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1. The price of an article is increased by $25\% $. By how much does this value be decreased to restore it to its former value?
2. The price of an article is reduced by $10\% $. By how much percent will this value be increased to restore?

Answer Verified Verified
Hint: 1. Consider the initial price of an article to be $x$. If the price of an article is increased by $25\% $, so the new price will become $x + 25\% $, we got the new price of the product. Now to restore the former value we need to find the value to be decreased and divide that value by the new price we got then multiply it by $100$ to get the percentage to restore it to its former value.
2. Consider the initial price of an article to be $x$. If the price of an article is decreased by $10\% $, so the new price will become $x - 10\% $, we got the new price of the product. Now to restore the former value we need to find the value to be added and divide that value by the new price we got then multiply it by $100$ to get the percentage to restore it to its former value.

Complete step-by-step answer:
$1.$ According to the question, the price of an article is increased by $25\% $ and we need to find value to be decreased to restore it to its former value.
Initial price of an article be $x$
And we need to find $25\% $ of $x$ = $\dfrac{{25x}}{{100}}$
So new cost will be $25\% $ increased, so it is equal = $ x + \dfrac{{25x}}{{100}}$
$\Rightarrow $ $\dfrac{{100x + 25x}}{{100}}$
$\Rightarrow $ $\dfrac{{125x}}{{100}}$
So here we get the new price = $\dfrac{{125x}}{{100}}$
To get back to the original price, how much we need to decrease = $\dfrac{{125x}}{{100}} - x$
$\Rightarrow $ $\dfrac{{25x}}{{100}}$
If we subtract this value from the new value we will get the former value.
The percentage decreased = $\dfrac{{{\text{amount decreased}}}}{{{\text{total amount}}}} \times 100$
We get = $\dfrac{{\dfrac{{25x}}{{100}}}}{{\dfrac{{125x}}{{100}}}} \times 100$
$\Rightarrow $ $\dfrac{{25}}{{125}} \times 100 = 20\% $
So here if we further decrease by $20\% $, then we will get the former value.
$2.$ If the price is reduced by $10\% $ and how much percent we increase to restore the value. So let us assume the initial price of an article is $x$.
Now it is decreased by $10\% $
$10\% $ of $x$ would be = $\dfrac{{10x}}{{100}}$
Now new price will become = $x - \dfrac{{10x}}{{100}}$
$\Rightarrow $ $\dfrac{{90x}}{{100}}$
So now we need to increase the price to the original price.
Subtract new price from initial price = $x - \dfrac{{90x}}{{100}}$
$\Rightarrow $ $\dfrac{{10x}}{{100}}$
So percentage increase would be = $\dfrac{{\dfrac{{10x}}{{100}}}}{{\dfrac{{90x}}{{100}}}} \times 100$
$\Rightarrow $ $\dfrac{{10x}}{{90x}} \times 100 = \dfrac{{100}}{9}\% $
$\Rightarrow $ $ 11.1\% $
So we need to increase it by $11.1\% $ to restore its value.

Note: In this type of question, instead of taking initial value as $x$, we can take it as $100$ also. So if it is said that the price has increased by $25\% $, then the increased price will be $125$. Now we want to decrease it to $100$, so percentage decrease = $\dfrac{{125 - 100}}{{125}} \times 100 = \dfrac{{25}}{{125}} \times 100 = 20\% $
In this way also, you can solve this.