Question

Out of the total income, X spends $20\%$ on house rent and $70\%$ of the remaining amount on household expenditure. X saves Rs $1800$, then the total income is
A. Rs $8000$
B. Rs $9500$
C. Rs $7500$
D. Rs $8500$

Answer 1

Hint: In this problem we will first assume the income of X as $a$. Now we have given that he spends $20\%$ on house rent, so we will calculate the $20\%$ of his income $a$ and also calculate the remaining amount by subtracting the calculated house rent from income of X. Again, they have mentioned that $70\%$ of remaining amount is used for household expenditure, so we will calculate the $70\%$ of remaining amount and calculate his savings by subtracting the $70\%$ of remaining amount from remaining amount after paying house rent. But in the problem they have given the saving of X as Rs $1800$, so we will equate the calculated savings and given savings and try to find the value of $a$.

Complete step by step answer:
Let us take the income of X as $a$.
We have been given that he spends $20\%$ of his income on house rent.
$\therefore $ House rent $=20\%a=\dfrac{20}{100}a$.
Now we get the remaining amount after paying house rent is given by
$\begin{align}
  & r=a-\dfrac{20}{100}a \\
 & \Rightarrow r=\dfrac{100a-20a}{100} \\
 & \Rightarrow r=\dfrac{80}{100}a...\left( \text{i} \right) \\
\end{align}$
We also know that he uses $70\%$ remaining amount on household expenditures.
$\therefore $ Household amount $=70\%r=\dfrac{70}{100}r$.
Now we get the remaining amount after spending money on household expenditures is given by
$\begin{align}
  & {{r}_{1}}=r-\dfrac{70}{100}r \\
 & \Rightarrow {{r}_{1}}=\dfrac{100r-70r}{100} \\
 & \Rightarrow {{r}_{1}}=\dfrac{30}{100}r \\
\end{align}$
In the problem we have given his savings as Rs $1800$. So, the value of ${{r}_{1}}$ is equal to the given savings.
$\begin{align}
  & \therefore {{r}_{1}}=1800 \\
 & \Rightarrow \dfrac{30}{100}r=1800 \\
\end{align}$
From equation $\left( \text{i} \right)$, substituting the value of $r$ in the above equation, then we will get
$\begin{align}
  & \dfrac{30}{100}\times \dfrac{80}{100}\times a=1800 \\
 & \Rightarrow a=\dfrac{1800\times 100\times 100}{30\times 80} \\
 & \Rightarrow a=7500 \\
\end{align}$

$\therefore $ The savings of X is Rs $7500$.

Note: In this problem they have clearly mentioned that X uses $70\%$ of the amount remained after paying his house rent on household expenditures. So, we have taken the $70\%$ of the value $r$. If X uses $70\%$ of his income on household expenditures then we need to take $70\%$ of the value $a$. To find the value of $a$, we will calculate all his spending by adding the house rent and amount for household expenditures and subtract it from his savings then we will equate the obtained value to the given savings.