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== 9.1#h(0.6em)Use a Problem Solving Strategy

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Translate #math.equation(block: false, alt: "“6")[$"“6"$] less than twice #math.equation(block: false, alt: "x ”")[$"x" "”"$] into an algebraic expression. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "2 x minus 6")[$2 x − 6$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 2 over 3 x equals 24 .")[$frac(2, 3) x = 24 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "36")[$36$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "3 x plus 8 equals 14 .")[$3 x + 8 = 14 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "2")[$2$]
]
]


=== Approach Word Problems with a Positive Attitude

The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will 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 in ?

#figure(figph[A cartoon image of a girl with a sad expression writing on a piece of paper is shown. There are 5 thought bubbles. They read, “I don't know whether to add, subtract multiply, or divide!,” then “I don't understand word problems!,” then “My teachers never explained this!,” then “If I just skip all the word problems, I can probably still pass the class,” and lastly, “I just can't do this!”], alt: "A cartoon image of a girl with a sad expression writing on a piece of paper is shown. There are 5 thought bubbles. They read, “I don't know whether to add, subtract multiply, or divide!,” then “I don't understand word problems!,” then “My teachers never explained this!,” then “If I just skip all the word problems, I can probably still pass the class,” and lastly, “I just can't do this!”", caption: [Negative thoughts about word problems can be barriers to success.])

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 like the student in . Read the positive thoughts and say them out loud.

#figure(figph[A cartoon image of a girl with a confident expression holding some books is shown.  There are 4 thought bubbles. They read, “While word problems were hard in the past, I think I can try them now,” then “I am better prepared now. I think I will begin to understand word problems,” then “I think I can! I think I can!,” and lastly, “It may take time, but I can begin to solve word problems.”], alt: "A cartoon image of a girl with a confident expression holding some books is shown. There are 4 thought bubbles. They read, “While word problems were hard in the past, I think I can try them now,” then “I am better prepared now. I think I will begin to understand word problems,” then “I think I can! I think I can!,” and lastly, “It may take time, but I can begin to solve word problems.”", caption: [When it comes to word problems, a positive attitude is a big step toward success.])

If we take control and believe we can be successful, we will be able to master word problems.

Think of something that you can do now but couldn't do three years ago. Whether it's driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!


=== Use a Problem-solving Strategy for Word Problems

In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then you've increased your math vocabulary as you learned about more algebraic procedures, and you've had more practice translating from words into algebra.

You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.

Now we'll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We'll demonstrate the strategy as we solve the following problem.

#examplebox("Example 1")[][
Pete bought a shirt on sale for #math.equation(block: false, alt: "$18 ,")[$"$18" ,$] which is one-half the original price. What was the original price of the shirt?

#solutionbox[
Step 1. #strong[Read] the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the Internet.

- #emph[In this problem, do you understand what is being discussed? Do you understand every word?]

Step 2. #strong[Identify] what you are 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!

- #emph[In this problem, the words “what was the original price of the shirt” tell you that what you are looking for: the original price of the shirt.]

Step 3. #strong[Name] what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents.

- #emph[Let #math.equation(block: false, alt: "p equals")[$p =$] the original price of the shirt]

Step 4. #strong[Translate] into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation.

#figure(figph[The top line reads: “18 is one-half of the original price.” Below 18 is a brace and the number 18. Below “is” is a brace and an equal sign. Below “one-half” is a brace and the fraction 1 over 2. Below “of” is a brace and a multiplication dot. Below “the original price” is a brace and an italicized p.], alt: "The top line reads: “18 is one-half of the original price.” Below 18 is a brace and the number 18. Below “is” is a brace and an equal sign. Below “one-half” is a brace and the fraction 1 over 2. Below “of” is a brace and a multiplication dot. Below “the original price” is a brace and an italicized p.", caption: none)

Step 5. #strong[Solve] the equation using good algebra techniques. Even if you know the answer right away, using algebra will better prepare you to solve problems that do not have obvious answers.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Write the equation.], [#figure(figph[A mathematical equation is presented, showing '18 = (1/2)p' in black font against a white background.], alt: "A mathematical equation is presented, showing '18 = (1/2)p' in black font against a white background.", caption: none)]),
  [Multiply both sides by 2.], [#figure(figph[A mathematical equation shows '2 \* 18 = 2 \* (1/2)p'. The number 2 is highlighted in red on both sides of the equals sign, indicating it is a common factor used in solving for 'p'.], alt: "A mathematical equation shows '2 * 18 = 2 * (1/2)p'. The number 2 is highlighted in red on both sides of the equals sign, indicating it is a common factor used in solving for 'p'.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a simple mathematical equation, '36 = p', where the number 36 is set equal to the variable 'p' on a plain white background.], alt: "The image displays a simple mathematical equation, '36 = p', where the number 36 is set equal to the variable 'p' on a plain white background.", caption: none)],
))

Step 6. #strong[Check] the answer in the problem and make sure it makes sense.

- #emph[We found that] #math.equation(block: false, alt: "p equals 36 ,")[$p = 36 ,$] #emph[which means the original price was] #math.equation(block: false, alt: "$36 .")[$"$36" .$] #emph[Does] #math.equation(block: false, alt: "$36")[$"$36"$] #emph[make sense in the problem? Yes, because] #math.equation(block: false, alt: "18")[$18$] #emph[is one-half of] #math.equation(block: false, alt: "36 ,")[$36 ,$] #emph[and the shirt was on sale at half the original price.]

Step 7. #strong[Answer] the question with a complete sentence.

- #emph[The problem asked “What was the original price of the shirt?” The answer to the question is: “The original price of the shirt was] #math.equation(block: false, alt: "$36 .”")[$"$36" .”$]

If this were a homework exercise, our work might look like this:

#figure(figph[The top reads, “Let p equal the original price. 18 is one-half the original price.” The next line shows the equation 18 equals one-half times p. The following line shows the same equation with each side being multiplied by 2. The next line shows 36 equals p. Below this, it reads, “Check: Is \$36 a reasonable price for a shirt? Yes. Is 18 one-half of 36? Yes. The original price of the shirt as \$36.], alt: "The top reads, “Let p equal the original price. 18 is one-half the original price.” The next line shows the equation 18 equals one-half times p. The following line shows the same equation with each side being multiplied by 2. The next line shows 36 equals p. Below this, it reads, “Check: Is $36 a reasonable price for a shirt? Yes. Is 18 one-half of 36? Yes. The original price of the shirt as $36.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Joaquin bought a bookcase on sale for #math.equation(block: false, alt: "$120 ,")[$"$120" ,$] which was two-thirds the original price. What was the original price of the bookcase?

