# The first bell rings every 20 minutes, second bell rings every 30 minutes, and the third bell rings every 50 minutes. If all three bells ring the same time at 12:00pm, when will be the next time the three bells will ring together?

Feb 20, 2017

$\text{5:00 pm}$

#### Explanation:

So first you find the LCM, or least common multiple, (can be called LCD, least common denominator).

The LCM of $20$, $30$, and $50$ is basically

$10 \cdot 2 \cdot 3 \cdot 5$

because you factor out the $10$ since that is a common factor.

$10 \cdot 2 \cdot 3 \cdot 5 = 300$

This is the number of minutes. To find the number of hours, you simply divide by $60$ and get $5$ hours. Then you count $5$ more hours from $\text{12:00 pm}$ and get $\text{5:00 pm}$.

Feb 20, 2017

5pm

#### Explanation:

$\textcolor{b l u e}{\text{Expanding on Ayushi's answer.}}$

Notice that we have:

$10 \times 2$
$10 \times 3$
$10 \times 5$

Each of 2, 3 and 5 are prime numbers. So the only common values they will divide exactly into is their product or some multiple of that product

So for 2,3 and 5 the least positive value they will divide into is:

$2 \times 3 \times 5 = 30$

but each of 2,3,and 5 is multiplied by 10 so we have to also multiply their product by 10 giving:

$10 \times 30 = 300$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{A different line of thinking that ends up in the same place}}$

3 and 5 are odd numbers but 2 is even.

As 2 is even then the $\textcolor{b r o w n}{\underline{\text{target value has to also be even}}}$. Otherwise 2 will not divide exactly into it

But some form of 3 and 5 have to be able to divide exactly into this even number as well.

$3 \times 5 = 15$ which is not even. However if we multiply 15 by 2 then 2 is automatically a factor:

$2 \times 15 = 2 \times 3 \times 5 = 30 \leftarrow \text{ even number}$

However we are counting in tens. In that we have 2 tens, 3 tens and 5 tens. So the answer is also counting in tens. Thus we have 30 tens $= 300$ IN MINUTES

$\text{1200 hours + "300/60}$$=$$\text{1200 hours + 5 hours}$$= \text{1700 hours}$

Alternatively written as 5 pm