Machine P can print one lakh books in 8 hours, machine Q can print the same number of books in 10 hours while machine R can print the same number of books in 12 hours. All the machines are started at 9 a.m. while machine P is closed at 11 a.m. and the remaining two machines complete the work. Approximately at what time will the work be finished ?

Options

A. `11:30` AM
B. 12 noon
C. 12 :30 p.m.
D. 1 p.m.

Answer

To solve the problem step by step, we will calculate the work done by each machine, the total work, and the time taken to complete the work.

### Step 1: Determine the work done by each machine per hour
- Machine P can print 1 lakh books in 8 hours.
- Work done by P in 1 hour = 100000 books / 8 hours = 12500 books/hour
- Machine Q can print 1 lakh books in 10 hours.
- Work done by Q in 1 hour = 100000 books / 10 hours = 10000 books/hour
- Machine R can print 1 lakh books in 12 hours.
- Work done by R in 1 hour = 100000 books / 12 hours = 8333.33 books/hour (approximately)

### Step 2: Calculate the combined work done by all machines in the first 2 hours
- All machines work together from 9 a.m. to 11 a.m. (2 hours).
- Combined work done in 1 hour = Work done by P + Work done by Q + Work done by R
- Combined work done in 1 hour = 12500 + 10000 + 8333.33 = 30833.33 books/hour
- Work done in 2 hours = 30833.33 books/hour * 2 hours = 61666.67 books

### Step 3: Calculate the remaining work
- Total work = 100000 books
- Work done in 2 hours = 61666.67 books
- Remaining work = Total work - Work done in 2 hours
- Remaining work = 100000 - 61666.67 = 38333.33 books

### Step 4: Calculate the work done by machines Q and R after P is closed
- After 11 a.m., only machines Q and R are working.
- Combined work done by Q and R in 1 hour = Work done by Q + Work done by R
- Combined work done by Q and R = 10000 + 8333.33 = 18333.33 books/hour

### Step 5: Calculate the time taken to finish the remaining work
- Time taken to finish remaining work = Remaining work / Combined work done by Q and R
- Time = 38333.33 books / 18333.33 books/hour ≈ 2.09 hours

### Step 6: Calculate the finishing time
- 2.09 hours is approximately 2 hours and 5 minutes.
- Since the remaining work starts at 11 a.m., we add 2 hours and 5 minutes to 11 a.m.
- Finishing time = 11:00 a.m. + 2 hours 5 minutes = 1:05 p.m.

### Conclusion
The work will be finished approximately at **1:05 p.m.**.

---