3.06 problem solving strategies units

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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

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.

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?

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.

Step 8: Check. Is [latex]\$36[/latex] a reasonable price for a purse? 

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].

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?

[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?

[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?

Step 1: Read t he 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]

[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.

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?

[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?

[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.

Step 1: Read the problem. Are all the words familiar?

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]

[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!

The number is [latex]19[/latex].

14) The difference of a number and eight is [latex]17[/latex]. Find the number.

[latex]25[/latex]

15) The difference of a number and eleven is [latex]-7[/latex]. Find the number.

[latex]4[/latex]

Example 3.7.4

The sum of twice a number and seven is [latex]15[/latex]. Find the number.

Step 1: Read the problem.

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]

[latex]2n\;+\;7\;=\;15[/latex]

[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.

16) The sum of four times a number and two is [latex]14[/latex]. Find the number.

[latex]3[/latex]

17) The sum of three times a number and seven is [latex]25[/latex]. Find the number.

[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.

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 1 st number and the 2 nd number is [latex]21[/latex].

[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]

[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]

The numbers are [latex]8[/latex] and [latex]13[/latex].

18) One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

[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.

[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.

Step 3: Name. Choose a variable.

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]

[latex]n\;+\;n-4\;=\;-14[/latex]

[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]

[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]

The numbers are [latex]−5[/latex] and [latex]−9[/latex].

20) The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

[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.

[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.

Step 2: Identify what you are looking for.

[latex]x=1^{st}\;number[/latex]

One number is [latex]10[/latex] more than twice another.

[latex]2x+10=2^{nd}\;number[/latex]

Their sum is one.

Restate as one sentence.

The sum of the two numbers is [latex]1[/latex].

[latex]x+2x+10=1[/latex]

[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]

[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]

The numbers are [latex]−3[/latex] and [latex]−4[/latex].

22) One number is eight more than twice another. Their sum is negative four. Find the numbers.

[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.

[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].

Example 3.7.8

The sum of two consecutive integers is [latex]47[/latex]. Find the numbers.

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].

[latex]n+n+1=47[/latex]

[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]

[latex]\begin{align*} 23+24&\overset?=47\\ 47&=47\checkmark\\ \end{align*}[/latex]

The two consecutive integers are [latex]23[/latex] and [latex]24[/latex].

24) The sum of two consecutive integers is [latex]95[/latex]. Find the numbers.

[latex]47[/latex] , [latex]48[/latex]

25) The sum of two consecutive integers is [latex]-31[/latex]. Find the numbers.

[latex]-16[/latex] , [latex]-15[/latex]

Example 3.7.9

Find three consecutive integers whose sum is [latex]-42[/latex].

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]

The sum of the three integers is [latex]−42[/latex].

[latex]n+n+1+n+2=-42[/latex]

[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]

[latex]\begin{align*} -13+(-14)+(-15)&\overset?=-42\\-42&=-42\checkmark \end{align*}[/latex]

The three consecutive integers are [latex]−13[/latex], [latex]−14[/latex], and [latex]−15[/latex].

26) Find three consecutive integers whose sum is [latex]-96[/latex].

[latex]-33[/latex], [latex]-32[/latex], [latex]-31[/latex]

27) Find three consecutive integers whose sum is [latex]-36[/latex].

[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:

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].

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]

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].

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]

The sum of the three even integers is [latex]84[/latex].

[latex]n+n+2+n+4=84[/latex]

[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]

[latex]\begin{align*}26+28+30&\overset?=84\\ 84&=84\checkmark \end{align*}[/latex]

The three consecutive integers are [latex]26[/latex], [latex]28[/latex], and [latex]30[/latex].

28) Find three consecutive even integers whose sum is [latex]102[/latex].

[latex]32[/latex], [latex]34[/latex], [latex]36[/latex]

29) Find three consecutive even integers whose sum is [latex]-24[/latex].

[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?

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 .

[latex]\underbrace{\text{The amount the husband earns}\;}+\;\underbrace{\text{the amount the wife earns}}\;\underbrace{\text{is}}\;\underbrace{\$110,000}[/latex]

[latex]h+2h-16,000=110,000[/latex]

[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]

If the wife earns [latex]\$68,000[/latex] and the husband earns [latex]\$42,000[/latex] is the total [latex]$110,000[/latex]? Yes!

The husband earns [latex]\$42,000[/latex] a year.

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?

[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?

[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]?

[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].

32) Translate and solve:

What number is [latex]45\%[/latex] of [latex]80[/latex]?

[latex]36[/latex]

33) Translate and solve:

What number is [latex]55\%[/latex] of [latex]60[/latex]?

[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]?

[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].

34) Translate and solve: [latex]7.5\%[/latex] of what number is [latex]\$1.95[/latex]?

[latex]\$26[/latex]

35) Translate and solve: [latex]8.5\%[/latex] of what number is [latex]\$3.06[/latex]?

[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]?

[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]

[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.

36) Translate and solve:

[latex]110[/latex] is what percent of [latex]88[/latex]?

[latex]125\%[/latex]

37) Translate and solve:

[latex]126[/latex] is what percent of [latex]72[/latex]?

[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.

Use a Problem-Solving Strategy to Solve an Application.

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?

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]

[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.

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?

[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?

[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?

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]

[latex]85=0.02\cdot a[/latex]

[latex]\begin{align*}85&=0.02\cdot a\\ 4,250&=a \end{align*}[/latex]

Yes, [latex]2\%[/latex] is a small percent and [latex]85[/latex] is a small part of [latex]4,250[/latex].

The amount of potassium that is recommended is [latex]4,250 mg[/latex].

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?

[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?

[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?

the percent of the total calories from fat

Let [latex]p=[/latex] percent of fat.

[latex]\underbrace{\text{What percent}}\;\underbrace{\text{of}}\;\underbrace{480}\;\underbrace{\text{is}}\;\underbrace{240?}[/latex]

[latex]p\cdot480=240[/latex]

[latex]\begin{align*}&\;&480p&=240\\ &\text{Divide by 480}\;\;&p&=0.5\\ &\text{Put in percent form}\;\;&p&=50\%\\ \end{align*}[/latex]

Yes, [latex]240[/latex] is half of [latex]480[/latex], so [latex]50\%[/latex] makes sense.

