P and Q together can complete a work in 24 days, while Q and R working together can complete the same job in 32 days. P and Q started the work and did it for 8 days, after that P left the work and R joined Q and after 12 more days, Q also left the work. Then, the remaining work was completed by R in 28 days. Find in how many days R will complete the work alone?

Options

A. 96 days
B. 72 days
C. 108 days
D. 90 days

Answer

To solve the problem step by step, we will break down the information provided and calculate the required values systematically.

### Step 1: Calculate Total Work
P and Q together can complete the work in 24 days. Therefore, their combined efficiency is:
\[ \text{Efficiency of P + Q} = \frac{1}{24} \text{ work/day} \]
Similarly, Q and R can complete the work in 32 days, so their combined efficiency is:
\[ \text{Efficiency of Q + R} = \frac{1}{32} \text{ work/day} \]

To find the total work, we can use the least common multiple (LCM) of the two time periods:
\[ \text{Total Work} = \text{LCM}(24, 32) = 96 \text{ units of work} \]

### Step 2: Calculate Individual Efficiencies
From the efficiencies calculated:
- Let the efficiency of P be \( p \), Q be \( q \), and R be \( r \).
- We have:
\[ p + q = \frac{96}{24} = 4 \text{ (units of work/day)} \]
\[ q + r = \frac{96}{32} = 3 \text{ (units of work/day)} \]

### Step 3: Set Up Equations
Now, we can set up the equations:
1. \( p + q = 4 \)
2. \( q + r = 3 \)

From the first equation, we can express \( p \) in terms of \( q \):
\[ p = 4 - q \]

Substituting \( p \) into the second equation:
\[ (4 - q) + r = 3 \]
This simplifies to:
\[ r = 3 - 4 + q = q - 1 \]

### Step 4: Calculate Work Done by P and Q
P and Q work together for 8 days:
\[ \text{Work done by P + Q in 8 days} = 8 \times 4 = 32 \text{ units} \]

### Step 5: Calculate Remaining Work
The total work is 96 units, so the remaining work after P and Q have worked for 8 days is:
\[ \text{Remaining Work} = 96 - 32 = 64 \text{ units} \]

### Step 6: Calculate Work Done by Q and R
Next, Q and R work together for 12 days:
\[ \text{Work done by Q + R in 12 days} = 12 \times 3 = 36 \text{ units} \]

### Step 7: Calculate Remaining Work After Q and R
After Q and R have worked for 12 days, the remaining work is:
\[ \text{Remaining Work} = 64 - 36 = 28 \text{ units} \]

### Step 8: Calculate Work Done by R
R completes the remaining 28 units of work in 28 days:
\[ \text{Efficiency of R} = \frac{28 \text{ units}}{28 \text{ days}} = 1 \text{ unit of work/day} \]

### Step 9: Calculate Time Taken by R Alone
To find out how many days R will take to complete the entire work alone:
\[ \text{Time taken by R} = \frac{\text{Total Work}}{\text{Efficiency of R}} = \frac{96}{1} = 96 \text{ days} \]

### Final Answer
R will complete the work alone in **96 days**.