We present examples on how to use the properties of commutativity, associativity and distributivity and the different rules of fractions to simplify expressions including fractions.
More questions and their answers are also included. Fractions with variables are also included.
Do NOT use the calculator to answer the questions.
Example 1
Write as a single fraction in reduced (simplified) form if possible. Justify your steps.
a) \( \quad \dfrac{5}{3} + \dfrac{1}{5} - \dfrac{1}{3} - \dfrac{3}{5} \)
b) \( \quad \dfrac{1}{3} \left( \dfrac{x}{2} + \dfrac{1}{8} \right) - \dfrac{x}{6} \)
c) \( \quad \dfrac{1}{3} \left( \dfrac{9}{2} - \dfrac{3}{8} \right) - \dfrac{1}{3} \left( - \dfrac{3}{8} + \dfrac{5}{2} \right) \)
d) \( \quad \dfrac{x}{2} \left( \dfrac{1}{x} + \dfrac{3}{2x} \right) \) for \( x \ne 0 \)
Solution to Example 1
There may be several ways to reduce (or simplify) the given fractions. In these examples, we apply the above properties to reduce the given fractions in order to explain the use of these properties.
a)
Given: \( \dfrac{5}{3} + \dfrac{1}{5} - \dfrac{1}{3} - \dfrac{3}{5} \)
Use commutativity of addition to write
\( \quad\quad \dfrac{5}{3} + \dfrac{1}{5} - \dfrac{1}{3} - \dfrac{3}{5} - \dfrac{3}{4} = \dfrac{5}{3} - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{3}{5} \)
Use associativity to write the above as
\( \quad\quad = (\dfrac{5}{3} - \dfrac{1}{3} ) + (\dfrac{1}{5} - \dfrac{3}{5}) \)
Add and subtract the fractions inside the brackets
\( \quad\quad = \dfrac{4}{3} - \dfrac{2}{5} \)
Rewrite with common denominator
\( \quad\quad = \dfrac{4}{3} \times \dfrac{5}{5} - \dfrac{2}{5} \times \dfrac{3}{3} \)
Simplify
\( \quad\quad = \dfrac{20}{15} - \dfrac{6}{15} \)
Subtract the fractions
\( \quad\quad = \dfrac{14}{15} \)
b)
Given: \( \dfrac{1}{3} \left( \dfrac{x}{2} + \dfrac{1}{8} \right) - \dfrac{x}{6} \)
Use distributivity to write
\( \quad\quad \dfrac{1}{3} \left( \dfrac{x}{2} + \dfrac{1}{8} \right) - \dfrac{x}{6} = \dfrac{1}{3} \times \dfrac{x}{2} + \dfrac{1}{3} \times \dfrac{1}{8} - \dfrac{x}{6} \)
Simplify
\( \quad\quad = \dfrac{x}{6} + \dfrac{1}{24} - \dfrac{x}{6} \)
Use commutativity to write the above as
\( \quad\quad = \dfrac{x}{6} - \dfrac{x}{6} + \dfrac{1}{24} \)
Use associativity to write the above as
\( \quad\quad = (\dfrac{x}{6} - \dfrac{x}{6}) + \dfrac{1}{24} \)
Simplify
\( \quad\quad = 0 + \dfrac{1}{24} \)
Simplify
\( \quad\quad = \dfrac{1}{24} \)
c)
Given: \( \dfrac{1}{3} \left( \dfrac{9}{2} - \dfrac{3}{8} \right) - \dfrac{1}{3} \left( - \dfrac{3}{8} + \dfrac{5}{2} \right) \)
Use distributivity ( from right to left) to factor out the fraction \( \dfrac{1}{3} \).
