344. The laws that have been established for ratios

should be remembered when ratios are expressed in fractional form.

This equation is satisfied when x = 0. For any other value of x,

EXERCISE 124.

1. Find a third proporţional to 21 and 28.

2. Find a mean proportional between 6 and 24.

3. Find a fourth proportional to 3, 5, and 42.

10. a-3b: a + 3b = c - 3d :c+3 d.

11. a2 + ab + b2: a2 - ab + b2 = c2 + cd + d2 : c2 - cd + d2.

Find x in the proportion:

12. 45:68 = 90: x.

13.6:3=x: 7.

14.x : 1 = 1 : 1층.

15.3: x = 7:42.

16. Find two numbers in the ratio 2: 3, the sum of whose squares is 325.

17. Find two numbers in the ratio 5:3, the difference of whose squares is 400.

18. Find three numbers which are to each other as 2:3:5, such that half the sum of the greatest and least exceeds the other by 25.

19. A and B trade with different sums. A gains $200 and B loses $50 and now A's stock: B's :: 2:14. But, if A had gained $100 and B lost $85, their stocks would have been as 15:34. Find the original stock of each.

proportions

20. Find x if 6x - a: 4x - b = 3x + b : 2 x + a. Find x and y from the proportionals

21.

x : y = x + y : 42 ; x : y = x - y : 6. 22. Find x and y from the proportionals 2x + y : y = 3y: 2 y-x; 2x + 1:2x + 6 = y : y + 2.

345. One quantity is said to vary as another, when the two quantities are so related that the ratio of any two values of the one is equal to the ratio of the corresponding values of the other.

Thus, if it is said that the weight of water varies as its volume, the meaning is, that one gallon of water is to any number of gallons as the weight of one gallon is to the weight of the given number of gallons.

346. Two quantities may be so related that when a value of one is given, the corresponding value of the other can be found. In this case one quantity is said to be a function of the other; that is, one quantity depends upon the other for its value. Thus, if the rate at which a man walks is known, the distance he walks can be found when the time is given; the distance is in this case a function of the time.

347. There is an unlimited number of ways in which two quantities may be related. We shall consider in this chapter only a few of these ways.

348. When x and y are so related that their ratio is constant, y is said to vary as x; this is abbreviated thus:

The sign∞, called the sign of variation, is read

varies as.

Thus, the area of a triangle with a given base varies as its altitude; for, if the altitude is changed in any ratio, the area will be changed in the same ratio.

In this case, if we represent the constant ratio by m,

Again, if y', x' and y", x" are two sets of corresponding

1

X

349. When x and y are so related that the ratio of y to is constant, y is said to vary inversely as x; this is written

Thus, the time required to do a certain amount of work varies inversely as the number of workmen employed; for, if the number of workmen is doubled, halved, or changed in any other ratio, the time required will be halved, doubled, or changed in the inverse ratio.

m

= m; ..y =, and xy = m; that is,

the product xy is constant.

X

350. If the ratio of y:xz is constant, then y is said to

vary jointly as x and z.

In this case,

and

y' : y" = x'z' : x"z".

y = mxz,

352. THEOREM 1. If yox, and x ∞ z, then y∞ z.

353. THEOREM 2. If yox, and z∞x, then (y + z) ∞ x. For

y = mx and z = nx.

.∵∴ y + z = (m ± n) x ;

since m±nis constant, y ± z varies as x.

354. THEOREM 3. If yox when z is constant, and y∞n when x is constant, then y∞ xz when x and z are both variable.

Let x', y', z', and x", y", z" be corresponding values of the variables.

Let x change from x' to x", z remaining constant, and let

the corresponding value of y be Y.

Now let z change from z' to z", x remaining constant.

... the ratio y: xz is constant, and y varies as xz.