Equivalent Fractions & Simplifying - Sixth Grade Mathematics

Equivalent fractions are fractions that name the exact same numerical quantity or proportion despite having different numerators and denominators.

The Principle of Equivalence

Fractions that appear different at first glance can represent the exact same value. Understanding equivalence enables you to scale fractions for addition and subtraction, as well as reduce them to simplest terms for clear communication.

Equivalent Fraction Visual Model (Shading 1/2 of an area):
1/2:  [ ***** | ..... ]                 --> 1 out of 2 equal parts
2/4:  [ ** | ** | .. | .. ]             --> 2 out of 4 equal parts
4/8:  [ * | * | * | * | . | . | . | . ] --> 4 out of 8 equal parts
Conclusion: 1/2 = 2/4 = 4/8 (Identical amount shaded!)

The Identity Property of Multiplication in Disguise

Why can we multiply or divide the numerator and denominator of a fraction without changing its value? The Multiplicative Identity Property states that multiplying any number by 1 leaves its value unchanged. Any non-zero number divided by itself equals 1 (for example, 2/2 = 1, 3/3 = 1, 5/5 = 1). When you multiply both the numerator and denominator by 3, you are multiplying the entire fraction by 3/3, which is simply multiplying by 1.

Algebraic Rule:
               a / b = (a * k) / (b * k)    (where k is not 0)
               a / b = (a / k) / (b / k)    (where k is a common factor)

Generating Equivalent Fractions

You can create an infinite family of equivalent fractions by multiplying or dividing the numerator and denominator by the same non-zero integer.

Multiplying to Scale Up:
3/5 * (2/2) = 6/10
3/5 * (4/4) = 12/20
3/5 * (10/10) = 30/50
Family of Equivalence: { 3/5, 6/10, 12/20, 30/50, ... }

Scaling Up for Common Denominators

Scaling up is primarily used when preparing fractions for addition or subtraction. If you need to convert 2/3 so that its denominator is 18, determine what factor turns 3 into 18: 18 / 3 = 6. Multiply both numerator and denominator by 6: (2 * 6) / (3 * 6) = 12/18.

Testing Equivalence Using Cross-Multiplication

To test whether two fractions a/b and c/d are equivalent, compute their cross-products: multiply the numerator of each fraction by the denominator of the other. The fractions are equivalent if and only if the cross-products are equal: a * d = b * c.

Testing Equivalence: Is 6/15 equivalent to 14/35?
Cross-Product 1: 6 * 35 = 210
Cross-Product 2: 15 * 14 = 210
Since 210 = 210, the fractions 6/15 and 14/35 are EQUIVALENT!

Simplifying Fractions to Simplest Form

A fraction is in its simplest form (also known as lowest terms) when its numerator and denominator are co-prime, meaning their Greatest Common Factor is 1.

+-------------------------------------------------------------------------+
|                  TWO PATHWAYS TO SIMPLIFY FRACTIONS                     |
+-------------------+-----------------------+-----------------------------+
| Method 1: The GCF | Divide numerator and  | 36/48: GCF(36, 48) = 12     |
| Shortcut          | denominator by their  | (36 / 12) / (48 / 12) = 3/4 |
| (Single Step)     | Greatest Common Factor| Single-step reduction!      |
+-------------------+-----------------------+-----------------------------+
| Method 2: Prime   | Divide sequentially   | 36/48 / (2/2) = 18/24       |
| Division Ladder   | by shared prime       | 18/24 / (2/2) = 9/12        |
| (Multi-Step)      | factors (2, 3, etc.)  | 9/12  / (3/3) = 3/4         |
+-------------------+-----------------------+-----------------------------+

Prime Factor Cancellation

Another visual method for simplifying fractions is writing the prime factorizations of both numerator and denominator and canceling common factors. For 36/48: the numerator is 2 * 2 * 3 * 3, and the denominator is 2 * 2 * 2 * 2 * 3. Canceling two 2s and one 3 from both top and bottom leaves 3 in the numerator and 2 * 2 = 4 in the denominator, resulting in 3/4.

Exercises

Exercise 1. Find three equivalent fractions for 5/9 by multiplying the numerator and denominator by 2, 3, and 5.

Exercise 2. Use cross-multiplication to verify whether 8/14 and 20/35 are equivalent.

Exercise 3. Reduce the fraction 42/70 to its simplest form using the Greatest Common Factor method.

Solutions

Solution to Exercise 1. Multiply numerator and denominator by each specified factor: Multiplying by 2: (5 * 2) / (9 * 2) = 10/18. Multiplying by 3: (5 * 3) / (9 * 3) = 15/27. Multiplying by 5: (5 * 5) / (9 * 5) = 25/45. Three equivalent fractions are 10/18, 15/27, and 25/45.

Solution to Exercise 2. Set up the two fractions: 8/14 and 20/35. Calculate the cross-products: 8 * 35 = 280, and 14 * 20 = 280. Because both cross-products equal 280, the fractions 8/14 and 20/35 are equivalent.

Solution to Exercise 3. Find the GCF of 42 and 70: prime factorization of 42 is 2 * 3 * 7; prime factorization of 70 is 2 * 5 * 7. The common prime factors are 2 and 7, so GCF(42, 70) = 2 * 7 = 14. Divide both numerator and denominator by 14: 42 / 14 = 3, and 70 / 14 = 5. The fraction in simplest form is 3/5.