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Core Mathematics

Katherine Yoshiwara

Section 2.4 Algebraic Expressions

  • Sums and Products
  • Differences and Quotients
  • Using Algebraic Expressions
  • Evaluating Algebraic Expressions
  • Writing Algebraic Expressions

Subsection 2.4.1 What is an Algebraic Expression?

In Section 2.3 we saw that we can use a letter to represent a variable quantity, such as temperature or age. Now we’ll see how to work with variables to describe changing situations and to solve problems.
Our first task is to describe variable situations in mathematical language. When we use operation symbols to combine numbers and variables, we create algebraic expressions. Some examples of algebraic expressions are
For example, the expression \(8 + b\) might stand for 8 dollars more than the bill, and \(r \times t\) is often used to represent rate (or speed) times time.

Note 2.4.1.

Remember that a variable really is a number, but one whose value is unknown or not specified.

Subsection 2.4.2 Sums and Products

  • When we add two numbers or variables together, the result is called the sum, and the things added together are called terms.
  • When we multiply two numbers or variables together, the result is called the product, and the things multiplied together are called factors.

Products.

In algebra, we can use a raised dot or parentheses instead of a cross to indicate multiplication. For example, we write "two times three" as
\begin{equation*} \alert{2 \cdot 3}~~~~~~\text{or}~~~~~~\alert{2(3)}~~~~~~\text{or}~~~~~~\alert{2 \times 3} \end{equation*}
We write the product of two variables, or a number and a variable, next to each other without any symbol between them. For example,
\begin{gather*} \alert{5g}~~~~~~\text{means}~~~~~~\alert{5 \times g}\\ \alert{xy}~~~~~~\text{means}~~~~~\alert{x \times y} \end{gather*}

Activity 2.4.1. Ways to Write Multiplication.

Write each product in two other ways.
Six times \(H\) \(\hphantom{Six times H}\) \(\hphantom{Six times H}\)
\(4 \times 9\) \(\hphantom{Six times H}\) \(\hphantom{Six times H}\)
\(lw\) \(\hphantom{Six times H}\) \(\hphantom{Six times H}\)
\(3z\) \(\hphantom{Six times H}\) \(\hphantom{Six times H}\)
\(12(25)\) \(\hphantom{Six times H}\) \(\hphantom{Six times H}\)
\(5 \cdot A\) \(\hphantom{Six times H}\) \(\hphantom{Six times H}\)

Example 2.4.2.

Write each phrase as an algebraic expression.
  1. The product of 0.05 and \(w\)
  2. The sum of \(D\) and \(\dfrac{3}{4}\)
  3. The product of \(P\) and \(r\)
  4. The sum of \(l\) and \(w\)

Solution.

  1. \(\displaystyle 0.05w\)
  2. \(\displaystyle D + \dfrac{3}{4}\)
  3. \(\displaystyle Pr\)
  4. \(\displaystyle l + w\)

Note 2.4.3.

  • In Example 2.4.2a, we do not write a dot or parentheses between 0.05 and \(w\text{.}\) The simplest way of writing an algebraic expression is usually best.
  • We could also write \(w(0.05)\text{,}\) because we can multiply the factors in either order. However, in algebra we prefer to write the number before the variable in a product.
  • In Example 2.4.2b, we could also write \(\dfrac{3}{4} + D\text{.}\) In a sum, it does not matter which term is written first.

Checkpoint 2.4.4.

Write each phrase as an algebraic expression.
  1. The sum of \(Q\) and 1.3
  2. The product of 1.07 and \(P\)
Answer.
  1. \(\displaystyle Q + 1.3\)
  2. \(\displaystyle 1.07 P\)

Subsection 2.4.3 Differences and Quotients

When we subtract one number or variable from another, the result is called the difference. Here are two common ways to describe subtraction.

