🍕Fractions

Cut a pizza into 8 equal slices and eat 3. How much did you eat? Not just "3 slices" — you ate $\frac{3}{8}$ of a pizza.

Whole numbers count complete things. Fractions count parts: how many pieces you took, out of how many equal pieces the whole was cut into. Sharing cake, reading the clock, checking a battery — fractions are everywhere.

Numerator and denominator, each with a job

The two numbers in $\frac{3}{8}$ have separate jobs:

• Denominator 8 (below): the whole was cut into 8 equal parts — it decides how big each slice is;
• Numerator 3 (above): you took 3 of them — it decides how much you get.

One pattern worth memorizing: the bigger the denominator, the smaller each slice. A pizza cut into 8 slices has slices half the size of one cut into 4, so $\frac{1}{8} < \frac{1}{4}$.

Equivalent fractions: different cuts, same amount

Take 4 slices out of 8, or 2 slices out of 4 — you eat exactly the same amount of pizza:

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

Multiply (or divide) the numerator and denominator by the same number, and the fraction keeps its size: $\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$. These are equivalent fractions — different outfits, same body.

Why must top and bottom move together? Multiplying the numerator by 2 is like slicing every piece in half again: twice as many pieces, each half as big — the total amount never changes. Move only one number and the pizza is a different pizza.

Shrinking a fraction instead is called simplifying: divide the top and bottom of $\frac{6}{9}$ by 3 to get $\frac{2}{3}$. Simplifying feels easy exactly when your times tables are sharp.

Comparing: same denominator first, then cross-multiply

If the denominators match, just compare numerators: $\frac{3}{8} > \frac{2}{8}$ — when every slice is the same size, three slices beat two.

Different denominators take one more move. Compare $\frac{3}{4}$ and $\frac{5}{7}$ by cross-multiplying:

$3 \times 7 = 21, \qquad 4 \times 5 = 20$

Since $21 > 20$, we know $\frac{3}{4} > \frac{5}{7}$. Each numerator multiplies the other fraction's denominator; the side with the bigger product has the bigger fraction.

There is a second route: rewrite both fractions with the same denominator, $\frac{3}{4} = \frac{21}{28}$ and $\frac{5}{7} = \frac{20}{28}$, and compare directly. Cross-multiplying is faster; common denominators are more visual. Both roads reach the same answer.

3/4

Numerator

−3+

Denominator

−4+

3/4 > 5/8

A greater than B

5/8

Numerator

−5+

Denominator

−8+

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

Fractions in daily life

• A quarter hour is $\frac{1}{4}$ of an hour; half past means $\frac{1}{2}$ has gone by;
• A battery at 75% holds $\frac{3}{4}$ of a charge;
• Recipes say "half a cup of sugar", "three quarters of a cup of flour";
• "Half price" means $\frac{1}{2}$ of the original price.

Fractions are also the foundation for what comes later: decimals, percentages and ratios are all close relatives. Get comfortable comparing $\frac{3}{4}$ with $\frac{5}{7}$ and all of them get easier.

Check yourself

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Quick quiz

↺ Try again

1. 1. Which is bigger: 3/8 or 5/8?

   3/85/8They are equal

2. 2. Which fraction equals 2/3?

   3/44/62/6

3. 3. Cross-multiply 3/4 vs 5/7: 3×7=21 and 4×5=20. Which is bigger?

   3/45/7They are equal