Complex Fractions

Show your work for each problem.

1.
Which of the following is a complex fraction?
1. a) $\frac{3}{4}$
2. b) $3 + \frac{1}{2}$
3. c) $\frac{\frac{1}{2}}{\frac{3}{4}}$
4. d) $\frac{5}{x}$
Answer: $\frac{\frac{1}{2}}{\frac{3}{4}}$
$\frac{\frac{1}{2}}{\frac{3}{4}}$ is a complex fraction because both the numerator ( $\frac{1}{2}$ ) and denominator ( $\frac{3}{4}$ ) are fractions.
2.
Simplify: $\frac{\frac{1}{2}}{\frac{1}{4}}$
Answer: 2
$\frac{\frac{1}{2}}{\frac{1}{4}} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$
3.
Simplify: $\frac{\frac{3}{5}}{\frac{9}{10}}$
1. a) $\frac{3}{2}$
2. b) $\frac{27}{50}$
3. c) $\frac{2}{3}$
4. d) $\frac{1}{3}$
Answer: $\frac{2}{3}$
$\frac{\frac{3}{5}}{\frac{9}{10}} = \frac{3}{5} \times \frac{10}{9} = \frac{30}{45} = \frac{2}{3}$
4.
Simplify: $\frac{\frac{2}{3}}{\frac{4}{9}}$
Answer: 3/2
1. What is the reciprocal of $\frac{4}{9}$ ? 9/4
2. Multiply: $\frac{2}{3} \times \frac{9}{4}$ = ? 18/12
3. $\frac{18}{12}$ in lowest terms is $\frac{\square}{2}$ . What number goes in the box? 3
5.
Simplify: $\frac{6}{\frac{3}{4}}$
Answer: 8
$\frac{6}{\frac{3}{4}} = 6 \times \frac{4}{3} = \frac{24}{3} = 8$
6.
Simplify: $\frac{\frac{1}{2} + \frac{1}{4}}{\frac{3}{4}}$
Answer: 1
1. First, add the fractions in the numerator. $\frac{1}{2} + \frac{1}{4}$ = ? 3/4
2. Now simplify $\frac{\frac{3}{4}}{\frac{3}{4}}$ 1
7.
Simplify: $\frac{\frac{1}{3} - \frac{1}{6}}{\frac{1}{2}}$
1. a) $\frac{2}{3}$
2. b) $1$
3. c) $\frac{1}{3}$
4. d) $\frac{1}{6}$
Answer: $\frac{1}{3}$
$\frac{1}{3} - \frac{1}{6} = \frac{1}{6}$ , then $\frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{6} \times 2 = \frac{2}{6} = \frac{1}{3}$
8.
Calculate the value: $\frac{\frac{2}{5}}{\frac{1}{5} + \frac{1}{10}}$
Answer: 4/3
$\frac{1}{5} + \frac{1}{10} = \frac{2}{10} + \frac{1}{10} = \frac{3}{10}$ . Then $\frac{\frac{2}{5}}{\frac{3}{10}} = \frac{2}{5} \times \frac{10}{3} = \frac{20}{15} = \frac{4}{3}$
9.
Simplify using the LCD method: $\frac{\frac{1}{2} + \frac{1}{3}}{\frac{1}{6}}$
Answer: 5
1. What is the LCD of 2, 3, and 6? 6
2. Multiply numerator by 6: $(\frac{1}{2} + \frac{1}{3}) \cdot 6$ = ? 5
3. Multiply denominator by 6: $\frac{1}{6} \cdot 6$ = ? 1
4. Final answer: $\frac{5}{1}$ = ? 5
10.
Simplify: $\frac{\frac{x}{2}}{\frac{x}{4}}$
1. a) $\frac{1}{2}$
2. b) $\frac{x}{2}$
3. c) $2$
4. d) $\frac{x^2}{8}$
Answer: $2$
$\frac{\frac{x}{2}}{\frac{x}{4}} = \frac{x}{2} \times \frac{4}{x} = \frac{4x}{2x} = 2$ (where $x \neq 0$ )
11.
Simplify: $\frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}$
Answer: (x+1)/(x-1)
1. What is the LCD of all fractions in this expression? x
2. Multiply the numerator by $x$ : $(1 + \frac{1}{x}) \cdot x$ = ? x+1
3. Multiply the denominator by $x$ : $(1 - \frac{1}{x}) \cdot x$ = ? x-1
4. The result is $\frac{x+1}{x-1}$ . Besides $x = 0$ , which value of $x$ is not allowed? 1
12.
Rewrite $\frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{xy}}$ (with $x, y \neq 0$ ) without any fractions. The result is $y - \square$ . What goes in the box?
Answer: x
Multiply by $xy$ : $\frac{\left(\frac{1}{x} - \frac{1}{y}\right) \cdot xy}{\frac{1}{xy} \cdot xy} = \frac{y - x}{1} = y - x$ , so the box holds $x$ .
13.
Simplify: $\frac{\frac{2}{x} - \frac{1}{x^2}}{\frac{3}{x}}$
1. a) $\frac{2x + 1}{3x}$
2. b) $\frac{2x - 1}{3x}$
3. c) $\frac{1}{3}$
4. d) $\frac{2 - x}{3}$
Answer: $\frac{2x - 1}{3x}$
Multiply by $x^2$ : Numerator: $\frac{2}{x} \cdot x^2 - \frac{1}{x^2} \cdot x^2 = 2x - 1$ . Denominator: $\frac{3}{x} \cdot x^2 = 3x$ . Answer: $\frac{2x - 1}{3x}$
14.
Two resistors of 6 ohms and 12 ohms are in parallel. Find the total resistance using $R_T = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$
Answer: 4
1. What is $\frac{1}{6}$ as a fraction with denominator 12? 2/12
2. Add $\frac{2}{12} + \frac{1}{12}$ = ? 3/12
3. $\frac{3}{12}$ in lowest terms is $\frac{1}{\square}$ . What number goes in the box? 4
4. Find $\frac{1}{\frac{1}{4}}$ = ? 4
15.
Write $\frac{\frac{a}{b}}{\frac{a^2}{b^2}}$ (with $a, b \neq 0$ ) as a single fraction. The result is $\frac{b}{\square}$ . What goes in the box?
Answer: a
$\frac{\frac{a}{b}}{\frac{a^2}{b^2}} = \frac{a}{b} \times \frac{b^2}{a^2} = \frac{ab^2}{ba^2} = \frac{b}{a}$ (where $a, b \neq 0$ )
16.
Simplify: $\frac{\frac{1}{x+1} + \frac{1}{x-1}}{\frac{2}{x^2-1}}$
Answer: x
1. Note that $x^2 - 1 = (x+1)(x-1)$ . What is the LCD? Write it as a product of two factors, like (x+2)(x-3). (x+1)(x-1)
2. Add the fractions in the numerator with LCD $(x+1)(x-1)$ : $\frac{(x-1) + (x+1)}{(x+1)(x-1)} = \frac{\square}{(x+1)(x-1)}$ . What goes in the box? 2x
3. Now divide: $\frac{\frac{2x}{x^2-1}}{\frac{2}{x^2-1}}$ = ? x