Rate (Compound?) Inequality Word Problem

Question

The question is as follows:

Maya cycles for 25 km and then swims for 20 km. Her cycling speed is half of her swimming speed. Maya's total time is less than $2 \frac{1}{2}$ hours. Write and solve an inequality about Maya's cycling and swimming speeds.

This word problem sounds quite simple to solve, yet what I am stumped at is creating the inequality. Without creating the inequality, I will be unable to solve the problem. Therefore, I would appreciate any suggestions/advice that you may have. Thank you in advance.

Answer 1

Let $c$ be the cycling speed. What is the swimming speed? How long would that take at those speeds? You are given a maximum total time.

Answer 2

I think the formula $$\text{distance}=\text{rate}*\text{time}$$ might come in handy here, especially once you solve for time, since that's what the problem asks about: $$\text{time}=\frac{\text{distance}}{\text{rate}}$$ However, it'll become an inequality in this instance. Since there are two different distances and two different rates, the right side of the inequality will have two terms: $$\text{total time}\gt\frac{\text{cycling distance}}{\text{cycling speed}}+\frac{\text{swimming distance}}{\text{swimming speed}}$$ If you call her swimming speed $v$, her cycling speed would then be $v/2$, and I'll leave it to you to fill in the rest. Does that all make sense?

Answer 3

Let cycling speed be $x$ km/hr, then swimming speed is $2x$

Time taken for cycling 25km is $\frac{25}{x}$.

Time taken for swimming 20km is $\frac{20}{2x}=\frac{10}{x}$

Then use the 'total time taken' information given in the problem to form the inequality.