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Difficulty: MediumFactors, Multiples, and Prime Factorization

Two cyclists, Alan and Beatrice, begin riding laps around a circular track at the same time, starting from the same line and traveling in the same direction. Alan completes one lap every 40 seconds, and Beatrice completes one lap every 50 seconds. What is the number of laps Beatrice will have completed the next time they both cross the starting line at the same instant?

  1. 4Answer
  2. B
    5
  3. C
    10
  4. D
    40
  5. E
    200

Answer

Beatrice will have completed 4 laps.
The correct answer is 4. The two cyclists will meet at the starting line after a period of time that is a multiple of both of their lap times. The first time they meet after starting is the least common multiple (LCM) of 40 and 50, which is 200 seconds. Dividing the total time of 200 seconds by Beatrice's lap time of 50 seconds yields 4 laps.

Step-by-Step Solution

1
Find the prime factorizations of both lap times to determine their least common multiple (LCM).
The prime factorization of 40 is 23×512^3 \times 5^1, and the prime factorization of 50 is 21×522^1 \times 5^2.
They will cross the starting line together at times that are common multiples of their individual lap times.
2
Calculate the LCM by taking the highest power of each prime factor present in either factorization.
LCM(40,50)=23×52=8×25=200\text{LCM}(40, 50) = 2^3 \times 5^2 = 8 \times 25 = 200 seconds.
The least common multiple represents the minimum amount of time that must pass before both cyclists reach the starting line at the same time.
3
Divide the total time elapsed by Beatrice's time per lap to find the number of laps she completes.
200 seconds÷50 seconds per lap=4200 \text{ seconds} \div 50 \text{ seconds per lap} = 4 laps.
This converts the total time until they meet into the number of laps completed specifically by Beatrice.

Key Concept

Least Common Multiple (LCM) and rate calculations
Estimated Time:1m 0s
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Topics

Factors, Multiples, and Prime Factorization

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