If 3:69:18; then, 3+6:6:: 9+18: 18, or 9:6:: 27: 18. In a similar manner it may be shown that a+b:a::c+d: c.

274. Proposition VIII.-If four quantities are in proportion, they will be in proportion by DIVISION; that is, the difference of the first and second will be to the first or second, as the difference of the third and fourth is to the third or fourth.

If 18:6:30:10; then, 18-6:6:30-10:10, or 12: 6:20:10. In a similar manner it may be shown that a-b: a::c-d: c.

275. Proposition IX.-If four quantities are in proportion, the sum of the first and second will be to their difference as the sum of the third and fourth is to their dif ference.

Let

Then will

a: b::c: d (1), a+ba-b::c+d: c-d.

From (1), by Composition and Division, (Arts. 273, 274,)

From which, (Art. 272), a+b: c+d::a-b: c-d.

Or, by alternation,

a+ba-b::c+d: c-d.

If 6:2::12:3; then, 6+2:6—2:: 12+4:12—4, or 8: 4 :: 16: 8.

276. Proposition X.-If four quantities are in proportion, like powers or roots of those quantities will also be in proportion.

Where n may be either a whole number or a fraction.

If 2: 6:10. 30; then, 22: 62: : 102: 302, or 4: 36: 100: 900. If 8: 27:: 64: 216; then, 8:27:: 7/64: 216, or 2:3::4:6.

277. Proposition XI.—If two sets of quantities are in proportion, the products of the corresponding terms will also be in proportion.

If 3:9.2:6, and 5:15::4:12; then, 15: 135 :: 8: 72.

278. Proposition XII.-In any number of proportions having the same ratio, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents.

Since abcd, we have bc=ad (Art. 267).
Since abm: n, we have bman,

ities gives.

Factoring,

ab+be+bm=ab+ad+an.

b(a+c+m)=a(b+d+n).

This gives (Art. 268), a: b:: a+c+m:b+d+n.

:

If 5 10 2: 4:3: 6, etc.; then, 5:10:5+2+3: 10+4+6, or 5 10 10: 20.

:

:

42?

EXERCISES IN RATIO AND PROPORTION.

1. Which is the greater ratio, that of 3 to 4, or 32 to

Ans. last.

2. Compound the duplicate ratio of 2 to 3; the triplicate ratio of 3 to 4; and the subduplicate ratio of 64 to 36. Ans. 1 to 4.

3. What quantity must be added to each of the terms of the ratio mn, that it may become equal to p: q?

4. If the ratio of a to b is 23, what is the ratio of 2a

to b, and of 3a to 4b?

Ans. 1, and 35.

5. If the ratio of a to b is 13, what is the ratio of a+b to b, and of b-a to a?

Ans.,

and 3.

6. If the ratio of m to n is 4, what is the ratio of m—n

to 6m, and also to 5n?

Ans. 14, and 63.

7. If the ratio of 5y-8x to 7x-5y is 6, what is the ratio of x to y? Ans. 7 to 11.

8. What eight proportions are deducible from the equation ab-a2-x2. Ans. aa+x:: a—x: b,

aa-x::a+x: b, ba+x:: a-x: a, etc.

х

9. If x2+y2=2ax, what is the ratio of x to y?

Ans. xy:: y: 2a-x.

10. Four given numbers are represented by a, b, c, d; what quantity added to each will make them proportionals?

11. If four numbers are proportionals, show that there is no number which being added to each, will leave the resulting four numbers proportionals.

12. Find x in terms of y from the proportions x:y :: a3: b3, and abc+x: /d+y.

13. Prove that equal multiples of two quantities are to each other as the quantities themselves, or that ma: mb:: a: b.

14. Prove that like parts of two quantities are to each b

α

other as the quantities themselves, or that : ::a: b.

n

15. If a : b : cd, prove that ma mb nc: nd, and also that ma: nb:: mc: nd, m and n being any multiples.

16. Prove that the quotients of the corresponding terms of two proportions are proportional.

279. The following examples are intended as exercises in application of the principles of proportion.

1. Resolve the number 24 into two factors, so that the sum of their cubes may be to the difference of their cubes as 35 to 19.

Let x and y denote the required factors; then, xy=24, and

From which y=x; then, substituting the value of y in the equation xy=24, we find x=±6; hence, y=±4.

2. Given

x+1+px- 1 3x+1-3x-1

=2, to find x.

Resolving this equation into a proportion, we have

x+1-x-1:0x+1+px- 1::1:2;

(Art. 275), 2x + 1 : 23 x−1 :: 3 :

8. It is required to find two numbers whose product is 320, and the difference of whose cubes is to the cube of their difference, as 61 is to 1.

Ans. 20 and 16.