3.7 Use a Problem-Solving Strategy and Applications
Learning Objectives
By the end of this section, you will be able to:
- Approach word problems with a positive attitude
- Use a problem-solving strategy for word problems
- Solve number problems
- Translate and solve basic percent equations
- Solve percent applications
- Find percent increase and percent decrease
- Solve simple interest applications
- Solve applications with discount or mark-up
Try It
Before you get started, take this readiness quiz:
1) Translate “[latex]6[/latex] less than twice [latex]x[/latex]” into an algebraic expression.
2) Solve: [latex]\frac{2}{3}x=24[/latex]
3) Solve: [latex]3x+8=14[/latex]
4) Convert [latex]4.5\%[/latex] to a decimal.
5) Convert [latex]0.6[/latex] to a percent.
6) Round [latex]0.875[/latex] to the nearest hundredth.
7) Multiply [latex](4.5)(2.38)[/latex]
8) Solve [latex]3.5=0.7n[/latex]
9) Subtract [latex]50-37.45[/latex]
Approach Word Problems with a Positive Attitude
“If you think you can… or think you can’t… you’re right.”—Henry Ford
The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?
How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?
Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?
When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.
Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure 3.7.2 and say them out loud.
Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!
Use a Problem-Solving Strategy for Word Problems
We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.
Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem-solving.
HOW TO
Use a Problem-Solving Strategy to Solve Word Problems.
- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.
Example 3.7.1
Pilar bought a purse on sale for [latex]\$18[/latex], which is one-half of the original price. What was the original price of the purse?
Solution
Step 1: Read the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.
In this problem, is it clear what is being discussed? Is every word familiar?
Step 2: Identify what you are looking for.
Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!
In this problem, the words “what was the original price of the purse” tell us what we need to find.
Step 3: Name what we are looking for.
Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.
Let [latex]p[/latex] = the original price of the purse.
Step 4: Translate into an equation.
It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.
Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.
Restate the problem in one sentence with all the important information.
[latex]\underbrace{18}\;\underbrace{\text{is}}\underbrace{\;\text{one-half the original price.}}[/latex]
Translate into an equation.
[latex]18\;=\;\frac12\;\times\;p[/latex]
Step 5: Solve the equation using good algebraic techniques.
Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.
Solve the equation:
[latex]\begin{align*}&\;&18&=\frac12p\\ &\text{Multiply both sides by 2.}\;\;&{\color{red}{2}}\;\times\;18&={\color{red}{2}}\;\times\;\frac12p\\ &\text{Simplify}\;\;&36&=p\\ \end{align*}[/latex]
Step 6: Check the answer in the problem to make sure it makes sense.
We solved the equation and found that [latex]p=36[/latex], which means “the original price” was [latex]\$36[/latex].
Does [latex]$36[/latex] make sense in the problem? Yes, because [latex]18[/latex] is one-half of [latex]36[/latex], and the purse was on sale at half the original price.
Step 7: Answer the question with a complete sentence. The problem asked “What was the original price of the purse?”
The answer to the question is: “The original price of the purse was [latex]$36[/latex].”
If this were a homework exercise, our work might look like this:
Pilar bought a purse on sale for [latex]\$18[/latex], which is one-half the original price. What was the original price of the purse?
Let [latex]p=[/latex] the original price.
[latex]18[/latex] is one-half the original price.
[latex]\begin{align*}&\;&18&=\frac12p\\ &\text{Multiply both sides by 2.}\;\;&{\color{red}{2}}\;\times\;18&={\color{red}{2}}\;\times\;\frac12p\\ &\text{Simplify}\;\;&36&=p\\ \end{align*}[/latex]
Step 8: Check. Is [latex]\$36[/latex] a reasonable price for a purse?
Yes.
Is [latex]18[/latex] one half of [latex]36[/latex]?
[latex]\begin{align*} 18&\overset?=\frac12\times36\\ 18&=18\checkmark \end{align*}[/latex]
The original price of the purse was [latex]\$36[/latex].
Try It
10) Joaquin bought a bookcase on sale for [latex]\$120[/latex], which was two-thirds of the original price. What was the original price of the bookcase?
Solution
[latex]\$180[/latex]
11) Two-fifths of the songs in Mariel’s playlist are country. If there are [latex]16[/latex] country songs, what is the total number of songs in the playlist?
Solution
[latex]40[/latex]
Let’s try this approach with another example.
Example 3.7.2
Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were [latex]11[/latex] girls in the study group. How many boys were in the study group?
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
How many boys were in the study group?
Step 3: Name. Choose a variable to represent the number of boys.
Let [latex]n=[/latex] the number of boys.
Step 4: Translate. Restate the problem in one sentence with all the important information.
[latex]\underbrace{\text{The number of girls}\;(11)}\;\underbrace{\text{was}}\underbrace{\;\text{three more than twice the number of boys.}}[/latex]
Translate into an equation.
[latex]11\;=\;2b\;+3[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&11\;&=\;2b\;+3\\ &\text{Subtract 3 from each side}&11\;{\color{red}{-}}{\color{red}{\;}}{\color{red}{3}}\;&=\;2b\;+3\;{\color{red}{-}}{\color{red}{\;}}{\color{red}{3}}\\ &\text{Simplify}&8&=2b\\ &\text{Divide each side by 2}&\frac8{\color{red}{2}}\;&=\;\frac{2b}{\color{red}{2}}\\ &\text{Simplify}&4&=b \end{align*}[/latex]
Step 6: Check. First, is our answer reasonable?
Yes, having [latex]4[/latex] boys in a study group seems OK. The problem says the number of girls was [latex]3[/latex] more than twice the number of boys. If there are four boys, does that make eleven girls? Twice [latex]4[/latex] boys is [latex]8[/latex]. Three more than [latex]8[/latex] is [latex]11[/latex].
Step 7: Answer the question.
There were [latex]4[/latex] boys in the study group.
Try It
12) Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was [latex]3[/latex] more than twice the number of notebooks. He bought [latex]7[/latex] textbooks. How many notebooks did he buy?
Solution
[latex]2[/latex]
13) Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed [latex]22[/latex] Sudoku puzzles. How many crossword puzzles did he do?
Solution
[latex]7[/latex]
Solve Number Problems
Now that we have a problem-solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem-solving strategy outlined above.
Example 3.7.3
The difference of a number and six is [latex]13[/latex]. Find the number.
Solution
Step 1: Read the problem. Are all the words familiar?
Step 2: Identify what we are looking for.
The number.
Step 3: Name. Choose a variable to represent the number.
Let [latex]n[/latex] = the number.
Step 4: Translate. Remember to look for clue words like “difference… of… and…”
Restate the problem as one sentence.
[latex]\underbrace{\text{The difference of the number and}\;6}\;\underbrace{\text{is}}\underbrace{\;13.}[/latex]
Translate into an equation.
