To solve the problem step by step, we will first determine the work done by A, B, and C, and then calculate how much A gets from the total payment of Rs 960.
### Step 1: Calculate the work done by A and B in one day.
- A can complete the work in 12 days, so A's work rate is \( \frac{1}{12} \) of the work per day.
- B can complete the work in 15 days, so B's work rate is \( \frac{1}{15} \) of the work per day.
**Hint:** To find the work rate, use the formula \( \text{Work rate} = \frac{1}{\text{Time taken}} \).
### Step 2: Calculate the combined work done by A and B in one day.
- The combined work rate of A and B is:
\[
\text{Combined work rate} = \frac{1}{12} + \frac{1}{15}
\]
- To add these fractions, find a common denominator. The LCM of 12 and 15 is 60.
\[
\frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}
\]
- Therefore,
\[
\text{Combined work rate} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20}
\]
**Hint:** When adding fractions, find a common denominator to combine them.
### Step 3: Calculate the work done by A, B, and C together in one day.
- They can finish the work in 5 days, so their combined work rate is:
\[
\text{Combined work rate of A, B, and C} = \frac{1}{5}
\]
**Hint:** The work rate for a group is the reciprocal of the time taken to complete the work.
### Step 4: Set up the equation to find C's work rate.
- We know the combined work rate of A, B, and C:
\[
\frac{3}{20} + \text{C's work rate} = \frac{1}{5}
\]
- Convert \( \frac{1}{5} \) to a fraction with a denominator of 20:
\[
\frac{1}{5} = \frac{4}{20}
\]
- Now, we can solve for C's work rate:
\[
\text{C's work rate} = \frac{4}{20} - \frac{3}{20} = \frac{1}{20}
\]
**Hint:** Isolate the variable (C's work rate) by subtracting the known work rates from the total work rate.
### Step 5: Calculate the total work done in terms of efficiency.
- The total efficiency of A, B, and C is:
\[
\text{Total efficiency} = \text{A's efficiency} + \text{B's efficiency} + \text{C's efficiency} = \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5}
\]
**Hint:** Efficiency is the sum of individual efficiencies.
### Step 6: Calculate A's share of the payment.
- The total payment for the work is Rs 960. A's share of the payment is proportional to A's efficiency:
\[
\text{A's share} = \text{Total payment} \times \frac{\text{A's efficiency}}{\text{Total efficiency}} = 960 \times \frac{5}{12}
\]
- Calculate A's share:
\[
\text{A's share} = 960 \times \frac{5}{12} = 960 \div 12 \times 5 = 80 \times 5 = 400
\]
**Hint:** To find A's share, multiply the total payment by the ratio of A's efficiency to total efficiency.
### Final Answer:
A gets Rs 400.
