Topic · Arithmetic

Fractions

A fraction is one number written as a ratio of two: a numerator over a denominator. That single notation supports three different intuitions — parts of a whole, a division waiting to happen, and a ratio — and the operations on fractions are easiest when you have the right intuition for each.

12 min read Prereqs: Multiplication & Division, Factors & Primes Updated 2026·05·17

What you'll leave with

  • Three intuitions for $\tfrac{a}{b}$ — parts of a whole, the unfinished division $a \div b$, the ratio $a : b$.
  • Why $\tfrac{2}{4}$ and $\tfrac{1}{2}$ are the same number, and how to reduce any fraction to lowest terms.
  • Addition and subtraction need a common denominator; multiplication and division do not.
  • Division by a fraction means multiplication by its reciprocal — and why.

1. What a fraction means

The notation $\tfrac{a}{b}$ is the same in every situation, but it tells a different story depending on what you're doing.

Fraction

A number written as $\tfrac{a}{b}$ where $a$ is the numerator, $b$ is the (nonzero) denominator, and the fraction bar means "divided by." The denominator tells you how many equal parts the whole is split into; the numerator counts how many of those parts you're taking.

The three intuitions

  • Parts of a whole. $\tfrac{3}{4}$ is three of four equal pieces of a pie. The denominator names the slice size ("quarters"); the numerator counts the slices.
  • A division. $\tfrac{3}{4}$ is also the answer to "$3$ divided by $4$." If you literally divide $3$ pizzas equally among $4$ people, each person gets $\tfrac{3}{4}$ of a pizza.
  • A ratio. $\tfrac{3}{4}$ is also the ratio $3 : 4$ — three of one thing for every four of another. A recipe with $\tfrac{3}{4}$ cups of flour per cup of sugar; a scale of $3 : 4$.

All three readings refer to the same number — just emphasising different aspects. When you find one operation hard, switching to a different reading often makes it obvious.

2. Equivalent fractions and lowest terms

The same fraction has many faces. $\tfrac{1}{2}$, $\tfrac{2}{4}$, $\tfrac{3}{6}$, $\tfrac{50}{100}$ — these are all the same number. They look different but locate the exact same point on the number line.

The rule for generating equivalents is the most important fact about fractions:

Multiplying (or dividing) the numerator and denominator by the same nonzero number doesn't change the value.
$$ \frac{a}{b} = \frac{a \cdot k}{b \cdot k} \quad \text{for any } k \neq 0. $$

Why? Because $\tfrac{k}{k} = 1$, and multiplying anything by $1$ leaves it alone: $\tfrac{a}{b} = \tfrac{a}{b} \cdot \tfrac{k}{k} = \tfrac{ak}{bk}$. So scaling both pieces in tandem is invisible.

Lowest terms

A fraction is in lowest terms (or simplest form) when its numerator and denominator share no common factors except $1$. To reduce a fraction, divide top and bottom by their greatest common divisor:

$$ \frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}, \quad \text{since } \gcd(18, 24) = 6. $$

You can also do it stepwise — divide by any common factor, then divide again, and again, until nothing more works. The two approaches always land at the same answer (by the uniqueness of prime factorization).

Always reduce at the end

An answer like $\tfrac{18}{24}$ isn't wrong, but $\tfrac{3}{4}$ is the canonical form — and lots of follow-on work (comparison, further operations, reading off a decimal) is easier when fractions are in lowest terms. Most graders and most textbooks expect reduced fractions as final answers.

3. Comparing fractions

If two fractions have the same denominator, comparison is easy: bigger numerator wins. $\tfrac{5}{8} > \tfrac{3}{8}$ because five eighths is more than three eighths.

If they have different denominators, the standard move is to rewrite both with a common denominator and then compare numerators:

$$ \tfrac{2}{3} \;\text{vs.}\; \tfrac{5}{7} \;\longrightarrow\; \tfrac{14}{21} \;\text{vs.}\; \tfrac{15}{21} \;\longrightarrow\; \tfrac{5}{7} > \tfrac{2}{3}. $$

A shortcut for the same comparison: cross-multiply. Multiply each numerator by the other denominator, then compare those products:

$$ \frac{2}{3} \;\text{vs.}\; \frac{5}{7} \;\longrightarrow\; 2 \cdot 7 = 14, \quad 5 \cdot 3 = 15, \quad 14 < 15, \;\text{so}\; \tfrac{2}{3} < \tfrac{5}{7}. $$

It works because cross-multiplying is exactly the act of clearing the denominators by multiplying both fractions by $3 \cdot 7 = 21$, without bothering to write the intermediate step.

4. Playground: build a fraction

Set the numerator and denominator. The pie chart fills accordingly, and you'll see the same fraction in its reduced form and as a decimal. Try $\tfrac{2}{4}$, then $\tfrac{1}{2}$, then $\tfrac{50}{100}$ — same shaded area every time, because they're the same number wearing different clothes.

