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# A number is increased by $20\%$ and the increased number is decreased by $20\%$ . Find the net increase or decrease percent.

Last updated date: 28th Feb 2024
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Hint: We approach the problem by first assuming the number to be an arbitrary $x$ . Thereafter, we go as per the statements of the problem, that is, increasing $x$ by $20\%$ or multiplying it by $1.2$ to get the increase, and then decreasing the latest value by $20\%$ or multiplying it by $0.8$ to get the decrease. Having done so, we can find the net change in $x$ and dividing the change by the initial value and multiplying with $100$ gives us the percentage change.

We assume that the initial number is $x$
If the number is increased by $20\%$ then actual increase in the number will be
$=x\times \dfrac{20}{100}$
$=0.2x$
Therefore, after getting increased by $20\%$ the number becomes
$=x+0.2x$
$=1.2x$
Now, the increased number $1.2x$ is getting further decreased by $20\%$
Hence, the actual decrease in the number will be
$=\left( 1.2x \right)\times \dfrac{20}{100}$
$=0.24x$
Therefore, after getting decreased by $20\%$ the number becomes
$=1.2x-0.24x$
$=0.96x$
The initial number is $x$ and the final number after getting increased and decreased becomes $0.96x$
As, the final number is lesser than the initial number we can say that the number has gone through an overall decrease.
The overall decrease in the number is equal to the difference between the initial and final number
$\therefore$ The net decrease in the number is
$=x-0.96x$
$=0.04x$
Hence, the percentage of decrease is equal to the ratio of the net decrease in the number to the initial number and multiplied to $100$
Hence, net percentage decrease in the number is
$=\dfrac{\text{net decrease in the number}}{\text{initial number}}\times 100\%$
$=\dfrac{0.04x}{x}\times 100\%$
$=4\%$
Therefore, we conclude that the net decrease in the number is $4\%$ .

Note: We can also solve this type of problem by assuming the initial number to be $100$ . For that case the entire process will be the same as the solution except we will use the number $100$ instead of $x$everywhere. Also, while calculating the percentage decrease, we must be careful as taking the initial number instead of the increased number will ruin the result of the problem.