Class Note: Ordering Fractions from Least to Greatest

Objective

Learn how to order fractions from least to greatest by finding common denominators.

---

Steps to Order Fractions

1. Identify Denominators

   • Example denominators: 2, 3, 5, and another 3 (do not need to multiply repeating denominators twice).

2. Find a Common Denominator

   • Multiply the unique denominators: (2 \times 3 \times 5 = 30)
   • The common denominator will be 30.

3. Rewrite Each Fraction with the Common Denominator

   • For ( \frac{1}{3} ): ( 30 / 3 = 10 ), so multiply by ( \frac{10}{10} ) → ( \frac{10}{30} )
   • For ( \frac{1}{2} ): ( 30 / 2 = 15 ), so multiply by ( \frac{15}{15} ) → ( \frac{15}{30} )
   • For ( \frac{2}{3} ): ( 30 / 3 = 10 ), so multiply by ( \frac{10}{10} ) → ( \frac{20}{30} )
   • For ( \frac{3}{5} ): ( 30 / 5 = 6 ), so multiply by ( \frac{6}{6} ) → ( \frac{18}{30} )

4. List Numerators

   • Now the fractions are: ( \frac{10}{30} ), ( \frac{15}{30} ), ( \frac{18}{30} ), ( \frac{20}{30} )
   • Numerators: 10, 15, 18, 20

5. Order the Fractions

   • From least to greatest (by numerator): ( \frac{10}{30} < \frac{15}{30} < \frac{18}{30} < \frac{20}{30} )
   • Translate back to original form:
     • ( \frac{1}{3} < \frac{1}{2} < \frac{3}{5} < \frac{2}{3} )

---

Confirmation Using Decimals

• ( \frac{1}{3} ) ≈ 0.333…
• ( \frac{1}{2} ) = 0.5
• ( \frac{3}{5} ) = 0.6
• ( \frac{2}{3} ) ≈ 0.666…

So, from least to greatest:
( \frac{1}{3} < \frac{1}{2} < \frac{3}{5} < \frac{2}{3} )

---

Key Takeaways

• Always find a common denominator to compare fractions easily.
• Rewriting each fraction with the common denominator allows for easy comparison using numerators.
• To double-check, convert each fraction to a decimal.
• Order is determined by the size of the numerators once denominators are the same.