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

The correct answer is 272.

Let's solve the problem step-by-step.

Step 1: Understand the initial scenario.
The initial ratio of coins in box A to box B is given as 17 : 7.
Let the initial number of coins in box A be $17x$ and in box B be $7x$, where $x$ is a common multiplier.

Step 2: Account for the first shift of coins.
After 108 coins are shifted from box A to box B:
The new number of coins in box A = $17x-108$
The new number of coins in box B = $7x+108$

Step 3: Set up the equation using the new ratio.
We are given that after this transfer, the ratio of coins in box A to box B becomes 37 : 20.

$\frac{17x-108}{7x+108}=\frac{37}{20}$

Cross-multiplying to solve for $x$:

$20\×(17x-108)=37\×(7x+108)$

$340x-2160=259x+3996$

Group the terms containing $x$ on one side and the numerical values on the other side:

$340x-259x=3996+2160$

$81x=6156$

$x=\frac{6156}{81}=76$

Step 4: Calculate the current number of coins in each box.
Using $x=76$:

Current coins in box A:

$17x-108=17\×76-108=1292-108=1184$

Current coins in box B:

$7x+108=7\×76+108=532+108=640$

Total number of coins in both boxes = $1184+640=1824$

Step 5: Find the further number of coins to be shifted to make the ratio 1 : 1.
To make the ratio of coins in box A to box B equal to 1 : 1, both boxes must have an equal number of coins.
Equal share for each box = $\frac{1824}{2}=912$ coins.

Let $y$ be the additional number of coins to be shifted from A to B.
New number of coins in box A = $1184-y=912$

$y=1184-912=272$

Thus, 272 more coins need to be shifted from box A to box B to make their ratio 1 : 1.