Inverse Proportion

Suppose that 20 men build a house in 6-days. If men are increased to 30 then they take 4-days to build the same house. If men become 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 no. of men is increased, the time taken to build the house is decreased in the same ratio.
In other words,
If increased in one quantity causes decrease in other quantity or decrease in one 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
Shows each product is constant or 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 by 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 28 workers working at the same rate take 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

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