Ram and Shyam undertook a work for Rs. 5600. Ram alone can do the work in 5 days and Shyam alone can do the work in 9 days. If they work together, what will be the difference (in Rs.) In the amount they receive?

Options

A. 1800
B. 2400
C. 2200
D. 1600

Answer

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

### Step 1: Determine the work done by Ram and Shyam
- Ram can complete the work in 5 days.
- Shyam can complete the work in 9 days.

### Step 2: Calculate the work done per day by each
- Work done by Ram in 1 day = \( \frac{1}{5} \) of the work
- Work done by Shyam in 1 day = \( \frac{1}{9} \) of the work

### Step 3: Find the combined work done per day
- Combined work done in 1 day = \( \frac{1}{5} + \frac{1}{9} \)
- To add these fractions, find a common denominator, which is 45.

\[ \frac{1}{5} = \frac{9}{45}, \quad \frac{1}{9} = \frac{5}{45} \]

- Therefore, combined work done in 1 day = \( \frac{9}{45} + \frac{5}{45} = \frac{14}{45} \)

### Step 4: Calculate the total work in terms of days
- Total work can be considered as 1 (whole work).
- Time taken to complete the work together = \( \frac{1}{\frac{14}{45}} = \frac{45}{14} \) days.

### Step 5: Calculate the ratio of their efficiencies
- Efficiency ratio of Ram to Shyam is the inverse of the time taken:
- Ram's efficiency = 9 (since he takes 5 days)
- Shyam's efficiency = 5 (since he takes 9 days)

- Therefore, the efficiency ratio of Ram to Shyam = 9:5.

### Step 6: Calculate the total parts of the payment
- Total parts = 9 + 5 = 14 parts.

### Step 7: Calculate the payment for each
- Total payment = Rs. 5600.
- Payment per part = \( \frac{5600}{14} = 400 \).

- Ram's share = 9 parts = \( 9 \times 400 = 3600 \).
- Shyam's share = 5 parts = \( 5 \times 400 = 2000 \).

### Step 8: Find the difference in their amounts
- Difference = Ram's share - Shyam's share = \( 3600 - 2000 = 1600 \).

### Final Answer
The difference in the amount they receive is Rs. 1600.

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