A and B started a business with respective investments of Rs25000 and Rs15000. The numbers of months for which B invested was 4 less than the number of months for which A had invested. If the ratio of the total profit to A’s share in profit was 7: 5, then what was the number of months for which B invested in the business?

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define the Variables
Let the number of months A invested be \( X \). Since B invested for 4 months less than A, the number of months B invested will be \( X - 4 \).

**Hint:** Define variables for unknowns to simplify the problem.

### Step 2: Calculate the Investment Ratios
A's investment is Rs 25,000 and B's investment is Rs 15,000. The profit is distributed based on the product of investment and time. Therefore, the profit shares can be expressed as:
- A's share of profit = \( 25000 \times X \)
- B's share of profit = \( 15000 \times (X - 4) \)

**Hint:** Remember that profit is proportional to the product of investment and time.

### Step 3: Set Up the Profit Ratio
According to the problem, the ratio of the total profit to A's share in profit is \( 7:5 \). This means:
\[ \frac{\text{Total Profit}}{A's \text{ Profit}} = \frac{7}{5} \]
Let’s denote A's profit as \( P_A \) and B's profit as \( P_B \). Thus, the total profit \( P_T = P_A + P_B \).

From the ratio, we can express:
\[ P_T = \frac{7}{5} P_A \]
Substituting for \( P_A \):
\[ P_T = \frac{7}{5} (25000 \times X) \]

**Hint:** Use the given ratios to express total profit in terms of A's profit.

### Step 4: Express Total Profit in Terms of B's Profit
Now, we can express the total profit in terms of B's profit:
\[ P_T = 25000X + 15000(X - 4) \]
This simplifies to:
\[ P_T = 25000X + 15000X - 60000 = 40000X - 60000 \]

**Hint:** Combine the profit expressions to find a single equation.

### Step 5: Set the Two Expressions for Total Profit Equal
Now we can set the two expressions for total profit equal to each other:
\[ \frac{7}{5} (25000X) = 40000X - 60000 \]

**Hint:** Equate the two expressions to solve for \( X \).

### Step 6: Solve for X
Cross-multiplying gives:
\[ 7 \times 25000X = 5(40000X - 60000) \]
This simplifies to:
\[ 175000X = 200000X - 300000 \]
Rearranging gives:
\[ 200000X - 175000X = 300000 \]
\[ 25000X = 300000 \]
\[ X = \frac{300000}{25000} = 12 \]

**Hint:** Isolate \( X \) to find the number of months A invested.

### Step 7: Calculate B's Investment Duration
Since B invested for \( X - 4 \):
\[ B's \text{ investment duration} = 12 - 4 = 8 \]

**Hint:** Use the value of \( X \) to find the duration for B.

### Final Answer
The number of months for which B invested in the business is **8 months**.

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