#solutionbox[
\$180
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Two-fifths of the people in the senior center dining room are men. If there are #math.equation(block: false, alt: "16")[$16$] men, what is the total number of people in the dining room?

#solutionbox[
40
]

We list the steps we took to solve the previous example.
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Problem-Solving Strategy]

+ #strong[Read] the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the internet.
+ #strong[Identify] what you are looking for.
+ #strong[Name] what you are looking for. Choose a variable to represent that quantity.
+ #strong[Translate] into an equation. It may be helpful to first restate the problem in one sentence before translating.
+ #strong[Solve] the equation using good algebra techniques.
+ #strong[Check] the answer in the problem. Make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.
]

Let's use this approach with another example.

#examplebox("Example 2")[][
Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought #math.equation(block: false, alt: "11")[$11$] apples to the picnic. How many bananas did he bring?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem.], []),
  [Step 2. #strong[Identify] what you are looking for.], [How many bananas did he bring?],
  [Step 3. #strong[Name] what you are looking for. #linebreak() Choose a variable to represent the number of bananas.], [Let #math.equation(block: false, alt: "b equals number of bananas")[$b = "number of bananas"$]],
  [Step 4. #strong[Translate.] Restate the problem in one sentence with all the important information. #linebreak() Translate into an equation.], [#linebreak() #figure(figph[An illustration demonstrating how to translate the word problem 'The number of apples was 3 more than twice the number of bananas' into the algebraic equation '11 = 3 + 2b'.], alt: "An illustration demonstrating how to translate the word problem 'The number of apples was 3 more than twice the number of bananas' into the algebraic equation '11 = 3 + 2b'.", caption: none)],
  [Step 5. #strong[Solve] the equation.], [#figure(figph[A mathematical equation is displayed, 11 = 2b + 3.], alt: "A mathematical equation is displayed, 11 = 2b + 3.", caption: none)],
  [Subtract 3 from each side.], [#figure(figph[A mathematical equation shows '11 - 3 = 2b + 3 - 3' in a black serif font, with the number '3' highlighted in red on both sides of the equals sign, indicating subtraction from each side.], alt: "A mathematical equation shows '11 - 3 = 2b + 3 - 3' in a black serif font, with the number '3' highlighted in red on both sides of the equals sign, indicating subtraction from each side.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation on a white background, displaying '8 = 2b' in black text. The numbers and letter are clearly visible, indicating a simple algebraic problem.], alt: "A mathematical equation on a white background, displaying '8 = 2b' in black text. The numbers and letter are clearly visible, indicating a simple algebraic problem.", caption: none)],
  [Divide each side by 2.], [#figure(figph[A mathematical equation shows '8 divided by 2 equals 2b divided by 2.' The denominator '2' is highlighted in red on both sides of the equation.], alt: "A mathematical equation shows '8 divided by 2 equals 2b divided by 2.' The denominator '2' is highlighted in red on both sides of the equation.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a mathematical equation, '4 = b,' indicating that the value of the variable 'b' is equal to 4, set against a plain white background.], alt: "The image displays a mathematical equation, '4 = b,' indicating that the value of the variable 'b' is equal to 4, set against a plain white background.", caption: none)],
  [Step 6. #strong[Check:] First, is our answer reasonable? Yes, bringing four bananas to a picnic seems reasonable. The problem says the number of apples was three more than twice the number of bananas. If there are four bananas, does that make eleven apples? Twice 4 bananas is 8. Three more than 8 is 11.], [],
  [Step 7. #strong[Answer] the question.], [Yash brought 4 bananas to the picnic.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was #math.equation(block: false, alt: "3")[$3$] more than the number of notebooks. He bought #math.equation(block: false, alt: "5")[$5$] textbooks. How many notebooks did he buy?

#solutionbox[
2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is seven more than the number of crossword puzzles. He completed #math.equation(block: false, alt: "14")[$14$] Sudoku puzzles. How many crossword puzzles did he complete?

#solutionbox[
7
]
]

In Solve Sales Tax, Commission, and Discount Applications, we learned how to translate and solve basic percent equations and used them to solve sales tax and commission applications. In the next example, we will apply our #strong[Problem Solving Strategy] to more applications of percent.

#examplebox("Example 3")[][
Nga's car insurance premium increased by #math.equation(block: false, alt: "$60 ,")[$"$60" ,$] which was #math.equation(block: false, alt: "8%")[$"8%"$] of the original cost. What was the original cost of the premium?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem. Remember, if there are words you don't understand, look them up.], []),
  [Step 2. #strong[Identify] what you are looking for.], [the original cost of the premium],
  [Step 3. #strong[Name.] Choose a variable to represent the original cost of premium.], [Let #math.equation(block: false, alt: "c equals the original cost")[$c = "the original cost"$]],
  [Step 4. #strong[Translate.] Restate as one sentence. Translate into an equation.], [#linebreak() #figure(figph[An image demonstrating how to translate a word problem into an algebraic equation. The phrase '\$60 was 8% of the original cost' is shown aligned with its mathematical translation: '60 = 0.08 \* c'.], alt: "An image demonstrating how to translate a word problem into an algebraic equation. The phrase '$60 was 8% of the original cost' is shown aligned with its mathematical translation: '60 = 0.08 * c'.", caption: none)],
  [Step 5. #strong[Solve] the equation.], [#figure(figph[The mathematical equation '60 = 0.08c' is displayed in black text on a white background, representing a calculation where 60 is equal to 0.08 times c.], alt: "The mathematical equation '60 = 0.08c' is displayed in black text on a white background, representing a calculation where 60 is equal to 0.08 times c.", caption: none)],
  [Divide both sides by 0.08.], [#figure(figph[Equation: 60/0.08 = 0.08c/0.08, demonstrating both sides divided by 0.08 (highlighted in red) to solve for the variable 'c'.], alt: "Equation: 60/0.08 = 0.08c/0.08, demonstrating both sides divided by 0.08 (highlighted in red) to solve for the variable 'c'.", caption: none)],
  [Simplify.], [#math.equation(block: false, alt: "c equals 750")[$c = 750$]],
  [Step 6. #strong[Check:] Is our answer reasonable? Yes, a \$750 premium on auto insurance is reasonable. Now let's check our algebra. Is 8% of 750 equal to 60? #linebreak() #figure(figph[A mathematical equation shows '750 = c' in black text on a white background, representing the value of c as 750.], alt: "A mathematical equation shows '750 = c' in black text on a white background, representing the value of c as 750.", caption: none)

 #linebreak() #figure(figph[A mathematical equation shows '0.08(750) = 60' in black text against a white background.], alt: "A mathematical equation shows '0.08(750) = 60' in black text against a white background.", caption: none)

 #linebreak() #figure(figph[The fundamental equality '60 = 60' is presented alongside a clear checkmark, visually confirming its undeniable truth and mathematical accuracy.], alt: "The fundamental equality '60 = 60' is presented alongside a clear checkmark, visually confirming its undeniable truth and mathematical accuracy.", caption: none)], [],
  [Step 7. #strong[Answer] the question.], [The original cost of Nga's premium was \$750.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Pilar's rent increased by #math.equation(block: false, alt: "4% .")[$"4%" .$] The increase was #math.equation(block: false, alt: "$38 .")[$"$38" .$] What was the original amount of Pilar's rent?