\( \quad\quad = \dfrac{1}{3} \left( \left( \dfrac{9}{2} - \dfrac{3}{8} \right) - \left( - \dfrac{3}{8} + \dfrac{5}{2} \right) \right) \)
Use distibutivity to write the above as
\( \quad\quad = \dfrac{1}{3} \left( \dfrac{9}{2} - \dfrac{3}{8} + \dfrac{3}{8} - \dfrac{5}{2} \right) \)
Use commutativity to write the above as
\( \quad\quad = \dfrac{1}{3} \left( \dfrac{9}{2} - \dfrac{5}{2} + \dfrac{3}{8} - \dfrac{3}{8} \right) \)
Use associativity to write the above as
\( \quad\quad = \dfrac{1}{3} \left( (\dfrac{9}{2} - \dfrac{5}{2}) + (\dfrac{3}{8} - \dfrac{3}{8}) \right) \)
Subtract fractions inside brackets
\( \quad\quad = \dfrac{1}{3} (\dfrac{4}{2} + 0) \)
Reduce the fraction \( \dfrac{4}{2} \) to \( \dfrac{2}{1} \)
\( \quad\quad = \dfrac{1}{3} \times \dfrac{2}{1} \)
Multiply fractions and simplify
\( = \dfrac{2}{3} \)
d)
Given: \( \dfrac{x}{2} \left( \dfrac{1}{x} + \dfrac{3}{2x} \right) \)
Use distributivity to write
\( \quad\quad = \dfrac{x}{2} \left( \dfrac{1}{x} + \dfrac{3}{2x} \right) = \dfrac{x}{2} \times \dfrac{1}{x} + \dfrac{x}{2} \times \dfrac{3}{2x} \)
Multiply the fractions in the above expression
\( \quad\quad = \dfrac{x}{2 x} + \dfrac{3 x}{4 x} \)
\( x \) is a common factor to both numerator and denominator and therefor the fractions may be reduced
\( \quad\quad = \dfrac{1}{2} + \dfrac{3 }{4 } \)
Rewrite the fraction \( \dfrac{1}{2} \) with denominator \( 4 \) as follows
\( \quad\quad = \dfrac{1}{2} \times \dfrac{2}{2} + \dfrac{3 }{4 } \)
Simplify
\( \quad\quad = \dfrac{2}{4} + \dfrac{3 }{4 } \)
Add the fractions and simplify
\( \quad\quad = \dfrac{5}{4} \)
Example 2
Expand and simplify the following expressions.
a) \( \quad \dfrac{1}{3} ( \dfrac{x}{2} - \dfrac{1}{2} ) + \dfrac{1}{2} ( \dfrac{2 x}{3} - \dfrac{4}{3} ) \)
b) \( \quad - \dfrac{1}{2} ( \dfrac{1}{5} - \dfrac{x}{5} ) + \dfrac{1}{5} ( \dfrac{3 x}{2} - \dfrac{3}{2} ) \)
Solution to Example 2
a)
Given: \( \quad \dfrac{1}{3} ( \dfrac{x}{2} - \dfrac{1}{2} ) + \dfrac{1}{2} ( \dfrac{2 x}{3} - \dfrac{4}{3} ) \)
Use distributivity to expand the given expressions
\( \quad = \dfrac{1}{3} \times \dfrac{x}{2} - \dfrac{1}{3} \times \dfrac{1}{2} + \dfrac{1}{2} \times \dfrac{2 x}{3} - \dfrac{1}{2} \times \dfrac{4}{3} \)
Multiply fractions and simplify
\( \quad = \dfrac{x}{6} - \dfrac{1}{6} + \dfrac{2x}{6} - \dfrac{4}{6} \)
The fractions have a common denominator and therefore the above may be written as
\( \quad = \dfrac{x + 2x - 1 - 4}{6} \)
Simplify
\( \quad = \dfrac{3x - 5}{6} \)
b)
Given: \( \quad - \dfrac{1}{2} ( \dfrac{1}{5} - \dfrac{x}{5} ) + \dfrac{1}{5} ( \dfrac{3 x}{2} - \dfrac{3}{2}) \)
Use distributivity to expand the given expressions
\( \quad = - \dfrac{1}{2} \times \dfrac{1}{5} - \dfrac{1}{2} \times (- \dfrac{x}{5}) + \dfrac{1}{5} \times \dfrac{3 x}{2} + \dfrac{1}{5} \times (- \dfrac{3}{2} ) \)
Multiply fractions and simplify
\( \quad = - \dfrac{1}{10} + \dfrac{x}{10} + \dfrac{3x }{10} - \dfrac{3}{10} \)
The fractions in the above expression have a common denominator and therefore the above may be written as
\( \quad = \dfrac{-1 + x + 3x - 3}{6} \)
Simplify
\( \quad = \dfrac{4x - 4}{10} \)
Factor numerator and denominator
\( \quad = \dfrac{2(2x - 2)}{2 \times 5} \)
Reduce
\( = \dfrac{2x - 2}{5} \)
Write as a single fraction in reduced (simplified) form if possible.
