Pascal and Rascal are two workers. Working together they can complete the whole work in 10 hours. If the Pascal worked for 2.5 hours and Rascal worked for 8.5 hours, still there was half of the work to be done. In how many hours Pascal working alone, can complete the whole work?

Options

A. 24 hours
B. `17 1/7` hours
C. 40 hours
D. can’t be determined

Answer

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

### Step 1: Determine the work done by Pascal and Rascal together
Pascal and Rascal can complete the whole work together in 10 hours. Therefore, their combined work rate is:
\[ \text{Combined work rate} = \frac{1 \text{ work}}{10 \text{ hours}} = 0.1 \text{ work/hour} \]

### Step 2: Calculate the work done by Pascal and Rascal individually
Let the work done by Pascal in one hour be \( P \) and the work done by Rascal in one hour be \( R \). Therefore, we have:
\[ P + R = 0.1 \]

### Step 3: Work done by Pascal and Rascal in the given hours
Pascal worked for 2.5 hours and Rascal worked for 8.5 hours. The work done by each can be expressed as:
- Work done by Pascal in 2.5 hours: \( 2.5P \)
- Work done by Rascal in 8.5 hours: \( 8.5R \)

The total work done by both is:
\[ 2.5P + 8.5R \]

### Step 4: Set up the equation based on the remaining work
According to the problem, after Pascal and Rascal worked for the specified hours, half of the work is still remaining. Therefore, the equation can be set up as:
\[ 2.5P + 8.5R = \frac{1}{2} \]

### Step 5: Substitute \( R \) from the combined work rate into the equation
From \( P + R = 0.1 \), we can express \( R \) as:
\[ R = 0.1 - P \]
Now substitute \( R \) into the work done equation:
\[ 2.5P + 8.5(0.1 - P) = \frac{1}{2} \]

### Step 6: Simplify the equation
Expanding the equation gives:
\[ 2.5P + 0.85 - 8.5P = \frac{1}{2} \]
Combining like terms:
\[ -6P + 0.85 = \frac{1}{2} \]

### Step 7: Solve for \( P \)
Convert \(\frac{1}{2}\) to a decimal:
\[ 0.5 \]
Now, rearranging gives:
\[ -6P = 0.5 - 0.85 \]
\[ -6P = -0.35 \]
\[ P = \frac{0.35}{6} = \frac{7}{120} \]

### Step 8: Calculate the total work
Now that we have \( P \), we can find the total work. Since \( P = \frac{7}{120} \) work/hour, the total work can be calculated as:
\[ \text{Total work} = 1 = \text{Total work units} \]

### Step 9: Find the time taken by Pascal to complete the work alone
To find the time taken by Pascal to complete the whole work alone, we use:
\[ \text{Time} = \frac{\text{Total work}}{\text{Work rate of Pascal}} = \frac{1}{P} = \frac{1}{\frac{7}{120}} = \frac{120}{7} \approx 17.14 \text{ hours} \]

### Final Answer
Pascal can complete the whole work alone in approximately **17.14 hours**.

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