Equivalent fractions, reducing to lowest terms
Middle School Bridge • 6-8
Have you ever split a pizza equally among friends? Whether you cut it into 4 slices and take 2, or cut it into 8 slices and take 4, you still get the same amount of pizza. That idea -- that different-looking fractions can represent the same quantity -- is one of the most important concepts in all of mathematics. Mastering equivalent fractions gives you the power to simplify complex problems and see hidden connections between numbers.
Two fractions are equivalent when they represent the same value. The key principle is this: multiplying or dividing both the numerator and denominator by the same nonzero number produces an equivalent fraction.
Each of these fractions names exactly half of a whole. We obtained them by multiplying the numerator and denominator of 1⁄2 by 2, 3, 4, and 5 respectively.
Write three fractions equivalent to 3⁄5.
So 3⁄5 = 6⁄10 = 9⁄15 = 12⁄20.
Simplifying (or "reducing") a fraction means rewriting it with the smallest possible numerator and denominator. To do this, divide both parts by their Greatest Common Factor (GCF) -- the largest number that divides evenly into both.
Finding the GCF: List the factors of each number and pick the largest one they share. For example, the factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 18 are 1, 2, 3, 6, 9, 18. The GCF is 6.
Simplify 18⁄24.
The simplified form is 3⁄4.
Simplify 7⁄15.
Factors of 7: 1, 7. Factors of 15: 1, 3, 5, 15. The only common factor is 1, so GCF = 1. The fraction is already in lowest terms.
Students sometimes add or subtract the same number from the numerator and denominator, thinking this creates an equivalent fraction. It does not. For example, 3⁄5 is NOT equal to 3 + 1⁄5 + 1 = 4⁄6. You must multiply or divide both parts by the same number -- never add or subtract.
1. Write two fractions equivalent to 2⁄7.
Multiply by 2: 4⁄14. Multiply by 3: 6⁄21. (Any correct multiplications are valid.)
2. Simplify 20⁄35.
GCF(20, 35) = 5. Divide both by 5: 20 ÷ 5⁄35 ÷ 5 = 4⁄7.
3. Simplify 36⁄48.
GCF(36, 48) = 12. Divide both by 12: 36 ÷ 12⁄48 ÷ 12 = 3⁄4.
4. Are 5⁄8 and 15⁄24 equivalent? Explain.
Yes. Simplify 15⁄24 by dividing both by 3: 5⁄8. Since both reduce to the same fraction, they are equivalent. You can also cross-multiply: 5 × 24 = 120 and 8 × 15 = 120, confirming equality.
5. Simplify 42⁄56.
GCF(42, 56) = 14. Divide both by 14: 42 ÷ 14⁄56 ÷ 14 = 3⁄4.
Equivalent fractions represent the same value and are created by multiplying or dividing both the numerator and denominator by the same nonzero number. To simplify a fraction, find the GCF of the numerator and denominator and divide both by it. A fraction is in lowest terms when its GCF is 1.