Fractions

A fraction is one number: the quotient of its numerator and nonzero denominator.

Meaning of a fraction

In

$a$ is the numerator and $b$ the denominator. The fraction can mean $a\div b$, $a$ copies of the unit fraction $1/b$, a ratio, or a point on the number line.

The denominator names equal-size parts; the numerator counts them.

Equivalent fractions

Multiplying or dividing numerator and denominator by the same nonzero number preserves value:

To simplify, divide by the greatest common factor. A fraction in lowest terms has numerator and denominator with no common factor greater than $1$.

Comparing fractions

Use a common denominator, benchmark values such as $0$, $1/2$, and $1$, or cross-products for positive denominators.

For example,

because $5\cdot5=25$ is greater than $3\cdot8=24$.

Adding and subtracting

Fractions need a common denominator because only equal-size parts can be combined:

Use the least common denominator when convenient, then simplify the result.

Worked example

The result is reasonable because subtracting less than $1$ from a little more than $2$ leaves between $1$ and $2$.

Multiplying fractions

Multiply numerators and denominators:

Cancel common factors before multiplying to keep numbers small. “Of” often signals multiplication: three quarters of two thirds is $(3/4)(2/3)=1/2$.

Dividing fractions

Multiply by the reciprocal of the divisor:

This asks how many groups of size $c/d$ fit into $a/b$. Only the divisor is inverted.

Improper fractions and mixed numbers

An improper fraction has numerator at least as large as its positive denominator. It is often easier for calculations. A mixed number separates whole and fractional parts and may be easier to interpret.

Convert based on the task, not because one form is always superior.

Fractions, decimals, and percents

Divide numerator by denominator to obtain a decimal. Multiply a decimal by $100\%$ to obtain a percent:

Some fractions terminate; others repeat. Exact fraction form avoids rounding.

Common mistakes

Adding numerators and denominators. $a/b+c/d$ is not $(a+c)/(b+d)$.

Finding a common denominator for multiplication. It is unnecessary.

Flipping both fractions in division. Invert only the divisor.

Cancelling across addition. Cancellation applies to factors.

Treating a larger denominator as a larger positive unit fraction. More equal pieces means smaller pieces.

Quick self-check

• What meaning does the fraction have in context?
• Are equivalent forms produced by multiplying/dividing both parts equally?
• For addition/subtraction, are parts the same size?
• For multiplication, can factors cancel first?
• For division, was only the divisor reciprocated?
• Is the result simplified and close to a benchmark estimate?

Related topics

• Decimals
• Percent
• Ratios
• Rational Numbers