CAT 2025Slot 2QAQuestion & Solution
Question
The ratio of expenditures of Lakshmi and Meenakshi is 2 : 3, and the ratio of income of Lakshmi to expenditure of Meenakshi is 6 : 7. If excess of income over expenditure is saved by Lakshmi and Meenakshi, and the ratio of their savings is 4 : 9, then the ratio of their incomes is
Options
3:5
5:6
2:1
7:8
Solution
1. Concept Used
- Topic: Ratio and Proportion — Setting up a common variable to link multiple ratios
- Formula: $$\text{Savings} = \text{Income} - \text{Expenditure}$$
2. Calculation
Let Lakshmi's income = ( A ), Lakshmi's expenditure = ( B ), Meenakshi's income = ( C ), and Meenakshi's expenditure = ( D ).
Step 1 — Assign a common variable to expenditures.
Given: ( B : D = 2 : 3 ). Let ( B = 2k ) and ( D = 3k ) for some positive constant ( k ).
Step 2 — Find Lakshmi's income using the given cross-ratio.
Given: ( A : D = 6 : 7 ), and since ( D = 3k ):
$$A = \frac{6}{7} \times D = \frac{6}{7} \times 3k = \frac{18k}{7}$$
Step 3 — Compute Lakshmi's savings.
$$\text{Lakshmi's savings} = A - B = \frac{18k}{7} - 2k = \frac{18k - 14k}{7} = \frac{4k}{7}$$
Step 4 — Use the savings ratio to find Meenakshi's income.
Given: ratio of savings of Lakshmi to Meenakshi ( = 4 : 9 ).
$$\frac{\frac{4k}{7}}{C - 3k} = \frac{4}{9}$$
Cross-multiplying:
$$\frac{4k}{7} \times 9 = 4 \times (C - 3k)$$
$$\frac{36k}{7} = 4C - 12k$$
$$4C = \frac{36k}{7} + 12k = \frac{36k + 84k}{7} = \frac{120k}{7}$$
$$C = \frac{30k}{7}$$
Step 5 — Find the ratio of incomes.
$$A : C = \frac{18k}{7} : \frac{30k}{7} = 18 : 30 = 3 : 5$$
3. Solution
Answer = Option A (3:5) ✅
The ratio of the incomes of Lakshmi and Meenakshi is 3 : 5.