[latex]n-6=13[/latex]
Step 5: Solve the equation.
Simplify.
[latex]\begin{align*}n-6&=13\\n&=19\end{align*}[/latex]
Step 6: Check.
The difference of [latex]19[/latex] and [latex]6[/latex] is [latex]13[/latex]. It checks!
Step 7: Answer the question.
The number is [latex]19[/latex].
Try It
14) The difference of a number and eight is [latex]17[/latex]. Find the number.
Solution
[latex]25[/latex]
15) The difference of a number and eleven is [latex]-7[/latex]. Find the number.
Solution
[latex]4[/latex]
Example 3.7.4
The sum of twice a number and seven is [latex]15[/latex]. Find the number.
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
The number.
Step 3: Name. Choose a variable to represent the number.
Let [latex]n[/latex] = the number.
Step 4: Translate.
Restate the problem as one sentence.
[latex]\underbrace{\text{The sum of twice a number and}\;7}\;\underbrace{\text{is}}\underbrace{\;13.}[/latex]
Translate into an equation.
[latex]2n\;+\;7\;=\;15[/latex]
Step 5: Solve the equation.
[latex]\begin{align*} &\text{Subtract 7 from each side and simplify.}&2n+7&=15\\ &\text{Divide each side by 2 and simplify.}&2n&=8\\ &\;&n&=4 \end{align*}[/latex]
Step 6: Check.
Is the sum of twice [latex]4[/latex] and [latex]7[/latex] equal to [latex]15[/latex]?
Step 7: Answer the question.
The number is [latex]4[/latex].
Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.
Try It
16) The sum of four times a number and two is [latex]14[/latex]. Find the number.
Solution
[latex]3[/latex]
17) The sum of three times a number and seven is [latex]25[/latex]. Find the number.
Solution
[latex]6[/latex]
Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.
Example 3.7.5
One number is five more than another. The sum of the numbers is [latex]21[/latex]. Find the numbers.
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
We are looking for two numbers.
Step 3: Name. We have two numbers to name and need a name for each.
Choose a variable to represent the first number.
[latex]n=1^{st}\;number[/latex]
What do we know about the second number?
[latex]n+5=2^{nd}\;number[/latex]
Step 4: Translate. Restate the problem as one sentence with all the important information.
The sum of the 1st number and the 2nd number is [latex]21[/latex].
Translate into an equation.
[latex]\underbrace{1^{st}\;\text{number}}\;+\;\underbrace{2^{nd}\;\text{number}}\;\underbrace=\;\underbrace{21}[/latex]
Substitute the variable expressions.
[latex]n+n+5=21[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&n+n+5&=21\\ &\text{Combine like terms.}&2n+5&=21\\ &\text{Subtract 5 from both sides and simplify.}&2n&=16\\ &\text{Divide by 2 and simplify.}&n&=8\;\;\;&1^{st}\;number\\ &\text{Find the second number, too}&{\color{red}{n}}&+5\;\;\;&2^{nd}\;number\\ &\;&{\color{red}{8}}+5&=13\;&\end{align*}[/latex]
Step 6: Check.
Do these numbers check in the problem?
[latex]\begin{align*}&\text{Is one number 5 more than the other?}\;&13&\overset?=8+5\\ &\text{Is thirteen 5 more than 8? Yes.}\;&13&=13\checkmark\\ &\text{Is the sum of the two numbers 21?}\;&8+13&\overset?=21\\ &\;&21&=21\checkmark \end{align*}[/latex]
Step 7: Answer the question.
The numbers are [latex]8[/latex] and [latex]13[/latex].
Try It
18) One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.
Solution
[latex]9[/latex], [latex]15[/latex]
19) The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.
Solution
[latex]27[/latex], [latex]31[/latex]
Example 3.7.6
The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
We are looking for two numbers.
Step 3: Name.
Choose a variable.
[latex]n=1^{st}\;number[/latex]
One number is [latex]4[/latex] less than the other.
[latex]n-4=2^{nd}\;number[/latex]
Step 4: Translate.
The sum of the [latex]2[/latex] numbers is negative [latex]14[/latex].
Write as one sentence.
[latex]\underbrace{1^{st}\;\text{number}}\;+\;\underbrace{2^{nd}\;\text{number}}\underbrace{\;\text{is}\;}\underbrace{\text{negative}\;14}[/latex]
Translate into an equation.
[latex]n\;+\;n-4\;=\;-14[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&n+n-4&=-14\\ &\text{Combine like terms.}\;&2n-4&=-14\\ &\text{Add 4 to each side and simplify.}\;&2n&=-10\\ &\;&n&=-5\;\;&1^{st}\;number\\&\text{Simplify.}\;&{\color{red}{n}}&-4\;\;&2^{nd}\;number\\ &\;&{\color{red}{-5}}-4&=-9\;&\end{align*}[/latex]
Step 6: Check.
[latex]\begin{align*} &\text{Is -9 four less than -5?}\;&-5-4&\overset?=-9\\ &\;&-9&=-9\checkmark\\ &\text{Is their sum -14?}\;&-5+(-9)&\overset?=-14\\ &\;&-14&=-14\checkmark\end{align*}[/latex]
Step 7: Answer the question.
The numbers are [latex]−5[/latex] and [latex]−9[/latex].
Try It
20) The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.
Solution
[latex]-15[/latex], [latex]-8[/latex]
21) The sum of two numbers is [latex]-18[/latex]. One number is [latex]40[/latex] more than the other. Find the numbers.
Solution
[latex]-29[/latex],[latex]11[/latex]
Example 3.7.7
One number is ten more than twice another. Their sum is one. Find the numbers.
Solution
Step 1: Read the problem.
Step 2: Identify what you are looking for.
We are looking for two numbers.
Step 3: Name.
Choose a variable.
[latex]x=1^{st}\;number[/latex]
One number is [latex]10[/latex] more than twice another.
[latex]2x+10=2^{nd}\;number[/latex]
Step 4: Translate.
Their sum is one.
Restate as one sentence.
The sum of the two numbers is [latex]1[/latex].
Translate into an equation.
[latex]x+2x+10=1[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&x+2x+10&=1\\ &\text{Combine like terms.}\;&3x+10&=1\\ &\text{Subtract 10 from each side.}\;&3x&=-9\\ &\text{Divide each side by 3.}\;&x&=-3\;\;\;&1^{st}\;number\\ &\;&2{\color{red}{x}}&+10\;\;\;&2^{nd}\;number\\ &\;&2({\color{red}{-3}})+10&=4&\;\end{align*}[/latex]
Step 6: Check.
