P and Q together can do a job in 6 days. Q and R can finish the same job in `(60)/(7)` days. P started the work and worked for 3 days. Q and R continued for 6 days. Then the difference of days in which R and P can complete the job is.

Options

A. 12
B. 10
C. 15
D. 8

Answer

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

### Step 1: Determine the work rates of P, Q, and R.

Given:
- P and Q together can complete the job in 6 days.
- Q and R can complete the job in \( \frac{60}{7} \) days.

**Work rate of P and Q:**
If P and Q can complete the job in 6 days, their combined work rate is:
\[ \text{Work rate of P and Q} = \frac{1}{6} \text{ (jobs per day)} \]

**Work rate of Q and R:**
If Q and R can complete the job in \( \frac{60}{7} \) days, their combined work rate is:
\[ \text{Work rate of Q and R} = \frac{7}{60} \text{ (jobs per day)} \]

### Step 2: Set up equations for individual work rates.

Let:
- Work rate of P = \( p \)
- Work rate of Q = \( q \)
- Work rate of R = \( r \)

From the above information, we have:
1. \( p + q = \frac{1}{6} \) (1)
2. \( q + r = \frac{7}{60} \) (2)

### Step 3: Solve the equations.

From equation (1):
\[ q = \frac{1}{6} - p \]

Substituting \( q \) in equation (2):
\[ \left(\frac{1}{6} - p\right) + r = \frac{7}{60} \]
\[ r = \frac{7}{60} - \frac{1}{6} + p \]

Convert \( \frac{1}{6} \) to a fraction with a denominator of 60:
\[ \frac{1}{6} = \frac{10}{60} \]
Thus,
\[ r = \frac{7}{60} - \frac{10}{60} + p = p - \frac{3}{60} = p - \frac{1}{20} \]

### Step 4: Find the total work done.

Now, we need to find how much work is done when P works for 3 days and then Q and R work for 6 days.

**Work done by P in 3 days:**
\[ \text{Work by P} = 3p \]

**Work done by Q and R in 6 days:**
\[ \text{Work by Q and R} = 6(q + r) = 6 \cdot \frac{7}{60} = \frac{42}{60} = \frac{7}{10} \]

### Step 5: Calculate total work done.

Total work done:
\[ \text{Total work} = 3p + \frac{7}{10} \]

### Step 6: Find the remaining work.

The total work is 1 (the whole job), so:
\[ 1 = 3p + \frac{7}{10} \]
\[ 3p = 1 - \frac{7}{10} = \frac{3}{10} \]
\[ p = \frac{1}{10} \]

### Step 7: Find Q's and R's work rates.

Using \( p \):
\[ q = \frac{1}{6} - p = \frac{1}{6} - \frac{1}{10} \]
Finding a common denominator (30):
\[ q = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15} \]

Now substituting \( q \) back to find \( r \):
\[ r = \frac{7}{60} - q = \frac{7}{60} - \frac{1}{15} = \frac{7}{60} - \frac{4}{60} = \frac{3}{60} = \frac{1}{20} \]

### Step 8: Calculate the time taken by P and R to complete the job.

**Time taken by P:**
\[ \text{Time by P} = \frac{1}{p} = \frac{1}{\frac{1}{10}} = 10 \text{ days} \]

**Time taken by R:**
\[ \text{Time by R} = \frac{1}{r} = \frac{1}{\frac{1}{20}} = 20 \text{ days} \]

### Step 9: Find the difference in days.

The difference in days between R and P:
\[ \text{Difference} = 20 - 10 = 10 \text{ days} \]

### Final Answer:
The difference in days in which R and P can complete the job is **10 days**.

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