Question

$36$ men can complete a piece of work in $18$ days. In how many days will $27$ men complete the same work?
A) $12$
B) $18$
C) $22$
D) $24$
E) None of these

Answer 1

Hint: This is an indirect proportion problem. Suppose $36$ men can complete a piece of work in $18$ days. Then in how many days the same work can be completed with the help of $27$ men. When men will increase then days will decrease. So we asked to find the number of days for the work to be complete with the least number of workers.

Complete step-by-step solution:
Given that, $36$ men can complete a piece of work in $18$ days.
Since $36$ men can complete $X$ work in $18$ days.
Then $27$ men can complete $X$ work in $Y$ days.
Since men are inversely proportional to days.
That is, when men will increase then days will decrease.
Therefore, Men $ = \dfrac{{X \times 1}}{{days}}$
Then the equation will be, $36 = \dfrac{{X \times 1}}{{18}} - - - - - \left( 1 \right)$
Now cross multiplying equation (1) we get,
$36 \times 18 = X \times 1$
Multiplying any number with $1$ we will get the same number, so the above equation becomes
$X = 36 \times 18 - - - - - \left( 2 \right)$
Now the second equation will be $27 = \dfrac{{X \times 1}}{Y} - - - - - \left( 3 \right)$
Cross multiplying equation $\left( 2 \right)$we get,
$Y \times 27 = X \times 1$
$ \Rightarrow X = Y \times 27 - - - - - \left( 4 \right)$
Since the left hand side of the equations $\left( 1 \right)\& \left( 2 \right)$ are same. So equating the equations $\left( 1 \right)\& \left( 2 \right)$ we get,
$36 \times 18 = 27 \times Y$
Keep the variables in one side implies, $Y = \dfrac{{36 \times 18}}{{27}}$
$ \Rightarrow Y = \dfrac{{36 \times 6}}{9}$
$ \Rightarrow Y = 4 \times 6$
$ \Rightarrow Y = 24$
Therefore, $27$ men can complete $X$ works in $24$ days.

Hence Option (D) is the correct answer.

Note: Indirect proportion refers to the relationship between two variables in which the product is a constant. Furthermore, when one variable increases the other decreases in proportion so that the product is unchanged.
There is another short method to solve this problem.
Let the required number of days be $x$
Less Men, more days (Indirect proportion)
$\therefore 27:36::18:x$
$ \Rightarrow 27 \times x = 36 \times 18$
Keep the variables one side,
$ \Rightarrow x = \dfrac{{36 \times 18}}{{27}}$
$ \Rightarrow x = 24$