Question

# A and B can do work in 8 days. B and C can do the same work in 12 days. A, B, and C can together finish it in 6 days. A and C together will do it in:(a) 4 days(b) 6 days(c) 8 days(d) 12 days

Hint: Calculate the one day work of A and B together, B and C together and A, B, and C together by using the fact that if someone finishes the work in ‘x’ days, then the one day work of that person is $\dfrac{1}{x}$. Simplify the equations by rearranging them to calculate the one day work of A and C, and thus, the number of days taken by A and C to complete the work.

We have data regarding the number of days taken by A and B, B and C and A, B, and C to finish the work. We have to calculate the number of days taken by A and C to finish the work.
We know that if someone finishes the work in ‘x’ days, then the one day work of that person is $\dfrac{1}{x}$.
We know that A and B can complete the work in 8 days.
Thus, one day work of A and B is $A+B=\dfrac{1}{8}.....\left( 1 \right)$.
We know that B and C can complete the work in 12 days.
Thus, one day work of B and C is $B+C=\dfrac{1}{12}.....\left( 2 \right)$.
We know that A, B, and C can complete the work in C days.
Thus, one day work of A, B, and C is $A+B+C=\dfrac{1}{6}.....\left( 3 \right)$.
Adding equation (1) and (2), we have $A+2B+C=\dfrac{1}{8}+\dfrac{1}{12}$.
Simplifying the above equation by taking LCM, we have $A+2B+C=\dfrac{1}{8}+\dfrac{1}{12}=\dfrac{3+2}{24}=\dfrac{5}{24}.....\left( 4 \right)$.
Subtracting equation (3) from equation (4), we have $\left( A+2B+C \right)-\left( A+B+C \right)=\dfrac{5}{24}-\dfrac{1}{6}$.
Simplifying the above equation by taking LCM, we have $B=\dfrac{5-4}{24}=\dfrac{1}{24}.....\left( 5 \right)$.
Substituting equation (5) in equation (3), we have $A+\dfrac{1}{24}+C=\dfrac{1}{6}$.
Rearranging the terms of the above equation, we have $A+C=\dfrac{1}{6}-\dfrac{1}{24}=\dfrac{4-1}{24}=\dfrac{3}{24}=\dfrac{1}{8}$.
Thus, one day work of A and C is $\dfrac{1}{8}$.
Hence, A and C take 8 days to complete the work, which is option (c).

Note: One must observe that this is not a question based on the unitary method. As the number of workers increases, the days taken to complete the given work decreases. This is because the number of days and amount of work are inversely proportional to each other. If we use a unitary method to solve this question, we will get an incorrect answer.