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 in their joint venture, we can follow these steps:

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

### Step 2: Determine the Time of Investment
- A invests his share at the start of the year, so A's investment is for 12 months.
- B joins after 9 months, which means B's investment is for the remaining 3 months of the year (12 - 9 = 3).

### Step 3: Set Up the Equation for Profit Sharing
The profit sharing can be expressed in terms of their investments and the time they invested:
\[ \frac{P_A}{P_B} = \frac{I_A \times T_A}{I_B \times T_B} \]
Where:
- \(P_A\) and \(P_B\) are the profits of A and B respectively.
- \(I_A\) and \(I_B\) are the investments of A and B respectively.
- \(T_A\) and \(T_B\) are the time periods for which A and B invested.

### Step 4: Substitute Known Values
From the problem, we know:
- \(P_A/P_B = 28/15\)
- \(T_A = 12\) months (for A)
- \(T_B = 3\) months (for B)

Substituting these values into the equation gives:
\[ \frac{28}{15} = \frac{I_A \times 12}{I_B \times 3} \]

### Step 5: Simplify the Equation
Cross-multiplying gives:
\[ 28 \times (I_B \times 3) = 15 \times (I_A \times 12) \]
This simplifies to:
\[ 84 I_B = 180 I_A \]

### Step 6: Rearrange to Find the Ratio of Investments
Rearranging the equation gives:
\[ \frac{I_A}{I_B} = \frac{84}{180} \]
Now, simplifying \( \frac{84}{180} \):
\[ \frac{I_A}{I_B} = \frac{7}{15} \]

### Step 7: State the Ratio of Investments
Thus, the ratio of the initial investments of A and B is:
\[ I_A : I_B = 7 : 15 \]

### Final Answer
The ratio of their initial investments respectively is **7:15**.

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