Of the total calories in each brownie, [latex]50\%[/latex] is fat.

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?

[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?

[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.

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.)

the percent increase

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]

[latex]10=p\cdot 26[/latex]

[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]

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].

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.

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.

[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].

[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.

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.

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.

First find the amount of decrease.

[latex]3.71 - 3.64 = 0.07[/latex]

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]

[latex]0.07=p\cdot3.71[/latex]

[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]

Yes, if the original price was [latex]\$4[/latex], a [latex]2\%[/latex] decrease would be [latex]8[/latex] cents.

The price of gas decreased [latex]1.9\%[/latex].

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.

[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].

[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

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]

the amount of interest earned

Let [latex]I=[/latex] the amount of interest.

[latex]\begin{align*} &\text{Write the formula}\;&I&=Prt\\ &\text{Substitute in the given information}\;&I&=(12,500)(0.04)(5)\\ \end{align*}[/latex]

[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.

The interest is [latex]\$2,500[/latex].

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?

[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?

[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]

the rate of interest

Let  [latex]r =[/latex] the rate of interest.

[latex]\begin{align*} &\text{Write the formula}\;&I&=Prt\\ &\text{Substitute in the given information}\;&660&=(3000)r(4)\\ \end{align*}[/latex]

[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]

[latex]\begin{align*}&\;&I&=Prt\\ &\;&660&\overset?=(3000)(0.55)(4)\\ &\;&660&=660\checkmark \end{align*}[/latex]

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.

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?

[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?

[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?

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.

[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]

[latex]\begin{align*} &\;&6,596&=P(0.075)(5)\\ &\text{Multiply.}\;&6,596.25&=0.375P\\ &\text{Divide.}\;&17,590&=P \end{align*}[/latex]

[latex]\begin{align*} I&=Prt\\ 6.596.25&\overset?=(14,590)(0.075)(5)\\ 6,596.25&=6,596.25\checkmark \end{align*}[/latex]

The principal was [latex]\$17,590[/latex].

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?

[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?

[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.

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?

a. Original Price =  [latex]\$140[/latex] Discount rate =  [latex]35\%[/latex] Discount  = ?

the amount of discount

Let [latex]d=[/latex] the amount of discount.

The discount is 35% of $140.

[latex]d=0.35(140)[/latex]

[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]

[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].

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].

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].

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.

a. Original price = [latex]\$31[/latex] Discount = ? Sale price = [latex]\$13.95[/latex]

The discount is the difference between the original price and the sale price.

[latex]d=31-13.95[/latex]

[latex]\begin{align*} d&=31-13.95\\ d&=17.05 \end{align*}[/latex]

Is [latex]17.05[/latex] less than [latex]31[/latex]? Yes.

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.

[latex]\underbrace{\text{The discount of}\;\$17.05}\underbrace{\;\text{is}}\;\underbrace{\text{what percent}}\;\underbrace{\text{of}}\;\underbrace{\$31?}[/latex]

[latex]17.05=r \cdot31[/latex]

[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]

The rate of discount was [latex]55\%[/latex].

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].

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].

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.

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.

the amount of mark-up

Let [latex]m=[/latex] the amount of markup.

[latex]\underbrace{\text{The mark-up}}\;\underbrace{\text{is}}\;\underbrace{40\%}\;\underbrace{\text{of the}\;\$250\;\text{original cost}}[/latex]

[latex]m=0.40\cdot250[/latex]

[latex]\begin{align*} m&=0.40\cdot250)\\ m&=100 \end{align*}[/latex]

Yes, [latex]40\%[/latex] is less than one-half and [latex]100[/latex] is less than half of [latex]250[/latex].

The mark-up on the photograph was [latex]\$100[/latex].

Step 1: Read the problem again.

the list price

Let [latex]p=[/latex] the list price.

[latex]\underbrace{\text{The list price}}\;\underbrace{\text{is}}\;\underbrace{\text{original cost}}\underbrace{\;\text{plus}}\;\underbrace{\text{the mark-up}}[/latex]

[latex]p=250+100[/latex]

[latex]\begin{align*} p&=250+100\\ p&=350 \end{align*}[/latex]

Is the list price more than the net price? Is [latex]\$350[/latex] more than [latex]\$250[/latex]? Yes

The list price of the photograph was [latex]\$350[/latex].

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].

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].

a. [latex]\$2,975[/latex] b. [latex]\$11,475[/latex]

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?

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.

Fanshawe Pre-Health Sciences Mathematics 1 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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3.1: Problem Solving and Unit Conversions

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Learning Objectives

• Given a quantity, convert from one set of units to another using dimensional analysis showing canceling on units. This includes one factor conversions.

During your studies of chemistry (and physics also), you will note that mathematical equations are used in many different applications. Many of these equations have a number of different variables with which you will need to work. You should also note that these equations will often require you to use measurements with their units. Using the correct units is critical!

Converting Between Units with Conversion Factors

A conversion factor is a ratio used to convert one unit of measurement into another. A simple conversion factor can convert meters into centimeters, for example. Since most calculations require measurements to be in certain units, you will find many uses for conversion factors. Always remember that a conversion factor has to represent a fact; this fact can either be simple or more complex. For instance, you already know that 12 eggs equal 1 dozen. A more complex fact is that the speed of light is \(1.86 \times 10^5\) miles/\(\text{sec}\). Either one of these can be used as a conversion factor depending on what type of calculation you are working with.

*Pounds and ounces are technically units of force, not mass, but this fact is often ignored by the non-scientific community.

Of course, there are other ratios which are not listed in Table \(\PageIndex{1}\). They may include:

• Ratios embedded in the text of the problem (using words such as per or in each , or using symbols such as / or %).
• Conversions in the SI system, as covered in the last chapter.
• Conversions in the English system (such as 12 inches \(=\) 1 foot or 4 quarts \(=\) 1 gallon).
• Time conversions (such as 60 seconds \(=\) 1 minute or 60 minutes \(=\) 1 hour).

