1 Fraction Skills

Learn what fractions represent and how to identify, simplify, compare, add, subtract, multiply, and divide them.

parts and types

A represents a number of equal parts. In $\frac{a}{b}$ , $a$ is the and $b$ is the ; the must not be zero. For example, $\frac{3}{4}$ means three of four equal parts.

A has a less than its , as in $\frac{3}{4}$ . An has a at least as large as its , as in $\frac{7}{4}$ . An can be written as a mixed number: $\frac{7}{4}=1\frac{3}{4}$ .

have the same value, even when their numerators and denominators differ. Multiply or divide the and by the same nonzero number to make an equivalent :

\frac{a}{b}=\frac{a\times k}{b\times k},\qquad k\ne 0.

For example, multiplying both parts of $\frac{2}{3}$ by $4$ gives:

\frac{2}{3}=\frac{8}{12}.

are useful for comparing fractions and for adding or subtracting them.

Simplifying fractions

A is in when its and have no common factor greater than $1$ . To simplify, divide both by their . For example:

\frac{18}{24}=\frac{18\div 6}{24\div 6}=\frac{3}{4}.

The value does not change because the and are divided by the same number. After simplifying, check that the resulting and have no common factor other than $1$ .

Adding and subtracting fractions

Fractions need a before they can be added or subtracted. If the denominators already match, add or subtract the numerators and keep the :

\frac{2}{7}+\frac{3}{7}=\frac{5}{7}.

If the denominators differ, rewrite the fractions using a . The least (LCD) is often convenient:

\frac{1}{3}+\frac{1}{4}=\frac{4}{12}+\frac{3}{12}=\frac{7}{12}.

For subtraction, use the same process:

\frac{5}{6}-\frac{1}{4}=\frac{10}{12}-\frac{3}{12}=\frac{7}{12}.

Simplify the final answer if possible. Do not add or subtract the denominators.

Multiplying fractions

To multiply fractions, multiply the numerators together and the denominators together. A is not needed:

\frac{2}{3}\times\frac{3}{5}=\frac{2\times3}{3\times5}=\frac{6}{15}=\frac{2}{5}.

You can simplify before multiplying by canceling common factors across a and a . In this example, the factor $3$ cancels, leaving $\frac{2}{5}$ .

Dividing fractions

To divide by a , multiply by its , which is formed by switching its and . Keep the first , change division to multiplication, and flip the second :

\frac{3}{4}\div\frac{2}{5}=\frac{3}{4}\times\frac{5}{2}=\frac{15}{8}=1\frac{7}{8}.

The divisor cannot be zero. Check the result by multiplying it by the divisor:

\frac{15}{8}\times\frac{2}{5}=\frac{3}{4}.

Checking calculations

Use these checks when working with fractions:

For addition and subtraction, first find a .
For multiplication and division, a is unnecessary.
Simplify answers when possible, and check that no is zero.
Estimate whether the result makes sense. Multiplying two positive proper fractions gives a product less than either factor.

Fraction parts and types

Equivalent fractions