Fraction Arithmetic

Addition, subtraction, multiplication and division each treat fractions differently: same denominators are easy, different denominators demand one extra step of preparation — almost every difficulty hides inside the denominator.

Recall the two ground rules of fractions: the denominator decides how big each slice is, the numerator decides how many slices you take. Every rule below grows out of those two sentences.

Same denominator: leave it alone, add on top

Pizzas cut the same way just need counting: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$ — 3 slices plus 2 slices is 5 slices, each still an eighth of a pizza. Subtraction works the same: $\frac{5}{8} - \frac{2}{8} = \frac{3}{8}$ .

The most common mistake

Denominators never take part in adding or subtracting. $\frac{3}{8} + \frac{2}{8}$ does not equal $\frac{5}{16}$ — if 3 eighths plus 2 eighths somehow shrank into sixteenths, pizza would vanish the more you eat.

Different denominators: match the cuts first

$\frac{1}{3} + \frac{1}{4}$ cannot be added directly: a third of a pizza and a quarter of a pizza are different sizes. The fix is finding a common denominator — rewrite both fractions as equivalent ones cut the same way. Of the common multiples of 3 and 4, the smallest is 12:

$\frac{1}{3} = \frac{4}{12}, \qquad \frac{1}{4} = \frac{3}{12}$

So $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$ . Hunting for that smallest common multiple is exactly the skill trained in factors and multiples.

InteractiveFraction Lab

3/4

Numerator

Denominator

3/4 > 5/8

A greater than B

5/8

Numerator

Denominator

Compare with: 3/4 = 6/8 —— Equivalent multiplier ×2

Compare $\frac{1}{3}$ and $\frac{1}{4}$ on the two pizzas: without matching the cuts, big or small is a guessing game; once both pizzas are cut the same way, adding and subtracting feel solid.

Multiply and divide: straight across, or flip

Multiplication is the easiest of the four: numerator times numerator, denominator times denominator.

$\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}$

It answers the question "what is half of three quarters?"

“Of” means multiply

"A third of half a kilo", "a quarter of the class", "a third of the price" — in everyday speech the word "of" is very often the multiplication sign in disguise. Spot it, and half of every word problem solves itself.

Division adds one move: dividing by a fraction equals multiplying by its upside-down twin (numerator and denominator swapped).

$\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2$

Check it in plain words: how many quarter-slices fit into half a pizza? Count them — exactly 2.

What about mixed numbers?

A number like $1\frac{1}{2}$ — a whole with a fraction attached — is called a mixed number, and it is really an addition in disguise: $1\frac{1}{2} = 1 + \frac{1}{2} = \frac{3}{2}$ . Before any operation, convert mixed numbers into improper fractions (where the numerator outgrows the denominator), then follow the rules above. For instance, $1\frac{1}{2} \times \frac{2}{3} = \frac{3}{2} \times \frac{2}{3} = 1$ .

Check yourself

Quick quiz

1. 3/8 + 2/8 = ?
2. 1/2 × 3/4 = ?
3. 1/2 ÷ 1/4 = ?