CAT 2025

Slot 2

QA

Question & SolutionQues & Sol

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

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.