Differences.

\begin{align*} \alert{b}~~\text{subtracted from}~~\alert{12}~~~~~~\amp \text{means}~~~~~~\alert{12-p}\\ \text{the difference of}~~\alert{12}~~\text{and}~~\alert{p} ~~~~~~\amp \text{means}~~~~~~\alert{12-p} \end{align*}
When we divide one number or variable by another, the result is called the quotient. In algebra, we often use a fraction bar instead of the symbol \(\div\) to denote division. For example,

Quotients.

\begin{equation*} \alert{\dfrac{20}{5}}~~~~~~\text{means}~~~~~~\alert{20 \div 5} \end{equation*}
The fraction bar and the symbol \(\div\) both mean "divided by."

Note 2.4.5.

  • Unlike addition, in subtraction it does matter which term appears first:
    \(\alert{7 - 3}~~~\) does not mean the same thing as \(~~~\alert{3 - 7}\text{.}\)
  • The order of the numbers also matters in division:
    \(\alert{20 \div 5 = \dfrac{20}{5} = 4},~~~\) but \(~~~\alert{5 \div 20 = \dfrac{5}{20} = \dfrac{1}{4}}\)
We can switch the order of the two numbers in a sum or in a product, but we cannot switch them in a difference or a quotient. We say that the operations of addition and multiplication are commutative, but subtraction and division are not commutative.

Division.

Here are three different ways to write the division "Fifteen divided by three":
\begin{equation*} ~~\alert{3)\overline{15}~~~~~~~~~~~ 15 \div 3 ~~~~~~~~~~~ \dfrac{15}{3}} \end{equation*}
Look carefully at the order of the numbers in each expression.

Activity 2.4.2. Ways to Write Division.

Write each quotient in three other ways. The first one is done for you.
94 divided by 6 \(6)\overline{94}\) \(94 \div 6\) \(\dfrac{94}{6}\)
\(\) \(\hphantom{00000000}\vphantom{\dfrac{94}{6}}\) \(60 \div 5\) \(\hphantom{00000000}\)
\(\) \(25)\overline{260}\) \(\hphantom{00000000}\vphantom{\dfrac{94}{6}}\) \(\hphantom{00000000}\)
\(M\) divided by 8 \(\hphantom{00000000}\vphantom{\dfrac{94}{6}}\) \(\hphantom{00000000}\) \(\hphantom{00000000}\)
\(\hphantom{00000000}\) \(\hphantom{00000000}\) \(\hphantom{00000000}\) \(\dfrac{T}{c}\)
\(\hphantom{00000000}\vphantom{\dfrac{94}{6}}\) \(\hphantom{00000000}\) \(Q \div 4\) \(\hphantom{00000000}\)
\(\hphantom{00000000}\) \(\hphantom{00000000}\) \(\hphantom{00000000}\) \(\dfrac{1.5}{v}\)
\(\hphantom{00000000}\vphantom{\dfrac{94}{6}}\) \(\hphantom{00000000}\) \(18 \div k\) \(\hphantom{00000000}\)

Example 2.4.6.

Write each phrase as an algebraic expression.
  1. 16 subtracted from \(z\)
  2. \(m\) divided by 6
  3. The difference of \(R\) and \(C\)
  4. The quotient of \(P\) and \(n\)

Solution.

  1. \(\displaystyle z - 16\)
  2. \(m \div 6\) or \(\dfrac{m}{6}\)
  3. \(\displaystyle R - C\)
  4. \(\dfrac{P}{n}\) or \(P \div n\)

Note 2.4.7.

Sometimes English phrases can be tricky. We must be especially careful with subtraction and division.
  • In part (a) of Example 2.4.6, the expression \(16 - z\) is not a correct answer.
  • In part (b) of Example 2.4.6, "\(m\) divided by 6" means \(\dfrac{m}{6}\text{,}\) but "\(m\) divided into 6" means \(\dfrac{6}{m}\text{.}\)
Another common way of referring to division uses the word ratio.

Definition.

The ratio of \(a\) to \(b\) is the quotient \(\dfrac{a}{b}\text{.}\)
For example, the "ratio of \(T\) to 100" means the quotient \(\dfrac{T}{100}\text{.}\)

Checkpoint 2.4.8.

