Fractions, explained
Numerator, denominator, unit fractions, equivalents, simplifying, common denominators, adding and comparing, improper fractions and mixed numbers — the whole map, built one idea at a time.
A fraction: part of a whole
A fraction is a way of talking about a piece of something. Cut a pizza into equal slices, take a few of them, and you have a fraction. The word comes from a Latin verb meaning to break, and the word that matters most in that sentence is equal.
The numerator on top
The top number is the numerator, and its job is simply to count. It tells you how many of the pieces you actually hold. If you took three slices, the numerator is three. On its own it says nothing whatever about how big those slices are.
The denominator below
The bottom number is the denominator, and its job is to name the size. Denominator shares a root with the word name. If it says eight, the whole was cut into eight equal parts, so a bigger bottom number means smaller pieces, which trips up almost everyone at first.
Unit fractions
A unit fraction has a one on top. One half, one third, one tenth. These are the building blocks, because every other fraction is just a count of them. Three quarters is nothing more mysterious than three copies of one quarter stacked side by side.
Top matches bottom means one whole
When the top number and the bottom number match, you are holding the whole thing again. Eight eighths is one. A hundred hundredths is one. That single fact quietly powers almost every trick that comes later, so it is worth saying out loud once.
Fractions on a number line
Fractions are not only slices of pie, they are addresses on a number line. Between zero and one there is room for one half, and between zero and one half there is room for one quarter, and you can keep halving for ever without running out of room.
Equivalent fractions
Cut every slice in half and you have twice as many pieces, each of them half the size, and exactly the same amount of pizza. One half, two quarters and eight sixteenths are the same number in different outfits. Multiply top and bottom by the same number and nothing really changes.
Simplifying
Simplifying runs that idea backwards. Divide the top and the bottom by any number that goes into both, and keep going until nothing does. Six twelfths becomes one half. The value never moves an inch, the fraction just gets easier to picture and easier to say aloud.
Comparing fractions
To compare two fractions, make the pieces the same size. One third against one quarter looks confusing until you remember that the more pieces you cut, the smaller each one gets, so one third wins. Turn both into twelfths and it becomes four against three, argument over.
Common denominators
A common denominator is just a shared piece size. To find one, look for a number that both bottom numbers divide into neatly. Halves and thirds both fit into sixths, and the moment everything is in sixths, fractions start behaving like ordinary counting again.
Adding and subtracting
Now adding is easy. One half plus one third becomes three sixths plus two sixths, which is five sixths. You add the top numbers and leave the bottom one alone, because you are counting pieces and all the pieces are finally the same size.
Improper fractions
Sometimes the top ends up bigger than the bottom, like seven quarters. Teachers once called that improper, which makes it sound naughty, but there is nothing wrong with it at all. It simply means more than one whole, and it is usually the easiest form to calculate with.
Mixed numbers
A mixed number splits that into a whole plus a leftover. Seven quarters is one whole with three quarters spare. Divide seven by four and you get one, remainder three, and that remainder becomes the new top number while the bottom number never changes.
Also decimals and percentages
The same amount can wear three different costumes. One half is also zero point five and also fifty in every hundred. Money is fractions in disguise, since a quarter of a pound is twenty five pence, and once you see that link, percentages stop being frightening.
Fractions all around you
Once you start noticing fractions you cannot stop. Half past three, a quarter of an inch, the three eighths marking on a spanner, two thirds of a cup of flour, and sixteenth notes in a drum beat. Music, cooking and building all run on the same handful of ideas.
Watch it explain itself
Every step above is narrated aloud. Play it, or open it in the editor and make it yours — no account needed.