Question

A and B together can do a piece of work in 12 days; B and C together can do it in 15 days. If A is twice as good a workman as C, in how many days A alone will do the same work?

Answer 1

We start solving the problem by assigning the variable for the total amount of work that needs all to be done. We then find the total amount of work done by A and B together in a single day to proceed through the problem. We then find the total amount of work done by B and C together in a single day and use the information that A is twice as good a workman as C. We then solve the equations obtained to get the required value.

Complete step-by-step answer :
According to the problem, we are given that A and B together can do a piece of work in 12 days; B and C together can do it in 15 days. We need to find how many days A alone will do the same work if he is twice as good a work man as C.
Let us assume the total amount of work that needs to be done is x. So, A and B together do x amount of work in 12 days.
So, we get $12\left( A+B \right)=x$. Let us find the amount of work done by A and B together in a single day.
So, we get $A+B=\dfrac{x}{12}$ ---(1).
We have B and C together does x amount of work in 15 days.
So, we get $15\left( B+C \right)=x$. Let us find the amount of work done by A and B together in a single day.
So, we get $B+C=\dfrac{x}{15}$ ---(2).
According to the problem, it is mentioned that A is twice as good a work man as C. This makes that C can do half the work that A can do in a day.
So, we have $C=\dfrac{A}{2}$. Let us substitute in equation (2).
$\Rightarrow B+\dfrac{A}{2}=\dfrac{x}{15}$.
$\Rightarrow B=\dfrac{x}{15}-\dfrac{A}{2}$ ---(3).
Let us substitute equation (3) in equation (1).
$\Rightarrow A+\dfrac{x}{15}-\dfrac{A}{2}=\dfrac{x}{12}$.
$\Rightarrow A-\dfrac{A}{2}=\dfrac{x}{12}-\dfrac{x}{15}$.
$\Rightarrow \dfrac{A}{2}=\dfrac{5x-4x}{60}$.
$\Rightarrow \dfrac{A}{2}=\dfrac{x}{60}$.
$\Rightarrow A=\dfrac{x}{30}$.
We have found that A can do a work of $\dfrac{x}{30}$ in a single day. This makes that A needed 30 days to complete the x amount of work.
We have found that A can alone complete work in 30 days.
∴ A can alone complete work in 30 days.

Note : We can also assume that the total amount of work done as 1. But this will create a confusion while making calculations. Whenever we get this type of problem, we start by assigning variables for the unknowns to get a better view and for avoiding confusion in calculations. We can calculate the days required for B and C alone to do the total amount of work. Similarly, we can expect problems to find the no. of days required for A and C together to complete given work.