Question

It is stated that a quantity varies directly as another if the two quantities always change in the same ratio. So, what would happen to the quantity y if the quantity x doubles in value?

Answer 1

If the quantity x doubles in value, the quantity y will also double in value for the ratio between x and y to remain the same.

Explanation:

The relationship between x and y can be represented as follows:

y = bx ............................... (1)

Where the constant b represents the ratio between x and y and can be written as follows:

b = y / x ............................ (2)

Since b is a constant and quantity y varies directly with quantity x, equation 2 implies that the ratio between x and y always remain the same.

Therefore, if the quantity x doubles in value, the quantity y will also double in value for the ratio between x and y to remain the same.

For example, if x = 3 and y = 6, we can substitute into equation (2) and solve as follows:

b = 6 / 3 = 2

Now, if the quantity x doubles in value from 3 to 6 (i.e. 3 * 2 = 6), y must also double in value from 6 to 12 (i.e 6 * 2 = 12) so that the ratio between x and y is still equal to 2 as follows:

b = 12 / 6 = 2