# 4.3 Multiply and divide mixed numbers and complex fractions  (Page 3/4)

 Page 3 / 4
$\frac{-1}{-3}=\frac{1}{3}\phantom{\rule{4em}{0ex}}\frac{\text{negative}}{\text{negative}}=\text{positive}$

## Placement of negative sign in a fraction

For any positive numbers $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b,$

$\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}$

Which of the following fractions are equivalent to $\frac{7}{-8}?$

$\phantom{\rule{0.2em}{0ex}}\frac{-7}{-8},\frac{-7}{8},\frac{7}{8},-\frac{7}{8}$

## Solution

The quotient of a positive and a negative is a negative, so $\frac{7}{-8}$ is negative. Of the fractions listed, $\phantom{\rule{0.2em}{0ex}}\frac{-7}{8}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-\frac{7}{8}$ are also negative.

Which of the following fractions are equivalent to $\frac{-3}{\phantom{\rule{0.2em}{0ex}}5}?$

$\phantom{\rule{0.2em}{0ex}}\frac{-3}{-5},\text{}\phantom{\rule{0.2em}{0ex}}\frac{3}{5},-\frac{3}{5},\frac{\phantom{\rule{0.2em}{0ex}}3}{-5}$

$\phantom{\rule{0.2em}{0ex}}-\frac{3}{5},\frac{\phantom{\rule{0.2em}{0ex}}3}{-5}$

Which of the following fractions are equivalent to $-\frac{2}{7}?$

$\phantom{\rule{0.2em}{0ex}}\frac{-2}{-7},\frac{-2}{7},\frac{2}{7},\frac{2}{-7}$

$\phantom{\rule{0.2em}{0ex}}\frac{-2}{7},\frac{2}{-7}$

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 $\left(4+8\right)÷\left(5-3\right).$ 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.

## Grouping symbols ## Simplify an expression with a fraction bar.

1. Simplify the numerator.
2. Simplify the denominator.
3. Simplify the fraction.

Simplify: $\frac{4+8}{5-3}.$

## Solution

 $\frac{4+8}{5-3}$ Simplify the expression in the numerator. $\frac{12}{5-3}$ Simplify the expression in the denominator. $\frac{12}{2}$ Simplify the fraction. 6

Simplify: $\frac{4+6}{11-2}.$

$\frac{10}{9}$

Simplify: $\frac{3+5}{18-2}.$

$\frac{1}{2}$

Simplify: $\frac{4-2\left(3\right)}{{2}^{2}+2}.$

## Solution

 $\frac{4-2\left(3\right)}{{2}^{2}+2}$ 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. $\frac{-2}{6}$ Simplify the fraction. $-\frac{1}{3}$

Simplify: $\frac{6-3\left(5\right)}{{3}^{2}+3}.$

$\frac{-3}{4}$

Simplify: $\frac{4-4\left(6\right)}{{3}^{3}+3}.$

$-\frac{2}{3}$

Simplify: $\frac{{\left(8-4\right)}^{2}}{{8}^{2}-{4}^{2}}.$

## Solution

 $\frac{{\left(8-4\right)}^{2}}{{8}^{2}-{4}^{2}}$ Use the order of operations (parentheses first, then exponents). $\frac{{\left(4\right)}^{2}}{64-16}$ Simplify the numerator and denominator. $\frac{16}{48}$ Simplify the fraction. $\frac{1}{3}$

Simplify: $\frac{{\left(11-7\right)}^{2}}{{11}^{2}-{7}^{2}}.$

$\frac{2}{9}$

Simplify: $\frac{{\left(6+2\right)}^{2}}{{6}^{2}+{2}^{2}}.$

$\frac{8}{5}$

Simplify: $\frac{4\left(-3\right)+6\left(-2\right)}{-3\left(2\right)-2}.$

## Solution

 $\frac{4\left(-3\right)+6\left(-2\right)}{-3\left(2\right)-2}$ Multiply. $\frac{-12+\left(-12\right)}{-6-2}$ Simplify. $\frac{-24}{-8}$ Divide. $3$

Simplify: $\frac{8\left(-2\right)+4\left(-3\right)}{-5\left(2\right)+3}.$

4

Simplify: $\frac{7\left(-1\right)+9\left(-3\right)}{-5\left(3\right)-2}.$

2

## Key concepts

• Multiply or divide mixed numbers.
1. Convert the mixed numbers to improper fractions.
2. Follow the rules for fraction multiplication or division.
3. Simplify if possible.
• Simplify a complex fraction.
1. Rewrite the complex fraction as a division problem.
2. Follow the rules for dividing fractions.
3. Simplify if possible.
• Placement of negative sign in a fraction.
• For any positive numbers $a$ and $b$ , $\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}$ .
• Simplify an expression with a fraction bar.
1. Simplify the numerator.
2. Simplify the denominator.
3. Simplify the fraction.

