CAT 2023

Slot 3

QA

Question & Solution

1. Concept Used

• Topic: Averages — Change in average when a new member joins a group
• Formula: $$\text{New Average} = \frac{\text{Sum of all members}}{\text{Number of members}}$$

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2. Calculation

Let the sum of weights of A, B, and C be $S$, so the average of the three is $\frac{S}{3}$.

Condition 1 — D joins the room (average decreases by $x$):

When D joins, the new average becomes $\frac{S + d}{4}$, and this is $x$ less than the original average:

$$\frac{S}{3} - \frac{S + d}{4} = x \quad \text{...(1)}$$

Condition 2 — E joins the room instead (average increases by $2x$):

When E joins, the new average becomes $\frac{S + e}{4}$, and this is $2x$ more than the original average:

$$\frac{S + e}{4} - \frac{S}{3} = 2x \quad \text{...(2)}$$

Adding equations (1) and (2):

$$\left(\frac{S}{3} - \frac{S + d}{4}\right) + \left(\frac{S + e}{4} - \frac{S}{3}\right) = x + 2x$$

The $\frac{S}{3}$ terms cancel out:

$$\frac{S + e}{4} - \frac{S + d}{4} = 3x$$

$$\frac{(S + e) - (S + d)}{4} = 3x$$

$$\frac{e - d}{4} = 3x$$

$$e - d = 12x$$

We are given that $e - d = 12$ kg, so:

$$12x = 12$$

$$x = 1$$

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3. Solution

Answer = Option C ✅

The final calculated value is $x = 1$.