Least Common Multiple

The least common multiple, or LCM, is the smallest positive multiple shared by two numbers. It can help combine fractions with different denominators.

Imagine a water tank that has two inlet pipes. One pipe fills the tank in 30 minutes, while the other fills it in 18 minutes.

If both pipes are opened together, how long will it take to fill the tank?

The first pipe fills one-thirtieth of the tank per minute, while the second fills one-eighteenth. Since both pipes add water simultaneously, their filling rates are added.

To add these rates, the fractions must have a common denominator. The LCM of 30 and 18 is 90, so 90 provides the smallest common denominator that allows the two rates to be combined.

Using 90, one-thirtieth becomes three-ninetieths, and one-eighteenth becomes five-ninetieths. Their combined rate is eight-ninetieths, or four-forty-fifths, of the tank per minute.

At this rate, the filling time is 1 divided by four-forty-fifths, which equals forty-five-fourths minutes, or 11.25 minutes.

The LCM provides the common denominator needed to combine the rates and calculate the filling time.