Expert: Josh - 11/6/2006

Question
A bell rings every 2 hours, a second bell rings every 3 hours, and a third bell rings every 4 hours. If all 3 bells ring at 9:00 AM. , at what time will all 3 bells next ring?  

Answer
I'll try:)

Solving problem mathematically:
Let A(n), B(n) and C(n) denote when the first, second and third bell, respectively, rings for the nth time after 9:00 a.m.

These moments are represented by:
A(n)=9+2*n
B(n)=9+3*n
C(n)=9+4*n.

That is to say, the time when the first bell rings for the first time (after 9 a.m) is given by A(1)=9+2*1=11 (i.e., 11:00 on a 24 hour clock), whereas the second bell rings for the first time at B(1)=9+3*1=12 (i.e., 12:00). From the formula B(n), we can easily deduce, for example, that the second bell rings for the second time (with n=2) at B(2)=9+3*2=15 (i.e., 15:00).

We want the kth time when the first bell rings, to coincide with, say, the mth time when the second bell rings, to also coincide with, the nth time when the third bell rings.

That is, we want A(k)=B(m)=C(n), for different integer values k, m and n.

Substituting the formulae for A(k),B(m) and C(n), this amounts to finding values of k, m and n, such that 2k=3m=4n. We need to know how to find the least common multiple between 2, 3 and 4.

If you are not sure about this, write back and ask me.

Basically, we need the smallest number divisible by 2, 3 and 4. This number is 12.

In fact, k=12/2=6, m=12/3=4 and n=12/4=3.

That is, when the first bell rings the sixth time (k=6), it will coincide with the second bell ringing for the fourth time (m=4), this coincides with the third bell ringing for the third time (n=3).

Notice that 2k=2*6=12, 3*m=3*4=12 and 4*n=4*3=12.

If you find this method too difficult, here's an alternative (graphical) approach:

Let each marking x divide the time line into 1 hour interval. Point P corresponds to 9:00 a.m. Instances when the first bell rings are marked by the letter A. Similarly, the letter "B" and "C" correspond to instances when the second and third bell ring.

P...x...x...x...x...x...x...x...x...x...x...x...x...x...x
A...x...A...x...A...x...A...x...A...x...A...x...A...x...A
B...x...x...B...x...x...B...x...x...B...x...x...B...x...x
C...x...x...x...C...x...x...x...C...x...x...x...C...x...x

Label the intervals hour by hour, you should see that the three bells ring at the same time after 12 hours have elasped (since 9:00 am).