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 by setting up the algebraic equations based on the ratios provided.

Step 1: Set up the initial conditions
Initially, the ratio of the number of coins in box A to box B is given as 17:7.
Let the number of coins in box A initially be $17x$ and the number of coins in box B initially be $7x$, where $x$ is a common multiplier.

Step 2: Translate the shift of coins into an equation
When 108 coins are shifted from box A to box B:
The new number of coins in box A becomes $17x-108$.
The new number of coins in box B becomes $7x+108$.
After this shift, the ratio of the coins becomes 37:20. Therefore, we can write the equation:

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

Step 3: Solve for $x$
Cross-multiplying the equation, we get:

$$20(17x-108)=37(7x+108)$$

Expanding both sides of the equation:

$$340x-2160=259x+3996$$

Group the terms with $x$ on one side and the constant terms on the other side:

$$340x-259x=3996+2160$$

$$81x=6156$$

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

Step 4: Find the current number of coins in each box
Now we calculate the number of coins in each box after the first shift:
Coins in box A:

$$A_{\text{current}}=17\left(76\right)-108=1292-108=1184$$

Coins in box B:

$$B_{\text{current}}=7\left(76\right)+108=532+108=640$$

The total number of coins in both boxes combined is:

$$1184+640=1824$$

Step 5: Determine the number of coins to be shifted further for a 1:1 ratio
A 1:1 ratio means both boxes must end up with an equal number of coins. Since the total number of coins remains unchanged, the number of coins in each box should be:

$$\frac{1824}{2}=912$$

Let $y$ be the number of coins that need to be shifted further from box A to box B. We can set up the relation:

$$1184-y=912$$

Solving for $y$:

$$y=1184-912=272$$

Thus, 272 more coins must be shifted from box A to box B to make the ratio of the coins 1:1.