Understanding fractions

You share a pizza with friends, break a chocolate bar into pieces, or wait a quarter of an hour for the bus: parts of a whole show up everywhere. That is exactly what fractions are for. They describe how much of a whole you mean, far more precisely than words like a bit or almost all.

Numerator and denominator

A fraction is made of two numbers. The number at the bottom is the denominator: it tells you into how many equal parts the whole is split. The number at the top is the numerator: it tells you how many of those parts you take. For 3/4 of a pizza, the pizza is cut into 4 equal slices and you get 3 of them. It matters that all parts are the same size, otherwise the fraction would not describe the share fairly.

Equivalent fractions

Different fractions can mean the same share. Whether you take half a pizza or 2 out of 4 slices: you get the same amount. That is why 1/2 = 2/4 = 3/6. Such fractions are called equivalent. You get them by multiplying numerator and denominator by the same number, a step called expanding. The share stays the same, the pieces are just cut finer.

Comparing fractions

Every fraction has its own spot on the number line between the whole numbers: 1/2 sits exactly halfway between 0 and 1, and 3/4 sits closer to 1. Two easy cases help when comparing. Same denominator: the fraction with the bigger numerator is bigger, because you have more equal pieces. Same numerator: the fraction with the smaller denominator is bigger, because the pieces themselves are bigger. 1/3 of a pizza is more than 1/8 of the same pizza. If neither case fits, give both fractions the same denominator: for 2/5 and 3/7 you rewrite both with denominator 35, which turns 2/5 into 14/35 and 3/7 into 15/35, so 3/7 is a touch bigger.

A fraction of an amount

Fractions work for any amount, not just for a pizza. 1/3 of 12 gummy bears means: split the 12 into 3 equal groups, each group holds 4, so 1/3 of 12 is 4. The rule behind it: divide by the denominator, multiply by the numerator. For 2/3 of 12 you take 2 such groups, that is 8. Percentages are just another way of writing shares: 20% is the fraction 20/100, which is equivalent to 1/5.