Question

A man spends $75\% $ of his income. This income is increased by $20\% $ and he increases his expenditure by $10\% $. By what percent are his savings increased?

Answer 1

Hint: Here in this question we need to simply assume the man’s income to be any number or the variable and then apply the above conditions and get the original and new values of the expenditure and his savings. For example: If we assume the man’s income as ${\text{Rs100}}$ then according to the given income we can calculate his original expenditure. Similarly if we increase the income by $20\% $ his new income becomes ${\text{Rs120}}$ and again find the same expenditure and the saving.
Now we can solve the percent increase in savings by the formula${\text{percent increase in savings}} = \dfrac{{{\text{new saving}} - {\text{old saving}}}}{{{\text{old saving}}}} \times 100\% $

Complete step-by-step answer:
Here we are given that man spends $75\% $ of his income. This income is increased by $20\% $ and he increases his expenditure by$10\% $.
Here we need to find the percentage increase in the savings. So first of all we need to assume the income as any amount or any variable and with respect to that we can calculate the percentage increase in the savings of a man.
So let the income of the man$ = {\text{Rs100}}$
He spends $75\% $ of his income which means he saves rest of the income which is $(100 - 75) = 25\% $ of his total income.
His expenditure$ = 75\% {\text{ of 100}} = {\text{Rs }}75$
So his savings$ = 25\% {\text{ of Rs100}}$$ = \dfrac{{25}}{{100}}(100) = {\text{Rs 25}}$
So his original savings is ${\text{Rs 25}}$
Now we know that his income is increased by $20\% $
So the new income
$
   = 100 + 20\% {\text{ of 100}} \\
   = 100 + \dfrac{{20}}{{100}}(100) = 100 + 20 = {\text{Rs120}} \\
 $
Now his expenditure is increased by $10\% $
So his new expenditure$ = {\text{original expenditure}} + {\text{10\% of original expenditure}}$
So we get new expenditure$ = 75 + \dfrac{{10}}{{100}}(75) = 75 + 7.5 = 82.5$
New saving
$
   = {\text{new income}} - {\text{new expenditure}} \\
   = 120 - 82.5 \\
   = 37.5 \\
 $
So percent increase$ = \dfrac{{{\text{new saving}} - {\text{original saving}}}}{{{\text{original saving}}}}(100)\% $
$\dfrac{{37.5 - 25}}{{25}} \times 100\% $$ = \dfrac{{12.5}}{{25}} \times 100\% = 50\% $
Hence his increase in saving is $50\% $

Note: Here in this type of the problem student can make a mistake in getting the statement properly. So students must analyse the question properly and for the new saving we need to subtract the new expenditure from the new income of the person not from the assumed income. So here the student must be aware of the situation.