<< Chapter < Page | Chapter >> Page > |
For any positive numbers $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b,$
Which of the following fractions are equivalent to $\frac{7}{\mathrm{-8}}?$
The quotient of a positive and a negative is a negative, so $\frac{7}{\mathrm{-8}}$ is negative. Of the fractions listed, $\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-7}}{8}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-\frac{7}{8}$ are also negative.
Which of the following fractions are equivalent to $\frac{\mathrm{-3}}{\phantom{\rule{0.2em}{0ex}}5}?$
$\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-3}}{\mathrm{-5}},\text{}\phantom{\rule{0.2em}{0ex}}\frac{3}{5},-\frac{3}{5},\frac{\phantom{\rule{0.2em}{0ex}}3}{\mathrm{-5}}$
$\phantom{\rule{0.2em}{0ex}}-\frac{3}{5},\frac{\phantom{\rule{0.2em}{0ex}}3}{\mathrm{-5}}$
Which of the following fractions are equivalent to $-\frac{2}{7}?$
$\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-2}}{\mathrm{-7}},\frac{\mathrm{-2}}{7},\frac{2}{7},\frac{2}{\mathrm{-7}}$
$\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-2}}{7},\frac{2}{\mathrm{-7}}$
Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, $\frac{4+8}{5-3}$ means $(4+8)\xf7(5-3).$ The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide.
We’ll add fraction bars to our set of grouping symbols from Use the Language of Algebra to have a more complete set here.
Simplify: $\frac{4+8}{5-3}.$
$\frac{4+8}{5-3}$ | |
Simplify the expression in the numerator. | $\frac{12}{5-3}$ |
Simplify the expression in the denominator. | $\frac{12}{2}$ |
Simplify the fraction. | 6 |
Simplify: $\frac{4-2(3)}{{2}^{2}+2}.$
$\frac{4-2(3)}{{2}^{2}+2}$ | |
Use the order of operations. Multiply in the numerator and use the exponent in the denominator. | $\frac{4-6}{4+2}$ |
Simplify the numerator and the denominator. | $\frac{\mathrm{-2}}{6}$ |
Simplify the fraction. | $-\frac{1}{3}$ |
Simplify: $\frac{6-3(5)}{{3}^{2}+3}.$
$\frac{\mathrm{-3}}{4}$
Simplify: $\frac{{(8-4)}^{2}}{{8}^{2}-{4}^{2}}.$
$\frac{{(8-4)}^{2}}{{8}^{2}-{4}^{2}}$ | |
Use the order of operations (parentheses first, then exponents). | $\frac{{(4)}^{2}}{64-16}$ |
Simplify the numerator and denominator. | $\frac{16}{48}$ |
Simplify the fraction. | $\frac{1}{3}$ |
Simplify: $\frac{{(11-7)}^{2}}{{11}^{2}-{7}^{2}}.$
$\frac{2}{9}$
Simplify: $\frac{{(6+2)}^{2}}{{6}^{2}+{2}^{2}}.$
$\frac{8}{5}$
Simplify: $\frac{4(\mathrm{-3})+6(\mathrm{-2})}{\mathrm{-3}(2)\mathrm{-2}}.$
$\frac{4(\mathrm{-3})+6(\mathrm{-2})}{\mathrm{-3}(2)\mathrm{-2}}$ | |
Multiply. | $\frac{\mathrm{-12}+(\mathrm{-12})}{\mathrm{-6}-2}$ |
Simplify. | $\frac{\mathrm{-24}}{\mathrm{-8}}$ |
Divide. | $3$ |
Simplify: $\frac{8(\mathrm{-2})+4(\mathrm{-3})}{\mathrm{-5}(2)+3}.$
4
Simplify: $\frac{7(\mathrm{-1})+9(\mathrm{-3})}{\mathrm{-5}(3)\mathrm{-2}}.$
2
Multiply and Divide Mixed Numbers
In the following exercises, multiply and write the answer in simplified form.
$4\frac{3}{8}\xb7\frac{7}{10}$
$\frac{15}{22}\xb73\frac{3}{5}$
$4\frac{2}{3}\phantom{\rule{0.2em}{0ex}}(\mathrm{-1}\frac{1}{8})$
$2\frac{2}{5}\phantom{\rule{0.2em}{0ex}}(\mathrm{-2}\frac{2}{9})$
$-\frac{16}{3}$
$\mathrm{-4}\frac{4}{9}\xb75\frac{13}{16}$
$\mathrm{-1}\frac{7}{20}\xb72\frac{11}{12}$
$-\frac{63}{16}$
In the following exercises, divide, and write your answer in simplified form.
$5\frac{1}{3}\xf7\phantom{\rule{0.2em}{0ex}}4$
$13\frac{1}{2}\xf7\phantom{\rule{0.2em}{0ex}}9$
$\frac{3}{2}$
$\mathrm{-12}\xf7\phantom{\rule{0.2em}{0ex}}3\frac{3}{11}$
$\mathrm{-7}\xf7\phantom{\rule{0.2em}{0ex}}5\frac{1}{4}$
$-\frac{4}{3}$
$6\frac{3}{8}\xf7\phantom{\rule{0.2em}{0ex}}2\frac{1}{8}$
$2\frac{1}{5}\xf7\phantom{\rule{0.2em}{0ex}}1\frac{1}{10}$
2
$\mathrm{-9}\frac{3}{5}\xf7\phantom{\rule{0.2em}{0ex}}(\mathrm{-1}\frac{3}{5})$
$\mathrm{-18}\frac{3}{4}\xf7\phantom{\rule{0.2em}{0ex}}(\mathrm{-3}\frac{3}{4})$
5
Translate Phrases to Expressions with Fractions
In the following exercises, translate each English phrase into an algebraic expression.
the quotient of $5u$ and $11$
the quotient of $p$ and $q$
the quotient of $r$ and the sum of $s$ and $10$
the quotient of $A$ and the difference of $3$ and $B$
$\frac{A}{3-B}$
Simplify Complex Fractions
In the following exercises, simplify the complex fraction.
