Question

Four persons \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) and \(\mathrm{S}\) start running around a circular track simultaneously. If they complete one round in \(10,8,12\) and 18 minutes respectively, after how much time will they next meet at the starting point? (1) 180 minutes (2) 270 minutes (3) 360 minutes (4) 450 minutes

Short answer

Answer: (3) 360 minutes

Step-by-step solution

Find the LCM for the given times

To find the LCM, we will list the prime factors of each time taken: 10, 8, 12, 18. \(10=2\times5;\) \(8=2^3;\) \(12=2^2\times3;\) \(18=2\times3^2\). Now we consider the highest powers of the common prime factors. \(2^3, 3^2,\) and \(5^1\).

Calculate the LCM

The LCM is the product of the highest powers of common prime factors. LCM = \(2^3\times3^2\times5^1\). LCM = \(8\times9\times5\). LCM = 360 minutes.

Choose the correct answer

From the given options, the correct answer is (3) 360 minutes.

Key concepts

Least Common Multiple

The Least Common Multiple (LCM) is a mathematical concept which is essential in solving problems involving repeated events or synchronized cycles, such as the circular track problem described in our exercise. LCM is the smallest number that is a multiple of two or more numbers, meaning it's the smallest number into which each of the given numbers can be divided without leaving a remainder. In the context of runners on a circular track, finding the LCM of the times taken by each runner to complete a lap tells us when they will all meet at the starting point again.

To determine the LCM, it can be helpful to first break down each number into its prime factors, which are the basic building blocks of all numbers. Once these prime factors are identified, the LCM is found by taking the highest power of each prime factor present in any of the numbers and multiplying them together. This ensures that the LCM will be divisible by each of the original numbers, hence they will all align at this common multiple.

Prime Factorization

Prime Factorization is a technique used to express a number as the product of its prime components. A prime number is a whole number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime number is one that has no divisors other than 1 and itself. Prime factorization plays a crucial role in finding the LCM because it simplifies complex multiplication problems into manageable pieces.

For example, in our circular track problem, we started with prime factorization of the numbers 10, 8, 12, and 18, which led to the discovery of the base primes 2, 3, and 5. Students should note that the power of a prime is just as important as the prime itself, as higher powers of a prime can dominate over the lower ones in determining the LCM. Thus, understanding prime factorization is key to solving problems that require the LCM, whether it's runners on a track or complex scheduling issues.

Circular Motion Mathematics

Circular motion mathematics comes into play when dealing with repetitions or cycles along a circular path, such as athletes running on a track, planets orbiting the sun, or gears turning in machinery. In these scenarios, the concept of LCM is often used to predict when events will coincide. The time it takes for a complete cycle is significant because it reveals the periodicity of an event.

In this particular exercise, we dealt with runners on a circular track each completing laps in different intervals. The challenge was to determine after what period of time they would meet up at the starting line, assuming they maintain a constant pace. The solution lies in understanding that they will next meet when they have each completed a whole number of laps, which occurs at the LCM of their lap times. This application of circular motion mathematics in everyday problems emphasizes its importance and utility in a wide range of disciplines, from sports to engineering.