the following ratios and give their values:

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13. If the ratio of x to 3 is equal to 5, what is the value of x?

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14. If the ratio of x to is equal to 2, what is the value of x?

15. What number bears to 5 the ratio .3?

(Solve =.3.)

16. Can you express a ratio between $12 and 4 ft.? 4 bu. and 2 qt.? 1 rd. and 1 in.? 10 sq. in. and 2 cu. in. ?

Simplify the following ratios by treating them as fractions and reducing them to their lowest terms:

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19. Which ratio is the greater, or 3? & or?

323. Proportion. An equality of two ratios is a proportion.

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are the same kind of numbers, and c and d are also the same kind of numbers. This proportion is read, the ratio of a to b equals the ratio of c to d. The proportion is also sometimes written a b c:d, or ab::c: d. These proportions may be read, a is to b as c is to d. The fractional form is, however, much more commonly used.

EXERCISE

324. 1. What value must be given to d, if a = 1, b = 2,

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2. What is the value of d if a = 2, b = 3, c=4?

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4. Divide 60 into two parts that are in the ratio of 2 to 3.

HINT. Let x and 60 -x be the two numbers.

325. Terms of a proportion. The four numbers, a, b, c, and d are the terms of the proportion a:b = c: d. The first and fourth terms, a and d, are the extremes, and the second and third terms, b and c, are the means. The first and third terms, a and c, are the antecedents, and the second and fourth terms, b and d, are the consequents.

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326. Fourth Proportional, Third Proportional, and Mean Pro

portional. The fourth term, d; of the proportion

a с

= is the b d

fourth proportional to the other three terms taken in the order

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third proportional to a and b, and b is the mean proportional between a and c.

ORAL EXERCISE

327. In the following proportions name the extremes, the means, the antecedents, the consequents, the fourth proportionals, the mean proportionals, and the third proportionals.

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328. A proportion may be treated as an ordinary frac

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3. 6.3 x 13:20. (Write in fractional form.)

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6. Find the fourth proportional to (a) 3, 4, 6 :

41, 91; (c) a, b, c.

7. Find the third proportional to (a) 9 and 6; (b) a2 — b2 and a ― b; (c) a and b.

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8. Divide 120 into two parts which are in the ratio of 2 to 3. HINT. Let x and 120

x represent the two parts. Why?

9. Divide 182 into two parts whose ratio equals

6

1

8

10. What number added to both terms of the ratio will

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11. Find a mean proportional between 2 and 8.

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13. Divide $180 between two men so that their shares will

be in the ratio of 13 to 5.

HINT. See example 8, or let 13 x and 5 x represent the two shares.

14. Divide $180 among three men so that their shares shall bear to each other the relation 2:3:

HINT.

: 5.

This notation means that the first man's share is to the second man's share as 2 is to 3. Also the first man's share is to the third man's share as 2 is to 5. The shares may be represented by 2x, 3x, and 5x.

A

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16. Solve for y, y-7:y-3y - 11 : y - 9.

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=

this gives 2.12 3.8. This illustrates the following important property of any proportion :

I. If four numbers are in proportion, the product of the means is equal to the product of the extremes.

PROOF. Let a, b, c, and d be four numbers in proportion.

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means in This is a

The last equation states that the product of the any proportion equals the product of the extremes. test of the correctness of a proportion, or of the equality of two ratios.

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