Write each phrase as an algebraic expression.
  1. The ratio of 100 to \(p\)
  2. \(d\) subtracted from 19.95
Answer.
  1. \(\displaystyle \dfrac{100}{p}\)
  2. \(\displaystyle 19.95 - d\)

Subsection 2.4.4 Using Algebraic Expressions

There are usually several different ways to write an algebraic expression in English. Here are some English phrases that are used for each of the four arithmetic operations.

Activity 2.4.3. English to Math.

  1. Addition
    English phrase Algebraic expression
    The sum of 5 and \(x\) \(\hphantom{000000}\)
    Three more than \(n\) \(\hphantom{000000}\)
    \(q\) increased by 5 \(\hphantom{000000}\)
    15 plus \(w\) \(\hphantom{000000}\)
    The total of 8 and \(D\) \(\hphantom{000000}\)
    \(M\) added to 12 \(\hphantom{000000}\)
    1.7 greater than \(B\) \(\hphantom{000000}\)
    Exceeds \(g\) by 3 \(\hphantom{000000}\)
  2. Subtraction
    English phrase Algebraic expression
    The difference of \(b\) and 2 \(\hphantom{000000}\)
    Five less than \(z\) \(\hphantom{000000}\)
    \(n\) minus 2.5 \(\hphantom{000000}\)
    Nine reduced by \(s\) \(\hphantom{000000}\)
    \(H\) subtracted from \(K\) \(\hphantom{000000}\)
    \(Q\) decreased by 16 \(\hphantom{000000}\)
  3. Multiplication
    English phrase Algebraic expression
    The product of 6 and \(w\) \(\hphantom{000000}\)
    Five times \(v\) \(\hphantom{000000}\)
    Three-quarters of \(c\) \(\hphantom{000000}\)
    Twice \(p\) \(\hphantom{000000}\)
    Eight percent of \(m\) \(\hphantom{000000}\)
  4. Division
    English phrase Algebraic expression
    The quotient of \(a\) and 7 \(\hphantom{000000}\)
    \(G\) divided by 10 \(\hphantom{000000}\)
    \(m\) divided into 15 \(\hphantom{000000}\)
    The ratio of \(V\) to 5 \(\hphantom{000000}\)
    \(S\) split 7 ways \(\hphantom{000000}\)
    Dollars, \(d\) per hour, \(h\) \(\hphantom{000000}\)

Example 2.4.9.

Write algebraic expressions for each phrase.
  1. Five-eighths of \(M\)
  2. Four less than \(w\)

Solution.

  1. \(\displaystyle \dfrac{5}{8}M\)
  2. \(\displaystyle w - 4\)

Note 2.4.10.

In part (a) of Example 2.4.9, notice that "of" means multiplication when used with fractions. The same is true for percents. For example, "35% of \(P\)" is written as \(0.35 P\)

Checkpoint 2.4.11.

Write algebraic expressions for each phrase.
  1. Six greater than \(h\)
  2. The product of 9 and \(c\)
Answer.
  1. \(\displaystyle h + 6\)
  2. \(\displaystyle 9c\)

Subsection 2.4.5 Evaluating Algebraic Expressions

Now we are ready to use algebraic expressions to describe variable situations. If we know the value of the variable, we can substitute this value into the expression and simplify the result. This process is called evaluating the algebraic expression.

Example 2.4.12.

If Martha puts \(g\) gallons of gas in her car, she can drive \(22g\) miles.
  1. How many miles can Martha drive if she buys 8 gallons of gas?
  2. Explain how to find the number of miles Martha can drive if she buys \(g\) gallons of gas.
  3. Evaluate the expression \(22g\) for the given values of \(g\text{.}\)
    \(g\) 4 7.5 9 10 12.5 14
    \(22g\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{00}\) \(\hphantom{000}\)

Solution.

  1. Martha can drive \(22 \cdot 8\text{,}\) or 176 miles.
  2. We multiply the number of gallons, \(g\text{,}\) by 22.
  3. We multiply each value of \(g\) by 22 and fill in the table.
    \(g\) 4 7.5 9 10 12.5 14
    \(22g\) 88 165 198 220 275 308

Checkpoint 2.4.13.

