Question

The ratio of income of two persons is 9:7 and the ratio of their expenditure is 4:3. If each of them manages to save Rs. 2000 per month, find their monthly income.
A. Rs. 18,000, Rs. 14,000
B. Rs. 1,000, Rs. 14,000
C. Rs. 18,000, Rs. 1,000
D. None of these

Answer 1

Hint: Assume two proportionality constants $\left( say\ {{K}_{1}}\ and\ {{K}_{2}} \right)$ for the ratio of income and the ratio of expenditure. Then form two equations, one for the saving of first person and second for the saving of 2nd person and solve the two equations to get ${{K}_{1}}\ and\ {{K}_{2}}$.

Complete step-by-step solution -
According to the question, the ratio of income of two persons is 9:7.
Let proportionality constant for this ratio is ${{K}_{1}}$.
Then, income of the 1st person will be $9{{K}_{1}}$ and income of the second person will be $7{{K}_{1}}$.
And the ratio of expenditure of two persons is 4:3.
Let proportionality constant for this ratio is ${{K}_{2}}$.
Then, income of the 1st person will be $4{{K}_{2}}$ and income of the second person will be$3{{K}_{2}}$.
According to the question, saving of both the persons is Rs. 2000 per month.
We know, saving = Income – Expenditure.
$\begin{align}
  & \Rightarrow Saving\ of\ {{1}^{st}}\ person=\left( Income\ of\ {{1}^{st}}\ person \right)-\left( Expenditure\ of\ {{1}^{st}}\ person \right) \\
 & \Rightarrow {{\left( Saving \right)}_{1}}=9{{K}_{1}}-4{{K}_{2}} \\
\end{align}$
And,
$\begin{align}
  & \Rightarrow Saving\ of\ {{2}^{nd}}\ person=\left( Income\ of\ {{2}^{nd}}\ person \right)-\left( Expenditure\ of\ {{2}^{nd}}\ person \right) \\
 & \Rightarrow {{\left( Saving \right)}_{2}}=7{{K}_{1}}-3{{K}_{2}} \\
\end{align}$
According to the question, ${{\left( Saving \right)}_{1}}={{\left( Saving \right)}_{2}}=2000$.
$\begin{align}
  & \Rightarrow 9{{K}_{1}}-4{{K}_{2}}=2000........\left( 1 \right) \\
 & \Rightarrow 7{{K}_{1}}-3{{K}_{2}}=2000........\left( 2 \right) \\
\end{align}$
Subtraction equation (2) from equation (1), we will get,
$\begin{align}
  & \dfrac{\begin{align}
  & 9{{K}_{1}}-4{{K}_{2}}=2000 \\
 & 7{{K}_{1}}-3{{K}_{2}}=2000 \\
 & -\ \ \ \ +\ \ \ \ \ \ \ \ - \\
\end{align}}{2{{K}_{1}}-{{K}_{2}}=0} \\
 & \Rightarrow 2{{K}_{1}}={{K}_{2}} \\
\end{align}$
Using this $''{{K}_{2}}=2{{K}_{1}}''$in equation (1), we will get,
\[\begin{align}
  & \Rightarrow 9{{K}_{1}}-4\left( 2{{K}_{1}} \right)=2000 \\
 & \Rightarrow 9{{K}_{1}}-8{{K}_{1}}=2000 \\
 & \Rightarrow {{K}_{1}}=2000 \\
\end{align}\]
We know, income of the 1st person is $9{{K}_{1}}$ and income of 2nd person is $7{{K}_{1}}$.
Hence, income of 1st person $=9\times 2000=Rs.18,000$
And income of 2nd person $=7\times 2000=Rs.14,000$
Hence, the answer is option (A).

Note: We can solve this question in short by using the options. There is only one option (A) in which ratio of the given two incomes is 9:7.
Use this option and calculate expenditure according to the ratio of expenditure and calculate saving.
If you will get saving Rs. 2000 then option (A) will be answered, otherwise option (D) will be answered.
Given ratio of expenditure = 4:3
By option A – Income of person 1 = Rs.18000 and income of person 2 = Rs.14000.
Let proportionality constant for ratio of expenditure be K.
So, expenditure of 1st person = 4K and expenditure of 2nd person = 3K.
$\Rightarrow Saving\ of\ {{1}^{st}}\ person=Rs.18,000-4K$
Equating this with 2000, we will get,
$\begin{align}
  & 18000-4K=2000 \\
 & \Rightarrow 18000-2000=4K \\
 & \Rightarrow K=\dfrac{16000}{4}=4000 \\
\end{align}$
Now, use this K for getting the saving of 2nd person, If we will get Rs. 2000 then option (A) will be answered, otherwise option (D) will be answered.
$\begin{align}
  & \Rightarrow Saving\ of\ {{2}^{nd}}\ person=\left( Income\ of\ {{2}^{nd}}\ person \right)-\left( Expenditure\ of\ {{2}^{nd}}\ person \right) \\
 & =14000-3\left( 4000 \right) \\
 & =14000-12000 \\
 & =2000 \\
\end{align}$
Hence, option (A) will be our answer.