#solutionbox[
\$950
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Steve saves #math.equation(block: false, alt: "12%")[$"12%"$] of his paycheck each month. If he saved #math.equation(block: false, alt: "$504")[$"$504"$] last month, how much was his paycheck?

#solutionbox[
\$4,200
]
]


=== Solve Number Problems

Now we will translate and solve #strong[number problems]. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the #strong[Problem Solving Strategy]. Remember to look for clue words such as #emph[difference], #emph[of], and #emph[and].

#examplebox("Example 4")[][
The difference of a number and six is #math.equation(block: false, alt: "13 .")[$13 .$] Find the number.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem. Do you understand all the words?], []),
  [Step 2. #strong[Identify] what you are looking for.], [the number],
  [Step 3. #strong[Name.] Choose a variable to represent the number.], [Let #math.equation(block: false, alt: "n equals the number")[$n = "the number"$]],
  [Step 4. #strong[Translate.] Restate as one sentence. #linebreak()  Translate into an equation.], [#figure(figph[Shows how to translate the phrase 'The difference of a number and 6 is 13' into the algebraic equation 'n - 6 = 13', breaking down the verbal statement into mathematical symbols.], alt: "Shows how to translate the phrase 'The difference of a number and 6 is 13' into the algebraic equation 'n - 6 = 13', breaking down the verbal statement into mathematical symbols.", caption: none)],
  [Step 5. #strong[Solve] the equation. #linebreak() Add 6 to both sides. #linebreak() Simplify.], [#figure(figph[A basic algebra equation is displayed, showing 'n - 6 = 13' in black text on a white background.], alt: "A basic algebra equation is displayed, showing 'n - 6 = 13' in black text on a white background.", caption: none)

 #linebreak() #figure(figph[A mathematical equation 'n - 6 + 6 = 13 + 6' is displayed on a white background, demonstrating the addition property of equality with the added '6' highlighted in red.], alt: "A mathematical equation 'n - 6 + 6 = 13 + 6' is displayed on a white background, demonstrating the addition property of equality with the added '6' highlighted in red.", caption: none)

 #linebreak() #figure(figph[The image displays the mathematical expression 'n = 19' centered on a white background.], alt: "The image displays the mathematical expression 'n = 19' centered on a white background.", caption: none)],
  [Step 6. #strong[Check:] #linebreak() The difference of 19 and 6 is 13. It checks.], [],
  [Step 7. #strong[Answer] the question.], [The number is 19.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The difference of a number and eight is #math.equation(block: false, alt: "17 .")[$17 .$] Find the number.

#solutionbox[
25
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The difference of a number and eleven is #math.equation(block: false, alt: "−7 .")[$−7 .$] Find the number.

#solutionbox[
4
]
]

#examplebox("Example 5")[][
The sum of twice a number and seven is #math.equation(block: false, alt: "15 .")[$15 .$] Find the number.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem.], []),
  [Step 2. #strong[Identify] what you are looking for.], [the number],
  [Step 3. #strong[Name.] Choose a variable to represent the number.], [Let #math.equation(block: false, alt: "n equals the number")[$n = "the number"$]],
  [Step 4. #strong[Translate.] Restate the problem as one sentence. #linebreak()  Translate into an equation.], [#figure(figph[The image demonstrates how to translate a word problem into an algebraic equation. The phrase 'The sum of twice a number and 7 is 15' is mapped to the equation 2 \* n + 7 = 15.], alt: "The image demonstrates how to translate a word problem into an algebraic equation. The phrase 'The sum of twice a number and 7 is 15' is mapped to the equation 2 * n + 7 = 15.", caption: none)],
  [Step 5. #strong[Solve] the equation.], [#figure(figph[The image shows a mathematical equation on a white background: '2n + 7 = 15'. The equation features the variable 'n', a multiplication operation (implied), addition, and an equality.], alt: "The image shows a mathematical equation on a white background: '2n + 7 = 15'. The equation features the variable 'n', a multiplication operation (implied), addition, and an equality.", caption: none)],
  [Subtract 7 from each side and simplify.], [#figure(figph[The mathematical equation '2n = 8' is displayed in black text against a white background.], alt: "The mathematical equation '2n = 8' is displayed in black text against a white background.", caption: none)],
  [Divide each side by 2 and simplify.], [#figure(figph[The mathematical expression 'n = 4' is displayed in the bottom right corner of a white background, with no other elements visible in the image.], alt: "The mathematical expression 'n = 4' is displayed in the bottom right corner of a white background, with no other elements visible in the image.", caption: none)],
  [Step 6. #strong[Check:] is the sum of twice 4 and 7 equal to 15? #linebreak() #figure(figph[A simple arithmetic equation, 2 \* 4 + 7 = 15, is presented on a white surface.], alt: "A simple arithmetic equation, 2 * 4 + 7 = 15, is presented on a white surface.", caption: none)

 #linebreak() #figure(figph[A mathematical equation displays '8 + 7 = 15' on a white background.], alt: "A mathematical equation displays '8 + 7 = 15' on a white background.", caption: none)

 #linebreak() #figure(figph[A mathematical equation '15 = 15' is displayed, followed by a checkmark, indicating its correctness.], alt: "A mathematical equation '15 = 15' is displayed, followed by a checkmark, indicating its correctness.", caption: none)], [],
  [Step 7. #strong[Answer] the question.], [The number is 4.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of four times a number and two is #math.equation(block: false, alt: "14 .")[$14 .$] Find the number.

#solutionbox[
3
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of three times a number and seven is #math.equation(block: false, alt: "25 .")[$25 .$] Find the number.

#solutionbox[
6
]
]

Some number word problems ask you 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. 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.

