Translating and Solving Algebraic Word Problems Practice Questions — ACT — Page 2

Question 21Question →

A landscaping company charges a flat equipment fee plus a fixed hourly rate for lawn maintenance. For a job that took $4$ hours, the company charged a total of $\$190$ . For a different job that took $7$ hours, the company charged a total of $\$295$ . The company also offers a package that includes a $15\%$ discount on the hourly rate, but the flat equipment fee remains the same. Under this discounted package, what is the total charge, in dollars, for a job that takes $8$ hours?

Show answer & explanation

Answer: 288

Answer

The total charge for an

8

-hour job under the discounted package is

288

dollars.

By translating the problem into the equations

F + 4H = 190

and

F + 7H = 295

, we find that the hourly rate

H

is

35

dollars and the flat fee

F

is

50

dollars. Applying a

15\%

discount to the hourly rate yields a new rate of

29.75

dollars per hour. The total cost for

8

hours is then

50 + 8(29.75) = 288

dollars.

Step-by-Step Solution

1

Define variables for the flat fee and hourly rate, and translate the given information into a system of linear equations.

F + 4H = 190

and

F + 7H = 295

, where

F

is the flat fee and

H

is the hourly rate.

To represent the cost structure algebraically.

2

Solve the system of equations by elimination or substitution.

H = 35

and

F = 50

.

To determine the individual cost components (flat fee and hourly rate).

3

Apply the

15\%

discount to the hourly rate.

H_{\text{discounted}} = 35 \times (1 - 0.15) = 29.75

.

To find the new hourly rate under the discounted package.

4

Calculate the total cost for

8

hours of work with the flat fee and the discounted hourly rate.

\text{Total Cost} = 50 + 8(29.75) = 288

.

To answer the question asking for the total charge of an

8

-hour job under the discounted package.

Key Concept

Translating verbal statements into a system of linear equations and solving them to evaluate a modified expression.

Question 22Question →

A digital marketing firm allocates a total monthly advertising budget of $B$ dollars between search engine ads and social media ads. The amount allocated to social media ads is $\$800$ less than twice the amount allocated to search engine ads. The firm then increases its total monthly budget by $20\%$ and allocates this entire increase to search engine ads. As a result, the new amount allocated to search engine ads is equal to the amount originally allocated to social media ads. What was the firm's original total monthly advertising budget, $B$ ?

\$1,600

\$2,200

\$2,400

\$4,000

\$8,000

Show answer & explanation

Answer: $\$4,000$

Answer

The original total monthly advertising budget was

\$4,000

.

By defining the original search engine advertising budget as

S

, the original social media advertising budget as

M = 2S - 800

, and the original total budget as

B = 3S - 800

, we can set up the equation

S + 0.20B = M

using the condition that the new search engine budget equals the original social media budget. Solving this equation yields

S = 1600

. Substituting

S = 1600

into the expression for the total budget gives

B = 3(1600) - 800 = 4000

dollars.

Step-by-Step Solution

1

Define the variables representing each budget allocation.

Let

S

represent the original amount allocated to search engine ads, and let

M

represent the original amount allocated to social media ads. The original total budget is

B = S + M

.

Establishing clear variables is necessary to translate the verbal relationships into equations.

2

Translate the relationship between the two advertising categories into an equation.

M = 2S - 800

The problem states that the social media ads budget is

\$800

less than twice the search engine ads budget.

3

Express the original total budget

B

in terms of a single variable,

S

.

B = S + (2S - 800) = 3S - 800

Substituting the expression for

M

into the total budget equation simplifies the system to a single variable.

4

Set up the equation for the new budget allocation after the

20\%

increase.

S + 0.20B = M

An increase of

20\%

of the total budget is

0.20B

, which is added entirely to the search engine ads budget, making it equal to the original social media ads budget.

5

Substitute the expressions for

B

and

M

in terms of

S

into the new budget equation.

S + 0.20(3S - 800) = 2S - 800

Substituting the relationships found in steps 2 and 3 allows us to solve for

S

directly.

6

Solve the equation for

S

.

S + 0.60S - 160 = 2S - 800 \Rightarrow 1.60S - 160 = 2S - 800 \Rightarrow 640 = 0.40S \Rightarrow S = 1,600

Distributing, combining like terms, and isolating the variable yields the original search engine ads allocation.

7

Calculate the original total budget,

B

.

