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}}

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

\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

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?

\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