Question

16 men can complete the work in 12 days, 24 children can complete the same work in 18 days, 12 men and 8 children start working and after 8 days, 3 more children join them. How many days will they take now to complete the work?
$\begin{align}
  & \text{a) 2 days} \\
 & \text{b) 4 days} \\
 & \text{c) 6 days} \\
 & \text{d) 8 days} \\
\end{align}$

Answer 1

Hint: Now we are given 16 men can complete the work in 12 days, 24 children can complete the same work in 18 days. Hence we will calculate the amount of work done by each man and each child in one day respectively. Now we will calculate the amount of work done by 12 men and 8 children in 8 days. Hence we can have the amount of work remaining as 1 – amount of work done. Now again we will calculate the amount of work done by 12 men and 8 + 3 = 11 children in one day. Now let us assume the number of days required are x. then we have x × amount of work done by 812 men and 11 children in one day = remaining work. Hence we will solve the equation to find the value of x.

Complete step-by-step answer:
Now we are given that 16 men can complete the work in 12 days.
Hence 16 men complete $\dfrac{1}{12}$ work in 1 day.
Hence 1 man can complete $\dfrac{1}{16\times 12}$ work in 1 day.
Now it is also given that 24 children complete the work in 18 days.
Hence 24 children complete $\dfrac{1}{18}$ of work in 1 day.
Hence 1 child completes $\dfrac{1}{18\times 24}$ work in 1 day.
Now let us calculate the amount of work done by 12 men and 8 children in one day.
Amount of work done by 12 men and 8 children in one day = amount of work done by one man in one day × 12 + amount of work done by one child in one day × 8.
Amount of work done by 12 men and 8 children in one day $=\dfrac{12}{16\times 12}+\dfrac{8}{18\times 24}$
$=\dfrac{1}{16}+\dfrac{1}{54}$
Taking LCM we get,
$\begin{align}
  & =\dfrac{27}{432}+\dfrac{8}{432} \\
 & =\dfrac{35}{432} \\
\end{align}$
Hence the amount of work done in 8 days is $=8\times \dfrac{35}{432}=\dfrac{35}{54}$
Now the remaining work is $1-\dfrac{35}{54}=\dfrac{54-35}{54}=\dfrac{19}{54}$ .
Hence in 8 days we have $\dfrac{19}{54}$ work is complete.
Now after 8 days we know 3 more children join in
Hence now there are 12 men and 11 children.
Now recall that the amount of work by 1 men in one day is $\dfrac{1}{16\times 12}$ and the amount of work done by 1 children in 1 day is $\dfrac{1}{18\times 24}$
Hence we have the amount of work done by 12 men and 11 children in one day is

$\begin{align}
  & =\dfrac{12}{16\times 12}+\dfrac{11}{18\times 24} \\
 & =\dfrac{1}{16}+\dfrac{11}{432} \\
 & =\dfrac{27+11}{432} \\
 & =\dfrac{38}{432} \\
\end{align}$
Now let us say they take x days to complete the work.
Hence we have,
$\begin{align}
  & \dfrac{38}{432}\times x=\dfrac{19}{54} \\
 & \Rightarrow x=\dfrac{19}{54}\times \dfrac{432}{38} \\
 & \Rightarrow x=4 \\
\end{align}$
Hence we need to work 4 more days.
Option b is the correct option.

So, the correct answer is “Option b”.

Note: Now note that in any question of these types try to find the amount per unit. Here we had two quantities: the number of days and number of people. Hence first we find the amount of work done by 1 person in 1 day. Once we have the unitary values we can easily put the conditions and solve further.