A sum of money is split among Amal, Sunil and Mita so that the ratio of the shares of Amal and Sunil is 3:2, while the ratio of the shares of Sunil and Mita is 4:5. If the difference between the largest and the smallest of these three shares is Rs 400, then Sunil’s share, in rupees, is

The correct answer is Rs 800.

We are given two separate ratios involving three people:

• Ratio of Amal's share to Sunil's share = 3 : 2
• Ratio of Sunil's share to Mita's share = 4 : 5

To compare all three shares together, we need to combine these two ratios into a single three-way ratio Amal : Sunil : Mita. The key is to make Sunil's part the same value in both ratios.

In the first ratio, Sunil's part is 2. In the second ratio, Sunil's part is 4. The LCM of 2 and 4 is 4. So we scale the first ratio by multiplying both parts by 2:

Amal : Sunil = 3 : 2   ⟹   6 : 4
Sunil : Mita = 4 : 5   ⟹   4 : 5

Now Sunil's part is 4 in both. We can write the combined ratio as:

Amal : Sunil : Mita = 6 : 4 : 5

Let the common multiplier be x. Then the three shares are:

• Amal's share = 6x
• Sunil's share = 4x
• Mita's share = 5x

Now we identify the largest and smallest shares among 6x, 4x, and 5x:

• Largest = Amal's share = 6x
• Smallest = Sunil's share = 4x

We are told the difference between the largest and smallest shares is Rs 400:

$$6x-4x=400$$

$$2x=400$$

$$x=200$$

Finally, we calculate Sunil's share:

$$\text{Sunil’s share}=4x=4\times 200=\mathbf{800}$$

∴ Sunil's share is Rs 800.