Adding and Subtracting Fractions

Why the parts must be the same size

Think about why $3 + 2 = 5$ never needs explaining. You are counting copies of a single unit, the apple, so you total the copies. A fraction is exactly the same kind of object: $\frac{2}{5}$ means two copies of the unit “one fifth.” In the same way, $\frac{1}{5}$ means one more copy of that same unit. Counting the copies gives three fifths:

The denominator stayed $5$ because the size of the piece never changed; only the count went up. Fractions add by counting parts, and counting only works when the parts are identical. A fifth and a fifth can be tallied together, the same way two apples and three more apples can. A half and a third cannot, for the same reason two apples and three oranges cannot just be added: they are not counts of the same-sized thing. Unlike apples and oranges, though, a half and a third can be forced to match: cut each into smaller, equally sized pieces until both are measured in that same small piece. That recutting is the whole trick, and it is what the next section does. So the rule splits into two cases: when the denominators already match, you add at once. When they do not match, your first job is to make them match.

Adding and subtracting with the same denominator

When two fractions share a denominator, they are built from identical pieces, so you combine them by working only with the numerators. For example,

Three eighths is three of those pieces, and one eighth is one more, so together you are holding four of them. Only the count changed, so the denominator is still $8$. The last step divides the top and bottom by their greatest common factor $4$, exactly the simplifying skill from the previous lesson.

Like-denominator rule. For any whole numbers $a$, $b$, and $c$ with $c$ not zero, $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \qquad\text{and}\qquad \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}.$ Here $a$ and $b$ are the two numerators, and $c$ is the denominator they share. (The subtraction examples in this lesson keep $a \ge b$, so the difference is $0$ or a positive fraction; you already know from the integers chapter what a negative result would mean.)

Both $\frac{a}{c}$ and $\frac{b}{c}$ are copies of the very same unit piece, $\frac{1}{c}$. Lining up $a$ copies with $b$ more copies of that piece gives $a + b$ copies in total, which is exactly $\frac{a + b}{c}$. Subtraction runs the same argument backwards: taking $b$ copies away from $a$ copies leaves $a - b$ copies.

So the procedure with like denominators is short: add (or subtract) the numerators and keep the common denominator. Then simplify the result to lowest terms. A common slip is to add the denominators too, writing $\frac{3}{8} + \frac{1}{8} = \frac{4}{16}$. That is wrong, because the denominator names the size of the piece. The pieces also did not get smaller when you put two groups of them together, so $8$ stays $8$.

Unlike denominators: make the pieces match first

When the denominators differ, the pieces are different sizes and cannot be counted together yet. The fix is the building rule from the last lesson: rewrite each fraction as an equivalent one over a denominator they share. Then the like-denominator rule finishes the job.

Take $\frac{1}{2} + \frac{1}{3}$. Halves and thirds are different sizes, so picture cutting both into a size they have in common. Sixths work: a half is three sixths, and a third is two sixths. Once both are written in sixths, they are made of identical pieces and you can count:

To rewrite $\frac{1}{2}$ in sixths, the denominator goes from $2$ to $6$ by multiplying by $3$, so the numerator does too: $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$. To rewrite $\frac{1}{3}$ in sixths, the denominator goes from $3$ to $6$ by multiplying by $2$: $\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$. Now both are sixths, so add the numerators:

Notice what the common denominator $6$ is: it is a multiple of both $2$ and $3$. Being a multiple of both is what lets each fraction be rebuilt over $6$. Any common multiple of the two denominators would serve. We could rewrite both over $12$ instead and still get the right answer, just with larger numbers to simplify at the end. The cleanest choice is the smallest common multiple, and you already know how to find it.

Try it with a harder pair: $\frac{2}{3} + \frac{1}{4}$. Fourths cannot be counted alongside thirds, so find the first denominator that is a multiple of both $3$ and $4$.

Cut each part of the bar below more finely, and watch the denominator on the bottom bar. Stop at the first cut where that denominator is also a whole number of fourths: that is the least common denominator of thirds and fourths.

Cutting each third into $4$ pieces gives twelfths, and a fourth can be rewritten in twelfths too, so that is the cut to use: $\frac{2}{3} = \frac{8}{12}$. Cutting into $8$ pieces would also work, giving $\frac{16}{24}$, just with larger numbers for the same sum.

Whichever cut you pick, keep an eye on the shaded length. It never moves, because the bar is still two thirds and only its name has changed. That is what makes rewriting a fraction safe in the middle of an addition.

The least common denominator

A common denominator of two fractions is any common multiple of their denominators. The least common denominator (LCD) is the smallest one. For $\frac{1}{2}$ and $\frac{1}{3}$ the denominators are $2$ and $3$, whose common multiples run $6, 12, 18$ and on. The smallest is $6$, which is why $6$ was the natural choice above.

So the LCD is exactly the least common multiple of the two denominators, from the factors chapter. Using the LCD keeps the numbers as small as possible, so there is less arithmetic and usually less simplifying at the end. The full method for unlike denominators is four steps:

1. Find the LCD, the least common multiple of the two denominators.
2. Rewrite each fraction as an equivalent fraction over the LCD, using the building rule.
3. Add or subtract the numerators, keeping the common denominator.
4. Simplify the result to lowest terms.

Each rewrite in step 2 is an equality, so nothing about the two amounts changes on the way to the common denominator. All that changes is the size of the pieces they are counted in. Once both are counted in the same pieces, step 3 is the like-denominator rule again.

When one denominator is already a multiple of the other

Sometimes the larger denominator is itself a multiple of the smaller one. Take $\frac{1}{6} + \frac{7}{12}$. Since $12$ is a multiple of $6$ (because $6 \times 2 = 12$), the least common multiple of $6$ and $12$ is $12$, so the LCD is $12$. The fraction $\frac{7}{12}$ is already over $12$ and stays as it is; only $\frac{1}{6}$ needs rebuilding:

The last step simplifies $\frac{9}{12}$ by dividing the top and bottom by their greatest common factor $3$. Whenever the larger denominator is a multiple of the smaller one, the LCD is that larger denominator. Only the fraction with the smaller denominator has to be rewritten, so spotting a pair like this saves half the work.

Adding a whole number to a fraction

A whole number can join a fraction the same way, once you remember that every whole number is a fraction with denominator $1$. To add $2 + \frac{3}{5}$, write $2$ as $\frac{2}{1}$, then give it the denominator $5$ so the pieces match. The LCD of $1$ and $5$ is $5$, and $\frac{2}{1} = \frac{2 \times 5}{1 \times 5} = \frac{10}{5}$, which just says two wholes is ten fifths. Then add:

The improper fraction $\frac{13}{5}$ is a complete and correct answer. (Rewriting it as a whole-number part plus a fraction is the subject of the later lesson on mixed numbers.) The same idea handles a subtraction like $1 - \frac{2}{7}$: write $1 = \frac{7}{7}$, so that $1 - \frac{2}{7} = \frac{7}{7} - \frac{2}{7} = \frac{5}{7}$.

Common mistakes