If you learned the SI units and prefixes described, then you know that 1 cm is 1/100th of a meter.

\[ 1\; \rm{cm} = \dfrac{1}{100} \; \rm{m} = 10^{-2}\rm{m}\]

\[100\; \rm{cm} = 1\; \rm{m}\]

Suppose we divide both sides of the equation by \(1 \text{m}\) (both the number and the unit):

\[\mathrm{\dfrac{100\:cm}{1\:m}=\dfrac{1\:m}{1\:m}}\]

As long as we perform the same operation on both sides of the equals sign, the expression remains an equality. Look at the right side of the equation; it now has the same quantity in the numerator (the top) as it has in the denominator (the bottom). Any fraction that has the same quantity in the numerator and the denominator has a value of 1:

\[ \dfrac{ \text{100 cm}}{\text{1 m}} = \dfrac{ \text{1000 mm}}{\text{1 m}}= \dfrac{ 1\times 10^6 \mu \text{m}}{\text{1 m}}= 1\]

We know that 100 cm is 1 m, so we have the same quantity on the top and the bottom of our fraction, although it is expressed in different units.

Performing Dimensional Analysis

Dimensional analysis is amongst the most valuable tools that physical scientists use. Simply put, it is the conversion between an amount in one unit to the corresponding amount in a desired unit using various conversion factors.

Here is a simple example. How many centimeters are there in 3.55 m? Perhaps you can determine the answer in your head. If there are 100 cm in every meter, then 3.55 m equals 355 cm. To solve the problem more formally with a conversion factor, we first write the quantity we are given, 3.55 m. Then we multiply this quantity by a conversion factor, which is the same as multiplying it by 1. We can write 1 as \(\mathrm{\frac{100\:cm}{1\:m}}\) and multiply:

\[ 3.55 \; \rm{m} \times \dfrac{100 \; \rm{cm}}{1\; \rm{m}}\]

The 3.55 m can be thought of as a fraction with a 1 in the denominator. Because m, the abbreviation for meters, occurs in both the numerator and the denominator of our expression, they cancel out:

\[\dfrac{3.55 \; \cancel{\rm{m}}}{ 1} \times \dfrac{100 \; \rm{cm}}{1 \; \cancel{\rm{m}}}\]

The final step is to perform the calculation that remains once the units have been canceled:

\[ \dfrac{3.55}{1} \times \dfrac{100 \; \rm{cm}}{1} = 355 \; \rm{cm}\]

In the final answer, we omit the 1 in the denominator. Thus, by a more formal procedure, we find that 3.55 m equals 355 cm. A generalized description of this process is as follows:

quantity (in old units) × conversion factor = quantity (in new units)

You may be wondering why we use a seemingly complicated procedure for a straightforward conversion. The conversion problems you encounter will not always be so simple . If you master the technique of applying conversion factors, you will be able to solve a large variety of problems.

In the previous example, we used the fraction \(\frac{100 \; \rm{cm}}{1 \; \rm{m}}\) as a conversion factor. Does the conversion factor \(\frac{1 \; \rm m}{100 \; \rm{cm}}\) also equal 1? Yes, it does; it has the same quantity in the numerator as in the denominator (except that they are expressed in different units). Why did we not use that conversion factor? If we had used the second conversion factor, the original unit would not have canceled, and the result would have been meaningless. Here is what we would have gotten:

\[ 3.55 \; \rm{m} \times \dfrac{1\; \rm{m}}{100 \; \rm{cm}} = 0.0355 \dfrac{\rm{m}^2}{\rm{cm}}\]

For the answer to be meaningful, we have to construct the conversion factor in a form that causes the original unit to cancel out . Figure \(\PageIndex{1}\) shows a concept map for constructing a proper conversion.

General Steps in Performing Dimensional Analysis

• Identify the " given " information in the problem. Look for a number with units to start this problem with.
• What is the problem asking you to " find "? In other words, what unit will your answer have?
• Use ratios and conversion factors to cancel out the units that aren't part of your answer, and leave you with units that are part of your answer.
• When your units cancel out correctly, you are ready to do the math . You are multiplying fractions, so you multiply the top numbers and divide by the bottom numbers in the fractions.

Significant Figures in Conversions

How do conversion factors affect the determination of significant figures?

• Numbers in conversion factors based on SI definitions, such as kilograms to grams, are not considered in the determination of significant figures in a calculation because the numbers in such conversion factors are exact.
• Conversions within the English system are also exact. The numbers in conversion factors from one English unit to another English unit do not effect the significant figures in the answer.
• Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact.
• In contrast, conversion factors that come from measurements (such as density, as we will see shortly) have a limited number of significant figures and should be considered in determining the significant figures of the final answer.
• Conversions between the SI and English systems are not exact. The significant figures in the conversion must be considered when determining the significant figures in the answer. The one exception is the conversion between inches and centimeters (1 in = 2.54 cm) which is exact.

Example \(\PageIndex{1}\)

Exercise \(\pageindex{1}\).

Perform each conversion.

• 101,000 ns to seconds
• 32.08 kg to grams
• 1.53 grams to cg

Contributions & Attributions

This page was constructed from content via the following contributor(s) and edited (topically or extensively) by the LibreTexts development team to meet platform style, presentation, and quality:

Marisa Alviar-Agnew  ( Sacramento City College )

Henry Agnew (UC Davis)

2.2 Use a Problem Solving Strategy

Learning objectives.

• Use a problem solving strategy for word problems
• Solve number word problems
• Solve percent applications
• Solve simple interest applications

Be Prepared 2.2

Before you get started, take this readiness quiz.

• Translate “six less than twice x ” into an algebraic expression. If you missed this problem, review Example 1.8 .
• Convert 4.5% to a decimal. If you missed this problem, review Example 1.40 .
• Convert 0.6 to a percent. If you missed this problem, review Example 1.40 .

Have you ever had any negative experiences in the past with word problems? When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. Realize that your negative experiences with word problems are in your past. To move forward you need to calm your fears and change your negative feelings.

