Fractions

What it is

A fraction a/b writes a number as a numerator a (on top) over a denominator b (below). The denominator says how many equal parts the whole is cut into, the numerator how many of those parts are taken. The denominator cannot be 0.

• 3/4 is three of four equal parts of a whole. As a division it is 3 ÷ 4 = 0.75.
• 1/2 = 2/4 = 3/6: different fractions for the same number, because they name the same part of the whole. You can scale up (multiply numerator and denominator by the same number) and simplify (divide both by the same number).
• 5/4 is more than 1: five quarters is one whole and one quarter, 1 1/4. A fraction whose numerator is larger than its denominator is called improper, see mixed numbers.
• 4/1 = 4 and 0/5 = 0: whole numbers can be written as fractions.
• Part of a quantity: 3/4 of 20 is 20 ÷ 4 · 3 = 15.

What is not a fraction, or does not work:

• 3/0 is not a number. No number multiplied by 0 gives 3.
• “Half” of different wholes is not the same amount: 1/2 of 10 is 5 but 1/2 of 100 is 50. A fraction is a part of something.
• 1/3 + 1/4 is not 2/7. Numerators and denominators are not added separately; the answer is 7/12, see below.

A number that can be written as a fraction with whole numbers above and below is a rational number. Whole numbers, negative fractions and decimals like 0.75 are rational, but √2 is not.

Why it matters

Fractions are the first way to write numbers that lie between the whole numbers, and they underlie percent, decimals, ratios and scale: 3/4 = 0.75 = 75%.

• Calculating with fractions: adding, subtracting, multiplying and dividing. Adding needs a common denominator: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
• Changing form: fraction to decimal (1/8 = 0.125) and fraction to percentage (3/4 = 75%).
• Simplifying: finding the largest number that goes into both numerator and denominator is the GCD, and it rests on divisibility.

The fraction calculator checks your own working step by step, and the worksheets practise fractions as part of a whole and of a set, comparing and scaling up.

Common misconceptions

• “A bigger denominator means a bigger fraction.” The opposite when the numerators are equal: 1/3 is larger than 1/4, because a whole cut in 3 gives bigger pieces than one cut in 4.
• “You add numerators and denominators.” 1/2 + 1/2 is 1, not 2/4. The parts must be the same size before they can be added, which is what a common denominator does.
• “2/4 and 1/2 are different numbers.” They are the same number written two ways.
• “A fraction is always less than 1.” 5/4 and 7/7 are not: 5/4 = 1.25 and 7/7 = 1.
• “A fraction is something other than division.” a/b is exactly a divided by b.

Common questions