# 5.4 Solve equations with decimals

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Solve: $a-3.93=-2.86.$

a = 1.07

Solve: $n-3.47=-2.64.$

n = 0.83

Solve: $-4.8=0.8n.$

## Solution

We will use the Division Property of Equality.

Use the Properties of Equality to find a value for $n.$

 We must divide both sides by 0.8 to isolate n . Simplify. Check:

Since $n=-6$ makes $-4.8=0.8n$ a true statement, we know we have a solution.

Solve: $-8.4=0.7b.$

b = −12

Solve: $-5.6=0.7c.$

c = −8

Solve: $\frac{p}{-1.8}=-6.5.$

## Solution

We will use the Multiplication Property of Equality .

 Here, p is divided by −1.8. We must multiply by −1.8 to isolate p Multiply. Check:

A solution to $\frac{p}{-1.8}=-6.5$ is $p=11.7.$

Solve: $\frac{c}{-2.6}=-4.5.$

c = 11.7

Solve: $\frac{b}{-1.2}=-5.4.$

b = 6.48

## Translate to an equation and solve

Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

Translate and solve: The difference of $n$ and $4.3$ is $2.1.$

## Solution

 Translate. Add $4.3$ to both sides of the equation. Simplify. Check: Is the difference of $n$ and 4.3 equal to 2.1? Let $n=4.3$ : Is the difference of 6.4 and 4.3 equal to 2.1? Translate. Simplify.

Translate and solve: The difference of $y$ and $4.9$ is $2.8.$

y − 4.9 = 2.8; y = 7.7

Translate and solve: The difference of $z$ and $5.7$ is $3.4.$

z − 5.7 = 3.4; z = 9.1

Translate and solve: The product of $-3.1$ and $x$ is $5.27.$

## Solution

 Translate. Divide both sides by $-3.1$ . Simplify. Check: Is the product of −3.1 and $x$ equal to $5.27$ ? Let $x=-1.7$ : Is the product of $-3.1$ and $-1.7$ equal to $5.27$ ? Translate. Simplify.

Translate and solve: The product of $-4.3$ and $x$ is $12.04.$

−4.3 x = 12.04; x = −2.8

Translate and solve: The product of $-3.1$ and $m$ is $26.66.$

−3.1 m = 26.66; m = −8.6

Translate and solve: The quotient of $p$ and $-2.4$ is $6.5.$

## Solution

 Translate. Multiply both sides by $-2.4$ . Simplify. Check: Is the quotient of $p$ and $-2.4$ equal to $6.5$ ? Let $p=-15.6:$ Is the quotient of $-15.6$ and $-2.4$ equal to $6.5$ ? Translate. Simplify.

Translate and solve: The quotient of $q$ and $-3.4$ is $4.5.$

$\frac{q}{-3.4}=4.5;\phantom{\rule{0.2em}{0ex}}q=-15.3$

Translate and solve: The quotient of $r$ and $-2.6$ is $2.5.$

$\frac{r}{-2.6}=2.5;\phantom{\rule{0.2em}{0ex}}r=-6.5$

Translate and solve: The sum of $n$ and $2.9$ is $1.7.$

## Solution

 Translate. Subtract $2.9$ from each side. Simplify. Check: Is the sum $n$ and $2.9$ equal to $1.7$ ? Let $n=-1.2:$ Is the sum $-1.2$ and $2.9$ equal to $1.7$ ? Translate. Simplify.

Translate and solve: The sum of $j$ and $3.8$ is $2.6.$

j + 3.8 = 2.6; j = −1.2

Translate and solve: The sum of $k$ and $4.7$ is $0.3.$

k + 4.7 = 0.3; k = −4.4

## Key concepts

• Determine whether a number is a solution to an equation.
• Substitute the number for the variable in the equation.
• Simplify the expressions on both sides of the equation.
• Determine whether the resulting equation is true.
If so, the number is a solution.
If not, the number is not a solution.
• Properties of Equality
 Subtraction Property of Equality Addition Property of Equality For any numbers $a$ , $b$ , and $c$ , $\begin{array}{cccc}\text{If}& \hfill a& =& b\hfill \\ \text{then}& \hfill a-c& =& b-c\hfill \end{array}$ For any numbers $a$ , $b$ , and $c$ , $\begin{array}{cccc}\text{If}& \hfill a& =& b\hfill \\ \text{then}& \hfill a+c& =& b+c\hfill \end{array}$ Division of Property of Equality Multiplication Property of Equality For any numbers $a$ , $b$ , and $c\ne 0$ , $\begin{array}{cccc}\text{If}& \hfill a& =& b\hfill \\ \text{then}& \hfill \frac{a}{c}& =& \frac{b}{c}\hfill \end{array}$ For any numbers $a$ , $b$ , and $c$ , $\begin{array}{cccc}\text{If}& \hfill a& =& b\hfill \\ \text{then}& \hfill a\cdot c& =& b\cdot c\hfill \end{array}$

## Practice makes perfect

Determine Whether a Decimal is a Solution of an Equation

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

$x-0.8=2.3$
$\phantom{\rule{0.2em}{0ex}}x=2$ $\phantom{\rule{0.2em}{0ex}}x=-1.5$ $\phantom{\rule{0.2em}{0ex}}x=3.1$

1. no
2. no
3. yes

$y+0.6=-3.4$
$\phantom{\rule{0.2em}{0ex}}y=-4$ $\phantom{\rule{0.2em}{0ex}}y=-2.8$ $\phantom{\rule{0.2em}{0ex}}y=2.6$

$\frac{h}{1.5}=-4.3$
$\phantom{\rule{0.2em}{0ex}}h=6.45$ $\phantom{\rule{0.2em}{0ex}}h=-6.45$ $\phantom{\rule{0.2em}{0ex}}h=-2.1$

1. no
2. yes
3. no

$0.75k=-3.6$
$\phantom{\rule{0.2em}{0ex}}k=-0.48$ $\phantom{\rule{0.2em}{0ex}}k=-4.8$ $\phantom{\rule{0.2em}{0ex}}k=-2.7$

Solve Equations with Decimals

In the following exercises, solve the equation.