Start with a fresh slate and begin to think positive thoughts. Repeating some of the following statements may be helpful to turn your thoughts positive. Thinking positive thoughts is a first step towards success.

  I think I can! I think I can!

  While word problems were hard in the past, I think I can try them now.

  I am better prepared now—I think I will begin to understand word problems.

  I am able to solve equations because I practiced many problems and I got help when I needed it—I can try that   with word problems.

  It may take time, but I can begin to solve word problems.

You are now well prepared and you are ready to succeed. If you take control and believe you can be successful, you will be able to master word problems.

Use a Problem Solving Strategy for Word Problems

Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.

Example 2.14

Normal yearly snowfall at the local ski resort is 12 inches more than twice the amount it received last season. The normal yearly snowfall is 62 inches. What was the snowfall last season at the ski resort?

Try It 2.27

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was three more than twice the number of notebooks. He bought seven textbooks. How many notebooks did he buy?

Try It 2.28

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 22 Sudoku puzzles. How many crossword puzzles did he do?

We summarize an effective strategy for problem solving.

Use a Problem Solving Strategy for word problems.

• Step 1. Read the problem. Make sure all the words and ideas are understood.
• Step 2. Identify what you are looking for.
• Step 3. Name what you are looking for. Choose a variable to represent that quantity.
• Step 4. 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 algebra equation.
• Step 5. Solve the equation using proper algebra techniques.
• Step 6. Check the answer in the problem to make sure it makes sense.
• Step 7. Answer the question with a complete sentence.

Solve Number Word Problems

We will now apply the problem solving strategy to “number word problems.” Number word problems give some clues about one or more numbers and we use these clues to write an equation. Number word problems provide good practice for using the Problem Solving Strategy.

Example 2.15

The sum of seven times a number and eight is thirty-six. Find the number.

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 2.29

The sum of four times a number and two is fourteen. Find the number.

Try It 2.30

The sum of three times a number and seven is twenty-five. Find the number.

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 2.16

The sum of two numbers is negative fifteen. One number is nine less than the other. Find the numbers.

Try It 2.31

The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

Try It 2.32

The sum of two numbers is negative eighteen. One number is forty more than the other. Find the numbers.

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. Therefore, if we define the first integer as n , the next consecutive integer is n + 1 . n + 1 . The one after that is one more than n + 1 , n + 1 , so it is n + 1 + 1 , n + 1 + 1 , which is n + 2 . n + 2 .

We will use this notation to represent consecutive integers in the next example.

Example 2.17

Find three consecutive integers whose sum is −54 . −54 .

Try It 2.33

Find three consecutive integers whose sum is −96 . −96 .

Try It 2.34

Find three consecutive integers whose sum is −36 . −36 .

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:

Notice each integer is two more than the number preceding it. If we call the first one n , then the next one is n + 2 . n + 2 . The one after that would be n + 2 + 2 n + 2 + 2 or n + 4 . n + 4 .

Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 63, 65, and 67.

Does it seem strange to have to add two (an even number) to get the next odd number? Do we get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you do not have to do anything different. The pattern is still the same—to get to the next odd or the next even integer, add two.

Example 2.18

Find three consecutive even integers whose sum is 120 120 .

Step 1. Read the problem. Step 2. Identify what you are looking for. three consecutive even integers Step 3. Name. Let n = 1 st even integer. n + 2 = 2 nd consecutive even integer n + 4 = 3 rd consecutive even integer Step 4. Translate. Restate as one sentence. The sum of the three even integers is 120 . Translate into an equation. n + n + 2 + n + 4 = 120 Step 5. Solve the equation. n + n + 2 + n + 4 = 120 Combine like terms. 3 n + 6 = 120 Subtract 6 from each side. 3 n = 114 Divide each side by 3. n = 38 1 st integer n + 2 2 nd integer 38 + 2 40 n + 4 3 rd integer 38 + 4 42 Step 6. Check. 38 + 40 + 42 = ? 120 120 = 120 ✓ Step 7. Answer the question. The three consecutive integers are 38, 40, and 42. Step 1. Read the problem. Step 2. Identify what you are looking for. three consecutive even integers Step 3. Name. Let n = 1 st even integer. n + 2 = 2 nd consecutive even integer n + 4 = 3 rd consecutive even integer Step 4. Translate. Restate as one sentence. The sum of the three even integers is 120 . Translate into an equation. n + n + 2 + n + 4 = 120 Step 5. Solve the equation. n + n + 2 + n + 4 = 120 Combine like terms. 3 n + 6 = 120 Subtract 6 from each side. 3 n = 114 Divide each side by 3. n = 38 1 st integer n + 2 2 nd integer 38 + 2 40 n + 4 3 rd integer 38 + 4 42 Step 6. Check. 38 + 40 + 42 = ? 120 120 = 120 ✓ Step 7. Answer the question. The three consecutive integers are 38, 40, and 42.

Try It 2.35

Find three consecutive even integers whose sum is 102.

Try It 2.36

Find three consecutive even integers whose sum is −24 . −24 .

When a number problem is in a real life context, we still use the same strategies that we used for the previous examples.

Example 2.19

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

Step 1. Read the problem. Step 2. Identify what you are looking for. How much does the husband earn? Step 3. Name. Choose a variable to represent Let h = the amount the husband earns. the amount the husband earns. The wife earns $16,000 less than twice that. Step 4. Translate. Restate the problem in one sentence with all the important information. Translate into an equation. 2 h − 16,000 = the amount the wife earns Together the husband and wife earn $110,000. h + 2 h − 16,000 = 110,000 Step 5. Solve the equation. Combine like terms. Add 16,000 to both sides and simplify. Divide each side by three. h + 2 h − 16,000 = 110,000 3 h − 16,000 = 110,000 3 h = 126,000 h = 42,000 $42,000 amount husband earns 2 h − 16,000 amount wife earns 2 ( 42,000 ) − 16,000 84,000 − 16,000 68,000 Step 6. Check: If the wife earns $68,000 and the husband earns $42,000, is that $110,000? Yes! Step 7. Answer the question. The husband earns $42,000 a year. Step 1. Read the problem. Step 2. Identify what you are looking for. How much does the husband earn? Step 3. Name. Choose a variable to represent Let h = the amount the husband earns. the amount the husband earns. The wife earns $16,000 less than twice that. Step 4. Translate. Restate the problem in one sentence with all the important information. Translate into an equation. 2 h − 16,000 = the amount the wife earns Together the husband and wife earn $110,000. h + 2 h − 16,000 = 110,000 Step 5. Solve the equation. Combine like terms. Add 16,000 to both sides and simplify. Divide each side by three. h + 2 h − 16,000 = 110,000 3 h − 16,000 = 110,000 3 h = 126,000 h = 42,000 $42,000 amount husband earns 2 h − 16,000 amount wife earns 2 ( 42,000 ) − 16,000 84,000 − 16,000 68,000 Step 6. Check: If the wife earns $68,000 and the husband earns $42,000, is that $110,000? Yes! Step 7. Answer the question. The husband earns $42,000 a year.

