### Math Notes

Subjects
Basic Mathematics Solved Problems | Topics - Contents

# Addition and Subtraction of Fractions

#### Addition and Subtraction of Similar Fractions

See Similar Fraction ...
1. Copy the denominator
2. Add or Subtract the numerator
3. Simplify the fraction if not in simplest form.
Examples:

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.
Examples:

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.
OR
3. Combine the whole and fraction part as a mixed fractoin.
Example:

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.
• $$\frac{7}{3}$$ = $$\frac{35}{15}$$
• $$\frac{27}{5}$$ = $$\frac{81}{15}$$
• $$\frac{7}{3}$$ + $$\frac{27}{5}$$ = $$\frac{35}{15}$$ + $$\frac{81}{15}$$ = $$\frac{116}{15}$$
• $$\frac{116}{15}$$ = $$7\frac{11}{15}$$
• $$2\frac{1}{3}$$ + $$5 \frac{2}{5}$$ =$$7 (\frac{1}{3} + \frac{2}{5})$$
• $$2\frac{1}{3}$$ + $$5 \frac{2}{5}$$ = $$7 (\frac{5}{15} + \frac{6}{15})$$
• $$2\frac{1}{3}$$ + $$5 \frac{2}{5}$$ = $$7 \frac{11}{15}$$
• $$7 \frac{11}{15}$$ = $$\frac{15x7 +11}{15}$$ = $$\frac{116}{15}$$