Correct Option

The total sum of Rs 2,500 is distributed among X, Y, and Z in the ratio $\frac{1}{2} : \frac{3}{4} : \frac{5}{6}$ .

To convert this fractional ratio into a whole number ratio, find the Least Common Multiple (LCM) of the denominators (2, 4, 6), which is 12.

Multiply each ratio component by the LCM:
For X: $\frac{1}{2} \times 12 = 6$
For Y: $\frac{3}{4} \times 12 = 9$
For Z: $\frac{5}{6} \times 12 = 10$

Thus, the simplified ratio of shares for X, Y, and Z is 6:9:10.

The sum of these ratio parts is $6 + 9 + 10 = 25$ .

Now, calculate the individual share for each person from the total sum of Rs 2,500:

Share of X = $(\frac{6}{25} \times 2500) = Rs. 600$
Share of Y = $(\frac{9}{25} \times 2500) = Rs. 900$
Share of Z = $(\frac{10}{25} \times 2500) = Rs. 1000$

From these calculations, the maximum share is Rs 1000 (Z's share), and the minimum share is Rs 600 (X's share).

The required difference between the maximum share and the minimum share is $1000 - 600 = Rs. 400$ .

Incorrect Options

Options 1 (Rs 300), 2 (Rs 350), and 4 (Rs 450) are incorrect. These values do not correspond to the accurate difference between the maximum and minimum shares as determined by the correct calculation method based on the given ratio and total sum.