#examplebox("Example 6")[][
One number is five more than another. The sum of the numbers is twenty-one. Find the numbers.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem.], [], []),
  [Step 2. #strong[Identify] what you are looking for.], [], [You are looking for two numbers.],
  [Step 3. #strong[Name.] #linebreak() Choose a variable to represent the first number. #linebreak() What do you know about the second number? #linebreak() Translate.], [], [#linebreak() Let #math.equation(block: false, alt: "n equals 1st number")[$n = "1st number"$] #linebreak() One number is five more than another. #linebreak() #math.equation(block: false, alt: "n plus 5 equals 2 to the power nd number")[$n + 5 = 2^("nd") " number"$]],
  [Step 4. #strong[Translate.] #linebreak() Restate the problem as one sentence with all the important information. #linebreak() Translate into an equation. #linebreak() Substitute the variable expressions.], [], [#linebreak() The sum of the numbers is 21. #linebreak() The sum of the 1st number and the 2nd number is 21. #linebreak() #figure(figph[An algebraic equation showing the sum of two numbers, 'n' and 'n+5', equaling 21. The first number is represented by 'n', and the second number is 'n+5', together summing up to 21.], alt: "An algebraic equation showing the sum of two numbers, 'n' and 'n+5', equaling 21. The first number is represented by 'n', and the second number is 'n+5', together summing up to 21.", caption: none)],
  [Step 5. #strong[Solve] the equation.], [], [#figure(figph[A mathematical equation is displayed on a white background, reading 'n + n + 5 = 21'.], alt: "A mathematical equation is displayed on a white background, reading 'n + n + 5 = 21'.", caption: none)],
  [Combine like terms.], [], [#figure(figph[A simple linear algebraic equation, '2x + 5 = 21' is displayed, demonstrating an unknown variable multiplied by a coefficient, added to a constant, and equated to another constant.], alt: "A simple linear algebraic equation, '2x + 5 = 21' is displayed, demonstrating an unknown variable multiplied by a coefficient, added to a constant, and equated to another constant.", caption: none)],
  [Subtract five from both sides and simplify.], [], [#figure(figph[The image shows the mathematical equation 2n = 16. This equation represents a linear equation where 'n' is the variable, '2' is the coefficient of 'n', and '16' is the constant term.], alt: "The image shows the mathematical equation 2n = 16. This equation represents a linear equation where 'n' is the variable, '2' is the coefficient of 'n', and '16' is the constant term.", caption: none)],
  [Divide by two and simplify.], [], [#figure(figph[The text 'n = 8 1st number' is displayed on a white background, indicating a variable assignment and its sequence.], alt: "The text 'n = 8 1st number' is displayed on a white background, indicating a variable assignment and its sequence.", caption: none)],
  [Find the second number too.], [], [#figure(figph[The image displays the mathematical expression 'n + 5' followed by the text '2nd number' on a white background.], alt: "The image displays the mathematical expression 'n + 5' followed by the text '2nd number' on a white background.", caption: none)],
  [Substitute #emph[n] = 8], [], [#figure(figph[A basic arithmetic expression, 8 + 5, is displayed on a white background. The number '8' is rendered in red, while the plus sign and the number '5' are in black.], alt: "A basic arithmetic expression, 8 + 5, is displayed on a white background. The number '8' is rendered in red, while the plus sign and the number '5' are in black.", caption: none)],
  [], [], [#figure(figph[The number '13' is displayed in a dark font against a plain white background, positioned towards the upper right side of the frame.], alt: "The number '13' is displayed in a dark font against a plain white background, positioned towards the upper right side of the frame.", caption: none)],
  [Step 6. #strong[Check:]], [], [],
  [Do these numbers check in the problem? #linebreak() Is one number 5 more than the other? #linebreak() Is thirteen, 5 more than 8? Yes. #linebreak()  #linebreak() Is the sum of the two numbers 21?], [#figure(figph[A mathematical equation asks if 13 is equal to 8 + 5, which is a true statement as both sides equal 13.], alt: "A mathematical equation asks if 13 is equal to 8 + 5, which is a true statement as both sides equal 13.", caption: none)

 #linebreak() #figure(figph[The image displays the equation '13 = 13' followed by a checkmark, indicating that the statement is correct.], alt: "The image displays the equation '13 = 13' followed by a checkmark, indicating that the statement is correct.", caption: none)

 #linebreak()  #linebreak() #figure(figph[A mathematical equation shows '8 + 13' followed by an equals sign with a question mark above it, then '21', posing whether 8 plus 13 equals 21.], alt: "A mathematical equation shows '8 + 13' followed by an equals sign with a question mark above it, then '21', posing whether 8 plus 13 equals 21.", caption: none)

 #linebreak() #figure(figph[The number 21 is equal to 21, validated with a checkmark, representing a correct mathematical statement or calculation.], alt: "The number 21 is equal to 21, validated with a checkmark, representing a correct mathematical statement or calculation.", caption: none)], [],
  [Step 7. #strong[Answer] the question.], [], [The numbers are 8 and 13.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

#solutionbox[
9, 15
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

#solutionbox[
27, 31
]
]

