Question

juan and joe start walking around a track at the same time and at the same place. it takes joe 8 minutes to walk around the track, and it takes juan 10 minutes. after how many minutes will they reach the starting point at the same time? 56 minutes 40 minutes 20 minutes 80 minutes

Explanation:

Step1: Identify the problem type

This is a problem about finding the least common multiple (LCM) of two numbers, 8 and 10, since we need to find the smallest time when both Juan and Joe are at the starting point simultaneously.

Step2: Prime factorize the numbers

• Prime factors of 8: $8 = 2 \times 2 \times 2 = 2^3$
• Prime factors of 10: $10 = 2 \times 5$

Step3: Calculate the LCM

To find the LCM, we take the highest power of each prime number present.

• For prime 2: the highest power is $2^3$ (from 8)
• For prime 5: the highest power is $5^1$ (from 10)

Then, LCM = $2^3 \times 5 = 8 \times 5 = 40$

Answer:

40 minutes