# 5.4 Solve equations with decimals

 Page 2 / 2

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 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 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 anyone know any internet site where one can find nanotechnology papers? Damian Reply research.net kanaga sciencedirect big data base Ernesto Introduction about quantum dots in nanotechnology Praveena Reply what does nano mean? Anassong Reply 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? Damian Reply absolutely yes Daniel how to know photocatalytic properties of tio2 nanoparticles...what to do now Akash Reply 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 How can I make nanorobot? Lily Do somebody tell me a best nano engineering book for beginners? s. Reply there is no specific books for beginners but there is book called principle of nanotechnology NANO how can I make nanorobot? Lily what is fullerene does it is used to make bukky balls Devang Reply 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 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.
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?