CAT 2022

Slot 3

QA

Question & Solution

1. Concept Used

• Topic: Mixtures and Alligations — successive transfer of liquids between containers
• Formula: When a mixture of total volume $$ V $$ has components in ratio $$ m : w $$, the fraction of milk in each $$ x $$ cc taken out is $$ \frac{m}{m+w} \times x $$ and fraction of water is $$ \frac{w}{m+w} \times x $$

2. Calculation

Initial State: Glass = 500 cc milk, Cup = 500 cc water.

Step 1 — Transfer 150 cc of pure milk from Glass to Cup: Glass now contains: $$ 500 - 150 = 350 \text{ cc milk} $$ Cup now contains: $$ 150 \text{ cc milk} + 500 \text{ cc water} = 650 \text{ cc total} $$ Ratio of milk to water in cup: $$ 150 : 500 = 3 : 10 $$

Step 2 — Transfer 150 cc of the mixture from Cup back to Glass: In the cup, milk fraction $$ = \frac{3}{13} $$ and water fraction $$ = \frac{10}{13} $$. So in 150 cc of mixture transferred: $$ \text{Milk transferred} = \frac{3}{13} \times 150 = \frac{450}{13} \text{ cc} $$ $$ \text{Water transferred} = \frac{10}{13} \times 150 = \frac{1500}{13} \text{ cc} $$

Water in the Glass after Step 2: Glass originally had zero water; the only water it receives is from this transfer: $$ \text{Water in glass} = \frac{1500}{13} \text{ cc} $$

Milk remaining in the Cup after Step 2: Cup had 150 cc milk before Step 2; milk taken out $$ = \frac{450}{13} $$ cc: $$ \text{Milk in cup} = 150 - \frac{450}{13} = \frac{1950 - 450}{13} = \frac{1500}{13} \text{ cc} $$

Required Ratio — Water in Glass : Milk in Cup: $$ \frac{1500}{13} : \frac{1500}{13} = 1 : 1 $$

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3. Solution

Answer = Option A ✅

The amount of water in the glass and the amount of milk in the cup are both equal to $$ \frac{1500}{13} $$ cc, giving a ratio of 1 : 1.