Same Value, Different Look: Equivalent Fractions
Equivalent fractions are different fractions that represent the same amount β for example, 1/2 and 2/4 both name exactly half of a whole. Recognizing and creating equivalent fractions is a critical skill for adding and subtracting fractions with unlike denominators, simplifying answers, and understanding proportional relationships.
Components of Equivalent Fractions
This section covers the core techniques for working with equivalent fractions:
- Multiplying to Find Equivalents: Multiply both numerator and denominator by the same nonzero number to create an equivalent fraction.
- Dividing to Simplify: Divide both numerator and denominator by their greatest common factor (GCF) to reduce a fraction to simplest form.
- Finding Common Denominators: Rewrite two or more fractions with the same denominator so they can be compared or combined.
- Applications of Equivalent Fractions: Using equivalence to solve real-world problems involving recipes, measurements, and fair sharing.
Examples of Equivalent Fractions
Multiplying to Find Equivalents Examples
- Start with 2/3 and multiply top and bottom by 4: 2 Γ 4 = 8 and 3 Γ 4 = 12, so 2/3 = 8/12.
- Start with 3/5 and multiply by 3: 3 Γ 3 = 9 and 5 Γ 3 = 15, so 3/5 = 9/15.
- A recipe calls for 1/4 cup of sugar. To double the recipe, find an equivalent: 1 Γ 2 = 2 and 4 Γ 2 = 8, but since you need twice as much, you need 2/4 = 1/2 cup.
Dividing to Simplify Examples
- Simplify 12/18: The GCF of 12 and 18 is 6, so divide both by 6 to get 2/3.
- Simplify 15/25: The GCF of 15 and 25 is 5, so 15 Γ· 5 = 3 and 25 Γ· 5 = 5, giving 3/5.
- Simplify 8/20: The GCF of 8 and 20 is 4, so 8 Γ· 4 = 2 and 20 Γ· 4 = 5, giving 2/5.
Finding Common Denominators Examples
- Rewrite 1/3 and 1/4 with a common denominator: The LCM of 3 and 4 is 12, so 1/3 = 4/12 and 1/4 = 3/12.
- Rewrite 2/5 and 3/10 with a common denominator: The LCM of 5 and 10 is 10, so 2/5 = 4/10 and 3/10 stays as 3/10.
- Rewrite 5/6 and 7/9 with a common denominator: The LCM of 6 and 9 is 18, so 5/6 = 15/18 and 7/9 = 14/18.
Applications Examples
- A recipe needs 2/3 cup of flour, but you only have a 1/6-cup scoop. Since 2/3 = 4/6, you need 4 scoops.
- Two boards measure 3/8 inch and 5/16 inch. Convert 3/8 = 6/16 to compare: the first board is thicker since 6/16 > 5/16.
- A class survey shows 12/30 of students prefer math. Simplify to 2/5 to report that two out of every five students prefer math.