# 10 men can build a house in 6 months, how many men would it take to build the house in 4 months?

Jul 19, 2017

$15$ men are needed to complete the same task in a shorter period of time.

#### Explanation:

This is an example of inverse proportion.

If the house is to be built in a SHORTER period of time, then MORE men will be needed.

As one quantity increases, the other decreases.

In an inverse proportion, the product of the two quantities will always give the same value, called the constant, $k$.

$k = x \times y$

The first step is to find that constant, because then you can use it to find any unknown value.

We know that $10$ men $\left(x\right)$ will take $6$ months $\left(y\right)$ to build the house.

$k = 10 \times 6 = 60$

How many men $\left(x\right)$ will be needed if the time is to be $4$ months $\left(y\right)$?

$x = \frac{k}{y} = \frac{60}{4} = 15$ men are needed.

On a higher level we have:
$x \propto \frac{1}{y}$, leading to

$x = \frac{k}{y}$

$k = x \times y$ as given above.

Jul 27, 2017

$15$ men would be needed to build the house in the reduced time of $4$ months.
This assumes the additional men have the same capabilities and capacities as the first $10$.

#### Explanation:

We see that 10 men can build the house in $6$ months, but now the house needs to be finished faster. We will probably need more men working at the same pace to complete it in $4$ months.

This is because $4$ months is exactly $\frac{4 \cancel{m}}{6 \cancel{m}} = \frac{2}{3}$ of the $6$ months.

The time to do the work has become a fraction of what it was, so we will definitely need more men.

To calculate how many men we will need, we take the original number $10$ and multiply it by the fraction (factor) we have just calculated. Don't panic! read on ...

But, before we do that, we need to tip over (invert) the fraction because we know as the time gets shorter, the number of men must increase. This relationship is called an inverse (like inverted) proportionality.

Then we have: $10$ men $\times \frac{3}{2} = 15$ men.