A started a business with Rs 10000 and B joined him with Rs 18000 after 4 months . After a year 'A' got 25 % of total profit for his office work while remaining profit is divided into A and B according to their investment . If at the end of the year 'A' got Rs 31,200 as his profit share , then find the total profit earned by them ?

Options

A. A)Rs 57,200
B. B)Rs 49,500
C. C)Rs 44,000
D. D)Rs 52,800

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Determine the investments of A and B
- A started the business with Rs 10,000.
- B joined later with Rs 18,000.

### Step 2: Calculate the time of investment
- A invested for 12 months (1 year).
- B invested for 8 months (since he joined after 4 months).

### Step 3: Calculate the ratio of their investments
- The investment ratio of A to B is:
\[ \text{Investment Ratio} = \frac{10000}{18000} = \frac{5}{9} \]

### Step 4: Calculate the ratio of time
- The time ratio of A to B is:
\[ \text{Time Ratio} = \frac{12}{8} = \frac{3}{2} \]

### Step 5: Calculate the overall profit-sharing ratio
- To find the overall profit-sharing ratio, we multiply the investment ratio by the time ratio:
\[ \text{Overall Ratio} = \left(5 \times 3\right) : \left(9 \times 2\right) = 15 : 18 \]
- Simplifying this gives:
\[ \text{Overall Ratio} = 5 : 6 \]

### Step 6: Determine A's share of the profit
- A receives 25% of the total profit for his office work.
- The remaining profit (75%) is divided according to their investment ratio (5:6).

### Step 7: Let the total profit be \( P \)
- A's share from the total profit:
\[ A's\ share = 0.25P + \left(0.75P \times \frac{5}{11}\right) \]
- Since A's total profit share is given as Rs 31,200:
\[ 0.25P + \left(0.75P \times \frac{5}{11}\right) = 31200 \]

### Step 8: Simplify the equation
- Calculate \( 0.75P \times \frac{5}{11} \):
\[ 0.75P \times \frac{5}{11} = \frac{3.75P}{11} \]
- Substitute back into the equation:
\[ 0.25P + \frac{3.75P}{11} = 31200 \]
- Convert \( 0.25P \) to a fraction with a common denominator:
\[ \frac{0.25P \times 11}{11} = \frac{2.75P}{11} \]
- Combine the terms:
\[ \frac{2.75P + 3.75P}{11} = 31200 \]
\[ \frac{6.5P}{11} = 31200 \]

### Step 9: Solve for \( P \)
- Multiply both sides by 11:
\[ 6.5P = 343200 \]
- Divide by 6.5:
\[ P = \frac{343200}{6.5} = 52800 \]

### Final Answer
- The total profit earned by A and B is Rs 52,800.