CAT 2020Slot 2QAQuestion & Solution
Question
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
Solution
1. Concept Used
- Topic: Ratio and Proportion — Combining Ratios using LCM
- Formula: $$\text{Combined Ratio} = \frac{\text{Amal}}{\text{Sunil}} \times \frac{\text{Sunil}}{\text{Mita}} \Rightarrow \text{Amal : Sunil : Mita}$$
2. Calculation
We are given two separate ratios: Amal : Sunil = 3 : 2, and Sunil : Mita = 4 : 5. To combine these into a single three-way ratio, we need Sunil's part to be the same in both. The LCM of 2 and 4 is 4, so we scale the first ratio by multiplying both parts by 2.
This gives us: Amal : Sunil = 6 : 4, and Sunil : Mita = 4 : 5.
Now the combined ratio is: $$\text{Amal : Sunil : Mita} = 6 : 4 : 5$$
Let the shares of Amal, Sunil, and Mita be $$6x,\ 4x,\ \text{and}\ 5x$$ respectively.
Identifying the largest and smallest shares: Amal has $$6x$$ (largest) and Sunil has $$4x$$ (smallest).
The difference between the largest and smallest shares is: $$6x - 4x = 2x = 400$$
Solving for $$x$$: $$x = \frac{400}{2} = 200$$
Therefore, Sunil's share is: $$4x = 4 \times 200 = \mathbf{800}$$
3. Solution
Answer = 800 ✅
The final calculated value is Rs. 800.