Question

The ratio of expenditure and savings is $3:2\;$. If the income increases by $15\%$ and the savings increases by $6\%$ , then by how much percent should his expenditure increase?
A. $25\;$
B. $21\;$
C. $12\;$
D. $24\;$

Answer 1

Hint: First find the total income by adding the expenditure and savings. Now increase the savings by $15\%$. Do the same with the savings but now increase it with $6\%$ . Subtract new savings from new income to get the new expenditure. Compare it with the old expenditure and find the percentage of increase in expenditure.

Complete step-by-step solution:
The given ratio of expenditure vs savings is $3:2\;$ .
Let the ratio be equal to a variable $x$
Now the expenditure becomes $3$ times of $x$
$\Rightarrow 3x$
The savings become $2$ times of $x$
$\Rightarrow 2x$
The income is the sum of expenditure and savings so,
$\Rightarrow$$3x + 2x = 5x$
Now given that the income increases by $15\%$
$\Rightarrow 15\%$ of $5\;x$
Now convert the percentage symbol into a fraction.
$\Rightarrow \dfrac{{15}}{{100}} \times 5x$
Simplify further.
$\Rightarrow \dfrac{3}{4}x$
This can be written in decimals as,
$\Rightarrow 0.75x$
The new income is now $5x + 0.75x = 5.75x$
Now it is also given that savings are also increased by $6\%$
$\Rightarrow 6\%$ of $2\;x$
Now convert the percentage symbol into a fraction.
$\Rightarrow \dfrac{6}{{100}} \times 2x$
Simplify further to get a simpler fraction.
$\Rightarrow \dfrac{3}{{25}}x$
It can also be written in decimals as,
$\Rightarrow 0.12x$
Now the new savings are given by $2x + 0.12x = 2.12x$
The new expenditure is given by subtracting new savings from new income which is,
$\Rightarrow 5.75x - 2.12x$
$\Rightarrow 3.63x$
The old expenditure was $3\;x$
The increase in expenditure is given by subtracting the old expenditure from the new expenditure and then dividing it again with the old expenditure.
$\Rightarrow \dfrac{{(3.63x - 3x)}}{{3x}}$
Since we need the percent of the increase, we will multiply it with $100\;$
$\Rightarrow \dfrac{{(3.63x - 3x)}}{{3x}} \times 100$
$\Rightarrow \dfrac{{(0.63x)}}{{3x}} \times 100$
On further canceling the variables in the numerator and denominator we get,
$\Rightarrow \dfrac{{0.63}}{3} \times 100$
$\Rightarrow 0.21 \times 100$
On performing the multiplication operation on the decimal, we get,
$\Rightarrow 21\%$
$\therefore$ The increase in expenditure is $21\%$

Option B is the correct answer.

Note: The ratio given in the question which is also known as proportion says the amount of one thing there is contrasted with something else. So, we write the ratio in fraction form and then equate it to a random variable. Now we write the quantities compared in terms of the variable to see how much one is comparable to the other.