[latex]\begin{align*} &\text{Is ten more than twice −3 equal to 4?}\;&2(-3)+10&\overset?=4\\ &\;&-6+10&\overset?=4\\ &\;&4&=4\checkmark\\ &\text{Is their sum 1?}\;&-3+4&\overset?=1\\ &\;&\;1&=1 \end{align*}[/latex]
Step 7: Answer the question.
The numbers are [latex]−3[/latex] and [latex]−4[/latex].
Try It
22) One number is eight more than twice another. Their sum is negative four. Find the numbers.
Solution
[latex]-4[/latex], [latex]0[/latex]
23) One number is three more than three times another. Their sum is [latex]-5[/latex]. Find the numbers.
Solution
[latex]-3[/latex], [latex]-2[/latex]
Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:
Notice that each number is one more than the number preceding it. So if we define the first integer as [latex]n[/latex], the next consecutive integer is [latex]n+1[/latex]. The one after that is one more than [latex]n+1[/latex], so it is [latex]n+1+1[/latex], which is [latex]n+2[/latex].
| [latex]n[/latex] | 1st integer |
| [latex]n+1[/latex] | 2nd consecutive integer |
| [latex]n+2[/latex] | 3rd consecutive integer… etc. |
Example 3.7.8
The sum of two consecutive integers is [latex]47[/latex]. Find the numbers.
Solution
Step 1: Read the problem.
Step 2: Identify what you are looking for.
Two consecutive integers.
Step 3: Name each number.
[latex]\begin{align*} n&=\;1^{st}\;integer \\ n+1&=\;2^{nd}\;next\;consecutive\;integer \end{align*}[/latex]
Step 4: Translate.
Restate as one sentence.
The sum of the integers is [latex]47[/latex].
Translate into an equation.
[latex]n+n+1=47[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&n+n+1&=47\\ &\text{Combine like terms.}\;&2x+1&=47\\ &\text{Subtract 1 from each side.}\;&2n&=46\\ &\text{Divide each side by 2.}\;&n&=23\;\;\;&1^{st}\;integer\\ &\;&{\color{red}{n}}&+1\;\;\;&2^{nd}\;integer\\ &\;&{\color{red}{23}}+1&=24\;&\end{align*}[/latex]
Step 6: Check.
[latex]\begin{align*} 23+24&\overset?=47\\ 47&=47\checkmark\\ \end{align*}[/latex]
Step 7: Answer the question.
The two consecutive integers are [latex]23[/latex] and [latex]24[/latex].
Try It
24) The sum of two consecutive integers is [latex]95[/latex]. Find the numbers.
Solution
[latex]47[/latex], [latex]48[/latex]
25) The sum of two consecutive integers is [latex]-31[/latex]. Find the numbers.
Solution
[latex]-16[/latex], [latex]-15[/latex]
Example 3.7.9
Find three consecutive integers whose sum is [latex]-42[/latex].
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
three consecutive integers
Step 3: Name each of the three numbers.
[latex]\begin{align*} n&=1^{st}\;integer \\ n+1&=2^{nd}\;consecutive\;integer\\ n+2&=3^{rd}\;consecutive\;integer \end{align*}[/latex]
Step 4: Translate.
Restate as one sentence.
The sum of the three integers is [latex]−42[/latex].
Translate into an equation.
[latex]n+n+1+n+2=-42[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&n+n+1+n+2&=-42\\ &\text{Combine like terms.}\;&3n+3&=-42\\ &\text{Subtract 3 from each side.}\;&3n&=-45\\ &\text{Divide each side by 3.}\;&n&=-15\;\;\;&1^{st}\;integer\\ &\;&{\color{red}{n}}&+1\\&\;&{\color{red}{-}}{\color{red}{15}}+1&=-14\;\;\;&2^{nd}\;integer&\\&\;&{\color{red}{-15}}+2&=-13\;\;\;&3^{rd}\;integer\;&\end{align*}[/latex]
Step 6: Check.
[latex]\begin{align*} -13+(-14)+(-15)&\overset?=-42\\-42&=-42\checkmark \end{align*}[/latex]
Step 7: Answer the question.
The three consecutive integers are [latex]−13[/latex], [latex]−14[/latex], and [latex]−15[/latex].
Try It
26) Find three consecutive integers whose sum is [latex]-96[/latex].
Solution
[latex]-33[/latex], [latex]-32[/latex], [latex]-31[/latex]
27) Find three consecutive integers whose sum is [latex]-36[/latex].
Solution
[latex]-13[/latex], [latex]-12[/latex], [latex]-11[/latex]
Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:
[latex]64[/latex], [latex]66[/latex], [latex]68[/latex]
[latex]-12[/latex], [latex]-10[/latex], [latex]-8[/latex]
Notice each integer is [latex]2[/latex] more than the number preceding it. If we call the first one [latex]n[/latex], then the next one is [latex]n+2[/latex]. The next one would be [latex]n+2+2[/latex] or [latex]n+4[/latex].
| [latex]n[/latex] | 1st even integer |
| [latex]n+2[/latex] | 2nd consecutive even integer |
| [latex]n+4[/latex] | 3rd consecutive even integer… etc. |
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers [latex]77[/latex], [latex]79[/latex], and [latex]81[/latex].
[latex]77[/latex], [latex]79[/latex], [latex]81[/latex]
[latex]n[/latex], [latex]n+2[/latex], [latex]n+4[/latex]
| [latex]n[/latex] | 1st odd integer |
| [latex]n+2[/latex] | 2nd consecutive odd integer |
| [latex]n+4[/latex] | 3rd consecutive odd integer… etc. |
Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add [latex]2[/latex] .
Example 3.7.10
Find three consecutive even integers whose sum is [latex]84[/latex].
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
three consecutive even integers
Step 3: Name the integers.
Let [latex]\begin{align*} n&=1^{st}\;integer \\ n+2&=2^{nd}\;consecutive\;even\;integer\\ n+4&=3^{rd}\;consecutive\;even\;integer \end{align*}[/latex]
Step 4: Translate.
Restate as one sentence.
The sum of the three even integers is [latex]84[/latex].
Translate into an equation.
[latex]n+n+2+n+4=84[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&n+n+2+n+4&=84\\ &\text{Combine like terms.}\;&3n+6&=84\\ &\text{Subtract 6 from each side.}\;&3n+6-6&=84-6\\ &\text{Divide each side by 3.}\;&3n&=78\\ &\;&n&=26\;&1^{st}\;integer\\ &\;&{\color{red}{n}}&+2\;\;\;&2^{nd}\;integer\\ &\;&{\color{red}{26}}+2&=28\\ &\;&{\color{red}{n}}&+4\;\;\;&3^{rd}\;integer\\ &\;&{\color{red}{26}}+4&=30 \end{align*}[/latex]
Step 6: Check.
[latex]\begin{align*}26+28+30&\overset?=84\\ 84&=84\checkmark \end{align*}[/latex]
Step 7: Answer the question.
