# It will take 100 people x hours to plant trees around a middle school. How many hours will it...

## Question:

It will take 100 people x hours to plant trees around a middle school. How many hours will it take 20 people to plant the trees?

## Proportions and Variation:

In real life, we observe inverse relationships on a daily basis. An example of an inverse-proportional case is the work done by a group of people in a given time. In this case, if we reduce the number of people who plant the same amount of trees it will take longer.

{eq}\eqalign{ & {\text{In this specific case}}{\text{, we have two proportional values }}\,x\,\left( {people} \right){\text{ and }} \cr & y\,\left( {hours} \right){\text{ that have a variation in inversely proportional form}}{\text{. }} \cr & {\text{So we have:}} \cr & \,\,\,\,{x_1} = 100\,people \cr & \,\,\,\,{y_1} = 1\,hour \cr & \,\,\,\,{x_2} = 20\,people \cr & \,\,\,\,{y_2} = ?\,\,hours \cr & {\text{Since}}{\text{, }}x{\text{ and }}y{\text{ vary inversely}}{\text{, then}}{\text{, when }}x{\text{ decreases }}y\,{\text{ increases}}{\text{. }} \cr & {\text{For this reason}}{\text{, it must be satisfied that:}} \cr & \,\,\,\,\frac{{{y_2}}}{{{x_1}}} = \frac{{{y_1}}}{{{x_2}}} \cr & {\text{So if we do cross - multiplying:}} \cr & \,\,\,\,{y_2} \cdot {x_2} = {y_1} \cdot {x_1} \cr & {\text{Now}}{\text{, solving for }}\,{y_2}{\text{:}} \cr & \,\,\,\,{y_2} = \frac{{{y_1} \cdot {x_1}}}{{{x_2}}} \cr & {\text{So}}{\text{, substituting the given values:}} \cr & \,\,\,\,{y_2} = \frac{{1 \times 100}}{{20}} = 5\,hours \cr & {\text{Therefore}}{\text{, it will take }}\boxed{{\text{5 hours}}}{\text{ for 20 people to plant the trees}}. \cr} {/eq}