Question

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

Answer 1

Solution:

Step 1: Finding the LCM to find the time after which the bells will toll together

Prime factors of:

$$\begin{gathered} 2=2 \\ 4=2\times 2 \\ 6=2\times 3 \\ 8=2\times 2\times 2 \\ 10=2\times 5 \\ 12=2\times 2\times 3 \end{gathered}$$

Therefore,

$\mathrm{LCM}\left(2,4,6,8,10,12\right)=2\times 2\times 2\times 3\times 5$

$\Rightarrow \mathrm{LCM}=120$

Therefore, after every $120$ seconds or $2$ minutes, the bells will toll together. $\left[\because 60\mathrm{seconds}=1\mathrm{minute}\right]$

Step 2: Frequency of bell:

In $30$ minutes the number of times the bell will toll $=\frac{30}{2}=15$

Initially, it tolls once, therefore the number of times bells will toll together is$=15+1=\mathbf{16}$

Final answer: Therefore, the bells will toll $\mathbf{16}$ times together in $\mathbf{30}$ minutes.