Fractions
A fraction describes equal parts of one whole: how many parts the whole was cut into, and how many of them you are talking about. The equal part is not a detail — without it, nothing is a fraction.
The fractions tool
Cut the bar into equal parts
Drag the handle to change the number of parts, tap a part to shade it
one quarter
Four equal parts, one shaded.
How fractions works
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Cut the whole into equal parts
The bar on this page is cut into parts of the same size, and that is the rule: three unequal slices are not thirds. Cut the whole into four equal parts and each part is a quarter, whatever the whole happens to be.
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Count the parts for the bottom number
The number underneath says how many equal parts the whole was cut into. A bar cut into eight parts has 8 underneath, which is why the bottom number is called the denominator — it names the size of the part.
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Count what you have for the top number
The number on top says how many of those parts you are talking about. Three parts shaded out of four is written 3/4 and said "three quarters". Shade all four and the fraction is 4/4, which is the whole thing back again.
Worked examples
3/4 of a bar
The bar is cut into four equal parts and three of them are shaded. Read the bottom number first — quarters — then the top: three quarters. One part is left unshaded, and that part is a quarter too.
1/8 or 1/4 — which is bigger?
One quarter is bigger. Cutting the same bar into eight parts makes each part smaller than cutting it into four, so one eighth is less than one quarter. A bigger number underneath means smaller parts, which is why 1/8 is the smaller fraction.
2/4 and 1/2
The same amount. Shade two quarters of a bar and you have covered exactly the top half of it, which is what one half means. Cutting the same bar more finely does not change how much is shaded: 2/4 = 1/2.
Where it goes wrong
- Cutting the whole into unequal parts A bar sliced down the middle is two halves; one cut near the end gives a big piece and a small one, and neither is a half. When the parts are not equal the lengths mean nothing, so the first check is always whether the pieces match.
- Thinking a bigger denominator means a bigger fraction The opposite is true. Eighths are smaller pieces than quarters, so 1/8 is less than 1/4 even though 8 is more than 4. The denominator describes how finely the whole was cut, not how much you have.
- Adding the numbers underneath 1/2 + 1/2 is 1, not 2/4. Adding halves does not change the size of the pieces, only how many of them there are: two halves make one whole. The denominator stays put until the pieces themselves change size.
Questions
Why do the parts have to be equal?
Because otherwise the fraction would not tell you how much you have. If a chocolate bar is broken into a large piece and a small piece, calling each one "a half" would be a lie about one of them. Equal parts are what let the top number count something meaningful.
What does the bottom number mean?
It names the size of the parts by saying how many of them make the whole. A 4 underneath means quarters, an 8 means eighths. That is why it is called the denominator: it gives the part its name, and the name is what makes 3/4 and 3/8 different amounts.
Why is 1/8 smaller than 1/4?
Because sharing the same whole between more people leaves everyone with less. Four people sharing a pizza get more each than eight people sharing the same pizza. More parts means smaller parts, and that idea is what makes comparing fractions with the same top number easy.
Can a fraction be more than one?
Yes. Cut two identical bars into quarters and you have eight quarters; five of them, written 5/4, is more than one bar. Fractions smaller than one are sometimes called proper fractions and those bigger than one improper, but both are just counts of equal parts.
Is a fraction the same as a division?
It is the same relationship seen from another angle: 3/4 means three shared between four, and it is also three quarters of a whole. That is why a fraction can always be turned into a decimal — 3 ÷ 4 = 0.75 — which is exactly what the next rung, decimals, is about.