Adding and Subtracting Fractions

Fractions can only be added or subtracted once they are expressed over the same denominator — you cannot combine thirds with quarters until both are written in the same units. Multiplication and division have no such requirement, which is why this step belongs to addition and subtraction alone.

Concept When a common denominator is needed

Needed — adding fractions.

Needed — subtracting fractions.

Not needed — multiplying fractions.

Not needed — dividing fractions.

The reason is straightforward: adding counts pieces of the same size. Multiplication does not compare pieces at all, so the denominators can differ without causing trouble.

Concept The general rule

\frac{a}{b} + \frac{c}{d} = \frac{a \times d + c \times b}{b \times d}

Multiply the first numerator by the second denominator.

Multiply the second numerator by the first denominator.

Add those two results to get the new numerator.

Multiply the two denominators together for the new denominator.

For subtraction the rule is identical, with a minus sign replacing the plus in the numerator.

Example One third plus one quarter, step by step

Work out $\dfrac{1}{3} + \dfrac{1}{4}$ .

Step 1 — label the parts:

a = 1

b = 3

c = 1

d = 4

Step 2 — new numerator:

a \times d + c \times b = 1 \times 4 + 1 \times 3 = 4 + 3 = 7

Step 3 — new denominator:

b \times d = 3 \times 4 = 12

⟹ 1/3 + 1/4 = 7/12

A quick sanity check: $\frac{7}{12}$ is about 0.58, and $0.33 + 0.25 = 0.58$ . Converting to decimals is a fast way to confirm an answer.

Example An addition and a subtraction

Work out $\dfrac{2}{5} + \dfrac{1}{3}$ and then $\dfrac{3}{4} - \dfrac{1}{6}$ .

\dfrac{2}{5} + \dfrac{1}{3} = \dfrac{2 \times 3 + 1 \times 5}{5 \times 3} = \dfrac{6 + 5}{15} = \dfrac{11}{15}

\dfrac{3}{4} - \dfrac{1}{6} = \dfrac{3 \times 6 - 1 \times 4}{4 \times 6} = \dfrac{18 - 4}{24} = \dfrac{14}{24}

\dfrac{14}{24}

simplifies by dividing both parts by 2.

⟹ 11/15 and 7/12

The second answer needed simplifying. Multiplying the denominators always works, but it does not necessarily give the smallest one — so check whether the result can be reduced.

Concept The shortcut when denominators already match

\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}

If the denominators are already the same, there is nothing to unify. Add or subtract the numerators and leave the denominator exactly as it is.

\dfrac{3}{7} + \dfrac{2}{7} = \dfrac{3+2}{7} = \dfrac{5}{7}

\dfrac{5}{9} - \dfrac{2}{9} = \dfrac{5-2}{9} = \dfrac{3}{9} = \dfrac{1}{3}

The denominator is never added. Both fractions already count pieces of the same size, so only the count changes.

Note Practical advice

Simplify first — reducing each fraction before you start keeps the numbers small.

Simplify last — always check whether the final answer can be reduced.

Check with decimals — converting both sides is a quick way to catch an error.

Use the shortcut — if the denominators already match, skip straight to adding the numerators.

Note Mistakes to avoid

Adding the denominators —

\frac{1}{3} + \frac{1}{4}

is not

\frac{2}{7}

Adding numerators while the denominators still differ.

Multiplying only one numerator by the opposite denominator and forgetting the other.

Getting the subtraction order backwards in the numerator.

Leaving the answer unsimplified when it can be reduced.

Looking for a common denominator when the operation is multiplication or division.

Summary

A common denominator is required for addition and subtraction, but not for multiplication or division.
The general rule is a/b + c/d = (a×d + c×b) / (b×d), with a minus sign for subtraction.
When the denominators already match, simply add or subtract the numerators and keep the denominator.
Never add the denominators themselves.
Simplify before and after, and check the result as a decimal.