Sainik School Entrance Exam 2020 Solved Paper

(b) Suppose that the income of $\mathrm{P}$ and $\mathrm{Q}$ is $5\mathit{p}$ and $4\mathit{p}$
and the expenditure of $\mathrm{P}$ and $\mathrm{Q}$ is $4\mathit{q}$ and $3\mathit{q}$
Then, the saving of $\mathrm{P}=5\mathit{p}-4\mathit{q}$
and, the saving of $\mathrm{Q}=4\mathit{p}-3\mathit{q}$
ATQ,
The saving of $\mathrm{P}$ is more than $\mathrm{Q}$ by $16\frac{2}{3}\%$
$\therefore 5\mathit{p}-4\mathit{q}=100+\frac{50}{3}$
$\Rightarrow 5\mathit{p}-4\mathit{q}=\frac{350}{3}$
$\Rightarrow 15\mathit{p}-12\mathit{q}=350........\left(\mathit{i}\right)$
and $4\mathit{p}-3\mathit{q}=100$
$\Rightarrow 16\mathit{p}-12\mathit{q}=400.......\left(\mathit{i}\mathit{i}\right)$
Subtracting (i) from (ii), we get $\mathit{p}=50$
from (ii), $\mathit{q}=\frac{100}{3}$
∴ The income of $\mathrm{P}=5\mathit{p}=5\times 50=250$
and, the expenditure of $\mathrm{P}=4\mathit{q}$
$=4\times \frac{100}{3}=\frac{400}{3}$
∴ Required percentage, $\mathrm{P}$ spend his income
$=\frac{\frac{400}{3}}{250}\times 100=\frac{400}{3\times 250}\times 100\%$
$=\frac{40\times 4}{3}=\frac{160}{3}\%=53\frac{1}{3}\%$