Question

A can do a piece of work in 7 days of 9 hr each and B can do it in 6 days of 7 hr each. How long will they take to do it working together $\dfrac{42}{5}$ hr a day?
(a) 3 days
(b) 4 days
(c) 4.5 days
(d) 6 days

Answer 1

Hint: Since in the given question first, we will find the total work done by A and B in 1 hr using the unitary method and then we will find the amount of work done by A and B together in 1hr, here we will use the unitary method. Finally, we can easily find the time taken by both A and B together to complete the work in $\dfrac{42}{5}$ hr a day.

Complete step-by-step solution:
We are given that,
A can do a piece of work in 7 days of 9 hr each. So, to do the calculation easier we will convert it in hr i.e.
$\begin{align}
  & =7days\times 9hr \\
 & =63hr \\
\end{align}$
So, if we write A’s total work in 1 day, we will write it as $=\dfrac{1}{63}$
Similarly, B can do a piece of work in 6 days of 7 hr each. So, to do the calculation easier we will convert it in hr i.e.
$\begin{align}
  & =6days\times 7hr \\
 & =42hr \\
\end{align}$
So, if we write B’s total work in 1 day, we will write it as $=\dfrac{1}{42}$
We have got the work done in one day for both A and B. So, now we will find their total work done in one day.
Therefore the work done together by A and B in one day is given by
 $\begin{align}
  & =\dfrac{1}{63}+\dfrac{1}{42} \\
 & =\dfrac{42+63}{63\times 42} \\
 & =\dfrac{105}{63\times 42} \\
\end{align}$
We have found out the total work done by A and B in one day. If we have to find the whole work done by A and B, we will take reciprocal of one day’s work done by both of them.
So together they (A and B) take $\dfrac{63\times 42}{105}$ days to complete the work.
Question provided us the time taken to do the work together $\dfrac{42}{5}$ hr a day and asked to find the number of days. So to find the number of days we will take product of total work done by A and B in one day, i.e. $\left( \dfrac{63\times 42}{105} \right)$ and the total time given, i.e $\left( \dfrac{5}{42} \right)$ hrs a day
$\begin{align}
  & =\dfrac{63\times 42}{105}\times \dfrac{5}{42}days \\
 & =3days \\
\end{align}$

Note: In the above question we calculate the capacity which is work done per unit time of each individual person then we take the LCM of their capacity and finally we easily calculate the time period of their work. Always keep in mind if we have to find anyone’s one day’s work or one hour’s work or say in one unit, we will take the reciprocal of the given unit as well. It is very important to keep all the units say (kilometer, second, hour, etc) the same for the whole calculation.