Fractions

Fractions can be expressed in two different ways:

• As a ratio of two integers: 2/3. This is the form used when a fraction is equal to a value less than 1 or when when a fraction is larger than 1 but does not simplify to a whole number: 5/2. The latter case is known as an improper fraction, or a fraction where the numerator (top half of the fraction) is greater than the denominator (bottom half of the fraction).
• Improper fractions can also be written as a mixture of whole numbers and fractions. For example, 5/2 can also be written as 2 1/2. Since these are a mixture of two different things, these are referred to as mixed numbers.

There are four different ways that fractions can interact: addition, subtraction, multiplication, and division. 

For the first two (addition and subtraction), each fraction needs to have the same denominator, known as a common denominator. Once the fractions have the same denominator, they can be added or subtracted as follows:

• Add: Add the numerator and keep the denominator (bottom half of the fraction) the same

• Subtract: Subtract the numerator of the second fraction from the first fraction and keep the denominator the same

• Multiply: To multiply fractions, multiply the numerators of the fractions together, and also multiply the denominators of the fraction together

• Divide: To divide fractions by each other, it’s easiest to turn division into multiplication. Dividing by a fraction is the same as multiplying by its reciprocal (swap the numerator and the denominator: what was on top will now be on bottom, and vice versa). For example, to divide something by 12, that’s the same as doubling it, or multiplying by 2/1!