Xiaohua takes 6 minutes to run a lap. Xiaogang takes 8 minutes. If they start at the same time, when will they meet at the starting point again?
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Let's solve this running track problem. Xiaohua takes 6 minutes to complete one lap, while Xiaogang takes 8 minutes per lap. They both start at the same time from the starting point. We need to find when they will meet at the starting point again.
Let's analyze their different speeds. Xiaohua completes one lap in 6 minutes, so her speed is one-sixth laps per minute. Xiaogang takes 8 minutes per lap, so his speed is one-eighth laps per minute. Since Xiaohua runs faster, she will complete more laps in any given time period. Let's visualize this with a progress chart showing how many laps each runner completes over time.
To solve this problem, we need to find when both runners will be at the starting point simultaneously. This happens when both have completed whole numbers of laps at the same time. We need to find the Least Common Multiple of 6 and 8 minutes. Using prime factorization: 6 equals 2 times 3, and 8 equals 2 to the power of 3. The LCM is 2 to the power of 3 times 3, which equals 24 minutes. Let's verify: in 24 minutes, Xiaohua completes 4 laps and Xiaogang completes 3 laps.
Let's verify our answer with detailed calculations. Using the division method: in 24 minutes, Xiaohua completes 24 divided by 6 equals 4 laps, and Xiaogang completes 24 divided by 8 equals 3 laps. We can also use the multiples method: multiples of 6 are 6, 12, 18, 24, and multiples of 8 are 8, 16, 24. The first common multiple is 24. Let's visualize this on a timeline showing when each runner completes their laps.
Now let's watch a complete simulation of the 24-minute period. Both runners start at the same point. Xiaohua, in blue, completes her first lap at 6 minutes, while Xiaogang, in green, takes 8 minutes for his first lap. As time progresses, we can see Xiaohua completing more laps due to her faster speed. At exactly 24 minutes, both runners meet again at the starting point - Xiaohua having completed 4 laps and Xiaogang having completed 3 laps.