A (working alone) can complete a task in 'x' days. A worked alone for `6 1/2` days and was then replaced by B and C who (working together) completed the remaining task in 3 days. Working alone, B and C take '0.5x' days and 8 days respectively to complete the entire task. What is the value of x?

Options

A. 8
B. 24
C. 20
D. 16

Answer

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

### Step 1: Define the total work
Let's assume the total work is 1 unit.

### Step 2: Calculate A's work rate
A can complete the entire task in 'x' days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{x} \text{ units per day} \]

### Step 3: Calculate the work done by A in 6.5 days
A worked alone for \(6 \frac{1}{2}\) days, which is \(6.5\) days. The work done by A in this time is:
\[ \text{Work done by A} = \text{Work rate of A} \times \text{Time} = \frac{1}{x} \times 6.5 = \frac{6.5}{x} \text{ units} \]

### Step 4: Calculate the remaining work
The remaining work after A has worked for \(6.5\) days is:
\[ \text{Remaining work} = 1 - \frac{6.5}{x} \]

### Step 5: Calculate the work rate of B and C
B can complete the task in \(0.5x\) days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{0.5x} = \frac{2}{x} \text{ units per day} \]
C can complete the task in \(8\) days, so C's work rate is:
\[ \text{Work rate of C} = \frac{1}{8} \text{ units per day} \]

### Step 6: Calculate the combined work rate of B and C
The combined work rate of B and C is:
\[ \text{Combined work rate of B and C} = \frac{2}{x} + \frac{1}{8} \]

### Step 7: Calculate the work done by B and C in 3 days
B and C completed the remaining work in \(3\) days. The work done by B and C in this time is:
\[ \text{Work done by B and C} = \left(\frac{2}{x} + \frac{1}{8}\right) \times 3 \]

### Step 8: Set up the equation
Since the work done by B and C is equal to the remaining work, we can set up the equation:
\[ \left(\frac{2}{x} + \frac{1}{8}\right) \times 3 = 1 - \frac{6.5}{x} \]

### Step 9: Simplify the equation
Distributing the \(3\) on the left side gives:
\[ \frac{6}{x} + \frac{3}{8} = 1 - \frac{6.5}{x} \]
Now, let's combine like terms:
\[ \frac{6}{x} + \frac{6.5}{x} = 1 - \frac{3}{8} \]
\[ \frac{12.5}{x} = 1 - \frac{3}{8} \]
Calculating \(1 - \frac{3}{8}\):
\[ 1 - \frac{3}{8} = \frac{8}{8} - \frac{3}{8} = \frac{5}{8} \]
So we have:
\[ \frac{12.5}{x} = \frac{5}{8} \]

### Step 10: Solve for x
Cross-multiplying gives:
\[ 12.5 \times 8 = 5x \]
\[ 100 = 5x \]
\[ x = \frac{100}{5} = 20 \]

### Final Answer
The value of \(x\) is \(20\).

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