Question

in a certain marathon, 45 percent of the runners were men, and the rest were women. the official timekeeper determined that 62 percent of the men and 66 percent of the women completed the marathon in under four hours. what percent of the runners who completed the marathon in under four hours were women? (disregard the percent sign when entering your answer.)

Explanation:

Step1: Calculate the percentage of women runners

Since 45% of the runners were men, the percentage of women runners is \(100 - 45=55\%\).

Step2: Use the weighted - average formula for percentage of runners under four hours

Let \(x\) be the percentage of women runners who completed the marathon in under four hours.
The formula for the overall percentage of runners under four hours is \((\text{Percentage of men}\times\text{Percentage of men under four hours})+(\text{Percentage of women}\times\text{Percentage of women under four hours})\).
We know that the overall percentage of runners under four hours is \(62\%\), the percentage of men is \(45\%\) (or \(0.45\)), the percentage of men under four hours is \(66\%\) (or \(0.66\)), and the percentage of women is \(55\%\) (or \(0.55\)).
Substitute into the formula: \(0.62=(0.45\times0.66)+(0.55\times x)\).
First, calculate \(0.45\times0.66 = 0.297\).
The equation becomes \(0.62=0.297 + 0.55x\).
Subtract \(0.297\) from both sides: \(0.62−0.297=0.55x\).
\(0.323 = 0.55x\).
Solve for \(x\): \(x=\frac{0.323}{0.55}\).
\(x = 0.58727\cdots\approx59\).

Answer:

\(59\)