Question

Two bells toll every 45 seconds and 60 seconds. If they toll together at \(8: 00 \mathrm{am}\), then which of the following is the probable time at which they can toll together? (1) \(8: 55 \mathrm{am}\) (2) \(8: 50 \mathrm{am}\) (3) \(8: 45\) am (4) \(8: 40 \mathrm{am}\)

Short answer

Answer: The probable time when both bells will toll together is 8:03 am.

Step-by-step solution

Find the LCM of 45 and 60 seconds intervals

To find the LCM of 45 and 60, let's represent each number as the product of its prime factors: \(45 = 3^2 \times 5\) \(60 = 2^2 \times 3 \times 5\) Now, let's find the LCM by choosing the highest power of each prime number from the factors: LCM = \(2^2 \times 3^2 \times 5\) LCM = 180 The LCM is 180 seconds, which means the bells will toll together every 180 seconds (3 minutes).

Find the time the bells toll together

The bells toll together at 8:00 am. After 180 seconds (3 minutes), they will toll together again. So, the time would be: \(8:00 \text{ am} + 180 \text{ seconds}\) \(8:00 \text{ am} + 3\text{ minutes}\) \(8:03 \text{ am}\)

Compare the found time with the given options

The found time is 8:03 am. This doesn't match any of the given options (1) \(8:55 \text{ am}\), (2) \(8:50 \text{ am}\), (3) \(8:45 \text{ am}\), and (4) \(8:40 \text{ am}\). The question may be flawed, or there might be a mistake in the options. However, based on the calculations, the probable time when both bells will toll together is 8:03 am.

Key concepts

Least Common Multiple (LCM)

To understand why we are finding the Least Common Multiple, or LCM, in time intervals problem solving, imagine two trains departing from the same station at different times. The LCM will tell us when both trains will depart at the same time again. In terms of our exercise with the bells, we're looking for that moment when both bells will toll together after their initial synchronization at 8:00 am.
Finding the LCM is essentially about determining the smallest number that two or more numbers can all divide into without leaving a remainder. It's the 'least common' because it's the smallest shared multiple among these numbers. When dealing with time, the LCM helps us synchronize events that occur in repeating patterns, such as the tolling of bells or the schedules of public transportation. This concept is vital in planning, logistics, and scheduling in real-world applications.

Prime Factorization

Prime factorization is a method we use to break down a number into its basic building blocks, which are prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. For example, the prime factors of 45 are 3, 3, and 5 because \(45 = 3^2 \times 5\).

The process involves dividing the number by the smallest possible prime number until the quotient is itself a prime number. This is helpful in finding the LCM because once you have the prime factors for two numbers, you can easily find their LCM by multiplying the highest powers of each prime number involved. In the bells problem, breaking 45 and 60 down into their prime factors simplifies the process of finding their LCM, ensuring that no bell tolleth at the wrong time!

Time Calculation

In problems involving time calculation, the goal is usually to find a specific moment in the future or past based on given intervals. We often work with units like seconds, minutes, and hours, and we may need to convert between these units to solve the problem. In our example with the bells, we calculated 180 seconds as the LCM of the two tolling intervals. That's the same as 3 minutes. Knowing how to add or subtract time intervals is like solving a puzzle where each piece represents a unit of time, and your job is to fit them together to see the bigger picture, which in this case is determining the next simultaneous toll after 8:00 am.

Arithmetic Sequences

An arithmetic sequence is a list of numbers where each term is found by adding a fixed number to the previous term. This fixed number is known as the common difference of the sequence. In our bells question, if we continued to add the LCM (180 seconds) to 8:00 am, we'd generate an arithmetic sequence of times when the bells toll together.
For example, starting from 8:00 am, the bells would toll together again at 8:03 am, 8:06 am, 8:09 am, and so on, with each time being 3 minutes apart, the common difference. Arithmetic sequences are everywhere: in finance to calculate interest, in computing for algorithm analysis, or even in architecture when designing a spiral staircase. Recognizing these patterns in the real world, just like with these tolling bells, significantly helps us understand and predict regular occurrences.