Question

A and B can together finish a work in 30 days. They worked together for 20 days and then B left. After another 20 days, A finished the remaining work. In how many days A alone can finish the work?

Answer 1

Hint: In this question, we first need to find A and B's work in 20 days. Then calculate the remaining work left and consider that work can be done A alone. Now, if A can do that part in 20 days we can calculate the time taken for him to do the complete work.

Complete step-by-step solution -
Now, given in the question that A & B together can complete the work in 30 days.
Now, let us calculate how much work is done by them in the 20 days
Here, we need to calculate the amount of work that can be done in 20days if they can complete the whole work in 30 days
We need to use the unitary method which is used to find the value of a single unit and then find the required value by multiplying it with the single unit value.
Now, as given that 1 work is done in 30 days we can find the work done is one day using the unitary method which is given by
\[\begin{align}
  & 30\to 1 \\
 & 1\to ? \\
\end{align}\]
Here, now the work done in one day can be further written as
\[\Rightarrow \dfrac{1}{30}\]
Here, let us assume that the work done in 20 days as x
Now, the work done in 20 days can be obtained by multiplying it with the work done in 1 day
\[\Rightarrow x=\dfrac{1}{30}\times 20\]
Now, on further simplification we get,
\[\therefore x=\dfrac{2}{3}\]
As given that after 20 days B left so the remaining part left after 20 days will be done by A alone.
Now, let us calculate the remaining work that A should do from the part we got that they do together
Let us assume remaining work as y which is done by A alone
\[\Rightarrow y=1-x\]
Now, on substituting the respective values we get,
\[\Rightarrow y=1-\dfrac{2}{3}\]
Now, on further simplification we get,
\[\Rightarrow y=\dfrac{1}{3}\]
Now, A takes 20 days to complete \[\dfrac{1}{3}\] of the work
Now, let us find how much time A takes to complete the whole work
Let us assume the time taken to complete the whole work by A as z
Now we know that A takes 20 days to complete \[\dfrac{1}{3}\] of work so we need to calculate how much he takes to complete the whole work.
\[\begin{align}
  & \dfrac{1}{3}\to 20 \\
 & 1\to z \\
\end{align}\]
Now, using the cross multiplication this can be further written as
\[\Rightarrow z\times \dfrac{1}{3}=20\]
Now, on rearranging the terms we get,
\[\Rightarrow z=20\times 3\]
Now, on further simplification we get,
\[\therefore z=60\]
Hence, A takes 60 days to complete the whole work alone.

Note: Instead of calculating the time taken to complete the whole work from the work done in 20 days we can also find it by adding the time taken to do part of work done by A & B to complete by A alone and add that to 20 days. Both the methods give the same result.
It is important to note that the remaining work is done by A in 20 days, not the part done by A & B. Because considering the other way changes the result completely.