# Inverse Proportion

Suppose that 20 men build a house in 6 days. If the number of men is increased to 30 then they take 4 days to build the same house. If the number of men becomes 40, they take 2 days to build the house; i.e.

\[\begin{array}{*{20}{c}} {{\text{No}}{\text{. of Men}}}&{}&{{\text{No}}{\text{. of Days}}} \\ {{\text{20}}}&{}&{\text{6}} \\ {{\text{30}}}&{}&{\text{4}} \\ {{\text{40}}}&{}&{\text{3}} \end{array}\]

It can be seen that as the number of men is increased, the time taken to build the house is decreased by the same ratio.

In other words, **if an increase in one quantity causes a decrease in another quantity, or a decrease in one quantity causes an increase in another quantity, then we say that both quantities are inversely related.**

More explicitly, if two quantities $$x$$ and $$y$$ are in inverse proportion, then their product will be constant.

i.e. $$xy = c$$ where $$c$$= constant

In the above example, we see that

20 x 6 = 120

30 x 4 = 120

40 x 3 = 120

This shows that each product is constant or the same. Therefore, if we are dealing with quantities which are related inversely, then we can use the following rule:

\[\begin{array}{*{20}{c}} {{\text{No}}{\text{. of Men}}}&{}&{{\text{No}}{\text{. of Days}}} \\ {{\text{20}}}& \leftrightarrow &{\text{6}} \\ {}&{\text{ = }}&{} \\ {{\text{30}}}& \leftrightarrow &{\text{4}} \end{array}\]

20 x 6 = 30 x 4

In general,

\[\boxed{\begin{array}{*{20}{c}} {{\text{No}}{\text{. of Men}}}&{}&{{\text{No}}{\text{. of days}}} \\ {\text{a}}& \leftrightarrow &{\text{c}} \\ {\text{b}}& \leftrightarrow &{\text{d}} \\ {{\text{ac = bd}}}&{{\text{or}}}&{\frac{{\text{a}}}{{\text{d}}} = \frac{{\text{b}}}{{\text{c}}}} \end{array}}\]

**Example:**

Four pipes can fill a tank in 70 minutes. How long will it take to fill the tank with 7 pipes?

**Solution:**

\[\begin{array}{*{20}{c}} {{\text{No}}{\text{. of Pipes}}}&{}&{{\text{Time Taken}}} \\ {{\text{Increase}} \downarrow \begin{array}{*{20}{c}} {\text{4}} \\ {\text{7}} \end{array}}&{}&{\begin{array}{*{20}{c}} {{\text{70}}} \\ x \end{array} \downarrow {\text{Decrease}}} \end{array}\]

By the principle of inverse proportion, we have

4 x 70 = 7 x $$x$$

$$x = \frac{{4{\text{ x 70}}}}{{\text{7}}}$$ = 40 minutes

**Example:**

Thirty-five workers can build a house in 16 days. How many days will it take 28 workers working at the same rate to build the same house?

**Solution:**

\[\begin{array}{*{20}{c}} {{\text{No}}{\text{. of Workers}}}&{}&{{\text{No}}{\text{. of Days}}} \\ {{\text{Decrease}} \downarrow \begin{array}{*{20}{c}} {{\text{35}}} \\ {{\text{28}}} \end{array}}&{}&{\begin{array}{*{20}{c}} {{\text{16}}} \\ x \end{array} \downarrow {\text{Increase}}} \end{array}\]

By the principle of inverse proportion, we have

28 x $$x$$ = 35 x 16

$$x = \frac{{{\text{35 x 16}}}}{{{\text{28}}}}$$ = 20 days