A can do a piece of work in 90 days, B in 40 days and C in 12 days. They work for a day each in turn, i.e., first day A does it alone, second day B does it alone and 3rd day C does it alone. After that the cycle is repeated till the work is finished. They get Rs. 240 for this job. If the wages are divided in proportion to the work each had done. Find the amount A will get?

Options

A. 14
B. 24
C. 34
D. 36

Answer

To solve the problem step by step, we will calculate the amount of work done by A, B, and C, and then determine how much A will earn from the total wages of Rs. 240.

### Step 1: Calculate the work done by A, B, and C in one day.
- A can complete the work in 90 days, so the work done by A in one day = \( \frac{1}{90} \).
- B can complete the work in 40 days, so the work done by B in one day = \( \frac{1}{40} \).
- C can complete the work in 12 days, so the work done by C in one day = \( \frac{1}{12} \).

### Step 2: Calculate the total work done in 3 days.
In 3 days, A, B, and C work in turns:
- Day 1: A works = \( \frac{1}{90} \)
- Day 2: B works = \( \frac{1}{40} \)
- Day 3: C works = \( \frac{1}{12} \)

Total work done in 3 days:
\[ \text{Total work} = \frac{1}{90} + \frac{1}{40} + \frac{1}{12} \]

### Step 3: Find a common denominator and sum the fractions.
The least common multiple (LCM) of 90, 40, and 12 is 360. Now we convert each fraction:
- \( \frac{1}{90} = \frac{4}{360} \)
- \( \frac{1}{40} = \frac{9}{360} \)
- \( \frac{1}{12} = \frac{30}{360} \)

Now, adding these:
\[ \text{Total work} = \frac{4 + 9 + 30}{360} = \frac{43}{360} \]

### Step 4: Calculate the total work done in 24 days.
Since the cycle repeats every 3 days, in 24 days (which is 8 cycles of 3 days):
\[ \text{Total work in 24 days} = 8 \times \frac{43}{360} = \frac{344}{360} \]

### Step 5: Calculate the remaining work.
The total work is considered as 1 (or \( \frac{360}{360} \)):
\[ \text{Remaining work} = 1 - \frac{344}{360} = \frac{16}{360} = \frac{2}{45} \]

### Step 6: Determine who completes the remaining work.
- On the 25th day, A works and completes \( \frac{1}{90} \).
- Remaining work after A's work on the 25th day:
\[ \text{Remaining work} = \frac{2}{45} - \frac{1}{90} = \frac{4}{90} - \frac{1}{90} = \frac{3}{90} = \frac{1}{30} \]

### Step 7: B works on the 26th day.
- B works and completes \( \frac{1}{40} \).
- Remaining work after B's work:
\[ \text{Remaining work} = \frac{1}{30} - \frac{1}{40} \]
Finding a common denominator (120):
\[ \text{Remaining work} = \frac{4}{120} - \frac{3}{120} = \frac{1}{120} \]

### Step 8: C works on the 27th day.
- C works and completes \( \frac{1}{12} \).
- Remaining work after C's work:
\[ \text{Remaining work} = \frac{1}{120} - \frac{1}{120} = 0 \]
Thus, the work is completed on the 27th day.

### Step 9: Calculate the total work done by each worker.
- A worked for 9 days (1 day in each cycle for 8 cycles + 1 additional day on the 25th).
- B worked for 9 days (1 day in each cycle for 8 cycles + 1 additional day on the 26th).
- C worked for 9 days (1 day in each cycle for 8 cycles + 1 additional day on the 27th).

### Step 10: Calculate the ratio of work done.
- Work done by A: \( 9 \times \frac{1}{90} = \frac{1}{10} \)
- Work done by B: \( 9 \times \frac{1}{40} = \frac{9}{40} \)
- Work done by C: \( 9 \times \frac{1}{12} = \frac{3}{4} \)

### Step 11: Convert to a common ratio.
To find the ratio of work done:
- Convert each to a common denominator (LCM of 10, 40, 4 is 40):
- A: \( \frac{4}{40} \)
- B: \( \frac{9}{40} \)
- C: \( \frac{30}{40} \)

Thus, the ratio of work done by A:B:C = 4:9:30.

### Step 12: Calculate A's share of the wages.
Total wages = Rs. 240.
Total parts = \( 4 + 9 + 30 = 43 \).

A's share:
\[ \text{A's share} = \frac{4}{43} \times 240 = \frac{960}{43} \approx 22.33 \]

### Final Answer:
A will get approximately Rs. 22.33.