$y+2.9=5.7$

y = 2.8

$m+4.6=6.5$

$f+3.45=2.6$

f = −0.85

$h+4.37=3.5$

$a+6.2=-1.7$

a = −7.9

$b+5.8=-2.3$

$c+1.15=-3.5$

c = −4.65

$d+2.35=-4.8$

$n-2.6=1.8$

n = 4.4

$p-3.6=1.7$

$x-0.4=-3.9$

x = −3.5

$y-0.6=-4.5$

$j-1.82=-6.5$

j = −4.68

$k-3.19=-4.6$

$m-0.25=-1.67$

m = −1.42

$q-0.47=-1.53$

$0.5x=3.5$

x = 7

$0.4p=9.2$

$-1.7c=8.5$

c = −5

$-2.9x=5.8$

$-1.4p=-4.2$

p = 3

$-2.8m=-8.4$

$-120=1.5q$

q = −80

$-75=1.5y$

$0.24x=4.8$

x = 20

$0.18n=5.4$

$-3.4z=-9.18$

z = 2.7

$-2.7u=-9.72$

$\frac{a}{0.4}=-20$

a = −8

$\frac{b}{0.3}=-9$

$\frac{x}{0.7}=-0.4$

x = −0.28

$\frac{y}{0.8}=-0.7$

$\frac{p}{-5}=-1.65$

p = 8.25

$\frac{q}{-4}=-5.92$

$\frac{r}{-1.2}=-6$

r = 7.2

$\frac{s}{-1.5}=-3$

Mixed Practice

In the following exercises, solve the equation. Then check your solution.

$x-5=-11$

x = −6

$-\frac{2}{5}=x+\frac{3}{4}$

$p+8=-2$

p = −10

$p+\frac{2}{3}=\frac{1}{12}$

$-4.2m=-33.6$

m = 8

$q+9.5=-14$

$q+\frac{5}{6}=\frac{1}{12}$

$q=-\frac{3}{4}$

$\frac{8.6}{15}=-d$

$\frac{7}{8}m=\frac{1}{10}$

$m=\frac{4}{35}$

$\frac{j}{-6.2}=-3$

$-\frac{2}{3}=y+\frac{3}{8}$

$y=-\frac{25}{24}$

$s-1.75=-3.2$

$\frac{11}{20}=-f$

$f=-\frac{11}{20}$

$-3.6b=2.52$

$-4.2a=3.36$

a = −0.8

$-9.1n=-63.7$

$r-1.25=-2.7$

r = −1.45

$\frac{1}{4}n=\frac{7}{10}$

$\frac{h}{-3}=-8$

h = 24

$y-7.82=-16$

Translate to an Equation and Solve

In the following exercises, translate and solve.

The difference of $n$ and $1.9$ is $3.4.$

$n-1.9=3.4;5.3$

The difference $n$ and $1.5$ is $0.8.$

The product of $-6.2$ and $x$ is $-4.96.$

−6.2 x = −4.96; 0.8

The product of $-4.6$ and $x$ is $-3.22.$

The quotient of $y$ and $-1.7$ is $-5.$

$\frac{y}{-1.7}=-5;\phantom{\rule{0.2em}{0ex}}8.5$

The quotient of $z$ and $-3.6$ is $3.$

The sum of $n$ and $-7.3$ is $2.4.$

n + (−7.3) = 2.4; 9.7

The sum of $n$ and $-5.1$ is $3.8.$

## Everyday math

Shawn bought a pair of shoes on sale for $78$ . Solve the equation $0.75p=78$ to find the original price of the shoes, $p.$

$104 Mary bought a new refrigerator. The total price including sales tax was $\text{1,350}.$ Find the retail price, $r,$ of the refrigerator before tax by solving the equation $1.08r=1,350.$ ## Writing exercises Think about solving the equation $1.2y=60,$ but do not actually solve it. Do you think the solution should be greater than $60$ or less than $60?$ Explain your reasoning. Then solve the equation to see if your thinking was correct. Answers will vary. Think about solving the equation $0.8x=200,$ but do not actually solve it. Do you think the solution should be greater than $200$ or less than $200?$ Explain your reasoning. Then solve the equation to see if your thinking was correct. ## Self check After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? #### Questions & Answers where we get a research paper on Nano chemistry....? Maira Reply nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review Ali what are the products of Nano chemistry? Maira Reply 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 Google 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 Hafiz Reply revolt da Application of nanotechnology in medicine what is variations in raman spectra for nanomaterials Jyoti Reply ya I also want to know the raman spectra Bhagvanji I only see partial conversation and what's the question here! Crow Reply what about nanotechnology for water purification RAW Reply 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 Brian Reply 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? LITNING Reply what is a peer LITNING Reply What is meant by 'nano scale'? LITNING Reply 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 ? Bob Reply 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? Damian Reply what king of growth are you checking .? Renato What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ? Stoney Reply why we need to study biomolecules, molecular biology in nanotechnology? Adin Reply ? Kyle yes I'm doing my masters in nanotechnology, we are being studying all these domains as well.. Adin why? Adin what school? Kyle biomolecules are e building blocks of every organics and inorganic materials. Joe how did you get the value of 2000N.What calculations are needed to arrive at it Smarajit Reply 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. Kimberly Reply 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.
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