## Practice makes perfect

Multiply and Divide Mixed Numbers

In the following exercises, multiply and write the answer in simplified form.

$4\frac{3}{8}·\frac{7}{10}$

$2\frac{4}{9}·\frac{6}{7}$

$\frac{44}{21}$

$\frac{15}{22}·3\frac{3}{5}$

$\frac{25}{36}·6\frac{3}{10}$

$\frac{35}{8}$

$4\frac{2}{3}\phantom{\rule{0.2em}{0ex}}\left(-1\frac{1}{8}\right)$

$2\frac{2}{5}\phantom{\rule{0.2em}{0ex}}\left(-2\frac{2}{9}\right)$

$-\frac{16}{3}$

$-4\frac{4}{9}·5\frac{13}{16}$

$-1\frac{7}{20}·2\frac{11}{12}$

$-\frac{63}{16}$

In the following exercises, divide, and write your answer in simplified form.

$5\frac{1}{3}÷\phantom{\rule{0.2em}{0ex}}4$

$13\frac{1}{2}÷\phantom{\rule{0.2em}{0ex}}9$

$\frac{3}{2}$

$-12÷\phantom{\rule{0.2em}{0ex}}3\frac{3}{11}$

$-7÷\phantom{\rule{0.2em}{0ex}}5\frac{1}{4}$

$-\frac{4}{3}$

$6\frac{3}{8}÷\phantom{\rule{0.2em}{0ex}}2\frac{1}{8}$

$2\frac{1}{5}÷\phantom{\rule{0.2em}{0ex}}1\frac{1}{10}$

2

$-9\frac{3}{5}÷\phantom{\rule{0.2em}{0ex}}\left(-1\frac{3}{5}\right)$

$-18\frac{3}{4}÷\phantom{\rule{0.2em}{0ex}}\left(-3\frac{3}{4}\right)$

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 $7v$ and $13$

$\frac{7v}{13}$

the quotient of $p$ and $q$

the quotient of $a$ and $b$

$\frac{a}{b}$

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.

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{8}{9}\phantom{\rule{0.2em}{0ex}}}$

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{4}{5}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{8}{15}\phantom{\rule{0.2em}{0ex}}}$

$\frac{3}{2}$

$\frac{-\frac{8}{21}}{\frac{12}{35}}$

$\frac{-\frac{9}{16}}{\frac{33}{40}}$

$-\frac{15}{22}$

$\frac{-\frac{4}{5}}{2}$

$\frac{-\frac{9}{10}}{3}$

$-\frac{3}{10}$

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{2}{5}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}8\phantom{\rule{0.2em}{0ex}}}$

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{5}{3}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}10\phantom{\rule{0.2em}{0ex}}}$

$\frac{1}{6}$

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{m}{3}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{n}{2}\phantom{\rule{0.2em}{0ex}}}$

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{r}{5}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{s}{3}\phantom{\rule{0.2em}{0ex}}}$

$\frac{3r}{5s}$

$\frac{-\frac{x}{6}}{-\frac{8}{9}}$

$\frac{-\frac{3}{8}}{-\frac{y}{12}}$

$\frac{9}{2y}$

$\frac{2\frac{4}{5}}{\frac{1}{10}}$

$\frac{4\frac{2}{3}}{\frac{1}{6}}$

28

$\frac{\frac{7}{9}}{-2\frac{4}{5}}$

$\frac{\frac{3}{8}}{-6\frac{3}{4}}$

$-\frac{1}{18}$

Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.

Which of the following fractions are equivalent to $\frac{5}{-11}?$

$\phantom{\rule{0.2em}{0ex}}\frac{-5}{-11},\frac{-5}{11},\frac{5}{11},-\frac{5}{11}$

Which of the following fractions are equivalent to $\frac{-4}{9}?$

$\phantom{\rule{0.2em}{0ex}}\frac{-4}{-9},\frac{-4}{9},\frac{4}{9},-\frac{4}{9}$

$\phantom{\rule{0.2em}{0ex}}\frac{-4}{9},\text{}\phantom{\rule{0.2em}{0ex}}-\frac{4}{9}$

