Reducing by Grouping Factors Equal to 1
Reducing fractions by grouping factors equal to 1 is a visual method taught in Grade 6 Saxon Math Course 1. After writing numerator and denominator as products of prime factors, pairs of identical factors in numerator and denominator form groups equal to 1 (e.g., 3/3 = 1) and cancel out. For 12/18: 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3; cancel one 2 and one 3 to get 2/3. This technique reinforces why canceling works (dividing by a form of 1 does not change value) and builds algebraic fraction skills.
Key Concepts
New Concept To divide by a fraction, you multiply by its reciprocal. This process answers the question of how many times the divisor fits into the dividend.
When the divisor is a fraction, we take two steps to find the answer. We first find how many of the divisors are in 1. This is the reciprocal of the divisor. Then we use the reciprocal to answer the original division problem by multiplying. What’s next You've just learned the core logic. Next, we’ll apply this two step division process and use factor grouping in worked examples to simplify the results.
Common Questions
What does 'grouping factors equal to 1' mean?
Writing matching prime factors in numerator and denominator as a fraction equal to 1 (e.g., 5/5 = 1), then removing them since multiplying by 1 does not change value.
Reduce 12/18 by grouping factors.
12 = 2 × 2 × 3, 18 = 2 × 3 × 3. Cancel one 2: ✓ and one 3: ✓. Remaining: 2/3.
Reduce 30/42 by grouping factors.
30 = 2 × 3 × 5, 42 = 2 × 3 × 7. Cancel 2 and 3. Remaining: 5/7.
Why does canceling a pair of identical factors equal 1?
Any nonzero number divided by itself = 1. So matching prime factors in numerator and denominator form a pair whose quotient is 1, leaving the fraction's value unchanged.
Is this method the same as dividing by the GCF?
Yes — the product of all canceled pairs equals the GCF. Both methods produce the same reduced fraction.