Lesson 5: Simplest Form of a Fraction

Property

We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.

Examples

• We can reduce $\frac{18}{30}$ by noting $\frac{18}{30} = \frac{6 \cdot 3}{6 \cdot 5}$. We divide out the common factor 6 to get $\frac{3}{5}$.

• The fraction $\frac{5x^2}{15x}$ can be written as $\frac{5 \cdot x \cdot x}{3 \cdot 5 \cdot x}$. Canceling the common factors of $5$ and $x$ gives $\frac{x}{3}$.

Property

A fraction is considered simplified if there are no common factors in its numerator and denominator.

How to Simplify a Fraction

1. Rewrite the numerator and denominator to show the common factors. If needed, factor them into prime numbers first.
2. Simplify using the equivalent fractions property by dividing out common factors.
3. Multiply the remaining factors, if necessary.

Examples

• To simplify $-\frac{24}{40}$, we find the common factor of 8. We rewrite it as $-\frac{3 \cdot 8}{5 \cdot 8}$. Dividing out the 8 gives us $-\frac{3}{5}$.
• To simplify $\frac{150}{225}$, we can see a common factor of 25. $\frac{6 \cdot 25}{9 \cdot 25} = \frac{6}{9}$. This can be simplified further by a factor of 3: $\frac{2 \cdot 3}{3 \cdot 3} = \frac{2}{3}$.
• To simplify $\frac{9x}{9y}$, we can divide out the common factor 9. This leaves us with $\frac{x}{y}$.