Three friends X, Y and Z started a business by investing a sum of money in the ratio of 4 : 2 : 5. After 6 months, X withdraws half of his capital. If the sum invested by Z is ₹23,000, then out of a total annual profit of ₹37,000, what is the difference between X's and Y's profit?

The correct option is ₹3700.

Step 1: Understand the ratio of initial investments

Let the initial investments of X, Y, and Z be in the ratio $4:2:5$.

We can assume their initial investments to be:

Investment of X = $4k$

Investment of Y = $2k$

Investment of Z = $5k$

Step 2: Calculate the effective monthly investments for 1 year (12 months)

For X:

X invests $4k$ for the first 6 months. After 6 months, X withdraws half of his capital, so his remaining capital becomes $\frac{4k}{2}=2k$ for the remaining 6 months.

Total effective investment of X = $(4k\times 6)+(2k\times 6)$

Total effective investment of X = $24k+12k=36k$

For Y:

Y keeps his investment unchanged for the entire 12 months.

Total effective investment of Y = $2k\times 12=24k$

For Z:

Z keeps his investment unchanged for the entire 12 months.

Total effective investment of Z = $5k\times 12=60k$

Step 3: Find the profit sharing ratio

The ratio of profit share among X, Y, and Z is equal to the ratio of their total effective investments:

Ratio = $36k:24k:60k$

Dividing each term by $12k$, we get the simplified profit ratio:

Profit Ratio of X : Y : Z = $3:2:5$

Step 4: Calculate the total ratio units and individual profit shares

Total sum of ratio units = $3+2+5=10$ units

Total annual profit = ₹37,000

Value of 1 unit of profit = $\frac{37000}{10}=\text{₹}3700$

Step 5: Calculate the difference between X's and Y's profit

Difference in profit ratio units between X and Y = $3-2=1$ unit

Difference in profit amount = $1\times 3700=\text{₹}3700$

Hence, the difference between X's and Y's profit is ₹3700.