# 5.5 Application ii - solving problems

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules (<link document="m21980"/>) and (<link document="m21979"/>)). Objectives of this module: be able to solve various applied problems.

## Overview

• Solving Applied Problems

## Solving applied problems

Let’s study some interesting problems that involve linear equations in one variable. In order to solve such problems, we apply the following five-step method:

## Five-step method for solving word problems

1. Let $x$ (or some other letter) represent the unknown quantity.
2. Translate the words to mathematical symbols and form an equation.
3. Solve this equation.
4. Ask yourself "Does this result seem reasonable?" Check the solution by substituting the result into the original statement of the problem.

If the answer doesn’t check, you have either solved the equation incorrectly, or you have developed the wrong equation. Check your method of solution first. If the result does not check, reconsider your equation.

5. Write the conclusion.

If it has been your experience that word problems are difficult, then follow the five-step method carefully. Most people have difficulty because they neglect step 1.

Always start by INTRODUCING A VARIABLE!

Keep in mind what the variable is representing throughout the problem.

## Sample set a

This year an item costs $44$ , an increase of $3$ over last year’s price. What was last year’s price?

$\begin{array}{l}\begin{array}{lllll}\text{Step}\text{\hspace{0.17em}}1:\hfill & \hfill \text{Let}\text{\hspace{0.17em}}x& =\hfill & \text{last}\text{\hspace{0.17em}}\text{year's}\text{\hspace{0.17em}}\text{price}\text{.}\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}2:\hfill & \hfill x+3& =\hfill & 44.\hfill & x+3\text{\hspace{0.17em}}\text{represents}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{3}\text{\hspace{0.17em}}\text{increase}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{price}\text{.}\hfill \\ \text{Step}\text{\hspace{0.17em}}3:\hfill & \hfill x+3& =\hfill & 44\hfill & \hfill \\ \hfill & \hfill x+3-3& =\hfill & 44-3\hfill & \hfill \\ \hfill & \hfill x& =\hfill & 41\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}4:\hfill & \hfill 41+3& =\hfill & \text{44}\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}\\ \begin{array}{ll}\text{Step}\text{\hspace{0.17em}}5:\hfill & \text{Last}\text{\hspace{0.17em}}\text{year's}\text{\hspace{0.17em}}\text{price}\text{\hspace{0.17em}}\text{was}\text{\hspace{0.17em}}41.\hfill \end{array}\end{array}$

## Practice set a

This year an item costs $23$ , an increase of $4$ over last year’s price. What was last year’s price?

1. Let $x=$
2. Last year's price was .

Last year's price was $19$

## Sample set b

The perimeter (length around) of a square is 60 cm (centimeters). Find the length of a side.

$\begin{array}{ll}\text{Step}\text{\hspace{0.17em}}\text{1:}\hfill & \text{Let}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{=}\text{\hspace{0.17em}}\text{length}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{side}\text{.}\hfill \\ \text{Step}\text{\hspace{0.17em}}\text{2:}\hfill & \text{We}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{draw}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{picture}\text{.}\hfill \end{array}$

$\begin{array}{l}\begin{array}{lllll}\text{Step}\text{\hspace{0.17em}}3:\hfill & \hfill x+x+x+x& =\hfill & 60\hfill & \hfill \\ \hfill & \hfill 4x& =\hfill & 60\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}4.\hfill \\ \hfill & \hfill x& =\hfill & 15.\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}4:\hfill & \hfill 4\left(15\right)& =\hfill & 60.\hfill & \hfill \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\end{array}\\ \begin{array}{ll}\text{Step}\text{\hspace{0.17em}}5:\hfill & \text{The}\text{\hspace{0.17em}}\text{length}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{side}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}15\text{\hspace{0.17em}}\text{cm}\text{.}\hfill \end{array}\end{array}$

## Practice set b

The perimeter of a triangle is 54 inches. If each side has the same length, find the length of a side.

1. Let $x=$
2. The length of a side is inches.

The length of a side is 18 inches.

## Sample set c

Six percent of a number is 54. What is the number?