The three consecutive integers are [latex]26[/latex], [latex]28[/latex], and [latex]30[/latex].
Try It
28) Find three consecutive even integers whose sum is [latex]102[/latex].
Solution
[latex]32[/latex], [latex]34[/latex], [latex]36[/latex]
29) Find three consecutive even integers whose sum is [latex]-24[/latex].
Solution
[latex]-10[/latex], [latex]-8[/latex], [latex]-6[/latex]
Example 3.7.11
A married couple together earns [latex]\$110,000[/latex] a year. The wife earns [latex]\$16,000[/latex] less than twice what her husband earns. What does the husband earn?
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
How much does the husband earn?
Step 3: Name.
Choose a variable to represent the amount the husband earns.
Let [latex]h=[/latex] the amount the husband earns.
The wife earns [latex]\$16,000[/latex] less than twice that.
[latex]2h-16,000[/latex] the amount the wife earns.
Together the husband and wife earn $110,000.
Step 4: Translate.
Restate the problem in one sentence with all the important information.
[latex]\underbrace{\text{The amount the husband earns}\;}+\;\underbrace{\text{the amount the wife earns}}\;\underbrace{\text{is}}\;\underbrace{\$110,000}[/latex]
Translate into an equation.
[latex]h+2h-16,000=110,000[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&h+2h-16,000&=110,000\\ &\text{Combine like terms.}\;&3h-16,000&=110,000\\ &\text{Add 16,000 to both sides and simplify.}\;&3h&=126,000\\ &\text{Divide each side by 3.}\;&h&=42,000\\ &\;&\;&\$42,000\;\;\;&the\;amount\;the\;husband\;earns\\ &\;&\;&2{\color{red}{h}}-16\;\;\;&the\;amount\;the\;wife\;earns\\ &\;&2({\color{red}{42,000}})&-16,000\\ &\;&84,000-16,000&=68,000 \end{align*}[/latex]
Step 6: Check.
If the wife earns [latex]\$68,000[/latex] and the husband earns [latex]\$42,000[/latex] is the total [latex]$110,000[/latex]? Yes!
Step 7: Answer the question.
The husband earns [latex]\$42,000[/latex] a year.
Try It
30) According to the National Automobile Dealers Association, the average cost of a car in 2014 was [latex]\$28,500[/latex]. This was [latex]\$1,500[/latex] less than [latex]6[/latex] times the cost in 1975. What was the average cost of a car in 1975?
Solution
[latex]\$5,000[/latex]
31) U.S. Census data shows that the median price of new home in the United States in November 2014 was [latex]\$280,900[/latex]. This was [latex]\$10,700[/latex] more than 14 times the price in November 1964. What was the median price of a new home in November 1964?
Solution
[latex]\$19,300[/latex]
Translate and Solve Basic Percent Equations
We will solve percent equations using the methods we used to solve equations with fractions or decimals. Without the tools of algebra, the best method available to solve percent problems was by setting them up as proportions. Now as an algebra student, you can just translate English sentences into algebraic equations and then solve the equations.
We can use any letter you like as a variable, but it is a good idea to choose a letter that will remind us of what you are looking for. We must be sure to change the given percent to a decimal when we put it in the equation.
Example 3.7.12
Translate and solve: What number is [latex]35\%[/latex] of [latex]90[/latex]?
Solution
[latex]\underbrace{\text{What number}}\;\underbrace{\text{is}}\;\underbrace{35\%}\underbrace{\text{of}}\;\underbrace{90?}[/latex]
Step 1: Translate into algebra. Let [latex]n=[/latex] the number.
Remember “of” means multiply, “is” means equals.
[latex]n=0.25\cdot90[/latex]
Step 2: Multiply.
[latex]n=31.5[/latex]
[latex]31.5[/latex] is [latex]35\%[/latex] of [latex]90[/latex].
Try It
32) Translate and solve:
What number is [latex]45\%[/latex] of [latex]80[/latex]?
Solution
[latex]36[/latex]
33) Translate and solve:
What number is [latex]55\%[/latex] of [latex]60[/latex]?
Solution
[latex]33[/latex]
We must be very careful when we translate the words in the next example. The unknown quantity will not be isolated at first, like it was in Example 3.7.12. We will again use direct translation to write the equation.
Example 3.7.13
Translate and solve: [latex]6.5\%[/latex] of what number is [latex]\$1.17[/latex]?
Solution
[latex]\underbrace{6.5\%}\underbrace{\text{of}}\;\underbrace{\text{what number}}\;\underbrace{\text{is}}\;\underbrace{\$1.17?}[/latex]
Step 1: Translate. Let [latex]n=[/latex] the number.
[latex]0.065\cdot n=1.17[/latex]
Step 2: Solve.
[latex]\begin{align*}&\;&0.065\cdot n&=1.17\\ &\text{Multiply.}\;&0.065n&=1.17\\ &\text{Divide both sides by 0.065 and simplify.}\;&n&=18\\ \end{align*}[/latex]
[latex]6.5\%[/latex] of [latex]\$18[/latex] is [latex]\$1.17[/latex].
Try It
34) Translate and solve: [latex]7.5\%[/latex] of what number is [latex]\$1.95[/latex]?
Solution
[latex]\$26[/latex]
35) Translate and solve: [latex]8.5\%[/latex] of what number is [latex]\$3.06[/latex]?
Solution
[latex]\$36[/latex]
In the next example, we are looking for the percent.
Example 3.7.14
Translate and solve: [latex]144[/latex] is what percent of [latex]96[/latex]?
Solution
[latex]\underbrace{144}\;\underbrace{\text{is}}\;\underbrace{\text{what percent}}\;\underbrace{\text{of}}\;\underbrace{96?}[/latex]
Step 1: Translate into algebra. Let [latex]p=[/latex] the percent.
[latex]144=p\cdot96[/latex]
Step 2: Solve.
[latex]\begin{align*}&\;&144&=p\cdot96\\ &\text{Multiply.}\;&144&=96p\\ &\text{Divide both sides by 96 and simplify.}\;&1.5&=p\\ &\text{Convert to percent.}\;&150\%&=p\\ \end{align*}[/latex]
[latex]144[/latex] is [latex]150\%[/latex] of [latex]96[/latex].
Note that we are asked to find percent, so we must have our final result in percent form.
Try It
36) Translate and solve:
[latex]110[/latex] is what percent of [latex]88[/latex]?
Solution
[latex]125\%[/latex]
37) Translate and solve:
[latex]126[/latex] is what percent of [latex]72[/latex]?
Solution
[latex]175\%[/latex]
Solve Applications of Percent
Many applications of percent—such as tips, sales tax, discounts, and interest—occur in our daily lives. To solve these applications we’ll translate to a basic percent equation, just like those we solved in previous examples. Once we translate the sentence into a percent equation, we know how to solve it.
