Using Prime Factorization to Reduce Fractions
Using prime factorization to reduce fractions finds the GCF by comparing prime factors of numerator and denominator, then canceling matching primes. In Grade 6 Saxon Math Course 1, students rewrite both numerator and denominator as products of primes, cancel common factors (each pair equals 1), and multiply the remaining factors. For 36/48: 36 = 2² × 3² and 48 = 2⁴ × 3, so cancel 2² and 3¹ to get 3/4. This systematic method is more reliable than guessing the GCF for large numbers.
Key Concepts
New Concept One way to reduce fractions with large terms is to factor the terms and then reduce the common factors. To reduce a fraction like $\frac{150}{750}$, we can begin by writing the prime factorizations of 150 and 750. $$ \frac{150}{750} = \frac{2 \cdot 3 \cdot 5 \cdot 5}{2 \cdot 3 \cdot 5 \cdot 5 \cdot 5} $$ What’s next This is just the foundation. Next, you'll apply this process in worked examples and practice problems to build your skill and confidence.
Common Questions
How do you use prime factorization to reduce 36/48?
36 = 2² × 3², 48 = 2⁴ × 3. Common factors: 2² and 3¹. Cancel: (2² × 3²)/(2⁴ × 3) = 3/2² = 3/4.
What does canceling a prime factor mean?
Any number divided by itself is 1. So canceling a prime factor in numerator and denominator removes it from both, reducing the fraction.
Why is prime factorization more reliable than guessing GCF?
It systematically identifies every shared factor so nothing is missed, unlike trial-and-error which may not find the complete GCF.
Reduce 30/42 using prime factorization.
30 = 2 × 3 × 5, 42 = 2 × 3 × 7. Cancel 2 and 3: 5/7.
What is the reduced form of 18/24?
18 = 2 × 3², 24 = 2³ × 3. GCF = 2 × 3 = 6. 18/24 = 3/4.