Question

Twenty four men can complete work in sixteen days. Thirty-two women can complete the same work in twenty-four days. Sixteen men and sixteen women started worked for 12 days. How many more men are to be added to complete the remaining work in 2 days?
A. 16
B.24
C.36
D.48

Answer 1

Hint: We will first take the number of extra men required to complete the work in the next 2 days as ‘x’ and then we will find out the amount of work that 1 man in 1 day and 1 woman in 1 day can do. Using this, we will find the amount of work 16 men and 16 women can do together in 12 days. Our next step will be to find the amount of work done remained which has to be completed in the next 2 days. Then, we will find the amount of work 16 men and 16 women can do in the next 2 days and with the help of that we will get the work done which is remained for ‘x’ number of men to do in 2 days. We will then find the amount of work done that can be done by ‘x’ men in 2 days and keep it equal to the amount of work done remained for ‘x’ men to do in the next days. Using this relation, we will find the value of x.

Complete step-by-step solution:
Now, let the number of extra men to be added to complete the remaining work in the next 2 days be ‘x’.
Now, we have been given that 24 men can complete the work in 16 days. This means that they work for 16 days and get one task done. Thus, the work done by them in one day will be equal to the total work in 16 days divided by the number of days required to do the work, i.e. 16.
Now, we know that they completed 1 task in 16 days. Thus, there total work done for 16 days can be taken as ‘1’.
Thus, the work done by 24 men in 1 day $=\dfrac{1}{16}$
Now, this work is done by 24 men in one day. So the work done by 1 man in 1 day will be equal to the work done by 24 men in 1 day divided by the total number of men doing the work, i.e. 24.
Thus, work done by 1 man in 1 day is given as:
$\begin{align}
  & \Rightarrow \dfrac{\dfrac{1}{16}}{24} \\
 & \Rightarrow \dfrac{1}{16\times 24} \\
 & \Rightarrow \dfrac{1}{384} \\
\end{align}$
Now, we have been given that 32 women complete the same work in 24 days. Thus, the work done by them in one day will be equal to the total work done by them in 24 days divided by the number of days taken to do that work, i.e. 24.
Thus, work done by 32 women in 1 day $=\dfrac{1}{24}$
Now, this work is done by 32 women in 1 day. So the work done by 1 woman in 1 day will be equal to the work done by 32 women in 1 day divided by the total number of women doing the work, i.e. 32.
Thus, work done by 1 woman in 1 day is given as:
$\begin{align}
  & \Rightarrow \dfrac{\dfrac{1}{24}}{32} \\
 & \Rightarrow \dfrac{1}{24\times 32} \\
 & \Rightarrow \dfrac{1}{768} \\
\end{align}$
Thus, the work done by 1 man in one day is $\dfrac{1}{384}$ and 1 woman in one day is $\dfrac{1}{768}$ .
Now, we have been given that 16 men and 16 women did the same work for 12 days.
Thus, work done by 16 men in 1 day is given as:
$\begin{align}
  & \Rightarrow 16\times \dfrac{1}{384} \\
 & \Rightarrow \dfrac{1}{24} \\
\end{align}$
Similarly, the work done by 16 women in 1 day is given as:
$\begin{align}
  & \Rightarrow 16\times \dfrac{1}{768} \\
 & \Rightarrow \dfrac{1}{48} \\
\end{align}$
 Now, the total work done by 16 men in 12 days in given by multiplying the work done by 16 men in 1 day by the number of days they work for, i.e. 12.
Thus, the total work done by 16 men in 12 days is given as:
$\begin{align}
  & \Rightarrow 12\times \dfrac{1}{24} \\
 & \Rightarrow \dfrac{1}{2} \\
\end{align}$
Similarly, the total work done by 16 women in 12 days is given as:
$\begin{align}
  & \Rightarrow 12\times \dfrac{1}{48} \\
 & \Rightarrow \dfrac{1}{4} \\
\end{align}$
Thus, the total work done by men and women in 12 days is given as:
$\begin{align}
  & \Rightarrow \dfrac{1}{2}+\dfrac{1}{4} \\
 & \Rightarrow \dfrac{3}{4} \\
\end{align}$
Now, we will find the amount of work remained which has to be done in the next 2 days.
The work remaining will be given by subtracting the total work done in 12 days from 1.
Thus, remaining work done is given as:
$\begin{align}
  & \Rightarrow 1-\dfrac{3}{4} \\
 & \Rightarrow \dfrac{1}{4} \\
\end{align}$
Now, we will find the amount of work 16 men and 16 women can do in the next two days.
Thus, work done by 16 men in the next two days will be:
$\begin{align}
  & \Rightarrow 2\times \dfrac{1}{24} \\
 & \Rightarrow \dfrac{1}{12} \\
\end{align}$
Similarly, work done by 16 women in the next 2 days will be:
$\begin{align}
  & \Rightarrow 2\times \dfrac{1}{48} \\
 & \Rightarrow \dfrac{1}{24} \\
\end{align}$
Thus, the total work done by 16 men and 16 women in the next 2 days will be:
$\begin{align}
  & \Rightarrow \dfrac{1}{12}+\dfrac{1}{24} \\
 & \Rightarrow \dfrac{3}{24} \\
 & \Rightarrow \dfrac{1}{8} \\
\end{align}$
Now, we already found out that the work that has to be done in the next 2 days is equal to $\dfrac{1}{4}$ and the work 16 men and 16 women can do in the next 2 days is equal to $\dfrac{1}{8}$ .
Thus, the amount of work for which ‘x’ number of men have to added to complete the work in 2 days will be equal to the remaining work done minus the work 16 men and 16 women can do in the next 2 days.
Thus, work to be done by ‘x’ number of men is given as:
$\begin{align}
  & \Rightarrow \dfrac{1}{4}-\dfrac{1}{8} \\
 & \Rightarrow \dfrac{1}{8} \\
\end{align}$
Now, we have been given that ‘x’ number of men have to do the work in 2 days.
The amount of work done by ‘x’ men in 1 day is given as:
$\begin{align}
  & \Rightarrow x\times \dfrac{1}{384} \\
 & \Rightarrow \dfrac{x}{384} \\
\end{align}$
Now, the total amount of work done by ‘x’ men in 2 days is given as:
$\begin{align}
  & \Rightarrow 2\times \dfrac{x}{384} \\
 & \Rightarrow \dfrac{x}{192} \\
\end{align}$
Now, we have already found out that the work that has to be done by ‘x’ men in 2 days is equal to $\dfrac{1}{8}$ .
Thus, $\dfrac{x}{192}$ and $\dfrac{1}{8}$ will be equal.
Thus, we can solve for ‘x’ by forming this equation.
Putting them equal, we get:
$\begin{align}
  & \Rightarrow \dfrac{x}{192}=\dfrac{1}{8} \\
 & \Rightarrow x=192\times \dfrac{1}{8} \\
 & \Rightarrow x=24 \\
\end{align}$
Thus, 24 more men have to be added to complete the remaining work in the next 2 days.
Thus, option (B) is the correct option.

Note: Don’t change the fractional forms of the work done in the decimal form. It will just contribute to more complicated calculations and increase the scope of committing mistakes. Thus, use all the values of different types of work done in fractional form only.