B = 3(1,600) - 800 = 4,800 - 800 = 4,000

The question asks for the original total budget

B

, so we must evaluate the expression for

B

using the value of

S

.

Key Concept

Translating verbal descriptions of algebraic relationships into systems of linear equations and solving them.

Question 23Question →

An agricultural cooperative packages a premium seed mixture containing rye grass, fescue, and bluegrass. The weight of the fescue in the mixture is $10$ pounds less than twice the weight of the rye grass. The weight of the bluegrass is $15$ pounds more than half the weight of the fescue. If the total weight of the mixture is $120$ pounds, how many pounds of bluegrass are in the mixture?

Show answer & explanation

Answer: 40

Answer

The weight of the bluegrass in the mixture is 40 pounds.

By translating the given relationships into equations (

f = 2r - 10

,

b = \frac{1}{2}f + 15

, and

r + f + b = 120

), we can express all variables in terms of

r

, yielding

r + (2r - 10) + (r + 10) = 120

. Solving this gives

r = 30

. Substituting this back gives the weight of bluegrass as

30 + 10 = 40

pounds.

Step-by-Step Solution

1

Define variables for each type of grass in the mixture.

Let

r

represent the weight of rye grass,

f

represent the weight of fescue, and

b

represent the weight of bluegrass.

Establishing variables is necessary to translate the verbal descriptions into algebraic terms.

2

Translate the given relationships into equations.

f = 2r - 10

and

b = \frac{1}{2}f + 15

The problem states the fescue is 10 pounds less than twice the rye grass, and the bluegrass is 15 pounds more than half the fescue.

3

Substitute the expression for f into the equation for b to express b solely in terms of r.

b = \frac{1}{2}(2r - 10) + 15 = r - 5 + 15 = r + 10

Reducing the number of variables simplifies the system of equations.

4

Set up the total weight equation and solve for r.

r + (2r - 10) + (r + 10) = 120 \implies 4r = 120 \implies r = 30

The sum of the three grass weights is given as 120 pounds.

5

Calculate the weight of the bluegrass using the value of r.

b = 30 + 10 = 40

The question asks specifically for the weight of the bluegrass.

Key Concept

Translating verbal relationships into linear equations and solving a system of equations

Question 24Question →

A manufacturing plant operates two machines, Machine X and Machine Y, to produce identical components. Machine Y produces 12 fewer components per hour than Machine X. On a certain day, Machine X operated for 5 hours and Machine Y operated for 7 hours. If the two machines produced a combined total of 636 components, how many components did Machine X produce on that day?

46

60

230

270

300

Show answer & explanation

Answer: 300

Answer

300

To find the number of components Machine X produced, let

x

represent Machine X's hourly production rate. This makes Machine Y's hourly rate

x - 12

. Since Machine X operated for 5 hours and Machine Y operated for 7 hours to produce a total of 636 components, we can write the equation:

5x + 7(x - 12) = 636

. Distributing the 7 yields

5x + 7x - 84 = 636

, which simplifies to

12x - 84 = 636

. Adding 84 to both sides gives

12x = 720

, and dividing by 12 yields

x = 60

components per hour for Machine X. To find the total produced by Machine X, we multiply its hourly rate by the 5 hours it operated:

5 \times 60 = 300

. This matches the correct option.

Step-by-Step Solution

1

Define variables for the hourly rates of both machines.

Let

x

be the number of components Machine X produces per hour. Since Machine Y produces 12 fewer components per hour, Machine Y's hourly rate is

x - 12

.

Establishing algebraic expressions for each machine's rate allows us to set up a linear equation.

2

Set up an equation representing the total combined production.

The total production is the sum of the components produced by Machine X in 5 hours and Machine Y in 7 hours:

5x + 7(x - 12) = 636

.

The sum of the products of each machine's rate and its operating time equals the total combined production of 636 components.

3

Solve the equation for the hourly rate of Machine X,

x

.

Distribute the 7:

5x + 7x - 84 = 636

. Combine like terms:

12x - 84 = 636

. Add 84 to both sides:

12x = 720

. Divide by 12:

x = 60

.

Solving for

x

gives the hourly rate of Machine X.

4

Calculate the total components produced by Machine X.

Multiply the hourly rate of Machine X by its hours of operation:

5 \text{ hours} \times 60 \text{ components/hour} = 300

.

The question asks for the total components produced by Machine X, not its hourly rate.

