Dinesh started a business with investing Rs 12000 and after some months , Sunny joined with investing Rs 9000. At the end of the year , total profit was Rs 8000 and share of Sunny is Rs 1600. Find after how many months did sunny joined in the business ?

Options

A. A)6 months
B. B)4 months
C. C)9 months
D. D)8 months

Answer

To solve the problem step-by-step, we can follow these instructions:

### Step 1: Understand the investments
Dinesh invested Rs 12,000 and Sunny invested Rs 9,000.

### Step 2: Set up the profit-sharing ratio
The profit-sharing ratio is based on the product of the investment and the time for which the investment was made.

Let’s denote:
- Dinesh's investment = 12,000
- Sunny's investment = 9,000

### Step 3: Determine the time Dinesh and Sunny were in business
- Dinesh was in the business for the entire year = 12 months.
- Let Sunny join after \( X \) months. Therefore, Sunny was in the business for \( 12 - X \) months.

### Step 4: Calculate the effective capital contribution
The effective capital contribution for Dinesh and Sunny can be calculated as:
- Dinesh's effective capital = Investment × Time = \( 12000 \times 12 = 144000 \)
- Sunny's effective capital = Investment × Time = \( 9000 \times (12 - X) = 9000(12 - X) \)

### Step 5: Set up the ratio of their effective capital contributions
The ratio of their effective contributions can be expressed as:
\[ \frac{Dinesh's \, effective \, capital}{Sunny's \, effective \, capital} = \frac{144000}{9000(12 - X)} \]

### Step 6: Simplify the ratio
This simplifies to:
\[ \frac{144000}{9000(12 - X)} = \frac{16}{(12 - X)} \]

### Step 7: Calculate the total profit and Sunny's share
The total profit at the end of the year is Rs 8000, and Sunny's share is Rs 1600.

### Step 8: Determine Dinesh's share
Dinesh's share of the profit can be calculated as:
\[ Dinesh's \, share = Total \, profit - Sunny's \, share = 8000 - 1600 = 6400 \]

### Step 9: Set up the ratio of their shares
Now, we can set up the ratio of their shares:
\[ \frac{Dinesh's \, share}{Sunny's \, share} = \frac{6400}{1600} = 4 \]

### Step 10: Set up the equation
Now we have two ratios:
\[ \frac{16}{(12 - X)} = \frac{4}{1} \]

### Step 11: Cross-multiply to solve for \( X \)
Cross-multiplying gives:
\[ 16 = 4(12 - X) \]
\[ 16 = 48 - 4X \]

### Step 12: Rearrange the equation
Rearranging gives:
\[ 4X = 48 - 16 \]
\[ 4X = 32 \]

### Step 13: Solve for \( X \)
Dividing by 4 gives:
\[ X = 8 \]

### Conclusion
Sunny joined the business after **8 months**.

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