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CAT 2025Slot 3QAQues & Sol

ArithmeticMedium

Question

The ratio of the number of coins in boxes A and B was 17:7. After 108 coins were shifted from box A to box B, this ratio became 37:20. The number of coins that needs to be shifted further from A to B, to make this ratio 1:1, is

Solution

1. Concept Used

  • Topic: Ratio & Proportion — Setting up linear equations from ratio conditions
  • Formula: $$\text{If ratio of A to B is } p:q, \text{ then } A = p \cdot x, \ B = q \cdot x \text{ for some constant } x$$

2. Calculation

Let the initial number of coins in box A and box B be $17x$ and $7x$ respectively, since the initial ratio is $17:7$.

After $108$ coins are shifted from A to B: $\text{Coins in A} = 17x - 108, \quad \text{Coins in B} = 7x + 108$

This new ratio equals $37:20$, so we write: $\frac{17x - 108}{7x + 108} = \frac{37}{20}$

Cross-multiplying: $20(17x - 108) = 37(7x + 108)$ $340x - 2160 = 259x + 3996$ $340x - 259x = 3996 + 2160$ $81x = 6156$ $x = 76$

So the initial coins in A and B are: $A = 17 \times 76 = 1292, \quad B = 7 \times 76 = 532$

Total coins in both boxes combined: $\text{Total} = 1292 + 532 = 1824$

For a $1:1$ ratio, each box must contain: $\frac{1824}{2} = 912 \text{ coins}$

Coins currently in box B (after the first transfer of 108 coins): $B_{\text{current}} = 7 \times 76 + 108 = 532 + 108 = 640$

Additional coins to be shifted from A to B: $912 - 640 = 272$

3. Solution

Answer = 272 ✅

The final calculated value is 272. After shifting 272 more coins from box A to box B, both boxes will contain exactly 912 coins each, achieving the 1:1 ratio.