We will restate the problem solving strategy we used earlier for easy reference.
HOW TO
Use a Problem-Solving Strategy to Solve an Application.
- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.
Now that we have the strategy to refer to, and have practised solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications will involve everyday situations, you can rely on your own experience.
Example 3.7.15
Dezohn and his girlfriend enjoyed a nice dinner at a restaurant and his bill was [latex]\$68.50[/latex]. He wants to leave an [latex]18\%[/latex] tip. If the tip will be [latex]18\%[/latex] of the total bill, how much tip should he leave?
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the amount of tip should Dezohn leave
Step 3: Name what we are looking for.
Choose a variable to represent it.
Let [latex]t =[/latex] amount of tip.
Step 4: Translate into an equation.
Write a sentence that gives the information to find it.
The tip is 18% of the total bill.
Translate the sentence into an equation.
[latex]\underbrace{\text{The tip}}\underbrace{\;\text{is}}\;\underbrace{18\%}\;\underbrace{\text{of}}\;\underbrace{\$68.50}[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&t&=0.18\cdot68.50\\ &\text{Multiply.}\;&t&=12.33\\ \end{align*}[/latex]
Step 6: Check. Does this make sense?
Yes, [latex]20\%[/latex] of [latex]\$70[/latex] is [latex]\$14[/latex].
Step 7: Answer the question with a complete sentence.
Dezohn should leave a tip of [latex]\$12.33[/latex].
Notice that we used [latex]t[/latex] to represent the unknown tip.
Try It
38) Cierra and her sister enjoyed a dinner in a restaurant and the bill was [latex]\$81.50[/latex]. If she wants to leave [latex]18\%[/latex] of the total bill as her tip, how much should she leave?
Solution
[latex]\$14.67[/latex]
39) Kimngoc had lunch at her favourite restaurant. She wants to leave [latex]15\%[/latex] of the total bill as her tip. If her bill was [latex]\$14.40[/latex], how much will she leave for the tip?
Solution
[latex]\$2.16[/latex]
Example 3.7.16
The label on Masao’s breakfast cereal said that one serving of cereal provides [latex]85[/latex] milligrams (mg) of potassium, which is [latex]2\%[/latex] of the recommended daily amount. What is the total recommended daily amount of potassium?
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the total amount of potassium that is recommended
Step 3: Name what we are looking for.
Choose a variable to represent it.
Let [latex]a=[/latex] total amount of potassium.
Step 4: Translate.
Write a sentence that gives the information to find it.
[latex]\underbrace{85mg}\;\underbrace{\text{is}}\;\underbrace{2\%}\;\underbrace{\text{of the}}\;\underbrace{\text{total amount}}[/latex]
Translate into an equation.
[latex]85=0.02\cdot a[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}85&=0.02\cdot a\\ 4,250&=a \end{align*}[/latex]
Step 6: Check. Does this make sense?
Yes, [latex]2\%[/latex] is a small percent and [latex]85[/latex] is a small part of [latex]4,250[/latex].
Step 7: Answer the question with a complete sentence.
The amount of potassium that is recommended is [latex]4,250 mg[/latex].
Try It
40) One serving of wheat square cereal has seven grams of fibre, which is [latex]28\%[/latex] of the recommended daily amount. What is the total recommended daily amount of fibre?
Solution
[latex]25 g[/latex]
41) One serving of rice cereal has [latex]190 mg[/latex] of sodium, which is [latex]8\%[/latex] of the recommended daily amount. What is the total recommended daily amount of sodium?
Solution
[latex]2,375 mg[/latex]
Example 3.7.17
Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was [latex]480[/latex] calories, and had [latex]240[/latex] calories of fat. What percent of the total calories in each brownie comes from fat?
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the percent of the total calories from fat
Step 3: Name what we are looking for.
Choose a variable to represent it.
Let [latex]p=[/latex] percent of fat.
Step 4: Translate. Write a sentence that gives the information to find it.
[latex]\underbrace{\text{What percent}}\;\underbrace{\text{of}}\;\underbrace{480}\;\underbrace{\text{is}}\;\underbrace{240?}[/latex]
Translate into an equation.
[latex]p\cdot480=240[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&480p&=240\\ &\text{Divide by 480}\;\;&p&=0.5\\ &\text{Put in percent form}\;\;&p&=50\%\\ \end{align*}[/latex]
Step 6: Check. Does this make sense?
Yes, [latex]240[/latex] is half of [latex]480[/latex], so [latex]50\%[/latex] makes sense.
Step 7: Answer the question with a complete sentence.
Of the total calories in each brownie, [latex]50\%[/latex] is fat.
Try It
42) Solve. Round to the nearest whole percent.
Veronica is planning to make muffins from a mix. The package says each muffin will be [latex]230[/latex] calories and [latex]60[/latex] calories will be from fat. What percent of the total calories is from fat?
Solution
[latex]26\%[/latex]
43) Solve. Round to the nearest whole percent.
The mix Ricardo plans to use to make brownies says that each brownie will be [latex]190[/latex] calories, and [latex]76[/latex] calories are from fat. What percent of the total calories are from fat?
Solution
[latex]40\%[/latex]
Find Percent Increase and Percent Decrease
People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.
To find the percent increase, first we find the amount of increase, the difference of the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.
HOW TO
Find the Percent Increase.
- Find the amount of increase.
new amount − original amount = increase - Find the percent increase.
The increase is what percent of the original amount?
Example 3.7.18
In 2011, the California governor proposed raising community college fees from [latex]\$26[/latex] a unit to [latex]\$36[/latex] a unit. Find the percent increase. (Round to the nearest tenth of a percent.)
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the percent increase
Step 3: Name what we are looking for.
Choose a variable to represent it.
Let [latex]p=[/latex] the percent.
Step 4: Translate.
Write a sentence that gives the information to find it.
new amount − original amount = increase
First find the amount of increase.
[latex]36-26=10[/latex]
Find the percent.
Increase is what percent of the original amount?
[latex]\underbrace{10}\underbrace{\;\text{is}}\;\underbrace{\text{what percent}}\;\underbrace{\text{of}}\underbrace{\;26?}[/latex]
Translate into an equation.
[latex]10=p\cdot 26[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&10&=26p\\ &\text{Divide by 26.}\;&0.384&=p\\ &\text{Change to percent form; round to the nearest tenth.}\;&38.4\%&=p\\ \end{align*}[/latex]
Step 6: Check. Does this make sense?
Yes, [latex]38.4\%[/latex] is close to [latex]\frac{1}{3}[/latex] and [latex]10[/latex] is close to [latex]\frac{1}{3}[/latex] of [latex]26[/latex].
Step 7: Answer the question with a complete sentence.
The new fees represent a [latex]38.4\%[/latex] increase over the old fees.
