Question

P and Q can do a piece of work in 12 days, Q and R can do a piece of work in 15 days, R and P in 20 days. In how many days R alone can do it?
A) $60$
B) $50$
C) $25$
D) $24$

Answer 1

Hint: According to the question we have to obtain that in how many days R alone can do the work when P and Q can do a piece of work in 12 days, Q and R can do a piece of work in 15 days, R and P in 20 days. So, first of all we have to let the one day work done by the person P, Q, and R.
Now, we will obtain the three linear equations in terms of two variables and after that we have to add all of the three expressions obtained to find the number of days that person R can do the work alone.

Complete step-by-step answer:
Step 1: First of all we have to let that,
Work done by person P is $ = P$days
Work done by person Q is $ = Q$ days
Work done by person R is $ = R$ days
Step 2: Work done by the person P and Q in one day is:
$ \Rightarrow P + Q = \dfrac{1}{{12}}..................(1)$
Step 3: Work done by the person Q and R in one day is:
$ \Rightarrow Q + R = \dfrac{1}{{15}}..................(2)$
Step 4: Work done by the person P and R in one day is:
$ \Rightarrow P + R = \dfrac{1}{{20}}..................(3)$
Step 5: Now, we have to add all the equations (1), (2), and (3) as obtained in the solution step 2, 3, and 4. Hence,
$
   \Rightarrow P + Q + Q + R + P + R = \dfrac{1}{{12}} + \dfrac{1}{{15}} + \dfrac{1}{{20}} \\
   \Rightarrow 2P + 2Q + 2R = \dfrac{{5 + 4 + 3}}{{60}} \\
   \Rightarrow P + Q + R = \dfrac{{12}}{{60 \times 2}}
 $
On solving the expression obtained just above,
$ \Rightarrow P + Q + R = \dfrac{1}{{10}}.................(4)$
Step 6: Now, to obtain the work done by R we have to subtract the equation (1) from the equation (4).
$
   \Rightarrow P + Q + R - P - Q = \dfrac{1}{{10}} - \dfrac{1}{{12}} \\
   \Rightarrow R = \dfrac{{6 - 5}}{{60}} \\
   \Rightarrow R = \dfrac{1}{{60}}
 $
Hence, work done by person R in one day is = $\dfrac{1}{{60}}$ therefore the number of days that R can do the complete work alone is = 60 days.
Final solution: Hence, if P and Q can do a piece of work in 12 days, Q and R can do a piece of work in 15 days, R and P in 20 days then R can do the complete work alone is = 60 days.

Therefore option (A) is the correct answer.

Note: The efficiency of work is the work done by one person in a day or a person can do in one day and a conventional way to solve these problems on efficiency and ratio is to use the concept of efficiency and ratio.