Try It 2.37

According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,400. This was $1,600 less than six times the cost in 1975. What was the average cost of a car in 1975?

Try It 2.38

US Census data shows that the median price of new home in the U.S. in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

Solve Percent Applications

There are several methods to solve percent equations. In algebra, it is easiest if we just translate English sentences into algebraic equations and then solve the equations. Be sure to change the given percent to a decimal before you use it in the equation.

Example 2.20

Translate and solve:

ⓐ What number is 45% of 84? ⓑ 8.5% of what amount is $4.76? ⓒ 168 is what percent of 112?

Try It 2.39

Translate and solve: ⓐ What number is 45% of 80? ⓑ 7.5% of what amount is $1.95? ⓒ 110 is what percent of 88?

Try It 2.40

Translate and solve: ⓐ What number is 55% of 60? ⓑ 8.5% of what amount is $3.06? ⓐ 126 is what percent of 72?

Now that we have a problem solving strategy to refer to, and have practiced 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 we will solve involve everyday situations, you can rely on your own experience.

Example 2.21

The label on Audrey’s yogurt said that one serving provided 12 grams of protein, which is 24% of the recommended daily amount. What is the total recommended daily amount of protein?

Try It 2.41

One serving of wheat square cereal has 7 grams of fiber, which is 28% of the recommended daily amount. What is the total recommended daily amount of fiber?

Try It 2.42

One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium?

Remember to put the answer in the form requested. In the next example we are looking for the percent.

Example 2.22

Veronica is planning to make muffins from a mix. The package says each muffin will be 240 calories and 60 calories will be from fat. What percent of the total calories is from fat?

Try It 2.43

Mitzi received some gourmet brownies as a gift. The wrapper said each 28% brownie was 480 calories, and had 240 calories of fat. What percent of the total calories in each brownie comes from fat? Round the answer to the nearest whole percent.

Try It 2.44

The mix Ricardo plans to use to make brownies says that each brownie will be 190 calories, and 76 calories are from fat. What percent of the total calories are from fat? Round the answer to the nearest whole percent.

It is often important in many fields—business, sciences, pop culture—to talk about how much an amount has increased or decreased over a certain period of time. This increase or decrease is generally expressed as a percent and called the percent change .

To find the percent change, first we find the amount of change, by finding the difference of the new amount and the original amount. Then we find what percent the amount of change is of the original amount.

Find percent change.

• Step 1. Find the amount of change. change = new amount − original amount change = new amount − original amount
• Step 2. Find what percent the amount of change is of the original amount. change is what percent of the original amount? change is what percent of the original amount?

Example 2.23

Recently, the California governor proposed raising community college fees from $36 a unit to $46 a unit. Find the percent change. (Round to the nearest tenth of a percent.)

Try It 2.45

Find the percent change. (Round to the nearest tenth of a percent.) In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents.

Try It 2.46

Find the percent change. (Round to the nearest tenth of a percent.) In 1995, the standard bus fare in Chicago was $1.50. In 2008, the standard bus fare was 2.25.

Applications of discount and mark-up 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.

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.

The sale price should always be less than the original price.

The list price should always be more than the original cost.

Example 2.24

Liam’s art gallery bought a painting at an original cost of $750. Liam marked the price up 40%. Find ⓐ the amount of mark-up and ⓑ the list price of the painting.

Try It 2.47

Find ⓐ the amount of mark-up and ⓑ the list price: Jim’s music store bought a guitar at original cost $1,200. Jim marked the price up 50%.

Try It 2.48

Find ⓐ the amount of mark-up and ⓑ the list price: The Auto Resale Store bought Pablo’s Toyota for $8,500. They marked the price up 35%.

Solve Simple Interest Applications

Interest is a part of our daily lives. From the interest earned on our savings to the interest we pay on a car loan or credit card debt, we all have some experience with interest in our lives.

The amount of money you initially deposit into a bank is called the principal , P , and the bank pays you interest , I. When you take out a loan, you pay interest on the amount you borrow, also called the principal.

In either case, the interest is computed as a certain percent of the principal, called the rate of interest , r . The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable t , (for time) represents the number of years the money is saved or borrowed.

Interest is calculated as simple interest or compound interest. Here we will use simple interest.

Simple Interest

If an amount of money, P , called the principal, is invested or borrowed for a period of t years at an annual interest rate r , the amount of interest, I , earned or paid is

Interest earned or paid according to this formula is called simple interest .

The formula we use to calculate interest is I = P r t . I = P r t . To use the formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information in a chart.

Example 2.25

Areli invested a principal of $950 in her bank account that earned simple interest at an interest rate of 3%. How much interest did she earn in five years?

I = ? P = $ 950 r = 3 % t = 5 years I = ? P = $ 950 r = 3 % t = 5 years

Identify what you are asked to find, and choose a What is the simple interest? variable to represent it. Let I = interest. Write the formula. I = P r t Substitute in the given information. I = ( 950 ) ( 0.03 ) ( 5 ) Simplify. I = 142.5 Check. Is $142.50 a reasonable amount of interest on $950? Yes. Write a complete sentence. The interest is $142.50. Identify what you are asked to find, and choose a What is the simple interest? variable to represent it. Let I = interest. Write the formula. I = P r t Substitute in the given information. I = ( 950 ) ( 0.03 ) ( 5 ) Simplify. I = 142.5 Check. Is $142.50 a reasonable amount of interest on $950? Yes. Write a complete sentence. The interest is $142.50.