Notice that we rounded the division to the nearest thousandth in order to round the percent to the nearest tenth.
Try It
44) Find the percent increase. (Round to the nearest tenth of a percent.)
In 2011, the IRS increased the deductible mileage cost to [latex]55.5[/latex] cents from [latex]51[/latex] cents.
Solution
[latex]8.8\%[/latex]
45) Find the percent increase.
In 1995, the standard bus fare in Chicago was [latex]\$1.50[/latex]. In 2008, the standard bus fare was [latex]\$2.25[/latex].
Solution
[latex]50\%[/latex]
Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference of the original amount and the new amount. Then we find what percent the amount of decrease is of the original amount.
HOW TO
Find the Percent Decrease.
- Find the amount of decrease.
original amount − new amount = decrease - Find the percent decrease.
Decrease is what percent of the original amount?
Example 3.7.19
The average price of a gallon of gas in one city in June 2014 was [latex]\$3.71[/latex]. The average price in that city in July was [latex]\$3.64[/latex]. Find the percent decrease.
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the percent decrease
Step 3: Name what we are looking for.
Choose a variable to represent that quantity.
Let [latex]p[/latex] = the percent decrease.
Step 4: Translate.
Write a sentence that gives the information to find it.
First find the amount of decrease.
[latex]3.71 - 3.64 = 0.07[/latex]
Find the percent.
Decrease is what percent of the original amount?
[latex]\underbrace{0.07}\underbrace{\;\text{is}}\;\underbrace{\text{what percent}}\;\underbrace{\text{of}}\underbrace{\;3.71?}[/latex]
Translate into an equation.
[latex]0.07=p\cdot3.71[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&0.07&=3.71p\\ &\text{Divide by 3.71.}\;&0.019&=p\\ &\text{Change to percent form; round to the nearest tenth.}\;&1.9\%&=p\\ \end{align*}[/latex]
Step 6: Check. Does this make sense?
Yes, if the original price was [latex]\$4[/latex], a [latex]2\%[/latex] decrease would be [latex]8[/latex] cents.
Step 7: Answer the question with a complete sentence.
The price of gas decreased [latex]1.9\%[/latex].
Try It
46) Find the percent decrease. (Round to the nearest tenth of a percent.)
The population of North Dakota was about [latex]672,000[/latex] in 2010. The population is projected to be about [latex]630,000[/latex] in 2020.
Solution
[latex]6.3\%[/latex]
47) Find the percent decrease.
Last year, Sheila’s salary was [latex]\$42,000[/latex]. Because of furlough days, this year, her salary was [latex]\$37,800[/latex].
Solution
[latex]10\%[/latex]
Solve Simple Interest Applications
Do you know that banks pay you to keep your money? The money a customer puts in the bank is called the principal, [latex]P[/latex], and the money the bank pays the customer is called the interest. The interest is computed as a certain percent of the principal; called the rate of interest, [latex]r[/latex]. We usually express rate of interest as a percent per year, and we calculate it by using the decimal equivalent of the percent. The variable [latex]t[/latex], (for time) represents the number of years the money is in the account.
To find the interest we use the simple interest formula, [latex]I=Prt[/latex].
Simple Interest
If an amount of money, [latex]P[/latex], called the principal, is invested for a period of [latex]t[/latex] years at an annual interest rate [latex]r[/latex], the amount of interest, [latex]I[/latex], earned is
| [latex]I = Prt[/latex] | where | [latex]I = interest[/latex]
[latex]P = principal[/latex] [latex]r = rate[/latex] [latex]t = time[/latex] |
Interest earned according to this formula is called simple interest.
Interest may also be calculated another way, called compound interest. This type of interest will be covered in later math classes.
The formula we use to calculate simple interest is [latex]I=Prt[/latex]. To use the formula, we substitute in the values the problem gives us for the variables, and then solve for the unknown variable. It may be helpful to organize the information in a chart.
Example 3.7.20
Nathaly deposited [latex]\$12,500[/latex] in her bank account where it will earn [latex]4\%[/latex] interest. How much interest will Nathaly earn in [latex]5[/latex] years?
[latex]I = ?[/latex]
[latex]P = $12,500[/latex]
[latex]r =4\%[/latex]
[latex]t = 5\;years[/latex]
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the amount of interest earned
Step 3: Name what we are looking for.
Choose a variable to represent that quantity.
Let [latex]I=[/latex] the amount of interest.
Step 4: Translate into an equation.
[latex]\begin{align*} &\text{Write the formula}\;&I&=Prt\\ &\text{Substitute in the given information}\;&I&=(12,500)(0.04)(5)\\ \end{align*}[/latex]
Step 5: Solve the equation.
[latex]I=2,500[/latex]
Step 6: Check: Does this make sense?
Is [latex]\$2,500[/latex] a reasonable interest on [latex]\$12,500[/latex]? Yes.
Step 7: Answer the question with a complete sentence.
The interest is [latex]\$2,500[/latex].
Try It
48) Areli invested a principal of [latex]\$950[/latex] in her bank account with interest rate [latex]3\%[/latex]. How much interest did she earn in [latex]5[/latex] years?
Solution
[latex]\$142.50[/latex]
49) Susana invested a principal of [latex]\$36,000[/latex] in her bank account with interest rate [latex]6.5\%[/latex]. How much interest did she earn in [latex]3[/latex] years?
Solution
[latex]\$7,020[/latex]
There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we don’t know the rate. To find the rate, we use the simple interest formula, substitute in the given values for the principal and time, and then solve for the rate.
Example 3.7.21
Loren loaned his brother [latex]\$3,000[/latex] to help him buy a car. In [latex]4[/latex] years his brother paid him back the [latex]\$3,000[/latex] plus [latex]\$660[/latex] in interest. What was the rate of interest?
[latex]I = \$660[/latex]
[latex]P = \$3,000[/latex]
[latex]r =[/latex]
[latex]t = 4\;years[/latex]
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the rate of interest
Step 3: Name what we are looking for.
Choose a variable to represent that quantity.
Let [latex]r =[/latex] the rate of interest.
Step 4: Translate into an equation.
[latex]\begin{align*} &\text{Write the formula}\;&I&=Prt\\ &\text{Substitute in the given information}\;&660&=(3000)r(4)\\ \end{align*}[/latex]
Step 5: Solve the equation.
[latex]\begin{align*}&\;&660&=(3,000)r(4)\\ &\text{Multiply.}\;&660&=(12,000)r \\&\text{Divide.}\;&0.055&=r\\ &\text{Change to percent form.}\;&5.5\%&=r\end{align*}[/latex]
Step 6: Check: Does this make sense?
[latex]\begin{align*}&\;&I&=Prt\\ &\;&660&\overset?=(3000)(0.55)(4)\\ &\;&660&=660\checkmark \end{align*}[/latex]
Step 7: Answer the question with a complete sentence.
