Fractions

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 $3\cdot8=24$.

Worked example

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

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?