Efficiency of A is two times more than efficiency of B. Both A & B start working alternatively, starting with A and they complete the work in total `74(2)/(3)` days. C can alone complete the same work in 100 days.
If A, B & C work for 36 days, 18 days & 16 days and get total wage of Rs. 2500, then find difference between wage of share of B & C together and wage share of A?

Options

A. 900 Rs.
B. 1500 Rs.
C. 1100 Rs.
D. 1800 Rs.

Answer

To solve the problem step by step, let's break it down as follows:

### Step 1: Determine the Efficiency of A and B
Given that the efficiency of A is two times more than the efficiency of B, we can denote the efficiency of B as \( x \). Therefore, the efficiency of A will be \( 2x \).

### Step 2: Calculate Total Work Done
A and B work alternately, starting with A, and they complete the work in \( 74 \frac{2}{3} \) days. We convert this mixed fraction into an improper fraction:
\[ 74 \frac{2}{3} = \frac{224}{3} \text{ days} \]

In two days, A and B together complete:
\[ \text{Work done in 2 days} = \text{A's work} + \text{B's work} = 2x + x = 3x \]

The number of complete 2-day cycles in \( \frac{224}{3} \) days is:
\[ \text{Number of cycles} = \frac{224/3}{2} = \frac{112}{3} \]

Thus, the total work done in these cycles is:
\[ \text{Total work} = \left(\frac{112}{3}\right) \times 3x = 112x \]

### Step 3: Calculate Total Work in Terms of Days
Now, we need to account for the extra day (since \( \frac{224}{3} \) is not a whole number). The extra work done by A on the last day is:
\[ \text{Extra work by A} = 2x \]

So, the total work \( W \) is:
\[ W = 112x + 2x = 114x \]

### Step 4: Relate Work to C's Efficiency
C can complete the same work in 100 days, so C's efficiency is:
\[ \text{C's efficiency} = \frac{W}{100} = \frac{114x}{100} = 1.14x \]

### Step 5: Calculate Work Done by A, B, and C
Now we calculate the work done by A, B, and C over the given days:
- A works for 36 days:
\[ \text{Work by A} = 36 \times 2x = 72x \]
- B works for 18 days:
\[ \text{Work by B} = 18 \times x = 18x \]
- C works for 16 days:
\[ \text{Work by C} = 16 \times 1.14x = 18.24x \]

### Step 6: Total Work Done
Adding the work done by A, B, and C:
\[ \text{Total work done} = 72x + 18x + 18.24x = 108.24x \]

### Step 7: Verify Total Work
Since we calculated earlier that the total work \( W = 114x \), we can see that:
\[ 108.24x < 114x \]
This indicates that the total work is indeed consistent with the original work calculation.

### Step 8: Calculate Wages
The total wage for the work done is Rs. 2500. The share of wages is proportional to the work done:
- Total work done by A, B, and C:
\[ \text{Total work} = 114x \]

- Wage per unit work:
\[ \text{Wage per unit} = \frac{2500}{114x} \]

### Step 9: Calculate Individual Wages
- Wage of A:
\[ \text{Wage of A} = 72x \times \frac{2500}{114x} = \frac{72 \times 2500}{114} = 1263.16 \]
- Wage of B:
\[ \text{Wage of B} = 18x \times \frac{2500}{114x} = \frac{18 \times 2500}{114} = 394.74 \]
- Wage of C:
\[ \text{Wage of C} = 18.24x \times \frac{2500}{114x} = \frac{18.24 \times 2500}{114} = 400.00 \]

### Step 10: Calculate the Difference in Wages
Now, we find the difference between the combined wage of B and C and the wage of A:
\[ \text{Combined wage of B and C} = 394.74 + 400.00 = 794.74 \]
\[ \text{Difference} = 1263.16 - 794.74 = 468.42 \]

### Conclusion
The difference between the wage share of B and C together and the wage share of A is approximately Rs. 468.42.