The rate of interest was [latex]5.5\%[/latex].
Notice that in this example, Loren’s brother paid Loren interest, just like a bank would have paid interest if Loren invested his money there.
Try It
50) Jim loaned his sister [latex]\$5,000[/latex] to help her buy a house. In [latex]3[/latex] years, she paid him the [latex]\$5,000[/latex], plus [latex]\$900[/latex] interest. What was the rate of interest?
Solution
[latex]6\%[/latex]
51) Hang borrowed [latex]\$7,500[/latex] from her parents to pay her tuition. In [latex]5[/latex] years, she paid them [latex]\$1,500[/latex] interest in addition to the [latex]\$7,500[/latex] she borrowed. What was the rate of interest?
Solution
[latex]4\%[/latex]
Example 3.7.22
Eduardo noticed that his new car loan papers stated that with a [latex]7.5\%[/latex] interest rate, he would pay [latex]\$6,596.25[/latex] in interest over [latex]5[/latex] years. How much did he borrow to pay for his car?
Solution
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the amount borrowed (the principal)
Step 3: Name what we are looking for.
Choose a variable to represent that quantity.
Let [latex]P[/latex] = principal borrowed.
Step 4: Translate into an equation.
[latex]\begin{align*} &\text{Write the formula}\;&I&=Prt\\ &\text{Substitute in the given information}\;&6,596.25&=P(0.075)(5)\\ \end{align*}[/latex]
Step 5: Solve the equation.
[latex]\begin{align*} &\;&6,596&=P(0.075)(5)\\ &\text{Multiply.}\;&6,596.25&=0.375P\\ &\text{Divide.}\;&17,590&=P \end{align*}[/latex]
Step 6: Check: Does this make sense?
[latex]\begin{align*} I&=Prt\\ 6.596.25&\overset?=(14,590)(0.075)(5)\\ 6,596.25&=6,596.25\checkmark \end{align*}[/latex]
Step 7: Answer the question with a complete sentence.
The principal was [latex]\$17,590[/latex].
Try It
52) Sean’s new car loan statement said he would pay [latex]\$4,866.25[/latex] in interest from an interest rate of [latex]8.5\%[/latex] over [latex]5[/latex] years. How much did he borrow to buy his new car?
Solution
[latex]\$11,450[/latex]
53) In [latex]5[/latex] years, Gloria’s bank account earned [latex]\$2,400[/latex] interest at [latex]5\%[/latex]. How much had she deposited in the account?
Solution
[latex]\$9,600[/latex]
Solve Applications with Discount or Mark-up
Applications of discount are very common in retail settings. When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount, we multiply the discount rate by the original price.
We summarize the discount model in the box below.
Discount
Keep in mind that the sale price should always be less than the original price.
Example 3.7.23
Elise bought a dress that was discounted [latex]35\%[/latex] off of the original price of [latex]\$140[/latex].
What was a. the amount of discount and b. the sale price of the dress?
Solution
a. Original Price = [latex]\$140[/latex] Discount rate = [latex]35\%[/latex] Discount = ?
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the amount of discount
Step 3: Name what we are looking for.
Choose a variable to represent that quantity.
Let [latex]d=[/latex] the amount of discount.
Step 4: Translate into an equation.
Write a sentence that gives the information to find it.
The discount is 35% of $140.
Translate into an equation.
[latex]d=0.35(140)[/latex]
Step 5: Solve the equation.
[latex]\begin{align*} d&=0.35(140)\\ d&=49 \end{align*}[/latex]
Step 6: Check: Does this make sense?
Is a [latex]\$49[/latex] discount reasonable for a [latex]\$140[/latex] dress? Yes.
Step 7: Write a complete sentence to answer the question.
The amount of discount was [latex]\$49[/latex].
b. Read the problem again.
Step 1: Identify what we are looking for.
the sale price of the dress
Step 2: Name what we are looking for.
Choose a variable to represent that quantity.
Let [latex]s=[/latex] the sale price.
Step 3: Translate into an equation.
Write a sentence that gives the information to find it.
[latex]\underbrace{\text{The sale price}}\;\underbrace{\text{is}}\;\underbrace{\text{the}\;\$140}\underbrace{\;\text{minus}}\;\underbrace{\text{the}\;\$49\;\text{discount}}[/latex]
Translate into an equation.
[latex]s=140-49[/latex]
Step 4: Solve the equation.
[latex]\begin{align*} s&=140-49\\ s&=91 \end{align*}[/latex]
Step 5: Check. Does this make sense?
Is the sale price less than the original price?
Yes, [latex]\$91[/latex] is less than [latex]\$140[/latex].
Step 6: Answer the question with a complete sentence.
The sale price of the dress was [latex]\$91[/latex].
Try It
54) Find
a. the amount of discount and
b. the sale price:
Sergio bought a belt that was discounted [latex]40\%[/latex] from an original price of [latex]\$29[/latex].
Solution
a. [latex]\$11.60[/latex]
b. [latex]\$17.40[/latex]
55) Find
a. the amount of discount and
b. the sale price:
Oscar bought a barbecue that was discounted [latex]65\%[/latex] from an original price of [latex]\$395[/latex].
Solution
a. [latex]\$256.75[/latex]
b. [latex]\$138.25[/latex]
There may be times when we know the original price and the sale price, and we want to know the discount rate. To find the discount rate, first we will find the amount of discount and then use it to compute the rate as a percent of the original price. Example 3.7.24 will show this case.
Example 3.7.24
Jeannette bought a swimsuit at a sale price of [latex]\$13.95[/latex]. The original price of the swimsuit was [latex]\$31[/latex]. Find the:
a. amount of discount and
b. discount rate.
Solution
a.
Original price = [latex]\$31[/latex]
Discount = ?
Sale price = [latex]\$13.95[/latex]
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the amount of discount
Step 3: Name what we are looking for.
Choose a variable to represent that quantity.
Let [latex]d=[/latex] the amount of discount.
Step 4: Translate into an equation.
Write a sentence that gives the information to find it.
The discount is the difference between the original price and the sale price.
Translate into an equation.
[latex]d=31-13.95[/latex]
Step 5: Solve the equation.
[latex]\begin{align*} d&=31-13.95\\ d&=17.05 \end{align*}[/latex]
Step 6: Check: Does this make sense?
Is [latex]17.05[/latex] less than [latex]31[/latex]? Yes.
Step 7: Answer the question with a complete sentence.
The amount of discount was [latex]\$17.05[/latex].
b. Read the problem again.