3 4
Reduced: 3/4
Decimal: 0.75
Type: proper
3
4

5. Adding and subtracting fractions

When two fractions have the same denominator, addition and subtraction are just numerator arithmetic:

$$ \frac{3}{8} + \frac{2}{8} = \frac{5}{8}, \qquad \frac{5}{8} - \frac{1}{8} = \frac{4}{8} = \frac{1}{2}. $$

The denominator names the slice size; adding doesn't change the slice size, only the count of slices.

When the denominators differ, you can't combine them directly — $\tfrac{1}{2}$ and $\tfrac{1}{3}$ are different-sized slices. The fix is to rewrite both with a common denominator first.

The procedure

  1. Find a common denominator. The cleanest choice is the LCM of the two denominators, but any common multiple works.
  2. Rewrite each fraction with that denominator, using the equivalent-fractions rule.
  3. Add or subtract the numerators. Keep the common denominator.
  4. Reduce to lowest terms.

Example: $\tfrac{1}{4} + \tfrac{2}{3}$. The denominators are $4$ and $3$; LCM is $12$.

$$ \frac{1}{4} + \frac{2}{3} = \frac{1 \cdot 3}{4 \cdot 3} + \frac{2 \cdot 4}{3 \cdot 4} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}. $$

The fraction $\tfrac{11}{12}$ is already in lowest terms ($11$ is prime), so we're done.

Don't add denominators!

$\tfrac{1}{2} + \tfrac{1}{3} \neq \tfrac{2}{5}$. Watch the slice sizes: a half plus a third can't be smaller than a half, but $\tfrac{2}{5}$ is. The correct answer is $\tfrac{3}{6} + \tfrac{2}{6} = \tfrac{5}{6}$. Adding numerators and denominators separately is the single most common arithmetic mistake adults still make.

6. Multiplying fractions

Multiplication is the operation where fractions are actually easier than the equivalents in mixed-number form. The rule is just:

$$ \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}. $$

Multiply the numerators; multiply the denominators. No common denominator needed.

$$ \frac{2}{3} \cdot \frac{3}{5} = \frac{2 \cdot 3}{3 \cdot 5} = \frac{6}{15} = \frac{2}{5}. $$

Why is this right? Lean on the area picture. The product $\tfrac{2}{3} \cdot \tfrac{3}{5}$ is "two thirds of three fifths." Imagine a unit rectangle split into $3$ columns and $5$ rows — $15$ small cells. "Three fifths of it" picks $3$ rows out of $5$. "Two thirds of that" picks $2$ columns out of $3$. The overlap is $2 \cdot 3 = 6$ cells out of the $15$ — exactly $\tfrac{6}{15}$.

Cancel before you multiply

$\tfrac{6}{7} \cdot \tfrac{7}{12}$ can be computed as $\tfrac{42}{84} = \tfrac{1}{2}$, but it's easier to cancel the $7$s first — and the $6$ against the $12$ — leaving $\tfrac{1}{2}$ directly. Whenever you see a factor appearing in a numerator and a denominator (even across different fractions in a product), you can cancel it. This is just doing the reduction step before the multiplication instead of after.

7. Dividing fractions

The famous rule: to divide by a fraction, multiply by its reciprocal.

$$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a \cdot d}{b \cdot c}. $$

The reciprocal of $\tfrac{c}{d}$ is $\tfrac{d}{c}$ — flip the fraction upside-down. So:

$$ \frac{3}{4} \div \frac{2}{9} = \frac{3}{4} \cdot \frac{9}{2} = \frac{27}{8}. $$

Why "multiply by the reciprocal" works

Dividing is the inverse of multiplying. To find $\tfrac{3}{4} \div \tfrac{2}{9}$ is to ask: "what number, when multiplied by $\tfrac{2}{9}$, gives $\tfrac{3}{4}$?" Multiply both sides of that equation by $\tfrac{9}{2}$ — the reciprocal of $\tfrac{2}{9}$ — and the $\tfrac{2}{9}$ on the left cancels (because $\tfrac{2}{9} \cdot \tfrac{9}{2} = 1$). What you're left with is $\tfrac{3}{4} \cdot \tfrac{9}{2}$ on the right, which is exactly the rule.

So "flip and multiply" isn't magic; it's just the inverse-of-multiplication move dressed up. It's the same trick you use to solve $2x = 6$ by multiplying both sides by $\tfrac{1}{2}$, only with a fractional coefficient.

8. Improper fractions and mixed numbers

A fraction is proper if the numerator is smaller than the denominator (so the value is less than $1$), and improper if the numerator is at least as large (so the value is $\geq 1$).

Improper fractions are sometimes rewritten as mixed numbers: a whole part plus a proper fractional part. $\tfrac{11}{4}$ becomes $2\tfrac{3}{4}$ — "two and three quarters."

