Question

# If 20 men can build a wall 56m long in 6 days, what length of a similar wall can be built by 35 men in 3 days?

Hint: We solve this problem by using time and work formula, which is work done by N people in time t is Nt. By using this formula, we find the amount of work done by the 20 men in 6 days which is equal to 56m long wall. Then we find the amount of work done by 35 men in 3 days and we find the length of wall by checking the proportionality using the formula $\dfrac{{{N}_{1}}{{D}_{1}}{{H}_{1}}}{{{W}_{1}}}=\dfrac{{{N}_{2}}{{D}_{2}}{{H}_{2}}}{{{W}_{2}}}$ where N, D, H, W, are number of people, number of days, number of hours worked and amount of work done respectively.

Let us discuss time and work formulas.
If N number of people can do a piece of work in T hours, then the total effort or total work done is equal to NT.
If a person can do a work in D days then, then the work done by that person in 1 day is $\dfrac{1}{D}$.
In number ${{N}_{1}}$ of people can do ${{W}_{1}}$ amount of work in ${{D}_{1}}$ days working ${{H}_{1}}$ hours per day and ${{N}_{2}}$ number of people can do ${{W}_{2}}$ amount of work in ${{D}_{2}}$ days then
$\dfrac{{{N}_{1}}{{D}_{1}}{{H}_{1}}}{{{W}_{1}}}=\dfrac{{{N}_{2}}{{D}_{2}}{{H}_{2}}}{{{W}_{2}}}$
We use this formula and solve the problem as below.
Here ${{N}_{1}}=20$, ${{D}_{1}}=6$, ${{W}_{1}}=56$, ${{N}_{2}}=35$ and ${{D}_{2}}=3$, as the number of hours they worked per day was not given in the question, we can neglect the number of hours per day on the both sides.
We were asked to find ${{W}_{2}}$.
By applying the above problem, we get
\begin{align} & \dfrac{20\times 6}{56}=\dfrac{35\times 3}{{{W}_{2}}} \\ & {{W}_{2}}=\dfrac{35\times 3\times 56}{20\times 6} \\ \end{align}
By solving this, we get
${{W}_{2}}=49$
Therefore, if 20 men can build a wall 56m long in 6 days, 35 men can build a 49m wall in 3 days.

Note: Here one must be careful while taking the formula. One might make a mistake of taking ${{N}_{1}}{{D}_{1}}{{H}_{1}}{{W}_{1}}={{N}_{2}}{{D}_{2}}{{H}_{2}}{{W}_{2}}$ instead of taking $\dfrac{{{N}_{1}}{{D}_{1}}{{H}_{1}}}{{{W}_{1}}}=\dfrac{{{N}_{2}}{{D}_{2}}{{H}_{2}}}{{{W}_{2}}}$. So, one needs to be careful while applying the formula.