aa

10. a bab; a2 + b2 : (a + b)2; (a2 — b2)2 : a* — ba.

11. Two numbers are in the ratio 2: 3, and if 9 is added to each, they are in the ratio 3:4. Find the numbers.

Let 2x and 3x represent the numbers.

12. Show that the ratio a: b is the duplicate of the ratio a+c:b+c, if c2 = ab.

Their sum is to Find the numbers.

13. Two numbers are in the ratio 3: 4. the sum of their squares as 7:50.

14. If five gold and four silver coins are worth as much as three gold and twelve silver coins, find the ratio of the value of a gold coin to that of a silver coin.

15. If eight gold and nine silver coins are worth as much as six gold and nineteen silver coins, find the ratio of the value of a silver coin to that of a gold coin.

16. There are two roads from A to B, one of them 14 miles longer than the other; and two roads from B to C, one of them 8 miles longer than the other. The distance from A to B is to the distance from B to C, by the shorter roads, as 1 to 2; by the longer roads, as 2 to 3. Find the distances.

17. What must be added to each of the terms of the ratio m:n, that it may become equal to the ratio p: q?

18. A rectangular field contains 5270 acres, and its length is to its breadth in the ratio of 31:17. Find its dimensions.

Proportion.

332. An equation consisting of two equal ratios is called. a proportion; and the terms of the ratios are called proportionals.

333. The algebraic test of a proportion is that the two fractions which represent the ratios of the quantities compared shall be equal.

Thus, the ratio a:b is equal to the ratio cd if the fraction that represents the ratio a :b is equal to the fraction that represents the ratio cd. Then the four quantities, a, b, c, d, are called proportionals, or are said to be in proportion.

334. If the ratios ab and c d form a proportion, the proportion is written

(read the ratio of a to b is equal to the ratio of c to d),

(read a is to b in the same ratio as c is to d).

The first and last terms, a and d, are called the extremes. The two middle terms, b and c, are called the means.

=

335. In the proportion a: b cd; d is called a fourth proportional to a, b, and c.

In the proportion a:bb: c; c is called a third proportional to a and b.

In the proportion a: b = b:c; b is called a mean proportional between a and c.

336. A continued proportion is a series of equal ratios in which each consequent is the same as the next antecedent.

Thus, a : b = b c = c : d=d:e=e:ƒ is a continued proportion.

337. When four quantities are in proportion, the product of the extremes is equal to the product of the means.

extreme may be found by dividing the product of the means by the other extreme; and a mean may be found by dividing the product of the extremes by the other mean.

NOTE. By the product of two quantities we mean the product of the two numbers that represent them when the quantities are expressed in a common unit.

338. If the product of two quantities is equal to the product of two others, either two may be made the extremes of a proportion and the other two the means.

339. Transformations of a Proportion. If four quantities, a, b, c, d, are in proportion, they will be in proportion by:

I. Inversion; that is, b will be to a as d is to c.

II. Composition; that is, a +b will be to b as c + d is to d.

III. Division; that is, a b will be to b as c

а

b

b

a

IV. Composition and Division; that is, a + b will be to

and from III,

Divide,

d

d

=c+d:c-d.

V. Alternation; that is, a will be to c as b is to d.

NOTE. In order for four quantities, a, b, c, d, to be in proportion, a and b must be of the same kind and c and d of the same kind; but c and d need not necessarily be of the same kind as a and b. In applying alternation, however, all four quantities must be of the same kind.

340. In a Series of Equal Ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

.. a + c +e+g: b + d + f + h = a; b.

In like manner it may be shown that

ma + nc + pe + qg : mb + nd + pf + qh = a : b.