Question

$45$ men can complete a work in $16$ days. Six days after they started working $30$ more men joined them. How many days will they now take to complete the remaining work?

Answer 1

Hint: In order to solve the question for the number of days needed to complete the whole work, first find the total units of work that was to be done by $45$ men in $16$ days. But some other men joined and will continue with the remaining work, so find the remaining work to be done and divide it by the total number of men present at this time to get the number of days.

Complete step by step solution:
It’s given that $45$ men can complete a work in $16$ days, which means that the whole work is divided into $45 \times 16$ units that is equal to $720$ units.
But, after six days some more men joined.
So, in the first six days, some of the work is already done.
Therefore, the part of work which is already completed is $45 \times 6$ units that is equal to $270$ units.
Now, after six days the unit of work left is (Total work to be done in $16$ days$ - $The work done in $6$ days) that is $\left( {720 - 270} \right) = 450$ units.
After the six days of work $30$ more men joined them.
So, the total number of workers at present is $30 + 45 = 75$ men.
Now, the amount of work $75$ men can do is $75 \times 1 = 75$ units.
But the remaining work is $450$ units.
So, the remaining work will finish in total work remaining divided by total men doing the work which is written as $\dfrac{\text{Work remaining }}{ 75}$ days that is equal to $\dfrac{{450}}{{75}} = 6$days.

Therefore, $75$ men will complete the remaining work in $6$ days.

Note:
> Do not find only the total work to be done by $75$ men directly, otherwise it would result into error because some work has already been done by $45$ men in the earlier six days.
> The men and days can be directly compared to find the number of days because the total work to be done was constant. This can be written as: $\left( {45 \times 16} \right) = \left( {45 \times 16} \right) + \left( {75 \times D} \right)$, solving for $D$ would give us the number of days.