Question

If 18 men can do a piece of work in 18 days, how many men would do it in 27 days?

Answer 1

Hint: Here, we need to find the number of men who would do the work in 27 days. Using the given information, we will find an equation for the amount of work done by one man in one day. Then, using the given information again, we will find another equation for the amount of work done by one man in one day. Finally, we will equate the two equations, and simplify to the number of men who would complete the work in 27 days.

Complete step-by-step answer:
This is a question of inverse proportion. The more the number of men working, the less number of days it would take to complete the same piece of work, and vice versa.
Let the number of men who can do the work in 27 days be $$x$$.
Now, we know that 18 men can complete the work in 18 days.
Dividing by 18, we get
Amount of work done by 18 men in 1 day $$={\displaystyle \frac{1}{18}}$$
Dividing the expression again by 18, we get
Amount of work done by 1 man in 1 day $$={\displaystyle \frac{1}{18\times 18}}$$……………….$$\left(1\right)$$
Now, we know that $$x$$ men can do the work in 27 days.
Dividing by 27, we get
Amount of work done by $$x$$ men in 1 day $$={\displaystyle \frac{1}{27}}$$
Dividing the expression by $$x$$, we get
Amount of work done by 1 man in 1 day $$={\displaystyle \frac{1}{27\times x}}$$……………..$$\left(2\right)$$
Now, comparing equation $$\left(1\right)$$ and equation $$\left(2\right)$$, we can observe that
$${\displaystyle \frac{1}{18\times 18}}={\displaystyle \frac{1}{27\times x}}$$
This is a linear equation in terms of $$x$$. We will simplify this equation to get the value of $$x$$.
Rewriting the equation by cross-multiplying, we get
$$\Rightarrow 27\times x=18\times 18$$
Multiplying the terms in the equation, we get
$$\Rightarrow 27x=324$$
Dividing both sides of the equation by 27, we get
$$\Rightarrow {\displaystyle \frac{27x}{27}}={\displaystyle \frac{324}{27}}$$
Thus, we get
$$\Rightarrow x=12$$
$$\therefore$$ The number of men that would complete the work in 27 days is 12 men.
Note: Here, the work done when completed is taken as 1. This is why we divided 1 by 18 to get the amount of work done by 18 men in 1 day. Similarly, we divided 1 by 27 to get the amount of work done by $$x$$ men in 1 day.
We have formed a linear equation in one variable in terms of $$x$$ in the solution. A linear equation in one variable is an equation that can be written in the form $$ax+b=0$$, where $$a$$ is not equal to 0, and $$a$$ and $$b$$ are real numbers. For example, $$x-100=0$$ and $$100P-566=0$$ are linear equations in one variable $$x$$ and $$P$$ respectively. A linear equation in one variable has only one solution and not more than that.