A is 1.5 times efficient than B therefore takes 8 days less than B to complete a work. If A and B work on alternate days and A works on first day, then in how many days the work will be completed?

Options

A. 17
B. 19
C. 19.5
D. 21

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Determine the time taken by A and B to complete the work.
Let the time taken by B to complete the work be \( x \) days. Since A is 1.5 times more efficient than B, the time taken by A to complete the work will be \( \frac{x}{1.5} = \frac{2x}{3} \).

According to the problem, A takes 8 days less than B to complete the work:
\[ \frac{2x}{3} = x - 8 \]

### Step 2: Solve the equation for \( x \).
To eliminate the fraction, multiply the entire equation by 3:
\[ 2x = 3x - 24 \]
Rearranging gives:
\[ 3x - 2x = 24 \implies x = 24 \]

### Step 3: Calculate the time taken by A.
Now that we have \( x = 24 \), we can find the time taken by A:
\[ \text{Time taken by A} = \frac{2x}{3} = \frac{2 \times 24}{3} = 16 \text{ days} \]

### Step 4: Determine the work done by A and B in one day.
The work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{16} \]
The work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{24} \]

### Step 5: Calculate the total work done in two days.
Since A and B work on alternate days, we can calculate the work done in 2 days (A works on the first day and B on the second day):
\[ \text{Work done in 2 days} = \frac{1}{16} + \frac{1}{24} \]
To add these fractions, find a common denominator (which is 48):
\[ \frac{1}{16} = \frac{3}{48}, \quad \frac{1}{24} = \frac{2}{48} \]
Thus,
\[ \text{Work done in 2 days} = \frac{3}{48} + \frac{2}{48} = \frac{5}{48} \]

### Step 6: Calculate the total number of days required to complete the work.
Let \( W \) be the total work, which we can consider as 1 unit of work. The number of 2-day cycles needed to complete the work is:
\[ \text{Number of 2-day cycles} = \frac{1}{\frac{5}{48}} = \frac{48}{5} = 9.6 \text{ cycles} \]
This means it will take 9 complete cycles (18 days) and a fraction of a cycle to complete the work.

### Step 7: Calculate the remaining work after 9 cycles.
After 9 cycles (18 days), the work done is:
\[ \text{Work done in 18 days} = 9 \times \frac{5}{48} = \frac{45}{48} \]
The remaining work is:
\[ 1 - \frac{45}{48} = \frac{3}{48} = \frac{1}{16} \]

### Step 8: Determine how long it will take to finish the remaining work.
Since A works on the first day of the next cycle, A will complete the remaining work of \( \frac{1}{16} \) in 1 day.

### Final Calculation: Total days taken.
Thus, the total days taken to complete the work is:
\[ \text{Total days} = 18 + 1 = 19 \text{ days} \]

### Conclusion:
The work will be completed in **19 days**.

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