Try It 2.49

Nathaly deposited $12,500 in her bank account where it will earn 4% simple interest. How much interest will Nathaly earn in five years?

Try It 2.50

Susana invested a principal of $36,000 in her bank account that earned simple interest at an interest rate of 6.5 % . 6.5 % . How much interest did she earn in three years?

There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we do not know the rate.

Example 2.26

Hang borrowed $7,500 from her parents to pay her tuition. In five years, she paid them $1,500 interest in addition to the $7,500 she borrowed. What was the rate of simple interest?

I = $ 1500 P = $ 7500 r = ? t = 5 years I = $ 1500 P = $ 7500 r = ? t = 5 years

Identify what you are asked to find, and choose What is the rate of simple interest? a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Change to percent form. Let r = rate of interest. I = P r t 1,500 = ( 7,500 ) r ( 5 ) 1,500 = 37,500 r 0.04 = r 4 % = r Check. I = P r t 1,500 = ? ( 7,500 ) ( 0.04 ) ( 5 ) 1,500 = 1,500 ✓ Write a complete sentence. The rate of interest was 4%. Identify what you are asked to find, and choose What is the rate of simple interest? a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Change to percent form. Let r = rate of interest. I = P r t 1,500 = ( 7,500 ) r ( 5 ) 1,500 = 37,500 r 0.04 = r 4 % = r Check. I = P r t 1,500 = ? ( 7,500 ) ( 0.04 ) ( 5 ) 1,500 = 1,500 ✓ Write a complete sentence. The rate of interest was 4%.

Try It 2.51

Jim lent his sister $5,000 to help her buy a house. In three years, she paid him the $5,000, plus $900 interest. What was the rate of simple interest?

Try It 2.52

Loren lent his brother $3,000 to help him buy a car. In four years, his brother paid him back the $3,000 plus $660 in interest. What was the rate of simple interest?

In the next example, we are asked to find the principal—the amount borrowed.

Example 2.27

Sean’s new car loan statement said he would pay $4,866,25 in interest from a simple interest rate of 8.5% over five years. How much did he borrow to buy his new car?

I = 4,866.25 P = ? r = 8.5 % t = 5 years I = 4,866.25 P = ? r = 8.5 % t = 5 years

Identify what you are asked to find, What is the amount borrowed (the principal)? and choose a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Let P = principal borrowed. I = P r t 4,866.25 = P ( 0.085 ) ( 5 ) 4,866.25 = 0.425 P 11,450 = P Check. I = P r t 4,866.25 = ? ( 11,450 ) ( 0.085 ) ( 5 ) 4,866.25 = 4,866.25 ✓ Write a complete sentence. The principal was $11,450. Identify what you are asked to find, What is the amount borrowed (the principal)? and choose a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Let P = principal borrowed. I = P r t 4,866.25 = P ( 0.085 ) ( 5 ) 4,866.25 = 0.425 P 11,450 = P Check. I = P r t 4,866.25 = ? ( 11,450 ) ( 0.085 ) ( 5 ) 4,866.25 = 4,866.25 ✓ Write a complete sentence. The principal was $11,450.

Try It 2.53

Eduardo noticed that his new car loan papers stated that with a 7.5% simple interest rate, he would pay $6,596.25 in interest over five years. How much did he borrow to pay for his car?

Try It 2.54

In five years, Gloria’s bank account earned $2,400 interest at 5% simple interest. How much had she deposited in the account?

Access this online resource for additional instruction and practice with using a problem solving strategy.

• Begining Arithmetic Problems

Section 2.2 Exercises

Practice makes perfect.

List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

There are 18 Cub Scouts in Troop 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?

Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?

In the following exercises, solve each number word problem.

The difference of a number and 12 is three. Find the number.

The difference of a number and eight is four. Find the number.

The sum of three times a number and eight is 23. Find the number.

The sum of twice a number and six is 14. Find the number.

The difference of twice a number and seven is 17. Find the number.

The difference of four times a number and seven is 21. Find the number.

Three times the sum of a number and nine is 12. Find the number.

Six times the sum of a number and eight is 30. Find the number.

One number is six more than the other. Their sum is 42. Find the numbers.

One number is five more than the other. Their sum is 33. Find the numbers.

The sum of two numbers is 20. One number is four less than the other. Find the numbers.

The sum of two numbers is 27. One number is seven less than the other. Find the numbers.

One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.

The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

The sum of two consecutive integers is 77. Find the integers.

The sum of two consecutive integers is 89. Find the integers.

The sum of three consecutive integers is 78. Find the integers.

The sum of three consecutive integers is 60. Find the integers.

Find three consecutive integers whose sum is −3 . −3 .

Find three consecutive even integers whose sum is 258.

Find three consecutive even integers whose sum is 222.

Find three consecutive odd integers whose sum is −213 . −213 .

Find three consecutive odd integers whose sum is −267 . −267 .

Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?

Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

In the following exercises, translate and solve.

ⓐ What number is 45% of 120? ⓑ 81 is 75% of what number? ⓐ What percent of 260 is 78?

ⓐ What number is 65% of 100? ⓑ 93 is 75% of what number? ⓐ What percent of 215 is 86?

ⓐ 250% of 65 is what number? ⓑ 8.2% of what amount is $2.87? ⓐ 30 is what percent of 20?

ⓐ 150% of 90 is what number? ⓑ 6.4% of what amount is $2.88? ⓐ 50 is what percent of 40?

In the following exercises, solve.

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16% of the total bill as a tip. How much should the tip be?

When Hiro and his co-workers had lunch at a restaurant near their work, the bill was $90.50. They want to leave 18% of the total bill as a tip. How much should the tip be?

One serving of oatmeal has 8 grams of fiber, which is 33% of the recommended daily amount. What is the total recommended daily amount of fiber?

