The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save ` 2000 per month, find their monthly incomes.
Step-by-Step Solution
Key Concept: Use the given ratios to express incomes and expenditures in terms of common multiples, then form two linear equations using the fact that (Income – Expenditure) = Savings = 2000 for each person. Solve the simultaneous equations by elimination (or substitution) to obtain the actual incomes.
1. Let the incomes of the two persons be $9x$ and $7x$ rupees per month respectively (since the ratio is $9:7$).
2. Let their expenditures be $4y$ and $3y$ rupees per month respectively (ratio $4:3$).
3. Savings for each person is given as Rs. $2000$, therefore
\[\begin{cases}\ 9x - 4y = 2000 \ 7x - 3y = 2000 \end{cases}\]
4. Subtract the second equation from the first to eliminate the constant term:
\[(9x-4y) - (7x-3y) = 0 \Rightarrow 2x - y = 0 \Rightarrow y = 2x.\]
5. Substitute $y = 2x$ in any one of the original equations, e.g. $9x - 4y = 2000$:
\[9x - 4(2x) = 2000 \Rightarrow 9x - 8x = 2000 \Rightarrow x = 2000.\]
6. Hence incomes are:
\[\text{First person's income} = 9x = 9 \times 2000 = \text{Rs. }18000,\]
\[\text{Second person's income} = 7x = 7 \times 2000 = \text{Rs. }14000.\]
7. (For verification) Expenditures become $y = 2x = 4000$, so $4y = 16000$ and $3y = 12000$.
Savings: $18000-16000 = 2000$, $14000-12000 = 2000$, which matches the given condition.
Correct Answer: The monthly incomes are Rs. 18,000 and Rs. 14,000 respectively.