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In this section, we learn how to add and subtract fractions. For this topic, we will have two different scenarios:
• Adding & Subtracting Fractions with a Common Denominator
• Adding & Subtracting Fractions without a Common Denominator
Adding & Subtracting Fractions with a Common Denominator:
1. Perform the given operation with the numerators - Add or Subtract the Numerators of the Fractions
2. Place the result over the common denominator
3. Simplify (reduce the fraction to lowest terms)
Example 1: Find each sum $$\frac{1}{12} + \frac{5}{12}$$ -Add the numerators: 1 + 5 = 6
-Place the result over the common denominator
$$\frac{6}{12}$$ -Simplify $$\frac{6}{12} = \frac{2 \cdot 3}{2 \cdot 2 \cdot 3}$$ $$\require{cancel}\frac{\cancel{2} \cdot \cancel{3}}{2 \cdot \cancel{2} \cdot \cancel{3}} = \frac{1}{2}$$ Adding & Subtracting Fractions without a Common Denominator:
1. Write equivalent fractions with the LCD as each denominator
2. Perform the given operation with the numerators - Add or Subtract the Numerators of the Fractions
3. Place the result over the common denominator
4. Simplify (reduce the fraction to lowest terms)
What does write equivalent fractions with the LCD as each denominator mean? What is an LCD? There are lots of acronyms that get thrown around in pre-algebra. It's easy to get confused between: LCD, LCM, GCF, GCD, ...etc. The LCD is an acronym for the Least Common Denominator. This is the LCM or Least Common Multiple of the denominators. Simply put, find the Least Common Multiple of the denominators, this will be the Least Common Denominator. How do we re-write each fraction as an equivalent fraction with the LCD as the denominator? Remember we can multiply by 1 and leave a number unchanged. Think about this scenario: $$\frac{2}{3} \cdot \frac{5}{5} = \frac{10}{15}$$ The value of 5/5 is 1, so multiplying 2/3 by 1 didn't change the value, it just changed the appearance of the fraction. If there is 10/15 of a pie remaining, it's the same as saying there is 2/3 of a pie remaining.
Example 1: Find each difference $$\frac{1}{3} - \frac{2}{7}$$ -LCD = LCM(3,7) = 21
-Rewrite each as an equivalent fraction with 21 as the denominator:
$$\frac{1}{3} \cdot \frac{7}{7} = \frac{7}{21}$$ $$\frac{2}{7} \cdot \frac{3}{3} = \frac{6}{21}$$ -Subtract the numerators: 7 - 6 = 1
-Place the result over the common denominator
$$\frac{1}{21}$$ -The fractions is already simplified $$\frac{1}{3} - \frac{2}{7} = \frac{1}{21}$$