Addition and Subtraction of Fractions

Addition and Subtraction of Similar Fractions

1. Copy the denominator
2. Add or Subtract the numerator
3. Simplify the fraction if not in simplest form.

1. \( \frac{5}{6} + \frac{1}{6}\) = \( \frac{5+1}{6}\) = \( \frac{6}{6}\) = 1

2. \( \frac{4}{15} + \frac{13}{15}\) = \( \frac{4+13}{15}\) = \( \frac{17}{15}\)

3. \( \frac{18}{3} - \frac{11}{3}\) = \(\frac{18-11}{3} = \frac{7}{3}\)

Addition and Subtraction of Proper and Improper Fractions

1. Find the Least Common Denominator(LCD) - This is the Least Common Multiple (LCM) of the Denominators.
2. Rewrite each fraction using the LCD or LCM of the denominator
1. The new denominator is the LCD
2. The new numerator - Divide the LCD by the denominator and multiply the numerator
3. Add as a similar fraction.

1. \( \frac{2}{3} + \frac{7}{6}\)

• LCM of 6 and 3 is 6 thus the LCD is 6
• \(\frac{2}{3} = \frac{(\frac{6}{3})(2)}{6}\) = \( \frac{4}{6} \)
• \(\frac{7}{6} = \frac{(\frac{6}{6})(7)}{6}\) = \( \frac{7}{6} \)
• Since they are similar just add the numerator
• \( \frac{2}{3} + \frac{7}{6}\) = \( \frac{4}{6} \) + \( \frac{7}{6} \) = \(\frac{11}{6}\)

2. \( \frac{8}{3} + \frac{11}{7}\)

• LCM of 7 and 3 is 21 thus the LCD is 21
• \(\frac{8}{3} = \frac{(\frac{21}{3})(8)}{21}\) = \( \frac{56}{21} \)
• \(\frac{11}{7} = \frac{(\frac{21}{7})(11)}{21}\) = \( \frac{33}{21} \)
• Since they are similar just add the numerator
• \( \frac{8}{3} + \frac{11}{7}\) = \( \frac{56}{21} + \frac{33}{21} \)= \(\frac{56+33}{21}\) = \(\frac{89}{21}\)

3. \( \frac{18}{3} - \frac{11}{7}\)

• LCM of 7 and 3 is 21 thus the LCD is 21
• \(\frac{18}{3} = \frac{(\frac{21}{3})(18)}{21}\) = \( \frac{126}{21} \)
• \(\frac{11}{7} = \frac{(\frac{21}{7})(11)}{21}\) = \( \frac{33}{21} \)
• Since they are similar just subtract the numerator
• \( \frac{18}{3} - \frac{11}{7}\) = \( \frac{126}{21} - \frac{33}{21} \)= \(\frac{126-33}{21}\) = \(\frac{93}{21}\)
• Reduce/Simplify the fraction
• \(\frac{93}{21}\) = \( \frac{31x3}{7x3}\) = \( \frac{31}{7}\)

Addition and Subtraction of Mixed Fractions

1. Convert the mixed fraction to its equivalent improper fraction.
2. Follow the procedures used in adding/subtracting improper and proper fractions.

1. Add/subtract the whole part.
2. Add/Subtract the fraction part.
3. Combine the whole and fraction part as a mixed fractoin.

1. \( 2\frac{1}{3}\) + \( 5 \frac{2}{5}\)

Solution 01. Using the first procedure
• Transform into mixed fraction: \( 2\frac{1}{3}\) = \(\frac{7}{3}\) and \( 5 \frac{2}{5}\) = \( \frac{27}{5}\)
• LCM of 3 and 5 is 15 thus the LCD is 15.
• \(\frac{7}{3}\) = \( \frac{35}{15}\)
• \( \frac{27}{5}\) = \(\frac{81}{15}\)
• Add
• \(\frac{7}{3}\) + \( \frac{27}{5}\) = \( \frac{35}{15}\) + \(\frac{81}{15}\) = \( \frac{116}{15}\)
• Change to mixed if nescessary.
• \( \frac{116}{15}\) = \( 7\frac{11}{15}\)

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Solution 02. Using the second procedure
• Add the whole part.
• \( 2\frac{1}{3}\) + \( 5 \frac{2}{5}\) =\( 7 (\frac{1}{3} + \frac{2}{5})\)
• Add the fraction part by rewriting the fraction using LCD.
• \( 2\frac{1}{3}\) + \( 5 \frac{2}{5}\) = \( 7 (\frac{5}{15} + \frac{6}{15})\)
• \( 2\frac{1}{3}\) + \( 5 \frac{2}{5}\) = \( 7 \frac{11}{15}\)
• Change to improper if nescessary.
• \( 7 \frac{11}{15}\) = \(\frac{15x7 +11}{15}\) = \( \frac{116}{15}\)