Converting between them

Improper → mixed: divide numerator by denominator. The quotient is the whole part; the remainder is the new numerator; the denominator stays.

$$ \tfrac{11}{4}: \;11 \div 4 = 2 \text{ remainder } 3, \;\text{so}\; \tfrac{11}{4} = 2\tfrac{3}{4}. $$

Mixed → improper: multiply whole by denominator, add numerator, keep denominator.

$$ 2\tfrac{3}{4} = \tfrac{2 \cdot 4 + 3}{4} = \tfrac{11}{4}. $$

Mixed numbers read more naturally in everyday speech ("I'll be there in two and a half hours"), but improper fractions are easier to compute with. Mathematicians almost always work with improper fractions and only convert to mixed at the end.

9. Common pitfalls

Adding numerators and denominators

$\tfrac{1}{2} + \tfrac{1}{3} \neq \tfrac{2}{5}$. The denominator names the slice size, and you can't add two different slice sizes together without first converting them. Always find a common denominator first.

Forgetting to reduce

$\tfrac{6}{15}$ is correct, but $\tfrac{2}{5}$ is canonical. Most textbooks and exams expect the simplest form. Always do a final pass to check whether the numerator and denominator share any factors.

Multiplying instead of flipping when dividing

$\tfrac{3}{4} \div \tfrac{2}{9}$ is not $\tfrac{3 \cdot 2}{4 \cdot 9} = \tfrac{6}{36}$. That would be multiplication. For division, flip the second fraction: $\tfrac{3}{4} \cdot \tfrac{9}{2} = \tfrac{27}{8}$. The mistake is easier to spot when you sanity-check: dividing by a number less than $1$ should make the answer bigger, not smaller.

Mixed-number arithmetic missteps

Adding $1\tfrac{1}{2} + 2\tfrac{3}{4}$ "by parts" is fine if you're careful, but easier still: convert to improper fractions first ($\tfrac{3}{2} + \tfrac{11}{4} = \tfrac{6}{4} + \tfrac{11}{4} = \tfrac{17}{4}$, which is $4\tfrac{1}{4}$). The number of small errors drops dramatically when you stop mixing wholes and fractions during the work.

10. Worked examples

Example 1 · Reduce $\tfrac{18}{24}$ to lowest terms

$\gcd(18, 24) = 6$. Divide top and bottom by $6$:

$$ \tfrac{18}{24} = \tfrac{18 \div 6}{24 \div 6} = \tfrac{3}{4}. $$

Check: $3$ and $4$ share no common factors except $1$ ✓.

Example 2 · $\tfrac{1}{4} + \tfrac{2}{3}$

LCM of $4$ and $3$ is $12$.

$$ \tfrac{1}{4} = \tfrac{3}{12}, \qquad \tfrac{2}{3} = \tfrac{8}{12}. $$

Add numerators:

$$ \tfrac{3}{12} + \tfrac{8}{12} = \tfrac{11}{12}. $$

$11$ is prime, so $\tfrac{11}{12}$ is already in lowest terms. Answer: $\boxed{\tfrac{11}{12}}$.

Example 3 · $\tfrac{2}{3} \times \tfrac{3}{5}$

Multiply numerators and denominators:

$$ \tfrac{2}{3} \cdot \tfrac{3}{5} = \tfrac{2 \cdot 3}{3 \cdot 5} = \tfrac{6}{15}. $$

Reduce: $\gcd(6,15) = 3$. So $\tfrac{6}{15} = \tfrac{2}{5}$.

Or — cancel the $3$ before multiplying: the $3$ in the numerator of the second fraction cancels with the $3$ in the denominator of the first, leaving $\tfrac{2}{1} \cdot \tfrac{1}{5} = \tfrac{2}{5}$ directly. Answer: $\boxed{\tfrac{2}{5}}$.

Example 4 · $\tfrac{3}{4} \div \tfrac{2}{9}$

Flip the second fraction, then multiply:

$$ \tfrac{3}{4} \div \tfrac{2}{9} = \tfrac{3}{4} \cdot \tfrac{9}{2} = \tfrac{27}{8}. $$

$\tfrac{27}{8}$ is in lowest terms ($27 = 3^3$, $8 = 2^3$, no shared primes). As a mixed number: $27 \div 8 = 3$ r $3$, so $\tfrac{27}{8} = 3\tfrac{3}{8}$. Either form is correct. Answer: $\boxed{\tfrac{27}{8}}$ (or $3\tfrac{3}{8}$).

Example 5 · Compare $\tfrac{5}{8}$ and $\tfrac{7}{12}$

Cross-multiply: $5 \cdot 12 = 60$ and $7 \cdot 8 = 56$. Since $60 > 56$, we have $\tfrac{5}{8} > \tfrac{7}{12}$.

Or verify with a common denominator: LCM$(8, 12) = 24$. $\tfrac{5}{8} = \tfrac{15}{24}$ and $\tfrac{7}{12} = \tfrac{14}{24}$. $15 > 14$, so the same answer. Answer: $\boxed{\tfrac{5}{8} > \tfrac{7}{12}}$.

Sources & further reading

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