$\begin{array}{ll}\text{Step}\text{\hspace{0.17em}}1:\hfill & \text{Let}\text{\hspace{0.17em}}x=\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{number}\hfill \\ \text{Step}\text{\hspace{0.17em}}2:\hfill & \text{We}\text{\hspace{0.17em}}\text{must}\text{\hspace{0.17em}}\text{convert}\text{\hspace{0.17em}}6%\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{decimal.}\hfill \end{array}\text{\hspace{0.17em}}$
$\begin{array}{l}\begin{array}{lllll}\hfill & \hfill 6%& =\hfill & .06\hfill & \hfill \\ \hfill & \hfill .06x& =\hfill & 54\hfill & .06x\text{\hspace{0.17em}}\text{occurs}\text{\hspace{0.17em}}\text{because}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{want}\text{\hspace{0.17em}}6%\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}x\text{.}\hfill \\ \text{Step}\text{\hspace{0.17em}}3:\hfill & \hfill .06x& =\hfill & 54.\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}.06.\hfill \\ \hfill & \hfill x& =\hfill & \frac{54}{.06}\hfill & \hfill \\ \hfill & \hfill x& =\hfill & 900\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}4:\hfill & .06\left(900\right)\hfill & =\hfill & 54.\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}\\ \begin{array}{ll}\text{Step}\text{\hspace{0.17em}}5:\hfill & \text{The}\text{\hspace{0.17em}}\text{number}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}900.\hfill \end{array}\end{array}$

## Practice set c

Eight percent of a number is 36. What is the number?

1. Let $x=$
2. The number is .

The number is 450.

## Sample set d

An astronomer notices that one star gives off about $3.6$ times as much energy as another star. Together the stars give off $55.844$ units of energy. How many units of energy does each star emit?

1. In this problem we have two unknowns and, therefore, we might think, two variables. However, notice that the energy given off by one star is given in terms of the other star. So, rather than introducing two variables, we introduce only one. The other unknown(s) is expressed in terms of this one. (We might call this quantity the base quantity.)

Let $x=$ number of units of energy given off by the less energetic star. Then, $3.6x=$ number of units of energy given off by the more energetic star.

$\begin{array}{l}\begin{array}{lllll}\text{Step}\text{\hspace{0.17em}}\text{2:}\hfill & \hfill x+3.6x& \text{=}\hfill & 55.844.\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}\text{3:}\hfill & \hfill x+3.6x& \text{=}\hfill & 55.844\hfill & \hfill \\ \hfill & \hfill 4.6x& \text{=}\hfill & 55.844\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{4}\text{.6}\text{.}\text{\hspace{0.17em}}\text{A}\text{\hspace{0.17em}}\text{calculator}\text{\hspace{0.17em}}\text{would}\text{\hspace{0.17em}}\text{be}\text{\hspace{0.17em}}\text{useful}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}\text{this}\hfill \\ \hfill & \hfill & \hfill & \hfill & \text{point}\text{.}\hfill \\ \hfill & \hfill x& \text{=}\hfill & \frac{55.844}{4.6}\hfill & \hfill \\ \hfill & \hfill x& \text{=}\hfill & 12.14\hfill & \text{The}\text{\hspace{0.17em}}\text{wording}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{problem}\text{\hspace{0.17em}}\text{implies}\text{\hspace{0.17em}}two\text{\hspace{0.17em}}\text{numbers}\text{\hspace{0.17em}}\text{are}\text{\hspace{0.17em}}\text{needed}\hfill \\ \hfill & \hfill & \text{=}\hfill & \hfill & \text{for}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{complete}\text{\hspace{0.17em}}\text{solution}\text{.}\text{\hspace{0.17em}}\text{We}\text{\hspace{0.17em}}\text{need}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{number}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{units}\text{\hspace{0.17em}}\text{of}\hfill \\ \hfill & \hfill & \hfill & \hfill & \text{energy}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{other}\text{\hspace{0.17em}}\text{star.}\hfill \\ \hfill & \hfill 3.6x& \text{=}\hfill & 3.6\left(12.14\right)\hfill & \hfill \\ \hfill & \hfill & \text{=}\hfill & 43.704\hfill & \hfill \\ \text{Step}\text{\hspace{0.17em}}\text{4:}\hfill & \hfill 12.14+43.704& \text{=}\hfill & 55.844.\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\text{\hspace{0.17em}}\hfill \end{array}\\ \begin{array}{ll}\text{Step}\text{\hspace{0.17em}}5:\hfill & \text{One}\text{\hspace{0.17em}}\text{star}\text{\hspace{0.17em}}\text{gives}\text{\hspace{0.17em}}\text{off}\text{\hspace{0.17em}}12.14\text{\hspace{0.17em}}\text{units}\text{​}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{energy}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{other}\text{\hspace{0.17em}}\text{star}\text{\hspace{0.17em}}\text{gives}\text{\hspace{0.17em}}\text{off}\text{\hspace{0.17em}}43.704\text{\hspace{0.17em}}\text{units}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{energy}\text{.}\hfill \end{array}\end{array}$

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
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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
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