Four bells begin to toll together and then each one at intervals of 6 s, 7 s,…
2023
Four bells begin to toll together and then each one at intervals of 6 s, 7 s, 8 s and 9 respectively. The number of times they will toll together in the next 2 hours is (approx)
- A.
14 times
- B.
15 times
- C.
13 times
- D.
11 times
Attempted by 3 students.
Show answer & explanation
Correct answer: A
To find out how many times the four bells will toll together in 2 hours, we must first determine how often they toll together, which is the Least Common Multiple (LCM) of their individual intervals.
Step-by-Step Calculation
1. Find the LCM of the intervals (6, 7, 8, 9 seconds):
Prime factorize each:
6 = 2¹ × 3¹
7 = 7¹
8 = 2³
9 = 3²
The LCM is the product of the highest power of each prime factor present:
Factors: 2³, 3², 7¹
LCM = 8 × 9 × 7 = 504 seconds.
2. Calculate the total time in seconds:
2 hours = 2 × 60 minutes = 120 minutes.
120 minutes = 120 × 60 seconds = 7200 seconds.
3. Determine how many times they toll together in this period:
The bells toll together every 504 seconds.
Number of intervals in 7200 seconds = 7200 / 504 ≈ 14.28.
Since they start tolling together at the very beginning (at time 0), we count the initial toll plus the number of times they toll within the 2-hour duration:
Total tolls = (Total time / LCM) + 1
Total tolls = 14 + 1 = 15 times.
Self-correction note based on the provided option A: While the math above suggests 15, let's look at the context. If the question implies how many times they toll after the initial start, the answer would be 14. Given option A is marked correct in your image, it is counting the tolls excluding the initial start.