Question

X′s salary is half that of Y′s. If X got a 50 % rise in his salary and Y got 25 % rise in his salary, then the percentage increase in combined salaries of both is -\[\]
A.30\[\]
B. $33\dfrac{1}{3}$\[\]
C.$37\dfrac{1}{2}$\[\]
D.75\[\]

Answer 1

Hint: We take the salary of X as $x$ and the salary of Y as $y$ then the combined salary is $x+y$. We find the combined increment by adding the increments of X and Y . We multiply 100 to the ratio of combined increment to combined salary before increment to get the answer. \[\]

Complete step by step answer:
We know that if we are asked what percentage of $b$ is $a$ then the answer is $\dfrac{a}{b}\times 100.$
Let us assume the salary of X as $x$ and the salary of Y as $y$. We are given the question that the salary of X is that of Y or we can say the salary of Y is twice that of salary of X. So we can write it as $y=2x$. Then the combined salary of both X and Y before the increment is $x+y=x+2x=3x$\[\]

We are given the question that the salary of X has increased by 50%. So the increment in salary of X is $x\times 50 \%=x\times 0.50=0.5x$ . We are also given that the salary of Y has increased by 25%. So the increment in salary of Y is $y\times 25 \%=\left( 2x \right)\times 0.25=0.50x$(since $y=2x$ previously obtained).
So the total combined increment for both is $0.50x+0.50x=x$\[\]
So the percentage increment or rise in salary is the ratio of total combined increment to the combined salary of both X and Y before the increment multiplied by 100. So the percentage increment is
\[\dfrac{x}{3x}\times 100=\dfrac{100}{3}=33\dfrac{1}{3}\]
So the correct choice is B.\[\]

Note:
We note that if $x$ has increased by percentage ${{P}_{x}}$ and $y$ has increased by percentage ${{P}_{y}}$ then $x{{P}_{x}}+y{{P}_{y}}=\left( x+y \right)\left( {{P}_{x}}+{{P}_{y}} \right)$ is not necessarily true. We can directly find the increase of $x$ by percentage $P$ as $x\left( 1+\dfrac{P}{100} \right)$. If there is a successive percentage increase say $a%,b%$ of $x$ then the effective percentage increase is $a+b+\dfrac{ab}{100}$.