# 4.7 Solve equations with fractions  (Page 5/5)

 Page 5 / 5

The quotient of $f$ and $-3$ is $-18.$

The quotient of $f$ and $-4$ is $-20.$

$\frac{f}{-4}=-20;f=80$

The quotient of $g$ and twelve is $8.$

The quotient of $g$ and nine is $14.$

$\frac{g}{9}=14;g=126$

Three-fourths of $q$ is $12.$

Two-fifths of $q$ is $20.$

$\frac{2}{5}q=20;q=50$

Seven-tenths of $p$ is $-63.$

Four-ninths of $p$ is $-28.$

$\frac{4}{9}p=-28;p=-63$

$m$ divided by $4$ equals negative $6.$

The quotient of $h$ and $2$ is $43.$

$\frac{h}{2}=43$

Three-fourths of $z$ is the same as $15.$

The quotient of $a$ and $\frac{2}{3}$ is $\frac{3}{4}.$

$\frac{\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\phantom{\rule{0.2em}{0ex}}}=\frac{3}{4}$

The sum of five-sixths and $x$ is $\frac{1}{2}.$

The sum of three-fourths and $x$ is $\frac{1}{8}.$

$\frac{3}{4}+x=\frac{1}{8};x=-\frac{5}{8}$

The difference of $y$ and one-fourth is $-\frac{1}{8}.$

The difference of $y$ and one-third is $-\frac{1}{6}.$

$y-\frac{1}{3}=-\frac{1}{6};y=\frac{1}{6}$

## Everyday math

Shopping Teresa bought a pair of shoes on sale for $\text{48}.$ The sale price was $\frac{2}{3}$ of the regular price. Find the regular price of the shoes by solving the equation $\frac{2}{3}p=48$

Playhouse The table in a child’s playhouse is $\frac{3}{5}$ of an adult-size table. The playhouse table is $18$ inches high. Find the height of an adult-size table by solving the equation $\frac{3}{5}h=18.$

30 inches

## Writing exercises

[link] describes three methods to solve the equation $-y=15.$ Which method do you prefer? Why?

Richard thinks the solution to the equation $\frac{3}{4}x=24$ is $16.$ Explain why Richard is wrong.

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

## Visualize Fractions

In the following exercises, name the fraction of each figure that is shaded.

$\frac{5}{9}$

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

$\frac{3}{2}$

In the following exercises, convert the improper fraction to a mixed number.

$\frac{58}{15}$

$\frac{63}{11}$

$5\frac{8}{11}$

In the following exercises, convert the mixed number to an improper fraction.

$12\frac{1}{4}$

$9\frac{4}{5}$

$\frac{49}{5}$

Find three fractions equivalent to $\frac{2}{5}.$ Show your work, using figures or algebra.

Find three fractions equivalent to $-\frac{4}{3}.$ Show your work, using figures or algebra.

In the following exercises, locate the numbers on a number line.

$\frac{5}{8},\frac{4}{3},3\frac{3}{4},4$

$\frac{1}{4},-\frac{1}{4},1\frac{1}{3},-1\frac{1}{3},\frac{7}{2},-\frac{7}{2}$

In the following exercises, order each pair of numbers, using $<$ or $>.$

$-1___-\frac{2}{5}$

$-2\frac{1}{2}___-3$

>

## Multiply and Divide Fractions

In the following exercises, simplify.

$-\frac{63}{84}$

$-\frac{90}{120}$

$-\frac{3}{4}$

$-\frac{14a}{14b}$

$-\frac{8x}{8y}$

$-\frac{x}{y}$

In the following exercises, multiply.

$\frac{2}{5}·\frac{8}{13}$

$-\frac{1}{3}·\frac{12}{7}$

$-\frac{4}{7}$

$\frac{2}{9}·\left(-\frac{45}{32}\right)$

$6m·\frac{4}{11}$

$\frac{24}{11}m$

$-\frac{1}{4}\left(-32\right)$

$3\frac{1}{5}·1\frac{7}{8}$

6

In the following exercises, find the reciprocal.

$\frac{2}{9}$

$\frac{15}{4}$

$\frac{4}{15}$

$3$

$-\frac{1}{4}$

−4

Fill in the chart.

Opposite Absolute Value Reciprocal
$-\frac{5}{13}$
$\frac{3}{10}$
$\frac{9}{4}$
$-12$

In the following exercises, divide.