Which of the following fractions are equivalent to $-\frac{11}{3}?$

$\phantom{\rule{0.2em}{0ex}}\frac{-11}{3},\frac{11}{3},\frac{-11}{-3},\text{}\phantom{\rule{0.2em}{0ex}}\frac{11}{-3}$

Which of the following fractions are equivalent to $-\frac{13}{6}?$

$\phantom{\rule{0.2em}{0ex}}\frac{13}{6},\frac{13}{-6},\frac{-13}{-6},\frac{-13}{6}\phantom{\rule{0.2em}{0ex}}$

$\phantom{\rule{0.2em}{0ex}}\frac{13}{-6},\frac{-13}{6}\phantom{\rule{0.2em}{0ex}}$

In the following exercises, simplify.

$\frac{4+11}{8}$

$\frac{9+3}{7}$

$\frac{12}{7}$

$\frac{22+3}{10}$

$\frac{19-4}{6}$

$\frac{5}{2}$

$\frac{48}{24-15}$

$\frac{46}{4+4}$

$\frac{23}{4}$

$\frac{-6+6}{8+4}$

$\frac{-6+3}{17-8}$

$-\frac{1}{3}$

$\frac{22-14}{19-13}$

$\frac{15+9}{18+12}$

$\frac{4}{5}$

$\frac{5\cdot 8}{-10}$

$\frac{3\cdot 4}{-24}$

$-\frac{1}{2}$

$\frac{4\cdot 3}{6\cdot 6}$

$\frac{6\cdot 6}{9\cdot 2}$

2

$\frac{{4}^{2}-1}{25}$

$\frac{{7}^{2}+1}{60}$

$\frac{5}{6}$

$\frac{8\cdot 3+2\cdot 9}{14+3}$

$\frac{9\cdot 6-4\cdot 7}{22+3}$

$\frac{26}{25}$

$\frac{15\cdot 5-{5}^{2}}{2\cdot 10}$

$\frac{12\cdot 9-{3}^{2}}{3\cdot 18}$

$\frac{11}{6}$

$\frac{5\cdot 6-3\cdot 4}{4\cdot 5-2\cdot 3}$

$\frac{8\cdot 9-7\cdot 6}{5\cdot 6-9\cdot 2}$

$\frac{5}{2}$

$\frac{{5}^{2}-{3}^{2}}{3-5}$

$\frac{{6}^{2}-{4}^{2}}{4-6}$

−10

$\frac{2+4\left(3\right)}{-3-{2}^{2}}$

$\frac{7+3\left(5\right)}{-2-{3}^{2}}$

−2

$\frac{7\cdot 4-2\left(8-5\right)}{9.3-3.5}$

$\frac{9\cdot 7-3\left(12-8\right)}{8.7-6.6}$

$\frac{51}{20}$

$\frac{9\left(8-2\right)-3\left(15-7\right)}{6\left(7-1\right)-3\left(17-9\right)}$

$\frac{8\left(9-2\right)-4\left(14-9\right)}{7\left(8-3\right)-3\left(16-9\right)}$

$\frac{18}{7}$

## Everyday math

Baking A recipe for chocolate chip cookies calls for $2\frac{1}{4}$ cups of flour. Graciela wants to double the recipe.

1. How much flour will Graciela need? Show your calculation. Write your result as an improper fraction and as a mixed number.
2. Measuring cups usually come in sets with cups for $\frac{1}{8},\frac{1}{4},\frac{1}{3},\frac{1}{2},\text{and}\phantom{\rule{0.2em}{0ex}}1$ cup. Draw a diagram to show two different ways that Graciela could measure out the flour needed 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.

1. How many cups of chocolate chips are in a half-pound of the fudge?
2. The owners of the booth make the fudge in $10$ -pound batches. How many chocolate chips do they need to make a $10$ -pound batch? Write your results as improper fractions and as a mixed numbers.

1. $\frac{4}{3}=1\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{cups}$
2. $\frac{80}{3}=26\frac{2}{3}\phantom{\rule{0.2em}{0ex}}\text{cups}$

## Writing exercises

Explain how to find the reciprocal of a mixed number.

Explain how to multiply mixed numbers.

Answers will vary.

Randy thinks that $3\frac{1}{2}·5\frac{1}{4}$ is $15\frac{1}{8}.$ Explain what is wrong with Randy’s thinking.

Explain why $-\frac{1}{2},\frac{-1}{2},$ and $\frac{1}{-2}$ are equivalent.

Answers will vary.

## Self check

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?

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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Kyle
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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s.
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for screen printed electrodes ?
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What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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