Key Concept

Setting up and solving single-variable linear equations from verbal descriptions

Alternative Method

Instead of defining the variable as Machine X's rate, we could define

y

as Machine Y's hourly rate. Then Machine X's rate is

y + 12

. The total equation becomes

5(y + 12) + 7y = 636 \implies 5y + 60 + 7y = 636 \implies 12y = 576 \implies y = 48

. Since Machine X's hourly rate is

y + 12 = 60

, it produced

5 \times 60 = 300

components.

Estimated Time:1m 30s

Question 25Question →

For the opening night of a school play, a total of $320$ tickets were sold, raising a total of $\$2{}140$ in ticket sales. Student tickets were sold for $\$5$ each, and adult tickets were sold for $\$8$ each. How many adult tickets were sold?

Show answer & explanation

Answer: 180

Answer

180

The correct answer is

180

. By setting up a single-variable linear equation where

a

represents the number of adult tickets, the problem translates to

8a + 5(320 - a) = 2140

. Simplifying this yields

3a + 1600 = 2140

, which simplifies to

3a = 540

and results in

a = 180

.

Step-by-Step Solution

1

Define variables for the unknown quantities based on the given total.

Let

a

represent the number of adult tickets sold. The number of student tickets sold is represented as

320 - a

.

Since the total number of tickets is

320

, subtracting the number of adult tickets from the total yields the number of student tickets.

2

Construct a linear equation using the ticket prices and the total revenue.

8a + 5(320 - a) = 2140

Multiplying the quantity of each ticket type by its respective price (

8

dollars for adult and

5

dollars for student) yields the total ticket sales revenue of

2,140

dollars.

3

Solve the equation for the variable

a

.

8a + 1600 - 5a = 2140 \implies 3a + 1600 = 2140 \implies 3a = 540 \implies a = 180

Apply the distributive property, group like terms, isolate the variable term, and divide to solve for the number of adult tickets.

Key Concept

Translating word problems into a single-variable linear equation and solving for the unknown quantity.

Question 26Question →

A local library has two types of books: fiction and non-fiction. The number of fiction books is $120$ more than twice the number of non-fiction books. If the library has a total of $1\text{}860$ books, how many more fiction books than non-fiction books does the library have?

580

780

700

870

1\text{}280

Show answer & explanation

Answer: $700$

Answer

The library has

700

more fiction books than non-fiction books.

The correct answer is

700

because defining the non-fiction books as

n

gives

2n + 120

fiction books. Summing these to equal

1860

yields

3n + 120 = 1860

, which solves to

n = 580

. The number of fiction books is therefore

2(580) + 120 = 1280

. The difference between the two quantities is

1280 - 580 = 700

.

Step-by-Step Solution

1

Define variables for the two categories of books.

Let

n

be the number of non-fiction books. Then the number of fiction books,

f

, can be written as

f = 2n + 120

.

This translates the statement 'the number of fiction books is

120

more than twice the number of non-fiction books' into an algebraic expression.

2

Set up an equation representing the total number of books.

n + f = 1860 \implies n + (2n + 120) = 1860 \implies 3n + 120 = 1860

.

The sum of the fiction and non-fiction books must equal the total library inventory of

1860

books.

3

Solve the equation for the number of non-fiction books,

n

.

3n = 1740 \implies n = 580

.

Subtracting

120

from both sides and then dividing by

3

isolates the variable

n

.

4

Calculate the number of fiction books and find the difference.

Fiction books:

f = 2(580) + 120 = 1280

. Difference:

f - n = 1280 - 580 = 700

.

The question asks for the difference between the number of fiction and non-fiction books.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 27Question →

A smart home heating system has two operating modes: Eco mode and Comfort mode. In a certain week, the system was active in one of these two modes for a total of $168$ hours. The number of hours the system was in Eco mode was $24$ hours less than three times the number of hours it was in Comfort mode. For how many hours was the heating system in Comfort mode during that week?

Show answer & explanation

Answer: 48

Answer

The heating system was in Comfort mode for 48 hours.

The correct answer is obtained by representing the hours in Comfort mode as

c

and Eco mode as

3c - 24

. Since their sum is

168

, we write the equation

c + 3c - 24 = 168

. Solving this linear equation gives

4c = 192

, which yields

c = 48

.

Step-by-Step Solution

1

Define variables for the Comfort and Eco modes.