#examplebox("Example 7")[][
The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem.], [], []),
  [Step 2. #strong[Identify] what you are looking for.], [], [two numbers],
  [Step 3. #strong[Name.] Choose a variable. #linebreak() What do you know about the second number? #linebreak() Translate.], [], [#linebreak() Let #emph[n] = 1#super[st] number #linebreak() One number is 4 less than the other. #linebreak() #emph[n] - 4 = 2#super[nd] number],
  [Step 4. #strong[Translate.] #linebreak() Write as one sentence. #linebreak() Translate into an equation. #linebreak() Substitute the variable expressions.], [], [#linebreak() The sum of two numbers is negative fourteen. #linebreak() #figure(figph[This image presents an algebraic equation where the sum of two numbers, 'n' and 'n-4', equals -14. It demonstrates the translation of a word problem into a solvable mathematical expression.], alt: "This image presents an algebraic equation where the sum of two numbers, 'n' and 'n-4', equals -14. It demonstrates the translation of a word problem into a solvable mathematical expression.", caption: none)],
  [Step 5. #strong[Solve] the equation.], [], [#figure(figph[A mathematical equation is displayed with a variable 'n'. The equation reads as 'n + n - 4 = -14'.], alt: "A mathematical equation is displayed with a variable 'n'. The equation reads as 'n + n - 4 = -14'.", caption: none)],
  [Combine like terms.], [], [#figure(figph[The image displays the equation 2n - 4 = -14, which is an algebraic expression involving a variable 'n', constants, and arithmetic operations.], alt: "The image displays the equation 2n - 4 = -14, which is an algebraic expression involving a variable 'n', constants, and arithmetic operations.", caption: none)],
  [Add 4 to each side and simplify.], [], [#figure(figph[A mathematical equation is displayed on a white background, reading '2n = -10' in black text.], alt: "A mathematical equation is displayed on a white background, reading '2n = -10' in black text.", caption: none)],
  [Divide by 2.], [], [#figure(figph[The image displays the mathematical expression 'n = -5 1st number' against a plain white background.], alt: "The image displays the mathematical expression 'n = -5 1st number' against a plain white background.", caption: none)],
  [Substitute #math.equation(block: false, alt: "n equals −5")[$n = −5$] to find the 2#super[nd] number.], [], [#figure(figph[The image displays the algebraic expression 'n - 4' followed by the text '2nd number', suggesting it represents a second number defined in relation to a variable 'n'.], alt: "The image displays the algebraic expression 'n - 4' followed by the text '2nd number', suggesting it represents a second number defined in relation to a variable 'n'.", caption: none)],
  [], [], [#figure(figph[The image displays a mathematical expression '-5 - 4' horizontally on a white background. The number -5 is in red, and -4 is in black.], alt: "The image displays a mathematical expression '-5 - 4' horizontally on a white background. The number -5 is in red, and -4 is in black.", caption: none)],
  [], [], [#figure(figph[A white image features a small, dark numeral '-9' positioned in the upper right portion of the frame.], alt: "A white image features a small, dark numeral '-9' positioned in the upper right portion of the frame.", caption: none)],
  [Step 6. #strong[Check:]], [], [],
  [Is −9 four less than −5? #linebreak()  #linebreak()  #linebreak() Is their sum −14?], [#figure(figph[The image displays the math problem -5 - 4 with a question mark positioned directly above the equals sign, before the result -9. It asks if the equation -5 - 4 = -9 is true.], alt: "The image displays the math problem -5 - 4 with a question mark positioned directly above the equals sign, before the result -9. It asks if the equation -5 - 4 = -9 is true.", caption: none)

 #linebreak() #figure(figph[A mathematical statement showing '-9 = -9' followed by a checkmark, indicating the equality is correct.], alt: "A mathematical statement showing '-9 = -9' followed by a checkmark, indicating the equality is correct.", caption: none)

 #linebreak() #figure(figph[A mathematical equation asking if -5 + (-9) equals -14.], alt: "A mathematical equation asking if -5 + (-9) equals -14.", caption: none)

 #linebreak() #figure(figph[The equation -14 = -14 is correctly shown with a checkmark, indicating it is a true statement.], alt: "The equation -14 = -14 is correctly shown with a checkmark, indicating it is a true statement.", caption: none)], [],
  [Step 7. #strong[Answer] the question.], [], [The numbers are −5 and −9.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two numbers is negative twenty-three. One number is #math.equation(block: false, alt: "7")[$7$] less than the other. Find the numbers.

#solutionbox[
−8, −15
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two numbers is negative eighteen. One number is #math.equation(block: false, alt: "40")[$40$] more than the other. Find the numbers.

#solutionbox[
−29, 11
]
]

#examplebox("Example 8")[][
One number is ten more than twice another. Their sum is one. Find the numbers.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem.], [], []),
  [Step 2. #strong[Identify] what you are looking for.], [], [two numbers],
  [Step 3. #strong[Name.] Choose a variable. #linebreak() One number is ten more than twice another.], [], [Let #emph[x] = 1#super[st] number #linebreak() 2#emph[x] + 10 = 2#super[nd] number],
  [Step 4. #strong[Translate.] Restate as one sentence.], [], [Their sum is one.],
  [Translate into an equation], [], [#figure(figph[Translating a word problem into an algebraic equation: 'The sum of the two numbers is 1' becomes x + (2x + 10) = 1.], alt: "Translating a word problem into an algebraic equation: 'The sum of the two numbers is 1' becomes x + (2x + 10) = 1.", caption: none)],
  [Step 5. #strong[Solve] the equation.], [], [#figure(figph[A mathematical equation is displayed against a white background: x + 2x + 10 = 1. The equation shows algebraic terms with variables and constants, combined with addition and equality signs.], alt: "A mathematical equation is displayed against a white background: x + 2x + 10 = 1. The equation shows algebraic terms with variables and constants, combined with addition and equality signs.", caption: none)],
  [Combine like terms.], [], [#figure(figph[A mathematical equation is displayed, showing '3x + 10 = 1' in black text against a white background.], alt: "A mathematical equation is displayed, showing '3x + 10 = 1' in black text against a white background.", caption: none)],
  [Subtract 10 from each side.], [], [#figure(figph[A mathematical equation, '3x = -9', is displayed against a white background.], alt: "A mathematical equation, '3x = -9', is displayed against a white background.", caption: none)],
  [Divide each side by 3 to get the first number.], [], [#figure(figph[The mathematical equation 'x = -3' is displayed in black text against a plain white background, presenting a simple algebraic expression.], alt: "The mathematical equation 'x = -3' is displayed in black text against a plain white background, presenting a simple algebraic expression.", caption: none)],
  [Substitute to get the second number.], [], [#figure(figph[The mathematical expression '2x + 10' is displayed in black text on a white background.], alt: "The mathematical expression '2x + 10' is displayed in black text on a white background.", caption: none)],
  [], [], [#figure(figph[The mathematical expression 2(-3) + 10 is displayed on a white background, representing a calculation where two is multiplied by negative three, and then ten is added to the product.], alt: "The mathematical expression 2(-3) + 10 is displayed on a white background, representing a calculation where two is multiplied by negative three, and then ten is added to the product.", caption: none)],
  [], [], [#figure(figph[The number 4 is displayed in black on a plain white background.], alt: "The number 4 is displayed in black on a plain white background.", caption: none)],
  [Step 6. #strong[Check.]], [], [],
  [Is 4 ten more than twice −3? #linebreak()  #linebreak()  #linebreak()  #linebreak() Is their sum 1?], [#figure(figph[A mathematical equation reads '2(-3) + 10 =? 4', asking if the expression on the left evaluates to 4. This challenges basic arithmetic operations including multiplication, addition, and negative numbers.], alt: "A mathematical equation reads '2(-3) + 10 =? 4', asking if the expression on the left evaluates to 4. This challenges basic arithmetic operations including multiplication, addition, and negative numbers.", caption: none)

 #linebreak()  #linebreak()  #linebreak() #figure(figph[The image shows the mathematical equation -6 + 10 = 4, displaying a basic arithmetic operation involving negative and positive integers.], alt: "The image shows the mathematical equation -6 + 10 = 4, displaying a basic arithmetic operation involving negative and positive integers.", caption: none)

 #linebreak() #figure(figph[A mathematical equation '4 = 4' with a checkmark, indicating that the statement is correct.], alt: "A mathematical equation '4 = 4' with a checkmark, indicating that the statement is correct.", caption: none)

 #linebreak() #figure(figph[An image displaying the arithmetic problem -3 + 4 with a question mark over the equals sign before the number 1, asking if the statement is true.], alt: "An image displaying the arithmetic problem -3 + 4 with a question mark over the equals sign before the number 1, asking if the statement is true.", caption: none)

 #linebreak() #figure(figph[The equation '1=1' with a checkmark, indicating a correct and self-evident mathematical truth.], alt: "The equation '1=1' with a checkmark, indicating a correct and self-evident mathematical truth.", caption: none)], [],
  [Step 7. #strong[Answer] the question.], [], [The numbers are −3 and 4.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One number is eight more than twice another. Their sum is negative four. Find the numbers.

