A sum of  ₹2,500 is distributed among X, Y and Z in the ratio $\frac{1}{2}:\frac{3}{4}:\frac{5}{6}$. What is the difference between the maximum share and the minimum share?

Correct Answer: ₹400

Step 1: Simplify the given ratio of shares

The total sum of money to be distributed is ₹2,500 among X, Y, and Z in the ratio:

$\frac{1}{2}:\frac{3}{4}:\frac{5}{6}$

To convert this fractional ratio into an integer ratio, we find the Least Common Multiple (LCM) of the denominators (2, 4, and 6).

LCM(2, 4, 6) = 12

Multiply each term of the ratio by 12:

$\text{X's ratio part}=\frac{1}{2}\times 12=6$

$\text{Y's ratio part}=\frac{3}{4}\times 12=9$

$\text{Z's ratio part}=\frac{5}{6}\times 12=10$

So, the ratio of shares of X, Y, and Z is 6 : 9 : 10.

Step 2: Calculate the total number of ratio parts

Total ratio parts = 6 + 9 + 10 = 25 parts.

Step 3: Identify maximum and minimum shares

From the ratio 6 : 9 : 10:

- Maximum share belongs to Z (10 parts)

- Minimum share belongs to X (6 parts)

Difference in ratio parts = 10 - 6 = 4 parts.

Step 4: Calculate the monetary difference

Value of 1 part = Total Amount / Total Parts = ₹2,500 / 25 = ₹100.

Therefore, the difference between the maximum share and the minimum share is:

$\text{Difference}=4\times \text{₹100}=\text{₹400}$