Simplify fraction \(\frac{90}{105}\)

$\frac{6}{7}$

Explanation

Strategy We begin by prime factoring the numerator, 90, and denominator, 105. Then we look for any factors common to the numerator and denominator and remove them.

WHY When the numerator and/or denominator of a fraction are large numbers, such as 90 and 105, writing their prime factorizations is helpful in identifying any common factors.

Solution

To prepare to simplify, write 90 and 105 in prime-factored form.

\(\frac{90}{105} = \frac{2 \cdot 3 \cdot 3 \cdot 5}{3 \cdot 5 \cdot 7}\)

Remove the common factors of 3 and 5 from the numerator and denominator. Slashes and 1's are used to show that \(\frac{3}{3}\) and \(\frac{5}{5}\) are replaced by the equivalent fraction \(\frac{1}{1}\). A factor equal to 1 in the form of \(\frac{3 \cdot 5}{3 \cdot 5} = \frac{15}{15}\) was removed.

\(\require{cancel}= \frac{2 \cdot \cancel{3}^1 \cdot 3 \cdot \cancel{5}^1}{\cancel{3}_1 \cdot \cancel{5}_1 \cdot 7}\)

Multiply the remaining factors in the numerator: 2 · 1 · 3 · 1 = 6. Multiply the remaining factors in the denominator: 1 · 1 · 7 = 7.

\(=\frac{6}{7}\)

Since 6 and 7 have no common factors (other than 1), \(\frac{6}{7}\) is in simplest form.