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Difficulty: MediumTranslating and Solving Algebraic Word Problems

Two cyclists start at opposite ends of a 9090-mile trail at the same time and ride toward each other. One cyclist rides at a constant speed that is 33 miles per hour faster than the other cyclist. If the two cyclists meet after exactly 33 hours, what is the constant speed, in miles per hour, of the faster cyclist?

Answer: 16.5 mph

Answer

The speed of the faster cyclist is 16.516.5 miles per hour.
The correct answer of 16.516.5 is found by setting the speed of the slower cyclist to ss and the faster cyclist to s+3s + 3. Since both cyclists ride toward each other for 33 hours, their combined distance is 3s+3(s+3)=903s + 3(s + 3) = 90. Solving for ss yields 6s+9=906s + 9 = 90, which simplifies to 6s=816s = 81, or s=13.5s = 13.5. Adding 33 to 13.513.5 gives the speed of the faster cyclist, which is 16.516.5 miles per hour.

Step-by-Step Solution

1
Define variables for the speeds of both cyclists in terms of a single variable.
Let ss be the speed of the slower cyclist. The speed of the faster cyclist is s+3s + 3.
Using a single variable simplifies the setup of a solvable linear equation.
2
Write a linear equation using the relationship Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}.
The equation is 3s+3(s+3)=903s + 3(s + 3) = 90.
The sum of the distances traveled by both cyclists when they meet must equal the total length of the trail, which is 9090 miles.
3
Solve the equation for ss.
6s+9=90  ⟹  6s=81  ⟹  s=13.56s + 9 = 90 \implies 6s = 81 \implies s = 13.5.
This yields the speed of the slower cyclist.
4
Calculate the speed of the faster cyclist.
13.5+3=16.513.5 + 3 = 16.5.
The question specifically asks for the speed of the faster cyclist, which is represented by s+3s + 3.

Key Concept

Translating distance-rate-time relationships from word problems into solvable linear equations.

Alternative Method

An alternative approach is to use the concept of relative speed. Since the two cyclists are moving directly toward each other, their relative speed of approach is the sum of their individual speeds. They cover a total of 9090 miles in 33 hours, which means their combined speed is 903=30\frac{90}{3} = 30 miles per hour. If the speed of the faster cyclist is ff and the slower is ss, then f+s=30f + s = 30 and f−s=3f - s = 3. Adding these two equations gives 2f=332f = 33, which yields f=16.5f = 16.5 miles per hour.
Estimated Time:1m 15s
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Topics

Translating and Solving Algebraic Word Problems

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