#solutionbox[
−4, 0
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
One number is three more than three times another. Their sum is negative five. Find the numbers.

#solutionbox[
−2, −3
]
]

Consecutive integers are integers that immediately follow each other. Some examples of #strong[consecutive integers] are:

#math.equation(block: true, alt: "; ; ; ; ... 1 , 2 , 3 , 4 ,...")[$#h(0.2em) \ #h(0.2em) \ #h(0.2em) \ #h(0.2em) \ "..." 1 , 2 , 3 , 4 ",..."$]#math.equation(block: true, alt: "... −10 , −9 , −8 , −7 ,...")[$"..." −10 , −9 , −8 , −7 ",..."$]#math.equation(block: true, alt: "... 150 , 151 , 152 , 153 ,...")[$"..." 150 , 151 , 152 , 153 ",..."$]Notice that each number is one more than the number preceding it. So if we define the first integer as #math.equation(block: false, alt: "n ,")[$n ,$] the next consecutive integer is #math.equation(block: false, alt: "n plus 1 .")[$n + 1 .$] The one after that is one more than #math.equation(block: false, alt: "n plus 1 ,")[$n + 1 ,$] so it is #math.equation(block: false, alt: "n plus 1 plus 1 ,")[$n + 1 + 1 ,$] or #math.equation(block: false, alt: "n plus 2 .")[$n + 2 .$]

#math.equation(block: true, alt: "n, 1st integer; n plus 1, 2nd consecutive integer; n plus 2, 3rd consecutive integer")[$n &  &  & "1st integer" \ n + 1 &  &  & "2nd consecutive integer" \ n + 2 &  &  & "3rd consecutive integer"$]#examplebox("Example 9")[][
The sum of two consecutive integers is #math.equation(block: false, alt: "47 .")[$47 .$] Find the numbers.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem.], [], []),
  [Step 2. #strong[Identify] what you are looking for.], [], [two consecutive integers],
  [Step 3. #strong[Name.]], [], [Let #emph[n] = 1#super[st] integer #linebreak() n + 1 = next consecutive integer],
  [Step 4. #strong[Translate.] #linebreak() Restate as one sentence. #linebreak() Translate into an equation.], [], [#figure(figph[An image showing the translation of the phrase 'The sum of the integers is 47' into the algebraic equation 'n + n + 1 = 47', with corresponding parts highlighted by brackets.], alt: "An image showing the translation of the phrase 'The sum of the integers is 47' into the algebraic equation 'n + n + 1 = 47', with corresponding parts highlighted by brackets.", caption: none)],
  [Step 5. #strong[Solve] the equation.], [], [#figure(figph[A mathematical equation is displayed, showing 'n + n + 1 = 47' in black text against a white background.], alt: "A mathematical equation is displayed, showing 'n + n + 1 = 47' in black text against a white background.", caption: none)],
  [Combine like terms.], [], [#figure(figph[A mathematical equation, '2n + 1 = 47', is displayed in black font against a white background, representing a linear equation to be solved for the variable 'n'.], alt: "A mathematical equation, '2n + 1 = 47', is displayed in black font against a white background, representing a linear equation to be solved for the variable 'n'.", caption: none)],
  [Subtract 1 from each side.], [], [#figure(figph[A simple mathematical equation is displayed on a white background, showing '2n = 46' in black text, representing a basic algebraic problem.], alt: "A simple mathematical equation is displayed on a white background, showing '2n = 46' in black text, representing a basic algebraic problem.", caption: none)],
  [Divide each side by 2.], [], [#figure(figph[The image displays mathematical notation 'n = 23' followed by the text '1st integer' on a white background.], alt: "The image displays mathematical notation 'n = 23' followed by the text '1st integer' on a white background.", caption: none)],
  [Substitute to get the second number.], [], [#figure(figph[The image shows the mathematical expression 'n + 1 2nd integer' in a horizontal layout.], alt: "The image shows the mathematical expression 'n + 1 2nd integer' in a horizontal layout.", caption: none)],
  [], [], [#figure(figph[The number 23 in red text is shown next to a plus sign and the number 1 in black text on a white background, forming the mathematical expression '23 + 1'.], alt: "The number 23 in red text is shown next to a plus sign and the number 1 in black text on a white background, forming the mathematical expression '23 + 1'.", caption: none)],
  [], [], [#figure(figph[The number 24 is displayed in black text on a white background.], alt: "The number 24 is displayed in black text on a white background.", caption: none)],
  [Step 6. #strong[Check:]], [#figure(figph[A mathematical expression reads '23 + 24 =? 47', indicating a query about whether 23 plus 24 indeed equals 47.], alt: "A mathematical expression reads '23 + 24 =? 47', indicating a query about whether 23 plus 24 indeed equals 47.", caption: none)

 #linebreak() #figure(figph[The equation '47 = 47' is displayed with a checkmark, indicating its correctness.], alt: "The equation '47 = 47' is displayed with a checkmark, indicating its correctness.", caption: none)], [],
  [Step 7. #strong[Answer] the question.], [], [The two consecutive integers are 23 and 24.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two consecutive integers is #math.equation(block: false, alt: "95 .")[$95 .$] Find the numbers.

#solutionbox[
47, 48
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sum of two consecutive integers is #math.equation(block: false, alt: "−31 .")[$−31 .$] Find the numbers.

