14. In a certain school the number of boys increased 10 per cent during the year, the number of girls increased 5 per cent, the total attendance increased 6 per cent. Find the ratio of the number of boys to the number of girls at the beginning of the year.

CHAPTER XXIII

VARIATION

169. Constants have been defined as numbers that remain the same throughout the discussion of a problem and variables as numbers whose values are constantly changing. Thus, the height of a room is a constant quantity, but the distance of a moving fly from the floor is a variable.

If two variables be so related, that when one changes, the other has a corresponding change, then one is said to be a function of the other, that is, the one quantity depends upon the other for its value.

Thus, if a train runs at a certain speed, the distance it travels depends upon the time during which it runs, that is, the distance is a function of the time.

One quantity is said to vary directly as another if the ratio of the two quantities remains constant throughout the change.

In the preceding illustration, the distance the train travels varies directly with the time.

The sign of variation is

and is read, varies as. Thus, xay is read x varies as y and is a brief method of writing the proportion,

when x is the value to which x has changed when y has changed to y1.

Since the ratio of x to y remains constant when x x y, this constant ratio may be represented by k, that is,

Therefore, if x varies as y, x is equal to a constant times y. 170. If one quantity varies as the reciprocal of another, one is said to vary inversely as the other.

Thus, the time required to do a certain piece of work varies inversely as the number of persons employed. For if it takes 5 men 4 days to do the work, it will take 4 men 5 days, 2 men 10 days, and 1 man 20 days to do it.

Inverse variation is expressed, xx

1

or the ratio of x to

y

Therefore, if x varies inversely as y, their product is

constant.

One quantity varies jointly as two others, when it varies as their product.

Thus, the area of a rectangle depends upon its base and altitude and therefore varies jointly as its base and altitude. If the base is doubled and the altitude trebled, the area would be six times as large.

If xyz, and the constant ratio of x to yz is k,

Hence, if x varies jointly as y and z, x is equal to a constant times their product.

One quantity varies directly as a second and inversely as a third when it varies jointly as the second and the reciprocal of the third.

Thus, the time required to do a piece of work varies directly as the amount of work to be done and inversely as the number of persons employed. If the work were 4 times as much and the number of persons twice as large, the time required to do the work would be twice as long.

Hence, if x varies directly as y and inversely as z, x is equal to a constant times.

2

171. In the case of three related variables x, y, and z, If x varies as y when z is constant, and x varies as z when y is constant, then x varies as yz when both y and z are variable.

Thus, the area of a rectangle varies as the altitude when the base remains constant; it varies as the base when the altitude is constant; and the area varies as the product of base and altitude when

both vary.

Proof:

The variation of ≈ depends upon the variations of y and z. Suppose the variations of y and z to take place successively. Let y change to y1, while z remains constant, causing x to change to x1.

Now let z change to z1 while y retains its value of y1, causing x1 to change to x2.

But as x2, yı and zı, are particular values of x, y, and z, they are

constants, and is

X2
Y121

a constant which may be represented by k.

Then x = kyz

And xyz

In a similar way, it may be proved that if x varies as each of three or more numbers, y, z, m, n, when the others are constant,

then x varies as their product when all vary.

Thus, the volume of a rectangular solid varies as the length when the breadth and thickness remain constant; it varies as the breadth when the length and thickness remain constant; it varies as the thickness when the length and breadth remain constant; and it varies as the product of length, breadth and thickness when all three vary.

EXERCISE 125

1. If x varies inversely as y, and x = 4 when y what is the value of x when y

=

6?

It is always advisable to find the constant of variation, if possible.

substituting the values of x and y, 4 =

4. If x varies jointly as y and z and inversely as the square y 5, 2= 6 and v = 3, what is

of v, and x

=

20,

when

=

the value of x expressed in terms of y, z, and v?

7. If x x y and z ∞ y, show that (x ± z)

αψ.

8. The circumference of a circle varies as its diameter. If the circumference of a circle, whose diameter is 1 foot, is 3.1416 feet, find the circumference of a circle 25 feet in diameter.

9. The area of a circle varies as the square of the radius. If the area of a circle whose radius is 1 foot is 3.1416 square feet, find the area of a circle whose radius is 10 feet.

10. The volume of a sphere varies as the cube of its radius. The volume of a sphere whose radius is 1 foot is 4.1888 cubic feet, find the volume of a sphere whose radius is 3 feet.

11. The velocity of a falling body varies as the time during which it has fallen from rest. If the velocity of a falling body is 96 feet per second at the end of 3 seconds, find its velocity at the end of 10 seconds.

12. The distance a body falls from rest varies as the square of the time of falling. If a body falls 144.72 ft. in 3 seconds, how far will it fall in 8 seconds?