X is 60% more efficient than Y. and Y alone can do a work in 80 days. Working together, X and Y will complete 52% of the same work in:

Options

A. 16 days
B. 15 days
C. 18 days
D. 20 days

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Determine the efficiency of Y
Y can complete the work in 80 days. Therefore, the efficiency of Y (E_Y) can be calculated as:
\[ E_Y = \frac{1}{80} \text{ (work per day)} \]

### Step 2: Calculate the efficiency of X
Since X is 60% more efficient than Y, we can express the efficiency of X (E_X) as:
\[ E_X = E_Y + 0.6 \times E_Y = 1.6 \times E_Y \]
Substituting the value of E_Y:
\[ E_X = 1.6 \times \frac{1}{80} = \frac{1.6}{80} = \frac{1}{50} \text{ (work per day)} \]

### Step 3: Calculate the combined efficiency of X and Y
The combined efficiency (E_XY) of X and Y working together is:
\[ E_{XY} = E_X + E_Y = \frac{1}{50} + \frac{1}{80} \]
To add these fractions, we need a common denominator. The least common multiple of 50 and 80 is 400:
\[ E_{XY} = \frac{8}{400} + \frac{5}{400} = \frac{13}{400} \text{ (work per day)} \]

### Step 4: Calculate the total work
The total work can be calculated based on Y's efficiency:
\[ \text{Total Work} = E_Y \times \text{Time taken by Y} = \frac{1}{80} \times 80 = 1 \text{ (whole work)} \]

### Step 5: Calculate the amount of work to be completed
We need to find out how long it will take for X and Y to complete 52% of the work:
\[ \text{Work to be completed} = 0.52 \times 1 = 0.52 \text{ (of the whole work)} \]

### Step 6: Calculate the time taken to complete 52% of the work
Using the combined efficiency, we can find the time (T) taken to complete 52% of the work:
\[ T = \frac{\text{Work to be completed}}{\text{Combined Efficiency}} = \frac{0.52}{\frac{13}{400}} = 0.52 \times \frac{400}{13} \]
Calculating this gives:
\[ T = \frac{208}{13} \approx 16 \text{ days} \]

### Final Answer
Thus, X and Y together will complete 52% of the work in approximately **16 days**.

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