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 $={\displaystyle \frac{1}{8}}$
Work done by machine Q in 1 hour $={\displaystyle \frac{1}{10}}$
Work done by machine R in 1 hour $={\displaystyle \frac{1}{12}}$
Now, therefore we have the work done by all the machines in 1 hour $={\displaystyle \frac{1}{8}}+{\displaystyle \frac{1}{10}}+{\displaystyle \frac{1}{12}}$
$\begin{array}{rl}& ={\displaystyle \frac{15+12+10}{120}}\\ & ={\displaystyle \frac{37}{120}}\end{array}$
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 $=2\times {\displaystyle \frac{37}{120}}$
$\begin{array}{rl}& ={\displaystyle \frac{74}{120}}\\ & ={\displaystyle \frac{37}{60}}\end{array}$
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,
$=1-{\displaystyle \frac{37}{60}}$
$={\displaystyle \frac{23}{60}}$
Now, we have the work done by Q and R in 1 hour $={\displaystyle \frac{1}{10}}+{\displaystyle \frac{1}{12}}$
$\begin{array}{rl}& ={\displaystyle \frac{6+5}{60}}\\ & ={\displaystyle \frac{11}{60}}\end{array}$
Now, in ${\displaystyle \frac{11}{60}}$ work is done by Q and R in 1 hour. Therefore, 1 work is done by Q and R in ${\displaystyle \frac{60}{11}}$ hour, so ${\displaystyle \frac{23}{60}}$ work is done by Q and R in ${\displaystyle \frac{60}{11}}\times {\displaystyle \frac{23}{60}}$ hour
$\begin{array}{rl}& ={\displaystyle \frac{23}{11}}\\ & \approx 2Hour\end{array}$
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 ${\displaystyle \frac{23}{11}}\approx 2hours$ approximately as given in the question.