Question

Two runners \(\mathrm{A}\) and \(\mathrm{B}\) are running on a circular track. A takes 40 seconds to complete every round and B takes 30 seconds to complete every round. If they start simultaneously at \(9: 00 \mathrm{am}\), then which of the following is the time at which they can meet at starting point? (1) \(9: 05\) a.m. (2) \(9: 10 \mathrm{a.m}\). (3) \(9: 15 \mathrm{a.m}\). (4) \(9: 13 \mathrm{am}\).

Short answer

Answer: The given options do not contain the correct answer. The correct time at which runners A and B will meet at the starting point is 9:02 am.

Step-by-step solution

Find the LCM of 40 and 30

To find the LCM, we first find the prime factors of both numbers: 40: 2 × 2 × 2 × 5 (2^3 × 5^1) 30: 2 × 3 × 5 (2^1 × 3^1 × 5^1) Now, we take the highest power of each prime factor: 2^3, 3^1, and 5^1. Multiply all these prime factors together to get the LCM: LCM(40, 30) = 2^3 × 3^1 × 5^1 = 8 × 3 × 5 = 120 seconds

Convert the LCM value to minutes

Since there are 60 seconds in a minute, we will divide the LCM value of 120 seconds by 60 to get it in minutes: 120 seconds ÷ 60 = 2 minutes.

Add the LCM value to the starting time

The runners start at 9:00 am. We calculated that they will meet at the starting point after 2 minutes. So, we will add 2 minutes to 9:00 am: 9:00 am + 2 minutes = 9:02 am.

Find the correct answer option

The calculated time at which runners A and B will meet at the starting point is 9:02 am. Looking at the answer options, none of them match the calculated time. Therefore, the given options do not contain the correct answer.

Key concepts

Least Common Multiple (LCM)

Understanding the Least Common Multiple (LCM) is essential when dealing with various mathematical problems, especially ones involving synchronization of events. LCM of two or more numbers is the smallest number that is a multiple of all the given numbers. For instance, when you have two runners on a circular track, determining when they will meet involves finding the LCM of their lap times.

To find the LCM, one generally starts by finding the prime factors of each number. Prime factors are the building blocks of a number—think of them like the smallest building pieces of a Lego structure. Once you have the prime factors, take the highest powers of each prime present in any of the numbers. Then, multiply these together to get the LCM. This procedure ensures that the LCM includes all the necessary factors to be a common multiple, but not more than needed, keeping it the least such multiple.

Mathematical problem-solving

Mathematical problem-solving is more than just crunching numbers; it's a process that requires understanding the problem, identifying the relevant math concepts, and applying them correctly. One crucial step in this process is to ensure your approach fits the context of the problem. In our circular track problem, after finding the LCM of the runners' lap times, it's crucial to interpret this LCM as the time in seconds when both runners will synchronously complete a lap.

Moreover, it's important to note that mathematical problem-solving involves checking your solutions against given alternatives. If an answer doesn't appear in the provided options—as in the case of our exercise—it's a cue to review your steps. Perhaps there's an alternative approach required or a mistake to correct, and being open to such scenarios is part of effective problem-solving.

Time conversion

In mathematical exercises, especially when calculating time events like races, we often need to convert units for clarity and comprehension. The most common time conversion within such problems is between seconds and minutes since traditional clocks measure time in hours and minutes.

Since 60 seconds make up a minute, converting from seconds to minutes involves dividing the number of seconds by 60. This helps us express time in a format that is more relatable to everyday life. For example, if the runners on our track meet every 120 seconds, converting this to minutes by dividing 120 by 60 gives us 2 minutes, which is easier to visualize and compare with a standard clock.

Circular track problem

Circular track problems are common in mathematics and often involve concepts like the LCM and time conversion. The goal is to calculate when participants—runners, cars, bikes, or even planets—traveling at different speeds will meet at a certain point.

These problems typically require understanding periodic events—the runners on our track complete a loop repeatedly after a fixed time. To solve these problems, one must first identify the time it takes for each participant to complete one loop. The LCM of these times gives you the interval at which these periodic events coincide—the runners meet. Finally, adjust this interval by converting it to a familiar time unit and adding it to the beginning of the event timeline, such as the race's start time, to determine the exact meeting time.