To solve the problem step by step, we will follow these calculations:
### Step 1: Determine the work rates of A, B, and C
- A can complete the work in 15 days, so A's work rate is \( \frac{1}{15} \) of the work per day.
- B can complete the work in 30 days, so B's work rate is \( \frac{1}{30} \) of the work per day.
- C can complete the work in 75 days, so C's work rate is \( \frac{1}{75} \) of the work per day.
### Step 2: Calculate the total work rate when A, B, and C work together
To find the combined work rate of A, B, and C, we add their individual work rates:
\[
\text{Combined work rate} = \frac{1}{15} + \frac{1}{30} + \frac{1}{75}
\]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 15, 30, and 75 is 150.
Converting each fraction:
- \( \frac{1}{15} = \frac{10}{150} \)
- \( \frac{1}{30} = \frac{5}{150} \)
- \( \frac{1}{75} = \frac{2}{150} \)
Now, adding these:
\[
\text{Combined work rate} = \frac{10}{150} + \frac{5}{150} + \frac{2}{150} = \frac{17}{150}
\]
### Step 3: Determine the total work done
Since they complete the work together, we can say that they complete 1 unit of work.
### Step 4: Calculate the time taken to complete the work together
Let the time taken to complete the work together be \( T \).
\[
\text{Total work} = \text{Combined work rate} \times T
\]
Setting this equal to 1 unit of work:
\[
1 = \frac{17}{150} \times T \implies T = \frac{150}{17} \text{ days}
\]
### Step 5: Calculate individual contributions of A, B, and C
Now we can find how much work each person does in that time:
- Work done by A:
\[
\text{Work by A} = A's \text{ rate} \times T = \frac{1}{15} \times \frac{150}{17} = \frac{10}{17}
\]
- Work done by B:
\[
\text{Work by B} = B's \text{ rate} \times T = \frac{1}{30} \times \frac{150}{17} = \frac{5}{17}
\]
- Work done by C:
\[
\text{Work by C} = C's \text{ rate} \times T = \frac{1}{75} \times \frac{150}{17} = \frac{2}{17}
\]
### Step 6: Calculate the total payment and shares
The total payment for the work is Rs. 1615. The shares of A, B, and C are proportional to the work they did:
- Total parts = \( \frac{10}{17} + \frac{5}{17} + \frac{2}{17} = \frac{17}{17} = 1 \)
Now we can calculate their shares:
- Share of A:
\[
\text{Share of A} = \frac{10}{17} \times 1615 = \frac{16150}{17} = 950
\]
- Share of B:
\[
\text{Share of B} = \frac{5}{17} \times 1615 = \frac{8075}{17} = 475
\]
- Share of C:
\[
\text{Share of C} = \frac{2}{17} \times 1615 = \frac{3230}{17} = 190
\]
### Step 7: Calculate the difference in shares of A and C
The difference in shares of A and C is:
\[
\text{Difference} = \text{Share of A} - \text{Share of C} = 950 - 190 = 760
\]
### Final Answer:
The difference in shares of A and C is Rs. 760.
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