A and B started the work alternatively starting with A. On last day A completed the work by doing 12.5% of the whole work. Which of the following can be the possible value of time taken by B alone to do that work if A alone can do the whole work in 6 days.

Options

A. 15 days
B. 8 days
C. 10 days
D. 6 days

Answer

To solve the problem step by step, we will analyze the information given and derive the necessary calculations.

### Step 1: Understanding the Work Done by A
A can complete the whole work in 6 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{6} \text{ of the total work} \]

### Step 2: Total Work Calculation
Let's assume the total work is 600 units (as suggested in the video). Hence, A can do:
\[ \text{Work done by A in one day} = \frac{600}{6} = 100 \text{ units} \]

### Step 3: Work Done by A on the Last Day
On the last day, A completed 12.5% of the total work:
\[ \text{Work done by A on the last day} = 12.5\% \text{ of } 600 = \frac{12.5}{100} \times 600 = 75 \text{ units} \]

### Step 4: Total Work Done Before the Last Day
Since A completed 75 units on the last day, the work done before the last day is:
\[ \text{Total work done before last day} = 600 - 75 = 525 \text{ units} \]

### Step 5: Work Alternating Between A and B
A and B work alternately, starting with A. Let's denote the time taken by B to complete the whole work as \( x \) days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{600}{x} \text{ units} \]

### Step 6: Total Days Calculation
Let’s say the total number of days taken to complete the work is \( n \). Since A starts the work, the sequence of work done will be:
- Day 1: A works
- Day 2: B works
- Day 3: A works
- Day 4: B works
- ...
- Last Day: A works

If \( n \) is odd, A will work on the last day. The number of days A works will be:
\[ \text{Days A works} = \frac{n + 1}{2} \]
And the number of days B works will be:
\[ \text{Days B works} = \frac{n - 1}{2} \]

### Step 7: Total Work Equation
The total work done by A and B can be expressed as:
\[ \text{Total Work} = \text{Work done by A} + \text{Work done by B} \]
This gives us:
\[ 600 = \left(\frac{n + 1}{2} \times 100\right) + \left(\frac{n - 1}{2} \times \frac{600}{x}\right) \]

### Step 8: Solving for \( n \) and \( x \)
Substituting the known values and simplifying:
\[ 600 = 50(n + 1) + \frac{300(n - 1)}{x} \]
This equation can be solved for \( n \) and \( x \) to find the possible values of time taken by B alone.

### Conclusion
From the above steps, we can derive the possible values for \( x \) based on the equation we formed.