$\frac{2}{3}÷\frac{1}{6}$

4

$\left(-\frac{3x}{5}\right)÷\left(-\frac{2y}{3}\right)$

$\frac{4}{5}÷3$

$\frac{4}{15}$

$8÷2\frac{2}{3}$

$8\frac{2}{3}÷1\frac{1}{12}$

8

## Multiply and Divide Mixed Numbers and Complex Fractions

In the following exercises, perform the indicated operation.

$3\frac{1}{5}·1\frac{7}{8}$

$-5\frac{7}{12}·4\frac{4}{11}$

$-\frac{268}{11}$

$8÷2\frac{2}{3}$

$8\frac{2}{3}÷1\frac{1}{12}$

8

In the following exercises, translate the English phrase into an algebraic expression.

the quotient of $8$ and $y$

the quotient of $V$ and the difference of $h$ and $6$

$\frac{V}{h-6}$

In the following exercises, simplify the complex fraction

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

$\frac{\frac{8}{9}}{-4}$

$-\frac{2}{9}$

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{n}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{3}{8}\phantom{\rule{0.2em}{0ex}}}$

$\frac{-1\frac{5}{6}}{-\frac{1}{12}}$

22

In the following exercises, simplify.

$\frac{5+16}{5}$

$\frac{8·4-{5}^{2}}{3·12}$

$\frac{7}{36}$

$\frac{8·7+5\left(8-10\right)}{9·3-6·4}$

## Add and Subtract Fractions with Common Denominators

In the following exercises, add.

$\frac{3}{8}+\frac{2}{8}$

$\frac{5}{8}$

$\frac{4}{5}+\frac{1}{5}$

$\frac{2}{5}+\frac{1}{5}$

$\frac{3}{5}$

$\frac{15}{32}+\frac{9}{32}$

$\frac{x}{10}+\frac{7}{10}$

$\frac{x+7}{10}$

In the following exercises, subtract.

$\frac{8}{11}-\frac{6}{11}$

$\frac{11}{12}-\frac{5}{12}$

$\frac{1}{2}$

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

$-\frac{31}{30}-\frac{7}{30}$

$-\frac{19}{15}$

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

$\frac{11}{15}-\frac{5}{15}-\left(-\frac{2}{15}\right)$

$\frac{8}{15}$

## Add and Subtract Fractions with Different Denominators

In the following exercises, find the least common denominator.

$\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{1}{12}$

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

15

$\frac{8}{15}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{11}{20}$

$\frac{3}{4},\frac{1}{6},\text{and}\phantom{\rule{0.2em}{0ex}}\frac{5}{10}$

60

In the following exercises, change to equivalent fractions using the given LCD.

$\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{1}{5},\phantom{\rule{0.2em}{0ex}}\text{LCD}=15$

$\frac{3}{8}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{5}{6},\phantom{\rule{0.2em}{0ex}}\text{LCD}=24$

$\frac{9}{24}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{20}{24}$

$-\frac{9}{16}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{5}{12},\phantom{\rule{0.2em}{0ex}}\text{LCD}=48$

$\frac{1}{3}\text{,}\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{4}{5},\phantom{\rule{0.2em}{0ex}}\text{LCD}=60$

$\frac{20}{60},\frac{15}{60}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{48}{60}$

In the following exercises, perform the indicated operations and simplify.

$\frac{1}{5}+\frac{2}{3}$

$\frac{11}{12}-\frac{2}{3}$

$\frac{1}{4}$

$-\frac{9}{10}-\frac{3}{4}$

$-\frac{11}{36}-\frac{11}{20}$

$-\frac{77}{90}$

$-\frac{22}{25}+\frac{9}{40}$

$\frac{y}{10}-\frac{1}{3}$

$\frac{3y-10}{30}$

$\frac{2}{5}+\left(-\frac{5}{9}\right)$

$\frac{4}{11}÷\frac{2}{7d}$

$\frac{14d}{11}$

$\frac{2}{5}+\left(-\frac{3n}{8}\right)\left(-\frac{2}{9n}\right)$

$\frac{{\left(\frac{2}{3}\right)}^{2}}{{\left(\frac{5}{8}\right)}^{2}}$

$\frac{256}{225}$

$\left(\frac{11}{12}+\frac{3}{8}\right)÷\left(\frac{5}{6}-\frac{1}{10}\right)$

In the following exercises, evaluate.

$y-\frac{4}{5}$ when

1. $y=-\frac{4}{5}$
2. $y=\frac{1}{4}$

1. $-\frac{8}{5}$
2. $-\frac{11}{20}$

$6m{n}^{2}$ when $m=\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n=-\frac{1}{3}$

## Add and Subtract Mixed Numbers

In the following exercises, perform the indicated operation.

