4.5 Add and subtract fractions with different denominators

Find the least common denominator

In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators?

Let’s think about coins again. Can you add one quarter and one dime? You could say there are two coins, but that’s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals $25$ cents and one dime equals $10$ cents, so the sum is $35$ cents. See [link] .

Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is $100.$ Since there are $100$ cents in one dollar, $25$ cents is $\frac{25}{100}$ and $10$ cents is $\frac{10}{100}.$ So we add $\frac{25}{100}+\frac{10}{100}$ to get $\frac{35}{100},$ which is $35$ cents.

You have practiced adding and subtracting fractions with common denominators. Now let’s see what you need to do with fractions that have different denominators.

First, we will use fraction tiles to model finding the common denominator of $\frac{1}{2}$ and $\frac{1}{3}.$

We’ll start with one $\frac{1}{2}$ tile and $\frac{1}{3}$ tile. We want to find a common fraction tile that we can use to match both $\frac{1}{2}$ and $\frac{1}{3}$ exactly.

If we try the $\frac{1}{4}$ pieces, $2$ of them exactly match the $\frac{1}{2}$ piece, but they do not exactly match the $\frac{1}{3}$ piece.

If we try the $\frac{1}{5}$ pieces, they do not exactly cover the $\frac{1}{2}$ piece or the $\frac{1}{3}$ piece.

If we try the $\frac{1}{6}$ pieces, we see that exactly $3$ of them cover the $\frac{1}{2}$ piece, and exactly $2$ of them cover the $\frac{1}{3}$ piece.

If we were to try the $\frac{1}{12}$ pieces, they would also work.

Even smaller tiles, such as $\frac{1}{24}$ and $\frac{1}{48},$ would also exactly cover the $\frac{1}{2}$ piece and the $\frac{1}{3}$ piece.

The denominator of the largest piece that covers both fractions is the least common denominator (LCD)    of the two fractions. So, the least common denominator of $\frac{1}{2}$ and $\frac{1}{3}$ is $6.$

Notice that all of the tiles that cover $\frac{1}{2}$ and $\frac{1}{3}$ have something in common: Their denominators are common multiples of $2$ and $3,$ the denominators of $\frac{1}{2}$ and $\frac{1}{3}.$ The least common multiple (LCM) of the denominators is $6,$ and so we say that $6$ is the least common denominator (LCD) of the fractions $\frac{1}{2}$ and $\frac{1}{3}.$

To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD.