So now that we know how to get the least common multiple (LCM) of two numbers, let’s apply this knowledge to add two fractions together that have different denominators.

So let’s look at:

\[\frac{3}{4}\hspace{0.33em}{+}\hspace{0.33em}\frac{3}{18}

\]

In my last post, we found the LCM of the two numbers 4 and 18 to be 36. Now that these numbers are in the denominator, we can also call 36 the least common denominator (LCD) – two different terms to describe the same thing depending on the country, school district, or era you find yourself in.

From Part 2 of the posts on Fractions, I covered the method to add two fractions together that have different denominators. The trick is to find a common denominator but now we are more wise and can find a least common denominator. So we will use the LCM we found in my last post to solve the problem.

So I want to convert the two fractions into equivalent ones that have a denominator of 36. I do this my multiplying the top and bottom of the fraction by the same number needed to make the denominator 36:

\[\begin{array}{c}

{\frac{3}{4}\hspace{0.33em}\times\hspace{0.33em}\frac{9}{9}\hspace{0.33em}{=}\hspace{0.33em}\frac{{3}\hspace{0.33em}\times\hspace{0.33em}{9}}{{4}\hspace{0.33em}\times\hspace{0.33em}{9}}\hspace{0.33em}{=}\hspace{0.33em}\frac{27}{36}}\\

{\frac{3}{18}\hspace{0.33em}\times\hspace{0.33em}\frac{2}{2}\hspace{0.33em}{=}\hspace{0.33em}\frac{{3}\hspace{0.33em}\times\hspace{0.33em}{2}}{{18}\hspace{0.33em}\times\hspace{0.33em}{2}}\hspace{0.33em}{=}\hspace{0.33em}\frac{6}{36}}

\end{array}

\]

Now that we have two equivalent fractions with the same denominator, the rest is easy as we just have to add the numerators together:

\[\frac{3}{4}\hspace{0.33em}{+}\hspace{0.33em}\frac{3}{18}\hspace{0.33em}{=}\hspace{0.33em}\frac{27}{36}\hspace{0.33em}{+}\hspace{0.33em}\frac{6}{36}\hspace{0.33em}{=}\hspace{0.33em}\frac{33}{36}

\]

So are we done? That is the answer but textbooks and teachers will always tell you to “simplify” your answer. That means get the numbers as small as possible. The method of getting equivalent fractions above can be done in reverse to get simpler (but equivalent) fractions. If you can identify factors common to the numerator and denominator, these can be cancelled. Notice in our answer that there is a common factor of 3. That is:

\[\frac{33}{36}\hspace{0.33em}{=}\hspace{0.33em}\frac{{11}\hspace{0.33em}\times\hspace{0.33em}\rlap{/}{3}}{{12}\hspace{0.33em}\times\hspace{0.33em}\rlap{/}{3}}\hspace{0.33em}{=}\hspace{0.33em}\frac{11}{12}

\]

So now we are done. In my next post I will do more examples.