How to simplify fractions by canceling common factors

If you’ve traveled 50 miles out of 100, you can say you’re halfway there. Both fractions describe the same part of the trip. Let’s see how to find the simpler fraction, even when the numbers are larger.

Divide the top and bottom by the same factor

A fraction stays the same when we divide its numerator and denominator by the same nonzero number. For $\frac{50}{100}$ , both are divisible by $50$ .

\frac{50 \div 50}{100 \div 50} = \frac{1}{2}

When a nonzero factor appears on both the top and bottom, it divides out because a nonzero number divided by itself equals $1$ . That’s what we mean by canceling a common factor.

Example: Simplify 630/945 using prime factors

Write $630$ and $945$ as products of prime numbers. The factors $3$ , $3$ , $5$ , and $7$ appear in both products.

\begin{array}{l} 630 = (3 \cdot 3 \cdot 5 \cdot 7) \cdot 2 \\ 945 = (3 \cdot 3 \cdot 5 \cdot 7) \cdot 3 \end{array}

The shaded groups match. Dividing each number by that group leaves 2 and 3.

The common factors multiply to $315$ . Divide the numerator and denominator by $315$ to get the fraction in lowest terms.

\begin{array}{l} \frac{630}{945} = \frac{315 \cdot 2}{315 \cdot 3} \\ = \frac{2}{3} \end{array}

The only positive factor that $2$ and $3$ share is $1$ , so there’s nothing left to divide out. The video works through different fractions using the same process.

How to simplify fractions by canceling common factors

Divide the top and bottom by the same factor

Example: Simplify 630/945 using prime factors

Courses