# 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

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

$\frac{5}{8}$

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

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

$\frac{3}{5}$

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

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

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

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 $p.$

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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
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?

#### Get Jobilize Job Search Mobile App in your pocket Now! By Stephen Voron By OpenStax By Janet Forrester By Edgar Delgado By OpenStax By OpenStax By Cath Yu By Edgar Delgado By Stephen Voron By Dindin Secreto