Fractions: Types, Addition, and Subtraction

Fractions are generally used to define any whole number into equal parts. While writing a fraction, there are two numbers involved. The number at the top is called the numerator, while the one at the bottom is called the denominator. There are three types of fractions:

  • Proper Fractions: In proper fractions, the denominator is greater than the numerator. For example: \(\frac{3}{7}, \frac{1}{3}\)
  • Improper Fractions: As the name suggests, these fractions are “top-heavy” where the numerator is greater than the denominator. For example: \(\frac{7}{3}, \frac{5}{2}\)
  • Mixed Fractions: Mixed fractions are another type of improper fraction where there is a whole number as well as a fractional part. For example: \(1 \frac{1}{3}, 2 \frac{3}{7}\)

Addition and Subtraction of Fractions

To add or subtract fractions, you need to follow certain criteria. First, you need to distinguish between like and unlike fractions.

Addition and Subtraction with Like Fractions

Two fractions whose denominators are the same are called like fractions. To add or subtract like fractions, simply add or subtract the numerators, and then write the result over the common denominator.

Example: \(\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}\)

So, for example: \(\frac{2}{4} + \frac{3}{4} = \frac{5}{4}\)

Addition and Subtraction with Unlike Fractions

For unlike fractions, or fractions with different denominators, perform the operation as follows:

Example: \(\frac{a}{d} + \frac{c}{b} = \frac{a \cdot b + c \cdot d}{b \cdot d}\)

Also, \(\frac{a}{d} – \frac{c}{b} = \frac{a \cdot b – c \cdot d}{b \cdot d}\)

So, for example: \(\frac{2}{5} – \frac{1}{6} = \frac{2 \cdot 6 – 1 \cdot 5}{5 \cdot 6} = \frac{12 – 5}{30} = \frac{7}{30}\)

Finally, you may have a fraction that can be reduced to a simpler form. It’s always best to reduce to the simplest form when possible.

Exercises for Add or Subtract Fractions

  1. \(\frac{7}{4} + \frac{5}{6} = \)
  2. \(\frac{8}{10} + \frac{2}{3} = \)
  3. \(\frac{9}{8} + \frac{8}{7} = \)
  4. \(\frac{5}{3} – \frac{3}{5} = \)
  5. \(\frac{6}{3} + \frac{8}{3} = \)
  6. \(\frac{5}{7} – \frac{5}{7} = \)
  7. \(\frac{3}{7} – \frac{8}{5} = \)
  8. \(\frac{9}{8} – \frac{3}{5} = \)
  9. \(\frac{7}{11} + \frac{16}{17} = \)
  10. \(\frac{8}{9} + \frac{15}{10} = \)

Answers

    1. \(\frac{7}{4} + \frac{5}{6} = \frac{7 \times 6 + 5 \times 4}{4 \times 6} = \frac{62}{24} = \frac{31}{12}\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{7}{4} + \frac{5}{6} = \frac{7 \times 6 + 5 \times 4}{4 \times 6} = \frac{62}{24} = \frac{31}{12}\)

      Then, simplify the result:

      \(\frac{31}{12}\)

    2. \(\frac{8}{14} + \frac{2}{5} = \frac{8 \times 5 + 2 \times 14}{14 \times 5} = \frac{44}{70} = \frac{22}{35}\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{8}{14} + \frac{2}{5} = \frac{8 \times 5 + 2 \times 14}{14 \times 5} = \frac{44}{70} = \frac{22}{35}\)

      Then, simplify the result:

      \(\frac{22}{35}\)

    3. \(\frac{9}{8} + \frac{7}{8} = \frac{9 \times 2 + 8 \times 8}{8 \times 2} = \frac{82}{16} = \frac{41}{8}\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{9}{8} + \frac{7}{8} = \frac{9 \times 2 + 8 \times 8}{8 \times 2} = \frac{82}{16} = \frac{41}{8}\)

      Then, simplify the result:

      \(\frac{41}{8}\)

    4. \(\frac{5}{3} – \frac{3}{5} = \frac{5 \times 5 – 3 \times 3}{3 \times 5} = \frac{10}{15} = \frac{2}{3}\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{5}{3} – \frac{3}{5} = \frac{5 \times 5 – 3 \times 3}{3 \times 5} = \frac{10}{15} = \frac{2}{3}\)

      Then, simplify the result:

      \(\frac{2}{3}\)

    5. \(\frac{6}{8} – \frac{3}{8} = \frac{6 \times 8 – 8 \times 3}{8 \times 8} = \frac{24}{24} = 1\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{6}{8} – \frac{3}{8} = \frac{6 \times 8 – 8 \times 3}{8 \times 8} = \frac{24}{24}\)

      Then, simplify the result:

      \(1\)

    6. \(\frac{5}{1} – \frac{5}{4} = \frac{5 \times 4 – 5 \times 1}{1 \times 4} = \frac{15}{4}\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{5}{1} – \frac{5}{4} = \frac{5 \times 4 – 5 \times 1}{1 \times 4} = \frac{15}{4}\)

    7. \(\frac{3}{7} – \frac{7}{8} = \frac{3 \times 8 – 7 \times 2}{2 \times 8} = \frac{10}{16} = \frac{5}{8}\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{3}{7} – \frac{7}{8} = \frac{3 \times 8 – 7 \times 2}{2 \times 8} = \frac{10}{16}\)

      Then, simplify the result:

      \(\frac{10}{16} = \frac{5}{8}\)

    8. \(\frac{9}{3} – \frac{3}{5} = \frac{9 \times 5 – 3 \times 3}{3 \times 5} = \frac{36}{15} = \frac{12}{5}\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{9}{3} – \frac{3}{5} = \frac{9 \times 5 – 3 \times 3}{3 \times 5} = \frac{36}{15}\)

      Then, simplify the result:

      \(\frac{36}{15} = \frac{12}{5}\)

    9. \(\frac{7}{11} + \frac{16}{17} = \frac{7 \times 17 + 16 \times 11}{11 \times 17} = \frac{295}{187}\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{7}{11} + \frac{16}{17} = \frac{7 \times 17 + 16 \times 11}{11 \times 17} = \frac{295}{187}\)

    10. \(\frac{8}{9} + \frac{15}{10} = \frac{8 \times 10 + 15 \times 9}{9 \times 10} = \frac{215}{90} = \frac{43}{18}\)Solution:

      For “unlike” fractions, the first step is to find the same denominator before you can add or subtract fractions with different denominators.

      To find the same denominator, multiply both denominators, and each numerator by the denominator of the other fraction:

      \(\frac{8}{9} + \frac{15}{10} = \frac{8 \times 10 + 15 \times 9}{9 \times 10} = \frac{215}{90}\)

      Then, simplify the result:

      \(\frac{215}{90} = \frac{43}{18}\)