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Chapter 5: Problem 98

Two people are in a bicycle race around a circular track. One rider can race completely around the track in 40 seconds. The other rider takes 45 seconds. If they both begin the race at a designated starting point, how long will it take them to be together at this starting point again if they continue to race around the track?

Short Answer

Expert verified
The racers will be together at the start again after 360 seconds.

Step by step solution

01

Identify the Cycle Times

Identify the times it takes each cyclist to complete a full loop of the track. Given in the exercise, the first cyclist takes 40 seconds and the second one takes 45 seconds.
02

Find the Least Common Multiple

The point at which the two cyclists will be together at the starting point again is the time that is a multiple of both 40 and 45 seconds. Use the prime factorization method to find the Least Common Multiple (LCM). For 40, the prime factors are \(2^3 \cdot 5^1\).For 45, the prime factors are \(3^2 \cdot 5^1\).Merge these two sets of factors, taking the greatest power of each prime, resulting in \(2^3 \cdot 3^2 \cdot 5^1\), which is 360.
03

Interpret the solution

The Least Common Multiple also corresponds to the time (in seconds) when the two people would be at the same point again. Thus, the two cyclists will be together at the starting point again after 360 seconds.

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