A starts a business with X 9000 and B joins him after 6 months with an investment of X 45000. What will be the ratio of the profits of A and B at the end of year ?

Options

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

Answer

To find the ratio of the profits of A and B at the end of the year, we need to consider the amount of capital each partner has invested and the duration for which they have invested that capital.

### Step 1: Calculate A's investment duration and capital contribution
- A invests X 9000 at the start of the year.
- A's investment duration is 12 months (from the start of the year to the end of the year).

**A's contribution = Investment × Duration = 9000 × 12 = 108000**

### Step 2: Calculate B's investment duration and capital contribution
- B joins after 6 months with an investment of X 45000.
- B's investment duration is 6 months (from the 6th month to the end of the year).

**B's contribution = Investment × Duration = 45000 × 6 = 270000**

### Step 3: Calculate the total contributions of A and B
- Total contribution of A = 108000
- Total contribution of B = 270000

### Step 4: Find the ratio of the contributions
- The ratio of A's contribution to B's contribution is:

\[ \text{Ratio} = \frac{A's \ contribution}{B's \ contribution} = \frac{108000}{270000} \]

### Step 5: Simplify the ratio
- To simplify the ratio, divide both numbers by their greatest common divisor (GCD), which is 9000:

\[ \text{Ratio} = \frac{108000 \div 9000}{270000 \div 9000} = \frac{12}{30} = \frac{2}{5} \]

### Conclusion
The ratio of the profits of A and B at the end of the year is **2:5**.

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