$\frac{\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{8}{9}\phantom{\rule{0.2em}{0ex}}}$
$\frac{\phantom{\rule{0.2em}{0ex}}\frac{4}{5}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{8}{15}\phantom{\rule{0.2em}{0ex}}}$
$\frac{3}{2}$
$\frac{-\frac{8}{21}}{\frac{12}{35}}$
$\frac{-\frac{4}{5}}{2}$
$\frac{\phantom{\rule{0.2em}{0ex}}\frac{2}{5}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}8\phantom{\rule{0.2em}{0ex}}}$
$\frac{\phantom{\rule{0.2em}{0ex}}\frac{5}{3}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}10\phantom{\rule{0.2em}{0ex}}}$
$\frac{1}{6}$
$\frac{\phantom{\rule{0.2em}{0ex}}\frac{m}{3}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{n}{2}\phantom{\rule{0.2em}{0ex}}}$
$\frac{\phantom{\rule{0.2em}{0ex}}\frac{r}{5}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{s}{3}\phantom{\rule{0.2em}{0ex}}}$
$\frac{3r}{5s}$
$\frac{-\frac{x}{6}}{-\frac{8}{9}}$
$\frac{2\frac{4}{5}}{\frac{1}{10}}$
$\frac{\frac{7}{9}}{\mathrm{-2}\frac{4}{5}}$
$\frac{\frac{3}{8}}{\mathrm{-6}\frac{3}{4}}$
$-\frac{1}{18}$
Simplify Expressions with a Fraction Bar
In the following exercises, identify the equivalent fractions.
Which of the following fractions are equivalent to $\frac{5}{\mathrm{-11}}?$
$\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-5}}{\mathrm{-11}},\frac{\mathrm{-5}}{11},\frac{5}{11},-\frac{5}{11}$
Which of the following fractions are equivalent to $\frac{\mathrm{-4}}{9}?$
$\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-4}}{\mathrm{-9}},\frac{\mathrm{-4}}{9},\frac{4}{9},-\frac{4}{9}$
$\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-4}}{9},\text{}\phantom{\rule{0.2em}{0ex}}-\frac{4}{9}$
Which of the following fractions are equivalent to $-\frac{11}{3}?$
$\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{-11}}{3},\frac{11}{3},\frac{\mathrm{-11}}{\mathrm{-3}},\text{}\phantom{\rule{0.2em}{0ex}}\frac{11}{\mathrm{-3}}$
Which of the following fractions are equivalent to $-\frac{13}{6}?$
$\phantom{\rule{0.2em}{0ex}}\frac{13}{6},\frac{13}{\mathrm{-6}},\frac{\mathrm{-13}}{\mathrm{-6}},\frac{\mathrm{-13}}{6}\phantom{\rule{0.2em}{0ex}}$
$\phantom{\rule{0.2em}{0ex}}\frac{13}{\mathrm{-6}},\frac{\mathrm{-13}}{6}\phantom{\rule{0.2em}{0ex}}$
In the following exercises, simplify.
$\frac{4+11}{8}$
$\frac{22+3}{10}$
$\frac{48}{24-15}$
$\frac{\mathrm{-6}+6}{8+4}$
$\frac{22-14}{19-13}$
$\frac{5\cdot 8}{\mathrm{-10}}$
$\frac{4\cdot 3}{6\cdot 6}$
$\frac{{4}^{2}-1}{25}$
$\frac{8\cdot 3+2\cdot 9}{14+3}$
$\frac{15\cdot 5-{5}^{2}}{2\cdot 10}$
$\frac{5\cdot 6-3\cdot 4}{4\cdot 5-2\cdot 3}$
$\frac{{5}^{2}-{3}^{2}}{3-5}$
$\frac{2+4(3)}{\mathrm{-3}-{2}^{2}}$
$\frac{7\cdot 4-2(8-5)}{9.3-3.5}$
$\frac{9(8-2)\mathrm{-3}(15-7)}{6(7-1)\mathrm{-3}(17-9)}$
$\frac{8(9-2)\mathrm{-4}(14-9)}{7(8-3)\mathrm{-3}(16-9)}$
$\frac{18}{7}$
Baking A recipe for chocolate chip cookies calls for $2\frac{1}{4}$ cups of flour. Graciela wants to double the recipe.
Baking A booth at the county fair sells fudge by the pound. Their award winning “Chocolate Overdose” fudge contains $2\frac{2}{3}$ cups of chocolate chips per pound.
Explain how to find the reciprocal of a mixed number.
Randy thinks that $3\frac{1}{2}\xb75\frac{1}{4}$ is $15\frac{1}{8}.$ Explain what is wrong with Randy’s thinking.
Explain why $-\frac{1}{2},\frac{\mathrm{-1}}{2},$ and $\frac{1}{\mathrm{-2}}$ are equivalent.
Answers will vary.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
Notification Switch
Would you like to follow the 'Prealgebra' conversation and receive update notifications?