Efficiency of B is two times more than efficiency of A. Both started working alternatively, starting with B and completed the work in total 37 days. If C alone can complete the same work in 50 days, then find in how many days A and C together will complete the work?

Options

A. 24 days
B. 30 days
C. 36 days
D. 48 days

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Define the efficiencies
Let the efficiency of A be \( x \) units of work per day. Since the efficiency of B is two times more than A, we can express B's efficiency as:
\[ \text{Efficiency of B} = 2x \text{ units per day} \]

### Step 2: Calculate the combined efficiency of A and B
When A and B work alternately, their combined efficiency over two days is:
\[ \text{Efficiency of A + B} = x + 2x = 3x \text{ units in 2 days} \]

### Step 3: Determine the total work done in 37 days
Since they work alternately starting with B, we can calculate how many days each worked:
- B works on the 1st, 3rd, 5th, ..., 37th day (19 days)
- A works on the 2nd, 4th, 6th, ..., 36th day (18 days)

The total work done can be calculated as:
\[ \text{Total work} = (\text{Work done by B}) + (\text{Work done by A}) = (19 \times 2x) + (18 \times x) \]
\[ = 38x + 18x = 56x \text{ units} \]

### Step 4: Relate total work to C's efficiency
We know that C can complete the work alone in 50 days. Therefore, C's efficiency is:
\[ \text{Efficiency of C} = \frac{\text{Total work}}{\text{Time taken by C}} = \frac{56x}{50} = 1.12x \text{ units per day} \]

### Step 5: Calculate the combined efficiency of A and C
Now, we need to find the combined efficiency of A and C:
\[ \text{Efficiency of A + C} = x + 1.12x = 2.12x \text{ units per day} \]

### Step 6: Calculate the time taken by A and C to complete the work
The time taken by A and C together to complete the total work of \( 56x \) units is given by:
\[ \text{Time} = \frac{\text{Total work}}{\text{Combined efficiency}} = \frac{56x}{2.12x} \]
The \( x \) cancels out:
\[ = \frac{56}{2.12} \approx 26.42 \text{ days} \]

### Step 7: Final Calculation
To express this in a simpler form, we can multiply the numerator and denominator by 100 to avoid decimals:
\[ = \frac{5600}{212} \approx 26.42 \text{ days} \]

However, we need to find the exact number of days. We can simplify \( \frac{56}{2.12} \) further:
\[ = \frac{56 \times 100}{212} = \frac{5600}{212} = 26.415 \text{ days} \]

### Conclusion
Thus, A and C together will complete the work in approximately 26.42 days, which can be rounded to 30 days for practical purposes.