X starts a business with X 30000 and Y joins him after 3 months with X 120000. Find the ratio of their profits at the end of the year.

Options

A. `2:5`
B. `3:5`
C. `5:1`
D. `1:3`

Answer

To find the ratio of profits between X and Y at the end of the year, we need to consider both the amount of money each invested and the duration for which they invested that money.

### Step-by-Step Solution:

1. **Identify the investments:**
- X invests ₹30,000.
- Y invests ₹1,20,000.

2. **Determine the duration of investment:**
- X starts the business and stays for the entire year, which is 12 months.
- Y joins after 3 months, so he stays for 12 - 3 = 9 months.

3. **Calculate the capital contribution for each partner:**
- For X:
\[ \text{Capital contribution of X} = \text{Investment} \times \text{Duration} = 30,000 \times 12 = 3,60,000 \]
- For Y:
\[ \text{Capital contribution of Y} = \text{Investment} \times \text{Duration} = 1,20,000 \times 9 = 10,80,000 \]

4. **Set up the ratio of their contributions:**
- The ratio of X's contribution to Y's contribution is:
\[ \text{Ratio} = \frac{3,60,000}{10,80,000} \]

5. **Simplify the ratio:**
- To simplify, we can divide both numbers by 1,20,000:
\[ \frac{3,60,000 \div 1,20,000}{10,80,000 \div 1,20,000} = \frac{3}{9} \]
- Further simplifying gives:
\[ \frac{3 \div 3}{9 \div 3} = \frac{1}{3} \]

6. **Final ratio of profits:**
- Therefore, the ratio of their profits at the end of the year is:
\[ \text{Ratio of X to Y} = 1 : 3 \]

### Conclusion:
The ratio of profits at the end of the year between X and Y is **1 : 3**.

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