Question

Four bells toll at intervals of 10 seconds, 15 seconds, 20 seconds and 30 seconds respectively. If they toll together at \(10: 00 \mathrm{a} \mathrm{m}\). at what time will they toll together for the first time after \(10 \mathrm{a} \cdot \mathrm{m}\).? (1) \(10: 01 \mathrm{a} . \mathrm{m}\). (2) \(10: 02 \mathrm{a} \cdot \mathrm{m}\). (3) \(10: 00: 30 \mathrm{a} \cdot \mathrm{m}\). (4) \(10: 00: 45\) a.m.

Short answer

Answer: \(10:01\) a.m.

Step-by-step solution

List the intervals

Write down the four tolling intervals for each bell: 10 seconds, 15 seconds, 20 seconds, and 30 seconds.

Find the least common multiple (LCM)

Find the LCM of 10, 15, 20, and 30 seconds. The LCM is the smallest positive integer that is a multiple of all the given numbers. We can prime factorize the numbers: 10 = 2 * 5 15 = 3 * 5 20 = 2^2 * 5 30 = 2 * 3 * 5 The highest powers of the prime factors found in these numbers are 2^2, 3^1, and 5^1. The LCM is the product of these highest powers: LCM = 2^2 * 3^1 * 5^1 LCM = 4 * 3 * 5 LCM = 60 seconds

Determine the time when the bells will toll together again

Add the least common multiple (60 seconds) to the initial time (10:00 AM) to get the time when the bells will toll together again: 10:00 AM + 60 seconds = 10:01 AM The answer is (1) \(10:01\) a.m.

Key concepts

Least Common Multiple

The Least Common Multiple (LCM) is a crucial concept in solving time interval problems where multiple events occur at regular intervals. The LCM helps determine when these events will coincide again. In our example with the bells tolling at different intervals, we seek the smallest time unit after they first toll together for it to happen again simultaneously.

To find the LCM, you need to identify the smallest positive integer that is divisible by each of the given numbers. This means finding a common timeline for all events, allowing them to occur together. The LCM is especially helpful in scenarios like scheduling or planning when dealing with repetitive events or cycles that need synchronization.

In practice, calculating the LCM involves breaking down each number into its prime factors and using these to determine the smallest shared multiple. This process ensures that you are capturing the interval that is evenly divisible by each original time period.

Prime Factorization

Prime Factorization is the method through which we break down composite numbers into their prime components. These components are the building blocks of numbers, only divisible by 1 and themselves, such as 2, 3, 5, and 7. Understanding this helps you work effortlessly with concepts like the Least Common Multiple.

To begin prime factorization, divide the number by the smallest possible prime number until only prime numbers remain. For instance, in our exercise, each bell's interval is expressed in terms of its prime factors:

• 10 can be broken down into 2 and 5.
• 15 is 3 and 5.
• 20 becomes 2² and 5.
• 30 factors into 2, 3, and 5.

Knowing the prime factorization for each number allows the calculation of the LCM, where the highest power of each prime number is taken into consideration. This way, you account for all necessary multiples to find the least amount of time for synchronized tolls.

Problem Solving Steps

Solving time interval problems efficiently involves a structured approach. Here, we'll outline clear, step-by-step problem-solving strategies that help in understanding and tackling similar exercises.

• **Identify Problem Requirements:** Start by clearly listing the intervals, as seen with the bell intervals of 10, 15, 20, and 30 seconds.
• **Find the LCM:** Break down each number using prime factorization and determine their LCM. This process gives insight into when the events synchronize again.
• **Apply the LCM:** Once you know the LCM, add this time to the starting point (e.g., 10:00 AM). This offers the precise time at which all events meet again, in this case at 10:01 AM.

This methodical approach ensures that no critical steps are overlooked, guiding you to the solution by systematically applying mathematical concepts. Thinking sequentially and logically about such problems can greatly enhance problem-solving skills across various domains.