Math Core

In algebra you will simplify fractions constantly, often with negative numbers mixed in. A fraction in simplest form is easier to read, easier to compare and much easier to compute with. This lesson reviews how to simplify quickly and how the negative sign behaves in a fraction.

Equivalent fractions

Multiplying or dividing the numerator and denominator by the same nonzero number does not change a fraction's value. It just cuts the whole into a different number of pieces.

\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \qquad\qquad \frac{15}{20} = \frac{15 \div 5}{20 \div 5} = \frac{3}{4}

Fractions with the same value, like $\dfrac{3}{4}$ and $\dfrac{15}{20}$ , are equivalent fractions. They sit at the same point on a number line.

3/4 and 15/20 name the same point

Definition

Simplest form

A fraction is in simplest form (or lowest terms) when its numerator and denominator have no common factor other than 1.

Simplifying with the GCF

To simplify in one step, divide the numerator and denominator by their greatest common factor (GCF).

Simplifying a fraction

Find the GCF of the numerator and denominator.
Divide both by the GCF.

\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}

If you can't spot the GCF, divide by any common factor you see and repeat until no common factor is left. You'll reach the same answer, just in more steps: $\dfrac{24}{36} = \dfrac{12}{18} = \dfrac{6}{9} = \dfrac{2}{3}$ .

Worked example: Using the GCF

Simplify $\dfrac{42}{56}$ .

Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56. The GCF is 14.

\frac{42}{56} = \frac{42 \div 14}{56 \div 14} = \frac{3}{4}

Worked example: Using prime factorization

Simplify $\dfrac{90}{126}$ .

Write each number as a product of primes and cancel the common factors:

\frac{90}{126} = \frac{2 \cdot 3 \cdot 3 \cdot 5}{2 \cdot 3 \cdot 3 \cdot 7} = \frac{5}{7}

The common factors $2 \cdot 3 \cdot 3 = 18$ cancel, which is the same as dividing by the GCF 18.

Negative fractions

A fraction is a division: $\dfrac{a}{b}$ means $a \div b$ . Since a negative divided by a positive is negative, and a positive divided by a negative is also negative, all three of these are equal:

-\frac{3}{5} = \frac{-3}{5} = \frac{3}{-5}

A negative divided by a negative is positive, so $\dfrac{-3}{-5} = \dfrac{3}{5}$ .

-3/5 and 3/5 are opposites

The usual way to write a negative fraction is with the sign in front: $-\dfrac{3}{5}$ . Simplify the numbers as usual and decide the sign separately.

Worked example: Simplifying with signs

Simplify each fraction.

$\dfrac{-18}{30}$
$\dfrac{28}{-8}$
$\dfrac{-45}{-60}$

Solutions.

One negative, so the result is negative. The GCF of 18 and 30 is 6: $\dfrac{-18}{30} = -\dfrac{3}{5}$ .
One negative, so negative. The GCF of 28 and 8 is 4: $\dfrac{28}{-8} = -\dfrac{7}{2}$ , which is $-3\dfrac{1}{2}$ as a mixed number.
Two negatives, so positive. The GCF of 45 and 60 is 15: $\dfrac{-45}{-60} = \dfrac{3}{4}$ .

Common mistake

Only cancel factors, never terms that are added. In $\dfrac{4 + 6}{4}$ you cannot cross out the 4s. Add first: $\dfrac{10}{4} = \dfrac{5}{2}$ . Crossing out would give 6, which is wrong. This matters even more in algebra: $\dfrac{x + 3}{3}$ is not $x$ .

Improper fractions and mixed numbers

In algebra, answers are usually left as improper fractions like $-\dfrac{7}{2}$ , because they are easier to multiply and divide. In word problems a mixed number is often clearer. You should be able to switch either way.

$\dfrac{17}{5}$ : $17 \div 5 = 3$ remainder 2, so $\dfrac{17}{5} = 3\dfrac{2}{5}$ .
$-4\dfrac{1}{3}$ : $4 \times 3 + 1 = 13$ , so $-4\dfrac{1}{3} = -\dfrac{13}{3}$ . The negative sign applies to the whole mixed number.

Tip

Quick divisibility checks help you find common factors fast: a number is divisible by 2 if it ends in an even digit, by 5 if it ends in 0 or 5, by 3 if its digits add to a multiple of 3, and by 9 if its digits add to a multiple of 9.

Practice

Practice 1

Which fraction is $\dfrac{16}{40}$ in simplest form?

Practice 2

What is the greatest common factor of 36 and 60?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Fill in the blank so the fractions are equivalent: $\dfrac{4}{7} = \dfrac{16}{?}$

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Which fraction is $\dfrac{-35}{50}$ in simplest form?

Practice 5

Which of these is not equal to $-\dfrac{2}{3}$ ?

Practice 6

Which fraction is $\dfrac{-84}{-126}$ in simplest form?

Practice 7

Write $-\dfrac{23}{6}$ as a mixed number.

Practice 8

Maya simplified $\dfrac{5 + 10}{5}$ by crossing out the 5s and got 10. What is the correct value?