# The time required to do a job varies inversely as the number of people working. It takes 5 hours...

## Question:

The time required to do a job varies inversely as the number of people working. It takes {eq}5 \ hr {/eq} for {eq}7 {/eq} bricklayers to build a park well.

How long will it take {eq}10 {/eq} bricklayers to complete the job?

## Solving Inverse Variation Problems:

If a variable, y, varies inversely as a variable, x, then {eq}y=\frac{k}{x} {/eq}, where k is a constant called the constant of variation. We can solve inverse variation problems by setting up the variation equation, using the information given to find the constant of variation, plugging that constant into the variation equation, and then using the variation equation to solve the problem.

It would take 3.5 hours for 10 bricklayers to build the park well. Let's start by letting t be the time required to do a job, and let's let n be the number of workers. We are given that the time required to do the job, t, varies inversely as the number of people working, n. Therefore, t varies inversely as n, and we have the following inverse variation equation.

• {eq}t=\frac{k}{n} {/eq}

The problem states that it takes 5 hours for 7 workers to complete the job. Thus, when t = 5, n = 7. Plugging these values into our variation equation gives {eq}5=\frac{k}{7} {/eq}, and we can solve this for k to find our constant of variation.

• {eq}5=\frac{k}{7} {/eq}

Multiply both sides of the equation by 7.

• {eq}35=k {/eq}

We get that our constant of variation is k = 35, so we plug this into our variation equation for k to get an equation representing our problem.

• {eq}t=\frac{35}{n} {/eq}

We want to know how long it would take 10 workers to compete the job, so we plug n = 10 into this equation, and solve for t.

• {eq}t=\frac{35}{10} {/eq}

Simplify.

• {eq}t=3.5 {/eq}

We get that it would take 3.5 hours for 10 workers to build the park well.