#solutionbox[
−15, −16
]
]

#examplebox("Example 10")[][
Find three consecutive integers whose sum is #math.equation(block: false, alt: "42 .")[$42 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem.], [], []),
  [Step 2. #strong[Identify] what you are looking for.], [], [three consecutive integers],
  [Step 3. #strong[Name.]], [], [Let #emph[n] = 1#super[st] integer #linebreak() #emph[n] + 1 = 2#super[nd] consecutive integer #linebreak() #emph[n] + 2 = 3#super[rd] consecutive integer],
  [Step 4. #strong[Translate.] #linebreak() Restate as one sentence. #linebreak() Translate into an equation.], [], [#figure(figph[Translating a word problem into an algebraic equation, where 'The sum of the three integers is 42' becomes 'n + n+1 + n+2 = 42' with visual cues.], alt: "Translating a word problem into an algebraic equation, where 'The sum of the three integers is 42' becomes 'n + n+1 + n+2 = 42' with visual cues.", caption: none)],
  [Step 5. #strong[Solve] the equation.], [], [#figure(figph[A mathematical equation is displayed on a white background, reading 'n + n + 1 + n + 2 = 42'. This equation can be simplified to 3n + 3 = 42, which implies 3n = 39, so n = 13.], alt: "A mathematical equation is displayed on a white background, reading 'n + n + 1 + n + 2 = 42'. This equation can be simplified to 3n + 3 = 42, which implies 3n = 39, so n = 13.", caption: none)],
  [Combine like terms.], [], [#figure(figph[A mathematical equation is displayed on a white background, reading '3n + 3 = 42'.], alt: "A mathematical equation is displayed on a white background, reading '3n + 3 = 42'.", caption: none)],
  [Subtract 3 from each side.], [], [#figure(figph[A mathematical equation is displayed against a white background, which reads '3n = 39'.], alt: "A mathematical equation is displayed against a white background, which reads '3n = 39'.", caption: none)],
  [Divide each side by 3.], [], [#figure(figph[The equation 'n = 13' is displayed next to the text '1st integer' on a white background.], alt: "The equation 'n = 13' is displayed next to the text '1st integer' on a white background.", caption: none)],
  [Substitute to get the second number.], [], [#figure(figph[The image shows the expression 'n + 1' alongside '2nd integer', indicating a numerical sequence or definition of an integer in relation to 'n'.], alt: "The image shows the expression 'n + 1' alongside '2nd integer', indicating a numerical sequence or definition of an integer in relation to 'n'.", caption: none)],
  [], [], [#figure(figph[The numbers '13 + 1' are displayed on a white background, with '13' in red and '+ 1' in black, indicating a simple addition problem.], alt: "The numbers '13 + 1' are displayed on a white background, with '13' in red and '+ 1' in black, indicating a simple addition problem.", caption: none)],
  [], [], [#figure(figph[The number '14' is displayed in black text against a plain white background.], alt: "The number '14' is displayed in black text against a plain white background.", caption: none)],
  [Substitute to get the third number.], [], [#figure(figph[A mathematical expression and text on a white background, showing 'n + 2' followed by '3rd integer' in blue text.], alt: "A mathematical expression and text on a white background, showing 'n + 2' followed by '3rd integer' in blue text.", caption: none)],
  [], [], [#figure(figph[A mathematical equation shows the number 13 in red, followed by a black plus sign and the number 2, all centered on a white background.], alt: "A mathematical equation shows the number 13 in red, followed by a black plus sign and the number 2, all centered on a white background.", caption: none)],
  [], [], [#figure(figph[The number 15 is prominently displayed in a simple, clear font against a white background.], alt: "The number 15 is prominently displayed in a simple, clear font against a white background.", caption: none)],
  [Step 6. #strong[Check:]], [#figure(figph[A mathematical equation displayed as 13 + 14 + 15 =? 42, posing the question of whether the sum of the three numbers equals 42.], alt: "A mathematical equation displayed as 13 + 14 + 15 =? 42, posing the question of whether the sum of the three numbers equals 42.", caption: none)

 #linebreak() #figure(figph[The number 42 is shown equal to itself, followed by a checkmark, indicating a verified mathematical truth or a solved equation.], alt: "The number 42 is shown equal to itself, followed by a checkmark, indicating a verified mathematical truth or a solved equation.", caption: none)], [],
  [Step 7. #strong[Answer] the question.], [], [The three consecutive integers are 13, 14, and 15.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three consecutive integers whose sum is #math.equation(block: false, alt: "96 .")[$96 .$]

#solutionbox[
31, 32, 33
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three consecutive integers whose sum is #math.equation(block: false, alt: "−36 .")[$−36 .$]

#solutionbox[
−11, −12, −13
]
]


=== Key Concepts

- #emph[Problem Solving Strategy]+ Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the internet.
+ Identify what you are looking for.
+ Name what you are looking for. Choose a variable to represent that quantity.
+ Translate into an equation. It may be helpful to first restate the problem in one sentence before translating.
+ Solve the equation using good algebra techniques.
+ Check the answer in the problem. Make sure it makes sense.
+ Answer the question with a complete sentence.


==== Practice Makes Perfect

#strong[Use a Problem-solving Strategy for Word Problems]

In the following exercises, use the problem-solving strategy for word problems to solve. Answer in complete sentences.

Two-thirds of the children in the fourth-grade class are girls. If there are #math.equation(block: false, alt: "20")[$20$] girls, what is the total number of children in the class?

#solutionbox[
There are 30 children in the class.
]

Three-fifths of the members of the school choir are women. If there are #math.equation(block: false, alt: "24")[$24$] women, what is the total number of choir members?

Zachary has #math.equation(block: false, alt: "25")[$25$] country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

#solutionbox[
Zachary has 125 CDs.
]

One-fourth of the candies in a bag of are red. If there are #math.equation(block: false, alt: "23")[$23$] red candies, how many candies are in the bag?

There are #math.equation(block: false, alt: "16")[$16$] girls in a school club. The number of girls is #math.equation(block: false, alt: "4")[$4$] more than twice the number of boys. Find the number of boys in the club.

#solutionbox[
There are 6 boys in the club.
]

There are #math.equation(block: false, alt: "18")[$18$] Cub Scouts in Troop #math.equation(block: false, alt: "645 .")[$645 .$] The number of scouts is #math.equation(block: false, alt: "3")[$3$] more than five times the number of adult leaders. Find the number of adult leaders.

Lee is emptying dishes and glasses from the dishwasher. The number of dishes is #math.equation(block: false, alt: "8")[$8$] less than the number of glasses. If there are #math.equation(block: false, alt: "9")[$9$] dishes, what is the number of glasses?

#solutionbox[
There are 17 glasses.
]

The number of puppies in the pet store window is twelve less than the number of dogs in the store. If there are #math.equation(block: false, alt: "6")[$6$] puppies in the window, what is the number of dogs in the store?

After #math.equation(block: false, alt: "3")[$3$] months on a diet, Lisa had lost #math.equation(block: false, alt: "12%")[$"12%"$] of her original weight. She lost #math.equation(block: false, alt: "21")[$21$] pounds. What was Lisa's original weight?

