A and B run a joint venture in which the profit earned by A and B are in the ratio 28 : 15. A invest his share.at the start of the year and B joins in after 9 months of the same year. What is the ratio of their initial investment, respectively?

Options

A. `7:15`
B. `8:13`
C. `5:17`
D. `15:7`

Answer

To find the ratio of the initial investments of A and B, we can follow these steps:

### Step 1: Understand the Profit Ratio
We know that the profit earned by A and B is in the ratio of 28:15. This means that for every 28 parts of profit A receives, B receives 15 parts.

### Step 2: Define the Variables
Let:
- A's investment = X
- B's investment = Y

### Step 3: Determine the Time of Investment
- A invests his amount at the start of the year, so A's investment is for 12 months.
- B joins after 9 months, so B's investment is for 3 months (12 - 9 = 3).

### Step 4: Calculate the Effective Contribution
The effective contribution to the profit from each partner can be calculated by multiplying their investment by the time for which it is invested:
- A's effective contribution = \( X \times 12 \)
- B's effective contribution = \( Y \times 3 \)

### Step 5: Set Up the Equation
According to the profit sharing ratio, we can set up the equation:
\[ \frac{X \times 12}{Y \times 3} = \frac{28}{15} \]

### Step 6: Simplify the Equation
Cross-multiply to eliminate the fraction:
\[ 15 \times (X \times 12) = 28 \times (Y \times 3) \]
This simplifies to:
\[ 180X = 84Y \]

### Step 7: Rearrange to Find the Ratio
Now, we can rearrange this equation to find the ratio of X to Y:
\[ \frac{X}{Y} = \frac{84}{180} \]
Simplifying this fraction:
\[ \frac{X}{Y} = \frac{14}{30} = \frac{7}{15} \]

### Step 8: Conclusion
Thus, the ratio of their initial investments, A to B, is:
\[ \text{Ratio of A's investment to B's investment} = 7:15 \]

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