Four operations, four rules — and each one has a picture that makes it obvious. Once you see why, you'll never forget how.
Adding and subtracting: same-size pieces first
Here is the whole secret to adding fractions: you can only add pieces that are the same size. If two fractions already share a denominator, the pieces match, and you simply add the numerators while keeping the denominator the same:
\[ \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \]One quarter plus two quarters is three quarters — the same way one apple plus two apples is three apples. The "apple" here is the size of the piece.
When the denominators are different, the pieces are different sizes and can't be combined yet. So we rewrite both fractions with a common denominator first. Take \( \frac{1}{2} + \frac{1}{3} \): both halves and thirds can be cut into sixths, so we use a common denominator of \( 6 \):
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]Now that every piece is a sixth, we just add the numerators. Subtraction works in exactly the same way — match the sizes first, then subtract the numerators.
Drag the slider below to see why \( \frac{3}{6} \) is the very same amount as \( \frac{1}{2} \) — the pieces are simply cut differently.
Why a common denominator? Only pieces of the same size can be added or subtracted — that's why we find a common denominator first, so every fraction is measured in matching pieces. Multiplication and division, as you'll see, don't need one at all.
Multiplying: multiply straight across
Multiplying is the friendliest of the four. There's no common denominator to hunt for — you just multiply the numerators together and the denominators together:
\[ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \]For example:
\[ \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \]The reason this works is hidden in a single word: "of" means multiply. Asking for \( \frac{1}{2} \) of \( \frac{1}{3} \) is the same as computing
\[ \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \]and that makes perfect sense — half of a third really is a sixth. Picture a third of a pizza, then cut that slice in half: each piece is one of six equal parts of the whole.
Dividing: flip and multiply
Division looks intimidating until you learn its one trick: dividing by a fraction is the same as multiplying by its reciprocal — you flip the second fraction upside down, then multiply. Watch:
\[ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} \]Why does flipping work? Division asks "how many of this fit into that?" So \( \frac{3}{4} \div \frac{1}{2} \) is really asking: how many halves fit into three-quarters? The answer is one and a half of them — exactly \( \frac{3}{2} \).
Tip Dividing by a fraction is the same as multiplying by its reciprocal — just "flip the second fraction and multiply." If you ever forget which fraction to flip, remember it's always the one you're dividing by, never the first.
Always simplify the answer
Whatever operation you used, finish by reducing the answer to its simplest form — divide the numerator and denominator by their greatest common factor until nothing more cancels. A handy shortcut for multiplication is to cancel common factors before you multiply; that keeps the numbers small and the arithmetic easy, and you arrive at the same simplest answer with less work.
- The denominators \( 3 \) and \( 4 \) differ, so find a common denominator: \( 12 \).
- Rewrite each fraction in twelfths: \( \frac{2}{3} = \frac{8}{12} \) and \( \frac{1}{4} = \frac{3}{12} \).
- Now the pieces match, so add the numerators: \( \frac{8}{12} + \frac{3}{12} = \frac{11}{12} \).
The answer is \( \mathbf{\frac{11}{12}} \), already in simplest form since \( 11 \) and \( 12 \) share no common factor.
- Multiply straight across: \( \frac{3 \times 10}{5 \times 9} = \frac{30}{45} \).
- Reduce by the greatest common factor, \( 15 \): \( \frac{30}{45} = \frac{2}{3} \).
The answer is \( \mathbf{\frac{2}{3}} \). Or cancel first to stay tidy: \( \frac{3}{9} = \frac{1}{3} \) and \( \frac{10}{5} = 2 \), giving \( \frac{1}{3} \times 2 = \frac{2}{3} \) directly.
Practice
Try each one yourself, then reveal the full solution.
1. Compute \( \frac{1}{2} + \frac{1}{6} \).
The denominators differ, so use a common denominator of \( 6 \).
Rewrite the first fraction: \( \frac{1}{2} = \frac{3}{6} \). The second is already in sixths.
Add the numerators: \( \frac{3}{6} + \frac{1}{6} = \frac{4}{6} \), then simplify by dividing top and bottom by \( 2 \).
So \( \frac{1}{2} + \frac{1}{6} = \mathbf{\frac{2}{3}} \).
2. Compute \( \frac{3}{4} \times \frac{2}{9} \).
Multiplication needs no common denominator — just multiply straight across.
\( \frac{3 \times 2}{4 \times 9} = \frac{6}{36} \).
Reduce by the greatest common factor, \( 6 \): \( \frac{6}{36} = \mathbf{\frac{1}{6}} \).
3. Compute \( \frac{2}{3} \div \frac{4}{5} \).
Dividing means multiplying by the reciprocal, so flip the second fraction: \( \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} \).
Multiply across: \( \frac{2 \times 5}{3 \times 4} = \frac{10}{12} \).
Reduce by dividing top and bottom by \( 2 \): \( \frac{10}{12} = \mathbf{\frac{5}{6}} \).