The initial
question gives the following information: Marcy takes two kinds of medicine; one every 4 hours,
and the other every 6 hours. We are told that Marcy took both medications at 10 a.m., and we are
asked to find when Marcy will take both medications.
This problem can be
solved in a number of ways. You can always use a brute force approach: list the times for each
medication and note when they are the same.
M1:
10a,2p,6p,10p,2a,6a,10a,2p,6p,10p,2a, etc.
M2: 10a,4p,10p,4a,10a,4p.10p,4a,10a,
etc
You should notice that the first time the medications
are taken together is 10pm; the next time is the next morning at 10am; then again the next day
at 10pm. So they are taken together every 12 hours.
A more efficient approach is to recognize that you need the least common multiple of 4
and 6. To find the LCM, we factor each number, then the LCM is the product of every factor in
either number raised to the highest power in either factorization. Ex:
`4: 2*2=2^2`
`6: 2*3`
The factors of the LCM are 2 and 3. 2 will be squared, as that is
the highest power, and 3 is to the first power.
So the
LCM(4,6)=`2^2*3=12`
It is not a coincidence that we get 12 and the times the
medications are taken together are 12 hours apart.
Other problems of this
type include:
1) Running/biking around a track at different speeds (easiest
if the speeds are in laps per hour).
2) Pizza ovens are dedicated to cooking
pizzas with different cooking times (e.g., 15 min and 20 min).
3) Replacing
fluids in a car (e.g., change oil every 5000 miles and flush/refill radiator every 40000
miles).
All of these problems have something in commontwo activities occur at
two different rates, and the situation involves cycles (e.g. time on a clock, going around a
cyclical path, etc.).
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