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 are started at 9 A.M, while machine P is closed at 11 A.M and the remaining two machines complete work. Approximately at what time will the work (to print one lakh books) be finished?
11 : 30 A.M
12 noon
12 : 30 P. M
1 : 00 P. M

Answer 1

Hint: Find the work done by machine P, Q and R in 1 hr. Calculate it for 2 hours. Find the remaining work done by Q and R after 11 P. M. Thus get the work done by Q and R in one hour and thus find the time taken by them to finish the remaining work.
Complete step by step answer:
The total number of books printed = 1 lakh books.
Machine P works for 8 hours, thus work done by machine P in 1 hr = \[\dfrac{1}{8}\].
Machine Q works for 10 hours, thus work done by machine Q in 1 hr = \[\dfrac{1}{10}\].
Machine R works for 12 hours, thus work done by machine R in 1 hr = \[\dfrac{1}{12}\].
The total work done by machines P, Q and R in 1 hr is,
(P + Q + R)’s work in 1 hr = \[\dfrac{1}{8}+\dfrac{1}{10}+\dfrac{1}{12}\]
                                             \[\begin{align}
  & =\dfrac{15+12+10}{120} \\
 & =\dfrac{37}{120} \\
\end{align}\]
\[\therefore \] Work done by (P + Q + R) in 1 hr \[=\dfrac{37}{120}\].
Work done by (P + Q + R) in 2hrs \[=\dfrac{37}{120}\times 2=\dfrac{37}{60}\].
Remaining work \[=1-\dfrac{37}{60}=\dfrac{23}{60}\].
It is said the machine P stops after 2 hours.
\[\therefore \] Work done by (Q + R) in 1 hr \[=\dfrac{1}{10}+\dfrac{1}{12}=\dfrac{12+10}{120}=\dfrac{22}{120}=\dfrac{11}{60}\].
Now \[\dfrac{11}{60}\] is the work done by (Q + R) in 1 hr.
Thus \[\dfrac{23}{60}\] of remaining work will be done by (Q + R) in how many hours,
\[=\dfrac{\dfrac{23}{60}}{\dfrac{11}{60}}=\dfrac{23}{60}\times \dfrac{60}{11}=\dfrac{23}{11}\approx 2\] hours.
So the work will be finished approximately 2 hours after 11 A.M i.e. around 1 P.M.
Thus the work will be finished at approximately 1 P.M.
\[\therefore \] Option (d) is the correct answer.

Note: Time and work is an important topic in quantitative aptitude. The basic concept of time and work is similar to that of arithmetic topics which is the concept of proportionality. Thus understand the question correctly and remember to find the remaining work done by Q and R after P leaves.

The total number of books printed = 1 lakh books.
Machine P works for 8 hours, thus work done by machine P in 1 hr = \[\dfrac{1}{8}\].
Machine Q works for 10 hours, thus work done by machine Q in 1 hr = \[\dfrac{1}{10}\].
Machine R works for 12 hours, thus work done by machine R in 1 hr = \[\dfrac{1}{12}\].
The total work done by machines P, Q and R in 1 hr is,
(P + Q + R)’s work in 1 hr = \[\dfrac{1}{8}+\dfrac{1}{10}+\dfrac{1}{12}\]
                                             \[\begin{align}
  & =\dfrac{15+12+10}{120} \\
 & =\dfrac{37}{120} \\
\end{align}\]
\[\therefore \] Work done by (P + Q + R) in 1 hr \[=\dfrac{37}{120}\].
Work done by (P + Q + R) in 2hrs \[=\dfrac{37}{120}\times 2=\dfrac{37}{60}\].
Remaining work \[=1-\dfrac{37}{60}=\dfrac{23}{60}\].
It is said the machine P stops after 2 hours.
\[\therefore \] Work done by (Q + R) in 1 hr \[=\dfrac{1}{10}+\dfrac{1}{12}=\dfrac{12+10}{120}=\dfrac{22}{120}=\dfrac{11}{60}\].
Now \[\dfrac{11}{60}\] is the work done by (Q + R) in 1 hr.
Thus \[\dfrac{23}{60}\] of remaining work will be done by (Q + R) in how many hours,
\[=\dfrac{\dfrac{23}{60}}{\dfrac{11}{60}}=\dfrac{23}{60}\times \dfrac{60}{11}=\dfrac{23}{11}\approx 2\] hours.
So the work will be finished approximately 2 hours after 11 A.M i.e. around 1 P.M.
Thus the work will be finished at approximately 1 P.M.
\[\therefore \] Option (d) is the correct answer.

Note: Time and work is an important topic in quantitative aptitude. The basic concept of time and work is similar to that of arithmetic topics which is the concept of proportionality. Thus understand the question correctly and remember to find the remaining work done by Q and R after P leaves.