We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Guides d'étude > Prealgebra

Simplifying and Evaluating Complex Fractions

Learning Outcomes

  • Simplify complex fractions that contain several different mathematical operations

In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,

3458=34÷58\frac{\frac{3}{4}}{\frac{5}{8}}=\frac{3}{4}\div \frac{5}{8} Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.

Simplify complex fractions

  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Divide the numerator by the denominator.
  4. Simplify if possible.

Example

Simplify: (12)24+32\frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}. Solution:
(12)24+32\frac{{\left(\frac{1}{2}\right)}^{2}}{4+{3}^{2}}
Simplify the numerator. 144+32\frac{\frac{1}{4}}{4+{3}^{2}}
Simplify the term with the exponent in the denominator. 144+9\frac{\frac{1}{4}}{4+9}
Add the terms in the denominator. 1413\frac{\frac{1}{4}}{13}
Divide the numerator by the denominator. 14÷13\frac{1}{4}\div 13
Rewrite as multiplication by the reciprocal. 14⋅113\frac{1}{4}\cdot \frac{1}{13}
Multiply. 152\frac{1}{52}

Try It

#146431 [ohm_question]146431[/ohm_question]

Example

Simplify: 12+2334−16\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}.

Answer: Solution:

12+2334−16\frac{\frac{1}{2}+\frac{2}{3}}{\frac{3}{4}-\frac{1}{6}}
Rewrite numerator with the LCD of 66 and denominator with LCD of 1212. 36+46912−212\frac{\frac{3}{6}+\frac{4}{6}}{\frac{9}{12}-\frac{2}{12}}
Add in the numerator. Subtract in the denominator. 76712\frac{\frac{7}{6}}{\frac{7}{12}}
Divide the numerator by the denominator. 76÷712\frac{7}{6}\div \frac{7}{12}
Rewrite as multiplication by the reciprocal. 76⋅127\frac{7}{6}\cdot \frac{12}{7}
Rewrite, showing common factors. )7‾⋅)6‾⋅2)6‾⋅)7‾⋅1\frac{\overline{)7}\cdot \overline{)6}\cdot 2}{\overline{)6}\cdot \overline{)7}\cdot 1}
Simplify. 22

Try It

#42079 [ohm_question]42079[/ohm_question] #146435 [ohm_question]146435[/ohm_question]
In the following video we sow more examples of simplifying complex expressions. https://youtu.be/lQCwze2w7OU

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Example

Evaluate x+13x+\frac{1}{3} when
  1. x=−13x=-\frac{1}{3}
  2. x=−34x=-\frac{3}{4}
Solution 1. To evaluate x+13x+\frac{1}{3} when x=−13x=-\frac{1}{3}, substitute −13-\frac{1}{3} for xx in the expression.
x+13x+\frac{1}{3}
Substitute −−13\color{red}{--\frac{1}{3}} for x. −−13+13\color{red}{--\frac{1}{3}} + \frac{1}{3}
Simplify. 00
2. To evaluate x+13x+\frac{1}{3} when x=−34x=-\frac{3}{4}, we substitute −34-\frac{3}{4} for xx in the expression.
x+13x+\frac{1}{3}
Substitute −−34\color{red}{--\frac{3}{4}} for x. −−34+13\color{red}{--\frac{3}{4}} + \frac{1}{3}
Rewrite as equivalent fractions with the LCD, 1212. −3⋅34⋅3+1⋅43⋅4-\frac{3\cdot 3}{4\cdot 3}+\frac{1\cdot 4}{3\cdot 4}
Simplify the numerators and denominators. −912+412-\frac{9}{12}+\frac{4}{12}
Add. −512-\frac{5}{12}

Try It

#146049 [ohm_question]146049[/ohm_question] #146090 [ohm_question]146090[/ohm_question]

Example

Evaluate y−56y-\frac{5}{6} when y=−23y=-\frac{2}{3}

Answer: Solution: We substitute −23-\frac{2}{3} for yy in the expression.

y−56y-\frac{5}{6}
Substitute −−23\color{red}{--\frac{2}{3}} for y. −−23−56\color{red}{--\frac{2}{3}} - \frac{5}{6}
Rewrite as equivalent fractions with the LCD, 66. −46−56-\frac{4}{6}-\frac{5}{6}
Subtract. −96-\frac{9}{6}
Simplify. −32-\frac{3}{2}

Try It

#146320 [ohm_question]146320[/ohm_question]

Example

Evaluate 2x2y2{x}^{2}y when x=14x=\frac{1}{4} and y=−23y=-\frac{2}{3}

Answer: Solution: Substitute the values into the expression. In 2x2y2{x}^{2}y, the exponent applies only to xx.

2x2y2{x}^{2}y
Substitute 14\color{red}{\frac{1}{4}} for x and −−23\color{blue}{--\frac{2}{3}} for y. 2(14)2(−−23)2(\color{red}{\frac{1}{4}})^{2}(\color{blue}{--\frac{2}{3}})
Simplify exponents first. 2(116)(−−23)2(\frac{1}{16})(--\frac{2}{3})
Multiply. The product will be negative. −−21⋅116⋅23--\frac{2}{1}\cdot\frac{1}{16}\cdot\frac{2}{3}
Simplify. −−448--\frac{4}{48}
Remove the common factors. −−1⋅44⋅ 12--\frac{1\cdot\color{red}{4}}{\color{red}{4} \cdot\ 12}
Simplify. −−112--\frac{1}{12}

try It

#146096 [ohm_question]146096[/ohm_question] #146093 [ohm_question]146093[/ohm_question]

Example

Evaluate p+qr\frac{p+q}{r} when p=−4,q=−2p=-4,q=-2, and r=8r=8

Answer: Solution: We substitute the values into the expression and simplify.

p+qr\frac{p+q}{r}
Substitute −−4\color{red}{--4} for p, −−2\color{blue}{--2} for q, and 8\color{red}{8} for r. −−4+(−−2)8\frac{\color{red}{--4} + \color{blue}{(--2)}}{\color{red}{8}}
Add in the numerator first. −68-\frac{6}{8}
Simplify. −34-\frac{3}{4}

Try It

#146106 [ohm_question]146106[/ohm_question] #146108 [ohm_question]146108[/ohm_question]

Licenses & Attributions