Math  /  Algebra

Question97. Two people are jogging around a circular track in the same direction. One person can run completely around the track in 15 minutes The second person takes 18 minutes If they both start running in the same place at the same time, how long will it take them to be together at this place if they contrue to run?

Studdy Solution

STEP 1

1. The track is circular, and both joggers start at the same point at the same time.
2. The first person completes one lap in 15 minutes.
3. The second person completes one lap in 18 minutes.
4. We need to find the time it takes for both joggers to meet at the starting point again.

STEP 2

1. Understand the concept of the least common multiple (LCM).
2. Define the problem in terms of LCM.
3. Calculate the LCM of the two times.
4. Interpret the result.

STEP 3

Understand the concept of the least common multiple (LCM).
The LCM of two numbers is the smallest number that is a multiple of both numbers. In this context, it represents the time at which both joggers will be at the starting point together.

STEP 4

Define the problem in terms of LCM.
We need to find the LCM of 15 minutes and 18 minutes to determine when both joggers will be at the starting point together.

STEP 5

Calculate the LCM of the two times.
First, find the prime factorization of each number: - 15=3×5 15 = 3 \times 5 - 18=2×32 18 = 2 \times 3^2
The LCM is found by taking the highest power of each prime number that appears in the factorizations: - Highest power of 2: 21 2^1 - Highest power of 3: 32 3^2 - Highest power of 5: 51 5^1
Thus, the LCM is: LCM=21×32×51=2×9×5=90 \text{LCM} = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90

STEP 6

Interpret the result.
The LCM of 15 and 18 is 90, which means that both joggers will meet at the starting point together after 90 minutes.
The time it will take them to be together at the starting point is:
90 \boxed{90} minutes

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