What is a fraction?

A part of a whole

Picture a chocolate bar that snaps into eight equal pieces. You break off three pieces and put them on a plate. The plate now holds a part of the whole bar. How much? Three of the eight equal pieces. Mathematicians write that amount as 3/8 and say "three eighths".

The bar is the whole. Each piece is a part. Because every piece is exactly the same size, one piece is one eighth of the bar, two pieces are two eighths, and all eight pieces are eight eighths, which is the whole bar again.

The top and the bottom number

Every fraction has two numbers with a line between them. The number below the line is the denominator. It tells you how many equal parts the whole is cut into, and it gives those parts their name: halves, thirds, quarters, fifths and so on. The number above the line is the numerator. It counts how many of those parts you mean.

In 3/8 the denominator is 8 and the numerator is 3. The 8 says the bar is cut into eighths. The 3 says you are talking about three of them. A helpful way to remember the jobs: the denominator is the sign on the door that names the room, and the numerator is the number of people who walk in.

The parts must be equal

There is one rule that makes fractions fair: the parts must all be the same size. If a pizza is cut into four pieces but one piece is huge and another is tiny, those pieces are not quarters. You cannot call the big piece 1/4, because a quarter means one of four equal pieces. Cutting is not enough. Equal cutting is what counts.

This is why the same whole can have many different fraction names on it. The same bar can be split into halves, quarters or eighths. The fraction changes with the number of equal parts, even though the bar does not grow or shrink.

Parts of a whole, and whole things

Some fractions are worth exactly one whole. If all of the parts are there, the numerator and the denominator are the same number: 4/4, 5/5 and 6/6 are all equal to 1. Two quarters plus two more quarters is four quarters, and four quarters fill the whole bar.

You already use fractions without thinking about them. Half a glass of water, a quarter of an hour, and three quarters of a cake are all everyday fractions. The words "half", "quarter" and "third" are the spoken names of 1/2, 1/4 and 1/3. A "quarter" and a "fourth" are the same thing.

Watch out for this

A common mistake is to think that 1/8 is bigger than 1/4 because 8 is bigger than 4. The opposite is true. Cutting a whole into 8 parts makes every part smaller than cutting the same whole into 4 parts. More parts, smaller pieces. When the numerators are the same, the fraction with the larger denominator is the smaller piece.

Worked example

A sandwich is cut into 6 equal pieces. Ava eats 2 pieces. What fraction of the sandwich did she eat, and what fraction is left?

Step 1: read the denominator. The sandwich is cut into 6 equal parts, so the denominator is 6. Step 2: count the eaten parts. Ava ate 2 pieces, so the numerator is 2. She ate 2/6 of the sandwich. Step 3: count what is left. Six pieces started on the plate, and 2 are gone, so 4 pieces remain. The fraction left is 4/6. Later lessons will show that 2/6 and 4/6 can be written in simpler ways, but both of these answers are correct as they stand.

Remember

• A fraction names one or more equal parts of a whole.
• The denominator is the bottom number. It names the parts.
• The numerator is the top number. It counts the parts.
• Parts must be equal, or they are not fractions of the whole.