A and B started a business with some amount .After 9 months B left the business & C joins the business with Rs 12,000 and remains in business till the end of year . At the end of the year , profit share of A,B and C is Rs . 48 ,Rs . 48 and Rs . 24 respectively .Find the sum of the amount (in Rs ) invested by A and B together in the business ?

Options

A. A)`8,000`
B. B)`10,000`
C. C)`15,000`
D. D)`14,000`

Answer

To solve the problem, we need to find the sum of the amounts invested by A and B together in the business. Let's break it down step by step.

### Step 1: Understand the Profit Sharing
At the end of the year, the profit shares are:
- A: Rs 48
- B: Rs 48
- C: Rs 24

### Step 2: Determine the Time Periods
- A invests for 12 months (the whole year).
- B invests for 9 months (leaves after 9 months).
- C invests for 3 months (joins after B leaves).

### Step 3: Calculate the Time Ratios
The time ratios for A, B, and C are:
- A: 12 months
- B: 9 months
- C: 3 months

Thus, the time ratio is:
- A : B : C = 12 : 9 : 3

This can be simplified by dividing each term by 3:
- A : B : C = 4 : 3 : 1

### Step 4: Set Up the Investment Ratios
Let the amounts invested by A and B be \( a \) and \( b \) respectively. The investment ratios, taking time into account, will be:
- A's contribution = \( 4a \) (since A invests for 12 months)
- B's contribution = \( 3b \) (since B invests for 9 months)
- C's contribution = \( 12000 \) (since C invests for 3 months)

### Step 5: Set Up the Profit Equation
According to the profit sharing:
- A's profit = Rs 48
- B's profit = Rs 48
- C's profit = Rs 24

### Step 6: Establish the Value of Each Profit Share
From C's profit, we can establish a value:
- If C's profit of Rs 24 corresponds to an investment of Rs 12000, we can find the value of Rs 1 profit:
\[ \text{Value of Rs 1 profit} = \frac{12000}{24} = 500 \]

### Step 7: Calculate A's and B's Investment Values
Now we can find the total profit values for A and B:
- A's profit of Rs 48 corresponds to:
\[ \text{Total investment value for A} = 48 \times 500 = 24000 \]
- B's profit of Rs 48 corresponds to:
\[ \text{Total investment value for B} = 48 \times 500 = 24000 \]

### Step 8: Set Up the Equations
From the profit values:
1. \( 4a = 24000 \) (for A)
2. \( 3b = 24000 \) (for B)

### Step 9: Solve for A and B's Investments
From the first equation:
\[ a = \frac{24000}{4} = 6000 \]
From the second equation:
\[ b = \frac{24000}{3} = 8000 \]

### Step 10: Calculate the Total Investment
Now, we can find the total amount invested by A and B together:
\[ \text{Total Investment} = a + b = 6000 + 8000 = 14000 \]

### Final Answer
The sum of the amounts invested by A and B together in the business is **Rs 14000**.

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