This means that we are going to have to multiply \(3\) and \(5\) together to get \(15\) in the denominator, but how will this affect the numerator?

Let's cut up our pies and see how to answer this question.

We want to cut our pies into equally-sized pieces.

So for our first pie, which is cut into three equal pieces, we cut each piece into five pieces. Then we have three-times-five equals fifteen pieces.

For our second pie, which is cut into five equal pieces, we cut each piece into three pieces. Then we have five-times-three equals fifteen pieces.

So let's say we have done this: now both pies are cut into fifteen equal pieces.

What about our original problem?

We wanted to add one-third to one-fifth.

Now we look at our pies.

We see that each of the three original pieces of the first pie has been cut into five pieces, so one third equals five fifteenths:

\[ \frac{1}{3} = \frac{5}{15}. \]

The process of cutting one third of the pie into five smaller is represented mathematically by the formula:

\[ \left( \frac{1}{5}\right) \left( \frac{1}{3}\right) = \frac{1}{15} ,\]

and then the process of adding together five of the resulting smaller pieces to reassemble a third of the pie is expressed mathematically by the formula:

\[ ( 5) \left( \frac{1}{15} \right) = \frac{5}{15} .\]

We can express both these operations at once by writing:

\[ \frac{1}{3} = \left(\frac{5}{5} \right) \left( \frac{1}{3} \right) = \frac{5}{15} .\]

Note that because \(\frac{5}{5} = 1\) we are not changing the value of the fraction--we are only changing the values of the numerator and the denominator.

We go through the same process with our second pie and so we see that each of the five original pieces has been cut into three pieces; so three fifteenths equal one fifths:

\[ \frac{1}{5} = \left( \frac{3}{3} \right) \left( \frac{1}{5} \right) = \frac{3}{15} .\]

Now we are ready to add the two pieces of the pies together. We have:

\[ \frac{1}{3} + \frac{1}{5} = \left( \frac{5}{5} \right) \left( \frac{1}{3} \right) + \left( \frac{3}{3} \right) \left( \frac{1}{5} \right) = \frac{5}{15} + \frac{3}{15} = \frac{8}{15}.\]

Which gives us our answer:

\[ \frac{1}{3} + \frac{1}{5} = \frac{8}{15}.\]

Notice that when we multiplied the denominators together to get a common denominator downstairs, we had to cross-multiply the numerators and the denominators upstairs.

We can express this in general terms as a formula for adding two fractions, each with numerator equal to one.

That is, let \(N\) and \(M\) be any two nonzero numbers, and suppose we are given the problem:

\[ \frac{1}{M} + \frac{1}{N} = ? \]

Then the answer is:

\[ \frac{1}{M} + \frac{1}{N} = \left( \frac{N}{N} \right) \left( \frac{1}{M} \right) + \left( \frac{M}{M} \right) \left( \frac{1}{N} \right) = \frac{(N + M)}{NM} .\]