Six Bells Commence Tolling Together and Toll at Intervals of 2, 4, 6, GMAT Problem Solving

Question:Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together ?

1. 10
2. 12
3. 15
4. 16
5. 4

Correct Answer: D

Solution and Explanation:
Approach Solution 1:
The given case scenario mentions that six bells commence tolling together and toll at intervals of 2,4,6,8,10 and 12 seconds respectively. Accordingly, it is required to find how many times the bell tolls together in 30 minutes.

The bells are supposed to toll together when there is one time interval which is divided by every single time interval.
In order to find how many times it tolls in 30 minutes, it is necessary to find how many times all the six bells toll together.

This can be found with the help of finding the Lowest Common Multiple or LCM of all the intervals. We need to find the LCM of 2,4,6,8,10,12
The LCM of all the numbers accounts for a total of 120.

Accordingly, based on the LCM, the number of tolls in 30 minutes can be evaluated by multiplying how many seconds are included in 30 minutes. This can be further divided by 120 which would help in evaluating the number of times the bell tolls in 1800 seconds.

Thus, in 30 minutes = 30 * 60 seconds, they will toll together:

30∗60/120
= 15

However, the question clearly states that the six bells commence tolling together which implies that at zero hours, all the six bells tolled once. Hence, the answer 15 needs to add 1 to it to give the solution 16.

Accordingly,

15+1= 16

Hence, the number of times the bell tolls together in 30 minutes= 16 times.

Approach Solution 2:
In the above problem, it is given that six bells commence their tolls together. It is required to find how many tolls at 30 minutes the bell commences at intervals of 2,4,6,8,10, and 12. Here it is clear that the bell tolls together once before the intervals of 2,4,6,8,10 and 12. This means that at 0 hours the bells together toll once.

To find out the number of times the bell tolls in 30 minutes, it is important to find the number of times it tolls in the given intervals.

The given intervals in which the bells toll are 2,4,6,8,10 and 12. To find out the number of times the bells toll in each of these intervals combined, the Lowest Common Multiple of the intervals are needed to be found. Accordingly, the LCM of the given intervals equals 120. Hence, for 120 seconds, the bells tolled.

Accordingly, 120 seconds implies 2 minutes for which the bells have tolled. For 30 minutes, the number of times, the bells have tolled can be evaluated as follows-

30 minutes * 60 seconds ÷ 120 seconds = 15

Hence, the number of times in 30 minutes that the bells would toll is 15. However, at 0 hours since the six bells tolled together once, 15 times of the bell tolling needs to be added along with the 0 hours. This equals 16 times.

Hence, the bell tolls together 16 times in 30 minutes.

Approach Solution 3:
The problem statement states that:
Given:

• Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively.

​Find Out:

•  The number of times the bells toll together in 30 minutes.

To solve the problem, we need to find the LCM of  2, 4, 6, 8 10 and 12.

Let’s find the LCM by using prime factorisation.

Let’s calculate first the prime factors of 2, 4, 6 and 8:
2 = 2
4 = 2 x 2
6 = 2 x 3
8 = 2 x 2 x 2
Now, let’s calculate the prime factors of 10 and 12:
10 = 2 x 5
12 = 2 x 2 x 3
Therefore, LCM of  2, 4, 6, 8 10 and 12 :
2³ x 3 x 5 = 120

Hence, the six bells toll together after every 120 seconds (i.e 120/60 = 2 minutes).

Therefore, the required number of times the bell tolls in 30 minutes = 30/2 = 15 times

But it is required to add 1 because at the start all bells will be rung once a time after that they ring 15 times.

⇒ 15 + 1

⇒ 16 times

Hence, the bell tolls together 16 times in 30 minutes.

“Six bells commence tolling together and toll at intervals of 2, 4, 6”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. To solve the GMAT Problem Solving questions, the candidates must have a solid knowledge of basic mathematics. The candidates can follow varieties of questions from the GMAT Quant practice papers that will help them to improve their mathematical knowledge.

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