Question

In the United States, individuals who earn an income in the top one percent can be referred to as "top earners." The percent of top earners who are men, , is more than four and a half times the percent of top earners who are women, . Alternatively, the percent of top earners who are men, , is less than five times the percent of top earners who are women, . Write and solve a compound inequality to determine the percent of top earners in the United States who are women, . Your final answer should not contain the variable .

Answer 1

Answer:

  Women comprise between 16.7% and 18.2% of top earners.

Step-by-step explanation:

You want the percent of top earners in the US who are women if the number of top earners who are men is 4.5 to 5 times the number who are women.

Setup

Let w represent the fraction of top earners who are women. Then (1-w) is the fraction who are men. The problem statement tells us ...

  4.5 < (1 -w)/w < 5

Solution

Multiplying by 5 gives ...

  4.5w < 1 -w < 5w

Adding w, we have ...

  5.5w < 1 < 6w

Solving each of these inequalities for w gives ...

  5.5w < 1   ⇒   w < 1/5.5 ≈ 18.2%

  1 < 6w   ⇒   w > 1/6 ≈ 16.7%

The percent of top earners in the US who are women se more than 16.7% and less than 18.2%.