Write fraction \(\frac{63}{36}\) in its simplest form

$\frac{7}{4}$

Explanation

Strategy We will prime factor the numerator and denominator. Then we will look for any factors common to the numerator and denominator and remove them.

WHY We need to make sure that the numerator and denominator have no common factors other than 1. If that is the case, then the fraction is in simplest form.

Solution

To prepare to simplify, write 63 and 36 in prime-factored form.

\(\frac{63}{36} = \frac{3 \cdot 3 \cdot 7}{2 \cdot 2 \cdot 3 \cdot 3}\)

Simplify by removing the common factors of 3 from the numerator and denominator.

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

Multiply the remaining factors in the numerator: 1 · 1 · 7 = 7.
Multiply the remaining factors in the denominator: 2 · 2 · 1 · 1 = 4.

\(=\frac{7}{4}\)

Success Tip If you recognized that 63 and 36 have a common factor of 9, you may remove that common factor from the numerator and denominator without writing the prime factorizations. However, make sure that the numerator and denominator of the resulting fraction do not have any common factors. If they do, continue to simplify.