$4\frac{1}{3}+9\frac{1}{3}$

$13\frac{2}{3}$

$6\frac{2}{5}+7\frac{3}{5}$

$5\frac{8}{11}+2\frac{4}{11}$

$8\frac{1}{11}$

$3\frac{5}{8}+3\frac{7}{8}$

$9\frac{13}{20}-4\frac{11}{20}$

$5\frac{1}{10}$

$2\frac{3}{10}-1\frac{9}{10}$

$2\frac{11}{12}-1\frac{7}{12}$

$\frac{10}{3}$

$8\frac{6}{11}-2\frac{9}{11}$

## Solve Equations with Fractions

In the following exercises, determine whether the each number is a solution of the given equation.

$x-\frac{1}{2}=\frac{1}{6}\text{:}$

• $x=1$
• $x=\frac{2}{3}$
• $x=-\frac{1}{3}$

1. no
2. yes
3. no

$y+\frac{3}{5}=\frac{5}{9}\text{:}$

1. $y=\frac{1}{2}$
2. $y=\frac{52}{45}$
3. $y=-\frac{2}{45}$

In the following exercises, solve the equation.

$n+\frac{9}{11}=\frac{4}{11}$

$n=-\frac{5}{11}$

$x-\frac{1}{6}=\frac{7}{6}$

$h-\left(-\frac{7}{8}\right)=-\frac{2}{5}$

$h=-\frac{51}{40}$

$\frac{x}{5}=-10$

$-z=23$

z = −23

In the following exercises, translate and solve.

The sum of two-thirds and $n$ is $-\frac{3}{5}.$

The difference of $q$ and one-tenth is $\frac{1}{2}.$

$q-\frac{1}{10}=\frac{1}{2};q=\frac{3}{5}$

The quotient of $p$ and $-4$ is $-8.$

Three-eighths of $y$ is $24.$

$\frac{3}{8}y=24;y=64$

## Chapter practice test

Convert the improper fraction to a mixed number.

$\frac{19}{5}$

Convert the mixed number to an improper fraction.

$3\frac{2}{7}$

$\frac{23}{7}$

Locate the numbers on a number line.

$\frac{1}{2},1\frac{2}{3},-2\frac{3}{4},\text{and}\phantom{\rule{0.2em}{0ex}}\frac{9}{4}$

In the following exercises, simplify.

$\frac{5}{20}$

$\frac{1}{4}$

$\frac{18r}{27s}$

$\frac{1}{3}·\frac{3}{4}$

$\frac{1}{4}$

$\frac{3}{5}·15$

$-36u\left(-\frac{4}{9}\right)$

16 u

$-5\frac{7}{12}·4\frac{4}{11}$

$-\frac{5}{6}÷\frac{5}{12}$

−2

$\frac{7}{11}÷\left(-\frac{7}{11}\right)$

$\frac{9a}{10}÷\frac{15a}{8}$

$\frac{12}{25}$

$-6\frac{2}{5}÷4$

$\left(-15\frac{5}{6}\right)÷\left(-3\frac{1}{6}\right)$

5

$\frac{-6}{\frac{6}{11}}$

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

$\frac{5p}{2q}$

$\frac{-\frac{4}{15}}{-2\frac{2}{3}}$

$\frac{{9}^{2}-{4}^{2}}{9-4}$

13

$\frac{2}{d}+\frac{9}{d}$

$-\frac{3}{13}+\left(-\frac{4}{13}\right)$

$-\frac{7}{13}$

$-\frac{22}{25}+\frac{9}{40}$

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

−1

$-\frac{3}{10}+\left(-\frac{5}{8}\right)$

$-\frac{3}{4}÷\frac{x}{3}$

$-\frac{9}{4x}$

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

$\frac{\frac{5}{14}+\frac{1}{8}}{\frac{9}{56}}$

3

Evaluate.

$x+\frac{1}{3}$ when

1. $x=\frac{2}{3}$
2. $x=-\frac{5}{6}$

In the following exercises, solve the equation.

$y+\frac{3}{5}=\frac{7}{5}$

$y=\frac{4}{5}$

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

$f+\left(-\frac{2}{3}\right)=\frac{5}{12}$

$f=\frac{13}{12}$

$\frac{m}{-2}=-16$

$-\frac{2}{3}c=18$

c = −27

Translate and solve: The quotient of $p$ and $-4$ is $-8.$ Solve for $p.$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
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?