There is a room tax on all hotel bills. When the bill is \(B\) dollars, the tax is \(0.08B\) dollars.
  1. How much is the tax on $100?
  2. Explain how to find the tax when the bill is \(B\) dollars.
  3. Evaluate the expression \(0.08B\) for the given values of \(B\text{.}\)
    \(B\) 60.00 80.00 120.00 250.00
    \(0.08B\)
Answer.
  1. $8
  2. Multiply \(B\) by 0.08.
  3. \(B\) 60.00 80.00 120.00 250.00
    \(0.08B\) 4.80 6.40 9.60 20.00

Example 2.4.14.

  1. Evaluate \(y + 14\) for \(y = \alert{7}\)
  2. Evaluate \(3xz\) for \(x = \alert{5}\) and \(z = \alert{4}\text{.}\)

Solution.

  1. We replace \(y\) by \(\alert{7}\) to get
    \begin{equation*} y + 14 = \alert{7} + 14 = 21~~~~~~~~ \blert{\text{Add 14 to 7.}} \end{equation*}
  2. We replace \(x\) by \(\alert{5}\) and \(z\) by \(\alert{4}\) to get
    \begin{equation*} 3xz = 3(\alert{5})(\alert{4}) = 60~~~~~~~~ \blert{\text{Multiply 3 by 5 by 4.}} \end{equation*}

Checkpoint 2.4.15.

Evaluate each expression for \(a = \alert{0.3},~b = \alert{1.2}\text{,}\) and \(c = \alert{6}\text{.}\)
  1. \(\displaystyle 8.4-b\)
  2. \(\displaystyle \dfrac{9}{a}\)
  3. \(\displaystyle 5bc\)
Answer.
  1. \(\displaystyle 7.2\)
  2. \(\displaystyle 30\)
  3. \(\displaystyle 36\)

Subsection 2.4.6 Writing Algebraic Expressions

Now it’s time to write our own algebraic expressions to describe variable situations.

Example 2.4.16.

Choose the correct algebraic expression from the box for each situation described below.
\(\hphantom{000} n+6 \hphantom{000}\) \(\hphantom{000} n-6 \hphantom{000}\) \(\hphantom{000} 6-n \hphantom{000}\)
\(6n\) \(\dfrac{n}{6}\) \(\dfrac{6}{n}\)
  1. Rashad is 6 years younger than Shelley. If Shelley is \(n\) years old, how old is Rashad?
  2. Each package of sodas contains 6 cans. If Antoine bought \(n\) packages of sodas, how many cans did he buy?
  3. Lizette and Patrick together own 6 cats. If Lizette owns \(n\) cats, how many cats does Patrick own?

Solution.

  1. We subtract 6 from Shelley’s age to get: \(~~n-6\)
  2. We multiply the number of packages by 6 to get: \(~~6n\)
  3. We subtract the number of Lizette’s cats from the total number of cats to get: \(~~6-n\)

Checkpoint 2.4.17.

Choose the correct algebraic expression from the box for each situation described below.
\(\hphantom{000} v+5 \hphantom{000}\) \(\hphantom{000} v-5 \hphantom{000}\) \(\hphantom{000} 5-v \hphantom{000}\)
\(5v\) \(\dfrac{v}{5}\) \(\dfrac{5}{v}\)
  1. The volume of a standard vat is \(v\) liters. What is the total volume of 5 standard vats?
  2. The maximum velocity of a jet ski is \(v\) miles per hour. How fast can the jet ski travel against a current, if the speed of the current is 5 miles per hour?
  3. Each batch of cookies requires \(v\) teaspoons of vanilla. How many batches can be made with 5 teaspoons of vanilla?
Answer.
  1. \(5v\) liters
  2. \(v - 5\) miles per hour
  3. \(\dfrac{5}{v}\) batches
The first step in solving problems with algebra is to describe the problem in mathematical terms. This usually involves writing algebraic expressions. We can think of this process in three steps.

