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

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

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.

## 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.
• 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.

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.

## 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?

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?