One serving of trail mix has 67 grams of carbohydrates, which is 22% of the recommended daily amount. What is the total recommended daily amount of carbohydrates?

A bacon cheeseburger at a popular fast food restaurant contains 2070 milligrams (mg) of sodium, which is 86% of the recommended daily amount. What is the total recommended daily amount of sodium?

A grilled chicken salad at a popular fast food restaurant contains 650 milligrams (mg) of sodium, which is 27% of the recommended daily amount. What is the total recommended daily amount of sodium?

The nutrition fact sheet at a fast food restaurant says the fish sandwich has 380 calories, and 171 calories are from fat. What percent of the total calories is from fat?

The nutrition fact sheet at a fast food restaurant says a small portion of chicken nuggets has 190 calories, and 114 calories are from fat. What percent of the total calories is from fat?

Emma gets paid $3,000 per month. She pays $750 a month for rent. What percent of her monthly pay goes to rent?

Dimple gets paid $3,200 per month. She pays $960 a month for rent. What percent of her monthly pay goes to rent?

Tamanika received a raise in her hourly pay, from $15.50 to $17.36. Find the percent change.

Ayodele received a raise in her hourly pay, from $24.50 to $25.48. Find the percent change.

Annual student fees at the University of California rose from about $4,000 in 2000 to about $12,000 in 2010. Find the percent change.

The price of a share of one stock rose from $12.50 to $50. Find the percent change.

A grocery store reduced the price of a loaf of bread from $2.80 to $2.73. Find the percent change.

The price of a share of one stock fell from $8.75 to $8.54. Find the percent change.

Hernando’s salary was $49,500 last year. This year his salary was cut to $44,055. Find the percent change.

In ten years, the population of Detroit fell from 950,000 to about 712,500. Find the percent change.

In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.

Janelle bought a beach chair on sale at 60% off. The original price was $44.95.

Errol bought a skateboard helmet on sale at 40% off. The original price was $49.95.

In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate (Round to the nearest tenth of a percent if needed.)

Larry and Donna bought a sofa at the sale price of $1,344. The original price of the sofa was $1,920.

Hiroshi bought a lawnmower at the sale price of $240. The original price of the lawnmower is $300.

In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

Daria bought a bracelet at original cost $16 to sell in her handicraft store. She marked the price up 45%. What was the list price of the bracelet?

Regina bought a handmade quilt at original cost $120 to sell in her quilt store. She marked the price up 55%. What was the list price of the quilt?

Tom paid $0.60 a pound for tomatoes to sell at his produce store. He added a 33% mark-up. What price did he charge his customers for the tomatoes?

Flora paid her supplier $0.74 a stem for roses to sell at her flower shop. She added an 85% mark-up. What price did she charge her customers for the roses?

Casey deposited $1,450 in a bank account that earned simple interest at an interest rate of 4%. How much interest was earned in two years?

Terrence deposited $5,720 in a bank account that earned simple interest at an interest rate of 6%. How much interest was earned in four years?

Robin deposited $31,000 in a bank account that earned simple interest at an interest rate of 5.2%. How much interest was earned in three years?

Carleen deposited $16,400 in a bank account that earned simple interest at an interest rate of 3.9% How much interest was earned in eight years?

Hilaria borrowed $8,000 from her grandfather to pay for college. Five years later, she paid him back the $8,000, plus $1,200 interest. What was the rate of simple interest?

Kenneth lent his niece $1,200 to buy a computer. Two years later, she paid him back the $1,200, plus $96 interest. What was the rate of simple interest?

Lebron lent his daughter $20,000 to help her buy a condominium. When she sold the condominium four years later, she paid him the $20,000, plus $3,000 interest. What was the rate of simple interest?

Pablo borrowed $50,000 to start a business. Three years later, he repaid the $50,000, plus $9,375 interest. What was the rate of simple interest?

In 10 years, a bank account that paid 5.25% simple interest earned $18,375 interest. What was the principal of the account?

In 25 years, a bond that paid 4.75% simple interest earned $2,375 interest. What was the principal of the bond?

Joshua’s computer loan statement said he would pay $1,244.34 in simple interest for a three-year loan at 12.4%. How much did Joshua borrow to buy the computer?

Margaret’s car loan statement said she would pay $7,683.20 in simple interest for a five-year loan at 9.8%. How much did Margaret borrow to buy the car?

Everyday Math

Tipping At the campus coffee cart, a medium coffee costs $1.65. MaryAnne brings $2.00 with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave?

Tipping Four friends went out to lunch and the bill came to $53.75 They decided to add enough tip to make a total of $64, so that they could easily split the bill evenly among themselves. What percent tip did they leave?

Writing Exercises

What has been your past experience solving word problems? Where do you see yourself moving forward?

Without solving the problem “44 is 80% of what number” think about what the solution might be. Should it be a number that is greater than 44 or less than 44? Explain your reasoning.

After returning from vacation, Alex said he should have packed 50% fewer shorts and 200% more shirts. Explain what Alex meant.

Because of road construction in one city, commuters were advised to plan that their Monday morning commute would take 150% of their usual commuting time. Explain what this means.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

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very successful.

more profitable.

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Labeling, faulty design, and mispricing

Poor service, inaccurate forms, and rudeness

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Adherence to policies

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3.1E: Exercises

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Practice Makes Perfect

Use the Approach Word Problems with a Positive Attitude

In the following exercises, prepare the lists described.

Exercise \(\PageIndex{1}\)

List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

Answers will vary

Exercise \(\PageIndex{2}\)

List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

Use a Problem-Solving Strategy for Word Problems

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

Exercise \(\PageIndex{3}\)

Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?

Exercise \(\PageIndex{4}\)

Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?

Exercise \(\PageIndex{5}\)

Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

Exercise \(\PageIndex{6}\)

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?

Exercise \(\PageIndex{7}\)

There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

Exercise \(\PageIndex{8}\)

There are 18 Cub Scouts in Pack 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

Exercise \(\PageIndex{9}\)

Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?

Exercise \(\PageIndex{10}\)

Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?

Exercise \(\PageIndex{11}\)

Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

Exercise \(\PageIndex{12}\)

Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?