Steps for Writing Algebraic Expressions.

  1. Identify the unknown quantity. Remember that it must have numerical values.
  2. Choose a variable to represent the unknown quantity.
  3. Translate the English phrase into an algebraic expression, using the variable and appropriate operation symbols.

Example 2.4.18.

Write an algebraic expression for "6 feet taller than the height of the roof."

Solution.

Step 1 The height of the roof is unknown.
Step 2 Height of the roof: \(~~~~h\)
Step 3 "Taller than" suggests addition: \(~~~~h+6\)

Checkpoint 2.4.19.

Write an algebraic expression for "two-thirds of the weight of the best-selling laptop."
Answer.
Step 1 The weight of the best-selling laptop is unknown
Step 2 Weight of best-selling laptop: \(~~~~w\)
Step 3 "Two-thirds of" indicates a product: \(~~~~\dfrac{2}{3}w\)

Subsection 2.4.7 Vocabulary

  • algebraic expression
  • evaluate
  • terms
  • factors
  • sum
  • product
  • difference
  • quotient
  • ratio
  • commutative

Exercises 2.4.8 Practice 2.4

1.

Choose the correct operation (addition, subtraction, multiplication, or division) associated with each word.
  1. product
  2. factors
  3. difference
  4. sum
  5. terms
  6. quotient
  7. ratio

2.

Write each phrase as an algebraic expression.
  1. The product of \(q\) and 6
  2. The ratio of \(t\) to 20
  3. Five subtracted form \(h\)
  4. The sum of 8 and \(z\)
  5. Four divided into \(d\)
  6. The difference of \(m\) and 15
  7. Sixteen divided by \(v\)
  8. The quotient of \(w\) and 12

3.

Describe two ways besides "\(\times\)" to indicate multiplication by giving an example of each.

4.

Describe three ways to show division by giving an example of each.

5.

When two numbers or variables are added together, the two numbers or variables are called . The result of the addition is called the .

6.

When two numbers or variables are multiplied together, the two numbers or variables are called . The result of the multiplication is called the .

Exercise Group.

For Problems 7-16, write an algebraic expression for the English phrase.
7.
  1. Eight less than \(v\)
  2. Four divided into \(c\)
8.
  1. \(H\) increased by 14
  2. Ten times \(g\)
9.
  1. \(t\) divided by 5
  2. Twenty more than \(J\)
10.
  1. Two-thirds of \(w\)
  2. Sixteen reduced by \(h\)
11.
  1. The ratio of 15 to \(b\)
  2. The total of 32 and \(f\)
12.
  1. The The difference of \(R\) and 26
  2. 17.5 divided by \(q\)
13.
  1. \(P\) split three ways
  2. \(Z\) reduced by \(\dfrac{3}{2}\)
14.
  1. 18% of \(N\)
  2. 4% of \(G\)
15.
  1. \(d\) subtracted from \(r\)
  2. The product of \(y\) and \(z\)
16.
  1. The total of \(P\) and \(I\)
  2. \(m\) less than \(k\)

Exercise Group.

For Problems 17-28, write an English phrase for the algebraic expression. (Many answers are possible.)
17.
\(7.1 - a\)
18.
\(s + 0.3\)
19.
\(15j\)
20.
\(\dfrac{36}{u}\)
21.
\(\dfrac{A}{c}\)
22.
\(M - D\)
23.
\(y + h\)
24.
\(ng\)
25.
\(H - 2.5\)
26.
\(\dfrac{x}{30}\)
27.
\(\dfrac{3}{5}b\)
28.
\(z + \dfrac{5}{2}\)

29.

Write an English phrase for each of the following, then simplify. Which expressions are the same?
  1. \(\displaystyle 24 \div 6\)
  2. \(\displaystyle 6 \div 24\)
  3. \(\displaystyle \dfrac{6}{24}\)
  4. \(\displaystyle \dfrac{24}{6}\)

30.

