P alone can do a work in 15 days and Q alone cando it in 20 days. With the help of R they finish the work in 5 days. How many days R alone will take to finish the work?

Let P will complete the work in 15 days

P can do the work in one day $$=\dfrac{1}{15}$$

Q can finish the work in 20 days.

Q can do the work in one day $$\dfrac{1}{20}$$

Let R can do the work in x days.

R can do the work in one day $$\dfrac{1}{x}$$

If P, Q and R working together, then they can finish the work in 5 days.

Hence, working together, then can do the work in one day $$\dfrac{1}{15}+\dfrac{1}{20}+\dfrac{1}{x}=\dfrac{1}{5}$$

$$\Rightarrow \dfrac{1}{x}=\dfrac{1}{5}-\dfrac{1}{15}-\dfrac{1}{20}$$

$$\Rightarrow \dfrac{1}{x}=\dfrac{1}{5}-\dfrac{7}{60}$$

$$\Rightarrow \dfrac{1}{x}=\dfrac{5}{60}=\dfrac{1}{12}$$

Hence x=12 days. So , R can finish the work in 12 days.