Exercise \(\PageIndex{13}\)

Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than four times the amount invested in bonds. How much does Laurie have invested in bonds?

Exercise \(\PageIndex{14}\)

Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

Solve Number Problems

In the following exercises, solve each number word problem.

Exercise \(\PageIndex{15}\)

The sum of a number and eight is 12. Find the number.

Exercise \(\PageIndex{16}\)

The sum of a number and nine is 17. Find the number.

Exercise \(\PageIndex{17}\)

The difference of a number and 12 is three. Find the number.

Exercise \(\PageIndex{18}\)

The difference of a number and eight is four. Find the number.

Exercise \(\PageIndex{19}\)

The sum of three times a number and eight is 23. Find the number.

Exercise \(\PageIndex{20}\)

The sum of twice a number and six is 14. Find the number.

Exercise \(\PageIndex{21}\)

The difference of twice a number and seven is 17. Find the number.

Exercise \(\PageIndex{22}\)

The difference of four times a number and seven is 21. Find the number.

Exercise \(\PageIndex{23}\)

Three times the sum of a number and nine is 12. Find the number.

Exercise \(\PageIndex{24}\)

Six times the sum of a number and eight is 30. Find the number.

Exercise \(\PageIndex{25}\)

One number is six more than the other. Their sum is 42. Find the numbers.

Exercise \(\PageIndex{26}\)

One number is five more than the other. Their sum is 33. Find the numbers.

Exercise \(\PageIndex{27}\)

The sum of two numbers is 20. One number is four less than the other. Find the numbers.

Exercise \(\PageIndex{28}\)

The sum of two numbers is 27. One number is seven less than the other. Find the numbers.

Exercise \(\PageIndex{29}\)

The sum of two numbers is −45. One number is nine more than the other. Find the numbers.

−18,−27

Exercise \(\PageIndex{30}\)

The sum of two numbers is −61. One number is 35 more than the other. Find the numbers.

Exercise \(\PageIndex{31}\)

The sum of two numbers is −316. One number is 94 less than the other. Find the numbers.

−111,−205

Exercise \(\PageIndex{32}\)

The sum of two numbers is −284. One number is 62 less than the other. Find the numbers.

Exercise \(\PageIndex{33}\)

One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

Exercise \(\PageIndex{34}\)

One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.

Exercise \(\PageIndex{35}\)

One number is five more than another. If their sum is increased by nine, the result is 60. Find the numbers.

Exercise \(\PageIndex{36}\)

One number is eight more than another. If their sum is increased by 17, the result is 95. Find the numbers.

Exercise \(\PageIndex{37}\)

One number is one more than twice another. Their sum is −5. Find the numbers.

−2,−3

Exercise \(\PageIndex{38}\)

One number is six more than five times another. Their sum is six. Find the numbers.

Exercise \(\PageIndex{39}\)

The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

Exercise \(\PageIndex{40}\)

The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

Exercise \(\PageIndex{41}\)

The sum of two consecutive integers is 77. Find the integers.

Exercise \(\PageIndex{42}\)

The sum of two consecutive integers is 89. Find the integers.

Exercise \(\PageIndex{43}\)

The sum of two consecutive integers is −23. Find the integers.

−11,−12

Exercise \(\PageIndex{44}\)

The sum of two consecutive integers is −37. Find the integers.

Exercise \(\PageIndex{45}\)

The sum of three consecutive integers is 78. Find the integers.

Exercise \(\PageIndex{46}\)

The sum of three consecutive integers is 60. Find the integers.

Exercise \(\PageIndex{47}\)

Find three consecutive integers whose sum is −36.

−11,−12,−13

Exercise \(\PageIndex{48}\)

Find three consecutive integers whose sum is −3.

Exercise \(\PageIndex{49}\)

Find three consecutive even integers whose sum is 258.

Exercise \(\PageIndex{50}\)

Find three consecutive even integers whose sum is 222.

Exercise \(\PageIndex{51}\)

Find three consecutive odd integers whose sum is 171.

Exercise \(\PageIndex{52}\)

Find three consecutive odd integers whose sum is 291.

Exercise \(\PageIndex{53}\)

Find three consecutive even integers whose sum is −36.

−10,−12,−14

Exercise \(\PageIndex{54}\)

Find three consecutive even integers whose sum is −84.

Exercise \(\PageIndex{55}\)

Find three consecutive odd integers whose sum is −213.

−69,−71,−73

Exercise \(\PageIndex{56}\)

Find three consecutive odd integers whose sum is −267.

Everyday Math

Exercise \(\pageindex{57}\).

Sale Price Patty paid $35 for a purse on sale for $10 off the original price. What was the original price of the purse?

Exercise \(\PageIndex{58}\)

Sale Price Travis bought a pair of boots on sale for $25 off the original price. He paid $60 for the boots. What was the original price of the boots?

Exercise \(\PageIndex{59}\)

Buying in Bulk Minh spent $6.25 on five sticker books to give his nephews. Find the cost of each sticker book.

Exercise \(\PageIndex{60}\)

Buying in Bulk Alicia bought a package of eight peaches for $3.20. Find the cost of each peach.

Exercise \(\PageIndex{61}\)

Price before Sales Tax Tom paid $1,166.40 for a new refrigerator, including $86.40 tax. What was the price of the refrigerator?

Exercise \(\PageIndex{62}\)

Price before Sales Tax Kenji paid $2,279 for a new living room set, including $129 tax. What was the price of the living room set?

Writing Exercises

Exercise \(\pageindex{63}\).

What has been your past experience solving word problems?

answers will vary

Exercise \(\PageIndex{64}\)

When you start to solve a word problem, how do you decide what to let the variable represent?

Exercise \(\PageIndex{65}\)

What are consecutive odd integers? Name three consecutive odd integers between 50 and 60.

Consecutive odd integers are odd numbers that immediately follow each other. An example of three consecutive odd integers between 50 and 60 would be 51, 53, and 55.

Exercise \(\PageIndex{66}\)

What are consecutive even integers? Name three consecutive even integers between −50 and −40.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.