To solve the problem step by step, we will follow these calculations:
### Step 1: Determine the work done by X and Y
- X can complete the work in 6 days, so the work done by X in one day is:
\[
\text{Work done by X in one day} = \frac{1}{6}
\]
- Y can complete the work in 8 days, so the work done by Y in one day is:
\[
\text{Work done by Y in one day} = \frac{1}{8}
\]
### Step 2: Calculate the combined work done by X and Y in one day
- The total work done by X and Y together in one day is:
\[
\text{Total work done by X and Y in one day} = \frac{1}{6} + \frac{1}{8}
\]
- To add these fractions, we need a common denominator. The LCM of 6 and 8 is 24.
\[
\frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24}
\]
- Therefore,
\[
\text{Total work done by X and Y in one day} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24}
\]
### Step 3: Determine the total work done by X, Y, and A in 3 days
- In 3 days, the total work done by X and Y is:
\[
\text{Work done by X and Y in 3 days} = 3 \times \frac{7}{24} = \frac{21}{24}
\]
- Since the total work is 1 (the whole work), the work done by A in 3 days is:
\[
\text{Work done by A in 3 days} = 1 - \frac{21}{24} = \frac{3}{24} = \frac{1}{8}
\]
### Step 4: Calculate A's work rate
- Since A completed \(\frac{1}{8}\) of the work in 3 days, the work done by A in one day is:
\[
\text{Work done by A in one day} = \frac{1}{8} \div 3 = \frac{1}{24}
\]
### Step 5: Determine the total efficiency of X, Y, and A
- The efficiencies are:
- Efficiency of X = \(\frac{1}{6}\)
- Efficiency of Y = \(\frac{1}{8}\)
- Efficiency of A = \(\frac{1}{24}\)
### Step 6: Calculate the total efficiency
- The total efficiency of X, Y, and A is:
\[
\text{Total efficiency} = \frac{1}{6} + \frac{1}{8} + \frac{1}{24}
\]
- Finding a common denominator (which is 24):
\[
\frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{24} = \frac{1}{24}
\]
- Therefore,
\[
\text{Total efficiency} = \frac{4}{24} + \frac{3}{24} + \frac{1}{24} = \frac{8}{24} = \frac{1}{3}
\]
### Step 7: Calculate A's share of the payment
- The total payment for the work is Rs. 1600.
- A's share of the payment is proportional to the work done by A.
- Since A's work contribution is \(\frac{1}{24}\) of the total work done in 3 days, we calculate A's share:
\[
\text{A's share} = 1600 \times \frac{1}{8} = 200
\]
### Final Answer
- A's share of the payment is Rs. 200.
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