#solutionbox[
Lisa's original weight was 175 pounds.
]

Tricia got a #math.equation(block: false, alt: "6%")[$"6%"$] raise on her weekly salary. The raise was #math.equation(block: false, alt: "$30")[$"$30"$] per week. What was her original weekly salary?

Tim left a #math.equation(block: false, alt: "$9")[$"$9"$] tip for a #math.equation(block: false, alt: "$50")[$"$50"$] restaurant bill. What percent tip did he leave?

#solutionbox[
18%
]

Rashid left a #math.equation(block: false, alt: "$15")[$"$15"$] tip for a #math.equation(block: false, alt: "$75")[$"$75"$] restaurant bill. What percent tip did he leave?

Yuki bought a dress on sale for #math.equation(block: false, alt: "$72 .")[$"$72" .$] The sale price was #math.equation(block: false, alt: "60%")[$"60%"$] of the original price. What was the original price of the dress?

#solutionbox[
The original price was \$120.
]

Kim bought a pair of shoes on sale for #math.equation(block: false, alt: "$40.50 .")[$"$40.50" .$] The sale price was #math.equation(block: false, alt: "45%")[$"45%"$] of the original price. What was the original price of the shoes?

#strong[Solve Number Problems]

In the following exercises, solve each number word problem.

The sum of a number and eight is #math.equation(block: false, alt: "12 .")[$12 .$] Find the number.

#solutionbox[
4
]

The sum of a number and nine is #math.equation(block: false, alt: "17 .")[$17 .$] Find the number.

The difference of a number and twelve is #math.equation(block: false, alt: "3 .")[$3 .$] Find the number.

#solutionbox[
15
]

The difference of a number and eight is #math.equation(block: false, alt: "4 .")[$4 .$] Find the number.

The sum of three times a number and eight is #math.equation(block: false, alt: "23 .")[$23 .$] Find the number.

#solutionbox[
5
]

The sum of twice a number and six is #math.equation(block: false, alt: "14 .")[$14 .$] Find the number.

The difference of twice a number and seven is #math.equation(block: false, alt: "17 .")[$17 .$] Find the number.

#solutionbox[
12
]

The difference of four times a number and seven is #math.equation(block: false, alt: "21 .")[$21 .$] Find the number.

Three times the sum of a number and nine is #math.equation(block: false, alt: "12 .")[$12 .$] Find the number.

#solutionbox[
−5
]

Six times the sum of a number and eight is #math.equation(block: false, alt: "30 .")[$30 .$] Find the number.

One number is six more than the other. Their sum is forty-two. Find the numbers.

#solutionbox[
18, 24
]

One number is five more than the other. Their sum is thirty-three. Find the numbers.

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

#solutionbox[
8, 12
]

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

A number is one more than twice another number. Their sum is negative five. Find the numbers.

#solutionbox[
−2, −3
]

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

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

#solutionbox[
4, 10
]

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

One number is fourteen less than another. If their sum is increased by seven, the result is #math.equation(block: false, alt: "85 .")[$85 .$] Find the numbers.

#solutionbox[
32, 46
]

One number is eleven less than another. If their sum is increased by eight, the result is #math.equation(block: false, alt: "71 .")[$71 .$] Find the numbers.

The sum of two consecutive integers is #math.equation(block: false, alt: "77 .")[$77 .$] Find the integers.

#solutionbox[
38, 39
]

The sum of two consecutive integers is #math.equation(block: false, alt: "89 .")[$89 .$] Find the integers.

The sum of two consecutive integers is #math.equation(block: false, alt: "−23 .")[$−23 .$] Find the integers.

#solutionbox[
−11, −12
]

The sum of two consecutive integers is #math.equation(block: false, alt: "−37 .")[$−37 .$] Find the integers.

The sum of three consecutive integers is #math.equation(block: false, alt: "78 .")[$78 .$] Find the integers.

#solutionbox[
25, 26, 27
]

The sum of three consecutive integers is #math.equation(block: false, alt: "60 .")[$60 .$] Find the integers.

Find three consecutive integers whose sum is #math.equation(block: false, alt: "−36 .")[$−36 .$]

#solutionbox[
−11, −12, −13
]

Find three consecutive integers whose sum is #math.equation(block: false, alt: "−3 .")[$−3 .$]


==== Everyday Math

#strong[Shopping] Patty paid #math.equation(block: false, alt: "$35")[$"$35"$] for a purse on sale for #math.equation(block: false, alt: "$10")[$"$10"$] off the original price. What was the original price of the purse?

#solutionbox[
The original price was \$45.
]

#strong[Shopping] Travis bought a pair of boots on sale for #math.equation(block: false, alt: "$25")[$"$25"$] off the original price. He paid #math.equation(block: false, alt: "$60")[$"$60"$] for the boots. What was the original price of the boots?

#strong[Shopping] Minh spent #math.equation(block: false, alt: "$6.25")[$"$6.25"$] on #math.equation(block: false, alt: "5")[$5$] sticker books to give his nephews. Find the cost of each sticker book.

#solutionbox[
Each sticker book cost \$1.25.
]

#strong[Shopping] Alicia bought a package of #math.equation(block: false, alt: "8")[$8$] peaches for #math.equation(block: false, alt: "$3.20 .")[$"$3.20" .$] Find the cost of each peach.

#strong[Shopping] Tom paid #math.equation(block: false, alt: "$1,166.40")[$"$1,166.40"$] for a new refrigerator, including #math.equation(block: false, alt: "$86.40")[$"$86.40"$] tax. What was the price of the refrigerator before tax?

#solutionbox[
The price of the refrigerator before tax was \$1,080.
]

#strong[Shopping] Kenji paid #math.equation(block: false, alt: "$2,279")[$"$2,279"$] for a new living room set, including #math.equation(block: false, alt: "$129")[$"$129"$] tax. What was the price of the living room set before tax?


==== Writing Exercises

Write a few sentences about your thoughts and opinions of word problems. Are these thoughts positive, negative, or neutral? If they are negative, how might you change your way of thinking in order to do better?

#solutionbox[
Answers will vary.
]

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


==== Self Check

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

#figure(figph[A self-assessment grid helps users rate their math skills: approaching word problems positively, using problem-solving strategies, and solving number problems, with options for Confidently, With some help, or No-I don't get it!], alt: "A self-assessment grid helps users rate their math skills: approaching word problems positively, using problem-solving strategies, and solving number problems, with options for Confidently, With some help, or No-I don't get it!", caption: none)

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives 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 because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom 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 a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.