Question

A 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 them in 12 hours. All the machines started at 9 a.m while machine P closed at 11 a.m. and the remaining two machines complete the work. Approximately at what time will the work be finished?
A. 11:30 a.m
B. 12 noon
C. 12:30 p.m
D. 1 p.m

Answer 1

We will first start by finding the per hour printing capacity of the machines P, Q, and R respectively. Then, we will calculate their collective work for 2 hours i.e. from 9 a.m to 11 a.m, and then finally we will find the time taken to finish the work by finding the work Q and R can do collectively in 1 hour.

Complete step-by-step solution:
Now, we have been given that a machine P can print one lakh books in 8 hours, machine Q can print the same number of books in 10 hours while R can print them in 12 hours.
Now, if we take the total work that is printing 1 lakh books a 1 therefore we have,
Work done by machine P in 1 hour $=\dfrac{1}{8}$
Work done by machine Q in 1 hour $=\dfrac{1}{10}$
Work done by machine R in 1 hour $=\dfrac{1}{12}$
Now, therefore we have the work done by all the machines in 1 hour $=\dfrac{1}{8}+\dfrac{1}{10}+\dfrac{1}{12}$
$\begin{align}
  & =\dfrac{15+12+10}{120} \\
 & =\dfrac{37}{120} \\
\end{align}$
Now, we have been given that all the machines are started at 9 a.m and closed at 11 a.m. So, the work done by the machines in 2 hours $=2\times \dfrac{37}{120}$
$\begin{align}
  & =\dfrac{74}{120} \\
 & =\dfrac{37}{60} \\
\end{align}$
Now, we have been given that at 11 a.m machine P is closed and we have to find the work remaining after 11 a. m as we know that the total work is taken as 1 therefore we will subtract the work done by all the machines in 2 hours from 1 to find the remaining work left so we have,
$=1-\dfrac{37}{60}$
$=\dfrac{23}{60}$
Now, we have the work done by Q and R in 1 hour $=\dfrac{1}{10}+\dfrac{1}{12}$
$\begin{align}
  & =\dfrac{6+5}{60} \\
 & =\dfrac{11}{60} \\
\end{align}$
Now, in $\dfrac{11}{60}$ work is done by Q and R in 1 hour. Therefore, 1 work is done by Q and R in $\dfrac{60}{11}$ hour, so $\dfrac{23}{60}$ work is done by Q and R in $\dfrac{60}{11}\times \dfrac{23}{60}$ hour
$\begin{align}
  & =\dfrac{23}{11} \\
 & \approx 2Hour \\
\end{align}$
So, approximately it will take 2 hours after 11 a.m for machines to complete work i.e. 1 p.m.
Hence, the correct option is (D).

Note: It is important to note that we have to find the time taken by Q and R to complete the work by the unitary method. Also, it is important to note that we have written $\dfrac{23}{11}\approx 2\ hours$ approximately as given in the question.