# Frank runs 10 miles in 75 minutes. At the same rate, how many miles would he run in 69 minutes?

## Question:

Frank runs 10 miles in 75 minutes.

At the same rate, how many miles would he run in 69 minutes?

## Proportions in Math:

In mathematics, a proportion is an equation that show that two ratios are equivalent. That is, if a:b is equivalent to c:d, then we write this as the proportion {eq}\frac{a}{b}=\frac{c}{d} {/eq}. We can use proportions to solve applications involving equivalent ratios.

In the scenario described, Frank could run 9.2 miles in 69 minutes. We can use proportions to solve this problem. We start by letting x be the number of miles that Frank can run in 69 minutes if he is running at the same rate as when he runs 10 miles in 75 minutes. Since the rate is the same, we have that x miles in 69 minutes is equivalent to 10 miles in 75 minutes. That is, x:69 is equivalent to 10:75, and we can set up the following proportion for these equivalent ratios.

• {eq}\frac{x}{69}=\frac{10}{75} {/eq}

Now, we can solve this proportion for x to find the number of miles that Frank can run in 69 minutes.

• {eq}\frac{x}{69}=\frac{10}{75} {/eq}

Cross multiply to eliminate fractions. This is the same as multiplying both sides of the equation by 69 and by 75.

• {eq}75x=69 \times 10 {/eq}

Simplify.

• {eq}75x=690 {/eq}

Divide both sides of the proportion by 75.

• {eq}x=9.2 {/eq}

We get that x = 9.2, so Frank can run 9.2 miles in 69 minutes at the rate he is running at.