Fractions

Why fractions exist

Three friends share two chocolate bars equally. How much does each person get? No whole number can answer that — the answer lives between $0$ and $1$ bar. Fractions are the numbers invented for exactly this gap: $\frac{2}{3}$ of a bar each. Any time something is shared, measured, or split — a bill, a recipe, a tank of gas — the whole numbers run out, and fractions take over.

Read a fraction like a measurement

A fraction is a number for part of a whole, and its two halves have different jobs. The bottom number — the denominator — names the size of piece you’re working with: cut the whole into $4$ equal parts and each part is “a fourth.” The top number — the numerator — just counts them. So $\frac{3}{4}$ reads like a measurement: three fourths, the same way “3 inches” is three of a unit called an inch.

That reading does real work. It tells you a proper fraction ($\frac{3}{4}$) is less than one whole, while an improper one ($\frac{7}{3}$ — seven thirds, more than two wholes) is not. And it will explain, in a moment, why adding fractions has a rule that multiplication doesn’t need: you can only count together pieces of the same size.

Equivalent fractions — the same amount, sliced differently

Cut every piece of a half-shaded bar in two and you get $\frac{2}{4}$: more pieces, smaller pieces, same shaded amount. Multiply top and bottom by the same number and the value never moves:

$\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}.$

Going the other way is simplifying: divide top and bottom by their greatest common factor — the GCF you built in F2 — to use the fewest, biggest pieces:

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

Which fraction is bigger?

Here’s the first place whole-number instinct betrays you: $\frac{1}{8}$ looks bigger than $\frac{1}{5}$, because $8 > 5$ and years of arithmetic trained you that bigger digits mean bigger numbers. But the denominator counts cuts, and more cuts make smaller pieces — an eighth of a pizza is the sad sliver. The instinct isn’t wrong, it’s aimed at the wrong number: it works on the tops, once the pieces match.

So to compare fairly, make the pieces match. For $\frac{2}{3}$ vs $\frac{3}{5}$, rename both in fifteenths: $\frac{10}{15}$ vs $\frac{9}{15}$ — now the tops decide, and $\frac{2}{3}$ wins. (Cross-multiplying — $2 \times 5 = 10$ against $3 \times 3 = 9$ — is that same renaming with the writing skipped.)

Adding: count pieces that match

Add $\frac{1}{2} + \frac{1}{3}$. The tempting move — add tops, add bottoms, get $\frac{2}{5}$ — feels right for a good reason: that is exactly how multiplying works, and “do the operation to everything you see” usually serves you well. But watch it break on the simplest case: $\frac{1}{2} + \frac{1}{2}$ would give $\frac{2}{4} = \frac{1}{2}$ — pour half a glass into half a glass and end up with… half a glass? Impossible. The move fails because halves and thirds are different units: “1 half + 1 third = 2 somethings” has no unit to count in, any more than 1 inch + 1 mile = 2 anythings.

The fix is the renaming trick you just learned — rewrite both in a unit they share:

That’s the whole rule: common denominator first, then add the tops — because the denominator is a unit, and only matching units can be counted together.

The widget opens on $\frac{3}{4} + \frac{1}{6}$ — predict the common denominator before looking (what’s the LCM of $4$ and $6$?). Then set $\frac{1}{2} + \frac{1}{2}$ and confirm the tops-and-bottoms answer $\frac{2}{4}$ is not what the bars show.

Multiplying: ”×” means “of”

A recipe calls for $\frac{3}{4}$ cup of flour and you’re making half a batch. You need half of three-quarters — and that of is what multiplication means, a thread that started with whole numbers ($3 \times 4$ is three groups of four). Picture the measuring cup: take the $\frac{3}{4}$, slice it in half, keep one layer: $\frac{3}{8}$. Tops multiplied, bottoms multiplied — and no common denominator needed, because you’re not counting two amounts in a shared unit; you’re re-slicing one amount.

The grid shows $\frac{2}{3} \times \frac{9}{10}$ as an overlap of shadings. Before you look: will the answer be bigger or smaller than $\frac{9}{10}$? Smaller — taking two-thirds of something leaves less than you started with. “Multiplying makes things bigger” is another piece of whole-number instinct that fractions retire.

Dividing: how many fit?

$3 \div \frac{1}{2}$ asks: how many half-cups fit in 3 cups? Six — dividing by a small number gives a big answer. Fit-counting is also why the famous keep · change · flip works: halves fit into things exactly twice as often as wholes do, so dividing by $\frac{1}{2}$ is multiplying by $2$ — and in general, dividing by $\frac{2}{5}$ is multiplying by $\frac{5}{2}$. The flip isn’t magic; it’s the fit-count turned into one multiplication. Switch the widget above to $\div$ and test it: $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$ — bigger than $\frac{3}{4}$, exactly because the divisor is smaller than $1$.

Mixed and improper are the same number

$\frac{7}{3}$ and $2\frac{1}{3}$ are one value in two outfits: improper form is easiest to calculate with, mixed form easiest to read (“a bit over two”). Convert freely.

Predict before you type: how does $\frac{7}{3}$ become a mixed number? (How many whole $3$s fit in $7$, and what’s left over?) Then go the other way with $2\frac{3}{4}$.

The one thing to remember

The denominator is a unit and the numerator counts it. Everything else follows: renaming a fraction changes the unit without changing the amount; adding needs matching units; multiplying means “of” and just re-slices; dividing counts how many times one amount fits into another.

The four rules

A worked sum, step by step