CAT 2025 Slot3 Quant: The ratio of the number of coins in boxes A and B was

Step 1: Set up the initial variables
Let the number of coins in box A and box B be $17x$ and $7x$ respectively.

Step 2: Apply the first shift
108 coins are moved from A to B:
$\frac{17x - 108}{7x + 108} = \frac{37}{20}$
$20(17x - 108) = 37(7x + 108)$
$340x - 2160 = 259x + 3996$
$81x = 6156 \implies \mathbf{x = 76}$.

Step 3: Find current counts
After the first shift:
Box A = $17(76) - 108 = 1292 - 108 = \mathbf{1184}$ coins.
Box B = $7(76) + 108 = 532 + 108 = \mathbf{640}$ coins.
Total coins = $1184 + 640 = 1824$.

Step 4: Calculate further shift for 1:1 ratio
For a $1:1$ ratio, both boxes must have $\frac{1824}{2} = 912$ coins.
Coins to be shifted from A to B = Current A - Target A
$= 1184 - 912 = \mathbf{272}$.