Arithmetic

Ratio, Proportion & Variation

Medium

The ratio of the rate of flow of water through pipes varies inversely, as the square of the radius of the pipes. What is the ratio of the rates of flow in two pipes of diameter 2 cm and 4 cm?

1:8

1:2

2:1

4:1

✅ Correct Option: 4

The rate of flow varies inversely as the square of the radius of the pipes.

This relationship is expressed as:

Rate $\propto \dfrac{1}{r^2}$

For two pipes:

$\dfrac{\text{Rate}_1}{\text{Rate}_2} = \dfrac{r_2^2}{r_1^2}$

The diameter of pipe 1 is 2 cm, so:

$r_1 = \dfrac{2}{2} = 1$ cm

The diameter of pipe 2 is 4 cm, so:

$r_2 = \dfrac{4}{2} = 2$ cm

Applying the inverse square relationship:

$\dfrac{\text{Rate}_1}{\text{Rate}_2} = \dfrac{r_2^2}{r_1^2}$

$\dfrac{\text{Rate}_1}{\text{Rate}_2} = \dfrac{(2)^2}{(1)^2}$

$\dfrac{\text{Rate}_1}{\text{Rate}_2} = \dfrac{4}{1}$

Therefore, the ratio of the rates of flow is $4:1$ .

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The ratio of the rate of flow of water through pipes varies inversely, as the square of the radius of the pipes. What is the ratio of the rates of flow in two pipes of diameter 2 cm and 4 cm?

Solution

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