Write an English phrase for each of the following, then simplify. Which expressions are the same?
  1. \(\displaystyle \dfrac{36}{4}\)
  2. \(\displaystyle \dfrac{1}{4}(36)\)
  3. \(\displaystyle 36 \div 4\)
  4. \(\displaystyle 0.25(36)\)

31.

Petite size skirts are hemmed shorter than regular size skirts in the same style. Write an expression for the length of a petite skirt if the regular skirt has length \(L\) inches.
Length of regular skirt (in) 24 26.5 31 34
Length of petite skirt (in) 21.5 24 28.5 31.5

32.

If Evyn writes a check for \(d\) dollars, write an expression for the amount of money left in her checking account.
Amount of check ($) 20 40 50 70
Amount left ($) 80 60 50 30

33.

The decorators for a new hotel are buying carpet. Write an expression for the cost of carpeting a room of area \(A\) square feet.
Area of room (sq ft) 100 150 200 360
Cost of carpet ($) 800 1200 1600 2880

34.

The engineers in Dana’s office are splitting the cost of a wedding gift for their secretary. If \(m\) people contribute, write an expression for each person’s share.
Number of people 2 3 4 8
Each person’s share ($) 24 16 12 5

35.

Write an expression for the sale price in terms of the regular price, \(p\text{.}\) (Hint: The sale price is a fraction of the regular price.)
Regular price ($) Sale price ($)
4 3
8 6
20 15
100 75

36.

Write an expression for the number of cups of flour in terms of the number of cups of milk, \(c\)
Cups of milk Cups of flour
2 3
5 7.5
6 9
8 12

Exercise Group.

For Problems 37 and 38, choose the correct algebraic expression from the options given.
\begin{equation*} ~~~~~~ m + 15 ~~~~~~~~~~~~ m - 15 ~~~~~~~~~~~~ 15 - m ~~~~~~ \end{equation*}
37.
  1. Carol weighs 15 pounds less than Garth. If Garth weighs \(m\) pounds, how much does Carol weigh?
  2. Amber and Beryl together planted 15 trees. If Amber planted \(m\) trees, how many trees did Beryl plant?
  3. Fred earned $15 more this week than last week. If he earned \(m\) dollars last week, how much did he earn this week?
38.
  1. Meg bicycled 15 miles farther than Kwan. If Kwan rode \(m\) miles, how far did Meg ride?
  2. There are 15 children in Amy’s swim class. If there are \(m\) girls, how many are boys?
  3. The sale price of a sweater is $15 less than the regular price. If the regular price is \(m\) dollars, what is the sale price?

Exercise Group.

For Problems 39 and 40, choose the correct algebraic expression from the options given.
\begin{equation*} ~~~~~~ 12p ~~~~~~~~~~~~ \dfrac{12}{p} ~~~~~~~~~~~~ \dfrac{p}{12} ~~~~~~ \end{equation*}
39.
  1. Julian earns $12 an hour. If he works for \(p\) hours, how much will he make?
  2. Farmer Brown collected \(p\) eggs this morning. How many dozen is that?
  3. Melissa bought 12 colored markers. If their total cost was \(p\) dollars, how much did each marker cost?
40.
  1. Rosalind is baby-sitting for \(p\) children. If she brings 12 puzzles, how many will each child get?
  2. Hector has to read 12 chapters in his history text. If he has \(p\) days to complete the assignment, how many chapters should he read per day?
  3. Roma swims 12 laps per day. After \(p\) days, how many laps will she have swum?

Exercise Group.

For Problems 41 and 42, choose the correct algebraic expression from the options given.
\begin{gather*} ~~~~~~ c+36 ~~~~~~~~~~~~ c-36 ~~~~~~~~~~~~ 36-c ~~~~~~\\ ~~ 36c ~~~~~~~~~~~~~~ \dfrac{36}{c} ~~~~~~~~~~~~~~~~ \dfrac{c}{36} ~~~~~~ \end{gather*}
41.
  1. Renee made $36 for \(c\) hours of work. How much does she earn per hour?
  2. A pair of shorts and a shirt cost $36. If the shorts cost \(c\) dollars, how much did the shirt cost?
  3. Elba’s Scrabble total was 36 points higher than Carla’s. If Carla’s total was \(c\) points, what was Elba’s total?
  4. Marc’s business will use \(c\) mailing envelopes this month. If there are 36 envelopes in a package, how many packages should he order?
42.
  1. Heather had 36 math problems to work. She finished \(c\) of them. How many more does she have to work?
  2. Takiji is 36 years older than Seiki. If Takiji is \(c\) years old, how old is Seiki?
  3. If the 36 members of the Ski Club each pay \(c\) dollars in dues, how much money will the club raise?
  4. If \(c\) roommates share the $36 electricity bill, how much should each pay?

