Adding and Subtracting Fractions and Mixed Numbers

- - Pre-Algebra

Before learning how to add and subtract fractions and mixed numbers, let us recall some terminologies like similar fractions, dissimilar fractions, proper fraction, improper fraction and mixed number. Similar fractions are fractions with like or the same denominator while dissimilar fractions are those with unlike or different denominators.

A proper fraction is a fraction whose numerator is less than its denominator while an improper fraction is a fraction whose numerator is greater than its denominator. Lastly, a mixed number contains a whole number and a proper fraction. The rules for adding and subtracting fractions would greatly depend on the kind of fractions to be added or subtracted.

Listed below are the different rules for adding or subtracting fractions and mixed numbers:

a. To add or subtract two similar fractions (proper or improper), add or subtract their numerators then copy the common denominator. Simplify the result.

Examples:

$1.\;\; \frac{2}{5}+\frac{1}{5}=\frac{2+1}{5}=\frac{3}{5}$
$2.\;\; \frac{3}{9}-\frac{1}{9}=\frac{3-1}{9}=\frac{2}{9}$
$3.\;\; \frac{12}{11}-\frac{3}{11}=\frac{12-3}{11}=\frac{9}{11}$

b. To add or subtract two dissimilar fractions, change the given fractions into similar fractions by using the least common multiple of the denominators as the common denominator. To convert, divide the least common denominator (LCD) by the denominator of the given fraction then multiply the result to its numerator. Then, follow rule 1.

Examples:

$1.\;\; \frac{1}{2}+\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{3+2}{6}=\frac{5}{6}$
$2.\;\; \frac{3}{4}-\frac{2}{8}=\frac{6}{8}-\frac{2}{8}=\frac{6-2}{8}=\frac{4}{8}=\frac{1}{2}$

c. To add or subtract a fraction and a mixed number whose proper fraction is similar to the given fraction, add or subtract the numerators of the proper fractions and prefix the whole number then copy the common denominator. In case of subtraction, if the numerator of the subtrahend is lesser than the numerator of the mixed number minuend, apply borrowing technique. After which, follow the rule for adding/subtracting similar fractions.

Examples:

$1.\;\; 1\frac{2}{3}+\frac{1}{3}=1+\frac{2+1}{3}=1+\frac{3}{3}=2$
$2.\;\; 2\frac{1}{4}-\frac{2}{4}=1\frac{5}{4}-\frac{2}{4}=1+\left ( \frac{5}{4}-\frac{2}{4} \right)=1\frac{3}{4}$
$3.\;\; 3\frac{5}{6}-\frac{2}{6}=3+\left ( \frac{5}{6}-\frac{2}{6} \right )=3\frac{3}{6}=3\frac{1}{2}$

d. To add or subtract a fraction and a mixed number whose proper fraction is dissimilar with the given fraction, convert the mixed number into an improper fraction and later convert it into a similar fraction. Follow rule number 1 and simplify the result.

Examples:

$1.\;\; 2\frac{2}{5}-\frac{1}{3}=\frac{12}{5}-\frac{1}{3}=\frac{36}{15}-\frac{5}{15}=\frac{31}{15}=2\frac{1}{5}$
$2.\;\; 2\frac{1}{5}+\frac{1}{3}=\frac{11}{5}+\frac{1}{3}=\frac{33}{15}+\frac{5}{15}=\frac{28}{15}=1\frac{13}{15}$
$3.\;\; 2\frac{3}{4}-\frac{5}{3}=\frac{11}{4}-\frac{5}{3}=\frac{33}{12}-\frac{20}{12}=\frac{13}{12}=1\frac{1}{12}$

e. To add or subtract two mixed numbers with similar proper fractions, add the whole numbers and their corresponding proper fractions then combine the result. In case, the numerator of the proper fraction of the subtrahend is smaller than the numerator of the minuend, apply borrowing technique. Simplify the final sum.

Examples:

$1.\;\; 2\frac{1}{3}+3\frac{2}{3}=\left ( 2+3 \right )+\left (\frac{1}{3}+\frac{2}{3} \right )=5+\frac{3}{3}=6$
$2.\;\; 3\frac{3}{4}-1\frac{1}{4}=\left ( 3-1 \right )+\left (\frac{3-1}{4}\right )=2\frac{2}{4}=2\frac{1}{2}$
$3.\;\; 3\frac{2}{4}-1\frac{3}{4}=2\frac{6}{4}-1\frac{3}{4}=\left (2-1 \right)+\left (\frac{6-3}{4}\right )=1\frac{3}{4}$

f. To add or subtract two mixed numbers with dissimilar proper fractions, change them into improper fractions and make sure that their denominators are the same before adding or subtracting.

Examples:

$1.\;\; 2\frac{1}{3}+3\frac{2}{3}=\left (2+3 \right)+\left (\frac{1}{3}+\frac{2}{3}\right )=5+\frac{3}{3}=6$
$2.\;\; 3\frac{1}{2}+1\frac{1}{3}=\frac{7}{2}+\frac{5}{3}=\frac{21}{6}+\frac{10}{6}=\frac{31}{6}=5\frac{1}{6}$
$2.\;\; 3\frac{1}{2}-2\frac{1}{3}=\frac{7}{2}-\frac{7}{3}=\frac{21}{6}-\frac{14}{6}=\frac{7}{6}=1\frac{1}{6}$