CAT 2025Slot 3QAQuestion & Solution
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.