Exercise Group.

For Problems 43-52, write an algebraic expression for each word phrase. Follow the three steps given in the lesson.
43.
Three times the cost of a water filter.
44.
Eight more than the number of students.
45.
The radius of the circle decreased by 6 centimeters.
46.
Ten times the height of the triangle.
47.
Two-fifths of the checking account balance.
48.
Three and one-quarter inches taller than last year’s height.
49.
The price of the pizza divided by four.
50.
Thirty dollars more than a bus ticket.
51.
The perimeter of the triangle diminished by 15 feet.
52.
The quotient of the volume of the sphere and 8.

Exercise Group.

For Problems 53-56,
  1. Write an algebraic expression.
  2. Evaluate the expression.
53.
  1. Amanda’s bread recipe calls for three cups of flour more than she has. Choose a variable for the amount of flour Amanda has, then write an expression for the amount of flour required.
  2. If Amanda has 8 cups of flour, how much flour does the recipe call for?
54.
  1. Garth received 432 fewer votes than his opponent in the election. Choose a variable for the number of votes Garth’s opponent received, then write an expression for the number of votes Garth received.
  2. If Garth’s opponent got 3297 votes, how many votes did Garth get?
55.
  1. To find his homework grade, Jamal should divide his total points by 600. Choose a variable for Jamal’s total points, then write an expression for Jamal’s homework grade.
  2. If Jamal earned 480 homework points, what is his homework grade?
56.
  1. Each member of the Forensics Club pays $5 in dues. Choose a variable for the number of members, then write an expression for the amount the club earns in dues.
  2. If the Forensics Club has 38 members, how much does it earn in dues?

Exercise Group.

For Problems 57 and 58, use the formula
\begin{equation*} \blert{\text{time} = \dfrac{\text{distance}}{\text{rate}}} \end{equation*}
where "rate" is another word for speed.
57.
  1. Write an expression for the time it takes to travel from Kansas City to Salt Lake City (approximately 1000 miles) at different speeds.
  2. How long would it take to travel from Kansas City to Salt Lake City by wagon train at 8 miles per hour?
58.
  1. Write an expression for the time it takes for the different members of the track team to run 800 meters at different speeds.
  2. How long would it take to run 800 meters at a speed of 5 meters per second?

Exercise Group.

For Problems 59-64, evaluate for \(x = 6,~ y = 3\text{,}\) and \(z = 8\text{.}\)
59.
  1. \(\displaystyle z-y\)
  2. \(\displaystyle x+x\)
  3. \(\displaystyle 9.4+x\)
60.
  1. \(\displaystyle 12.7-z\)
  2. \(\displaystyle 5x\)
  3. \(\displaystyle \dfrac{1}{2}z\)
61.
  1. \(\displaystyle xz\)
  2. \(\displaystyle 2xyz\)
  3. \(\displaystyle 0.4y\)
62.
  1. \(\displaystyle xxx\)
  2. \(\displaystyle \dfrac{12}{x}\)
  3. \(\displaystyle \dfrac{z}{4}\)
63.
  1. \(\displaystyle \dfrac{y}{9x}\)
  2. \(\displaystyle \dfrac{x}{z}\)
  3. \(\displaystyle \dfrac{2.4}{x}\)
64.
  1. \(\displaystyle 0.8z\)
  2. \(\displaystyle z-0.8\)
  3. \(\displaystyle \dfrac{z}{0.8}\)
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