1. When we translate this into an equation, we obtain [latex]17.05[/latex] equals [latex]r[/latex] times [latex]31[/latex]. We are told to solve the equation [latex]17.05[/latex] equals [latex]31r[/latex]. We divide by [latex]31[/latex] to obtain [latex]0.55[/latex] equals [latex]r[/latex]. We put this in percent form to obtain [latex]r[/latex] equals [latex]55%[/latex]. We are told to check: does this make sense? Is [latex]7.05[/latex] equal to [latex]55%[/latex] of [latex]>1[/latex]? Below this, we have [latex]17.05[/latex] equals with a question mark over it [latex]0.55[/latex] times [latex]31[/latex]. Below this, we have [latex]17.05[/latex] equals [latex]17.05[/latex] with a checkmark next to it. Then we are told to answer the question with a complete sentence: The rate of discount was [latex]55\%[/latex].
Step 1: Identify what we are looking for.
the discount rate
Step 2: Name what we are looking for.
Choose a variable to represent it.
Let [latex]r=[/latex] the discount rate.
Step 3: Translate into an equation.
Write a sentence that gives the information to find it.
[latex]\underbrace{\text{The discount of}\;\$17.05}\underbrace{\;\text{is}}\;\underbrace{\text{what percent}}\;\underbrace{\text{of}}\;\underbrace{\$31?}[/latex]
Translate into an equation.
[latex]17.05=r \cdot31[/latex]
Step 4: Solve the equation.
[latex]\begin{align*} &\;&17.05&=31r\\ &\text{Divide both sides by 31.}\;&0.55&=r\\ &\text{Change to percent form}\;&r&=55\% \end{align*}[/latex]
Step 5: Check. Does this make sense?
Is [latex]$17.05[/latex] equal to [latex]55\%[/latex] of [latex]\$31[/latex]?
[latex]\begin{align*} 17.05&=0.55\times(31)\\ 17.05&=17.05\checkmark \end{align*}[/latex]
Step 6: Answer the question with a complete sentence.
The rate of discount was [latex]55\%[/latex].
Try It
56) Find
a. the amount of discount and
b. the discount rate.
Lena bought a kitchen table at the sale price of [latex]\$375.20[/latex]. The original price of the table was [latex]\$560[/latex].
Solution
a. [latex]\$184.80[/latex]
b. [latex]33\%[/latex]
57) Find
a. the amount of discount and
b. the discount rate.
Nick bought a multi-room air conditioner at a sale price of [latex]\$340[/latex]. The original price of the air conditioner was [latex]\$400[/latex].
Solution
a. [latex]\$60[/latex]
b. [latex]15\%[/latex]
Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the original cost. The retailer then adds a mark-up to the original cost to get the list price, the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.
We summarize the mark-up model in the box below.
Mark-Up
Keep in mind that the list price should always be more than the original cost.
Example 3.7.25
Adam’s art gallery bought a photograph at original cost [latex]\$250[/latex]. Adam marked the price up [latex]40\%[/latex]. Find the:
a. amount of mark-up and
b. the list price of the photograph.
Solution
a.
Step 1: Read the problem.
Step 2: Identify what we are looking for.
the amount of mark-up
Step 3: Name what we are looking for.
Choose a variable to represent it.
Let [latex]m=[/latex] the amount of markup.
Step 4: Translate into an equation.
Write a sentence that gives the information to find it.
[latex]\underbrace{\text{The mark-up}}\;\underbrace{\text{is}}\;\underbrace{40\%}\;\underbrace{\text{of the}\;\$250\;\text{original cost}}[/latex]
Translate into an equation.
[latex]m=0.40\cdot250[/latex]
Step 5: Solve the equation.
[latex]\begin{align*} m&=0.40\cdot250)\\ m&=100 \end{align*}[/latex]
Step 6: Check. Does this make sense?
Yes, [latex]40\%[/latex] is less than one-half and [latex]100[/latex] is less than half of [latex]250[/latex].
Step 7: Answer the question with a complete sentence.
The mark-up on the photograph was [latex]\$100[/latex].
b.
Step 1: Read the problem again.
Step 2: Identify what we are looking for.
the list price
Step 3: Name what we are looking for.
Choose a variable to represent it.
Let [latex]p=[/latex] the list price.
Step 4: Translate into an equation.
Write a sentence that gives the information to find it.
[latex]\underbrace{\text{The list price}}\;\underbrace{\text{is}}\;\underbrace{\text{original cost}}\underbrace{\;\text{plus}}\;\underbrace{\text{the mark-up}}[/latex]
Translate into an equation.
[latex]p=250+100[/latex]
Step 5: Solve the equation.
[latex]\begin{align*} p&=250+100\\ p&=350 \end{align*}[/latex]
Step 6: Check. Does this make sense?
Is the list price more than the net price?
Is [latex]\$350[/latex] more than [latex]\$250[/latex]? Yes
Step 7: Answer the question with a complete sentence.
The list price of the photograph was [latex]\$350[/latex].
Try It
58) Find
a. the amount of mark-up and
b. the list price.
Jim’s music store bought a guitar at the original cost [latex]\$1,200[/latex]. Jim marked the price up [latex]50\%[/latex].
Solution
a. [latex]\$600[/latex]
b. [latex]\$1,800[/latex]
59) Find
a. the amount of mark-up and
b. the list price.
The Auto Resale Store-bought Pablo’s Toyota for [latex]\$8,500[/latex]. They marked the price up [latex]35\%[/latex].
Solution
a. [latex]\$2,975[/latex]
b. [latex]\$11,475[/latex]
Self Check
After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
What does this checklist tell you about your mastery of this section? What steps will you take to improve?
Glossary
- amount of discount
- The amount of discount is the amount resulting when a discount rate is multiplied by the original price of an item.
- discount rate
- The discount rate is the percent used to determine the amount of a discount, common in retail settings.
- interest
- Interest is the money that a bank pays its customers for keeping their money in the bank.
- list price
- The list price is the price a retailer sells an item for.
- mark-up
- A mark-up is a percentage of the original cost used to increase the price of an item.
- original cost
- The original cost in a retail setting, is the price that a retailer pays for an item.
- principal
- The principal is the original amount of money invested or borrowed for a period of time at a specific interest rate.
- rate of interest
- The rate of interest is a percent of the principal, usually expressed as a percent per year.
- simple interest
- Simple interest is the interest earned according to the formula [latex]I=Prt[/latex].
Interest is the money that a bank pays its customers for keeping their money in the bank.
The rate of interest is a percent of the principal, usually expressed as a percent per year.
Simple interest is the interest earned according to the formula I=Prt.
The discount rate is the percent used to determine the amount of a discount, common in retail settings.
The amount of discount is the amount resulting when a discount rate is multiplied by the original price of an item.
A mark-up is a percentage of the original cost used to increase the price of an item.
The list price is the price a retailer sells an item for.
The original cost in a retail setting, is the price that a retailer pays for an item.
The principal is the original amount of money invested or borrowed for a period of time at a specific interest rate.
