Simplify with Fractions

Solution to Example 1
There may be several ways to reduce (or simplify) the given fractions. In these examples, we apply the 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:
\( \quad = \dfrac{5}{3} - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{3}{5} \)
Use associativity to write the above as:
\( \quad = \left(\dfrac{5}{3} - \dfrac{1}{3}\right) + \left(\dfrac{1}{5} - \dfrac{3}{5}\right) \)
Add and subtract the fractions inside the brackets:
\( \quad = \dfrac{4}{3} - \dfrac{2}{5} \)
Rewrite with a common denominator:
\( \quad = \dfrac{4}{3} \times \dfrac{5}{5} - \dfrac{2}{5} \times \dfrac{3}{3} = \dfrac{20}{15} - \dfrac{6}{15} \)
Subtract the fractions:
\( \quad = \dfrac{14}{15} \)

b)
Given: \( \dfrac{1}{3} \left( \dfrac{x}{2} + \dfrac{1}{8} \right) - \dfrac{x}{6} \)
Use distributivity to write:
\( \quad = \dfrac{1}{3} \times \dfrac{x}{2} + \dfrac{1}{3} \times \dfrac{1}{8} - \dfrac{x}{6} = \dfrac{x}{6} + \dfrac{1}{24} - \dfrac{x}{6} \)
Use commutativity:
\( \quad = \dfrac{x}{6} - \dfrac{x}{6} + \dfrac{1}{24} \)
Use associativity:
\( \quad = \left(\dfrac{x}{6} - \dfrac{x}{6}\right) + \dfrac{1}{24} = 0 + \dfrac{1}{24} = \dfrac{1}{24} \)

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 \( \dfrac{1}{3} \):
\( \quad = \dfrac{1}{3} \left[ \left( \dfrac{9}{2} - \dfrac{3}{8} \right) - \left( - \dfrac{3}{8} + \dfrac{5}{2} \right) \right] \)
Remove parentheses and distribute the negative sign:
\( \quad = \dfrac{1}{3} \left( \dfrac{9}{2} - \dfrac{3}{8} + \dfrac{3}{8} - \dfrac{5}{2} \right) \)
Use commutativity and associativity:
\( \quad = \dfrac{1}{3} \left[ \left(\dfrac{9}{2} - \dfrac{5}{2}\right) + \left(\dfrac{3}{8} - \dfrac{3}{8}\right) \right] \)
Subtract fractions inside brackets:
\( \quad = \dfrac{1}{3} \left(\dfrac{4}{2} + 0\right) = \dfrac{1}{3} \times \dfrac{2}{1} = \dfrac{2}{3} \)

d)
Given: \( \dfrac{x}{2} \left( \dfrac{1}{x} + \dfrac{3}{2x} \right) \) for \( x \ne 0 \)
Use distributivity:
\( \quad = \dfrac{x}{2} \times \dfrac{1}{x} + \dfrac{x}{2} \times \dfrac{3}{2x} = \dfrac{x}{2x} + \dfrac{3x}{4x} \)
Reduce fractions by canceling \( x \):
\( \quad = \dfrac{1}{2} + \dfrac{3}{4} \)
Rewrite with common denominator 4:
\( \quad = \dfrac{1}{2} \times \dfrac{2}{2} + \dfrac{3}{4} = \dfrac{2}{4} + \dfrac{3}{4} = \dfrac{5}{4} \)

Solution to Example 2

a)
Use distributivity to expand:
\( \quad = \dfrac{1}{3} \times \dfrac{x}{2} - \dfrac{1}{3} \times \dfrac{1}{2} + \dfrac{1}{2} \times \dfrac{2x}{3} - \dfrac{1}{2} \times \dfrac{4}{3} \)
Multiply fractions and simplify:
\( \quad = \dfrac{x}{6} - \dfrac{1}{6} + \dfrac{2x}{6} - \dfrac{4}{6} \)
Combine over common denominator:
\( \quad = \dfrac{x + 2x - 1 - 4}{6} = \dfrac{3x - 5}{6} \)

b)
Use distributivity to expand:
\( \quad = - \dfrac{1}{2} \times \dfrac{1}{5} - \dfrac{1}{2} \times \left(- \dfrac{x}{5}\right) + \dfrac{1}{5} \times \dfrac{3x}{2} + \dfrac{1}{5} \times \left(- \dfrac{3}{2}\right) \)
Multiply fractions and simplify:
\( \quad = - \dfrac{1}{10} + \dfrac{x}{10} + \dfrac{3x}{10} - \dfrac{3}{10} \)
Combine over common denominator:
\( \quad = \dfrac{-1 + x + 3x - 3}{10} = \dfrac{4x - 4}{10} \)
Factor numerator and denominator and reduce:
\( \quad = \dfrac{2(2x - 2)}{2 \times 5} = \dfrac{2x - 2}{5} \)