Let

c

represent the hours in Comfort mode and

e

represent the hours in Eco mode.

To represent the unknown quantities algebraically.

2

Translate the given information into a system of equations.

c + e = 168

and

e = 3c - 24

The sum of the hours in both modes is

168

, and the relationship between Eco and Comfort hours is described.

3

Substitute the expression for

e

into the total hours equation.

c + (3c - 24) = 168

To create a single linear equation with one variable.

4

Solve for the variable

c

.

4c - 24 = 168 \Rightarrow 4c = 192 \Rightarrow c = 48

Simplifying the equation gives the value of

c

, which is the hours spent in Comfort mode.

Key Concept

Translating and Solving Algebraic Word Problems

Question 28Question →

A commercial printing press charges a business a setup fee of $100$ plus $0.15$ per brochure printed. To encourage green practices, the press offers a recycling discount equal to $30$ less than $0.05$ per brochure printed. If the business has a budget of at most $380$ for their brochure order, what is the maximum number of brochures they can print?

1,250

1,550

2,500

3,100

3,500

Show answer & explanation

Answer: 2,500

Answer

2,500

The correct answer is 2,500 brochures. The total cost is the setup fee of

100 plus the printing cost of

0.15 per brochure, minus the discount of

30 less than

0.05 per brochure. Setting

b

as the number of brochures, the discount is

0.05b - 30

. The total cost equation is

100 + 0.15b - (0.05b - 30) = 130 + 0.10b

. Setting this less than or equal to

380

yields

130 + 0.10b \leq 380 \Rightarrow 0.10b \leq 250 \Rightarrow b \leq 2,500

.

Step-by-Step Solution

1

Define the variable and write the algebraic expressions for the cost components.

Let

b

be the number of brochures printed. The total cost before the discount is

100 + 0.15b

. The recycling discount is represented as

0.05b - 30

.

Establishing clear variables and translating the textual relationships into mathematical expressions is the first step in solving word problems.

2

Set up the inequality representing the budget constraint.

100 + 0.15b - (0.05b - 30) \leq 380

The total cost (original cost minus the discount) must be at most the budget of

380

.

3

Simplify the left side of the inequality.

100 + 0.15b - 0.05b + 30 \leq 380 \Rightarrow 130 + 0.10b \leq 380

Distributing the subtraction sign across the discount expression and combining like terms simplifies the inequality.

4

Solve for the variable

b

.

0.10b \leq 250 \Rightarrow b \leq 2500

Isolating

b

gives the maximum number of brochures that can be printed.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 29Question →

A boutique chocolatier packages two types of gift boxes: Standard and Deluxe. Each Standard box contains $4$ dark chocolates, and each Deluxe box contains $8$ dark chocolates. The chocolatier plans to prepare a batch of boxes such that the number of Standard boxes is exactly twice the number of Deluxe boxes. If the chocolatier has a total of $160$ dark chocolates available and uses all of them for this batch, what is the total number of boxes (both Standard and Deluxe combined) they can make?

Show answer & explanation

Answer: 30

Answer

The total number of boxes they can make is 30.

By defining the number of Deluxe boxes as

x

, the number of Standard boxes is

2x

. The equation for the total number of dark chocolates is

4(2x) + 8x = 160

. Solving this gives

16x = 160

, which simplifies to

x = 10

. Thus, there are

10

Deluxe boxes and

20

Standard boxes, for a combined total of

30

boxes.

Step-by-Step Solution

1

Define variables for the number of boxes of each type.

Let

x

be the number of Deluxe boxes and

2x

be the number of Standard boxes.

We are given that the number of Standard boxes is exactly twice the number of Deluxe boxes.

2

Set up an equation representing the total number of dark chocolates used.

4(2x) + 8x = 160

Each Standard box requires

4

dark chocolates, each Deluxe box requires

8

, and the total used is

160

.

3

Solve the equation for

x

.

16x = 160 \Rightarrow x = 10

Combining like terms gives

16x = 160

, and dividing both sides by

16

yields

x = 10

.

4

Calculate the total number of boxes.

x + 2x = 30

The total number of boxes is the sum of Deluxe boxes (

10

) and Standard boxes (

20

).

Key Concept

Translating verbal descriptions of relationships and totals into linear equations to solve algebraic word problems.

Estimated Time:1m 30s

Question 30Question →

An online streaming service offers a basic monthly plan that costs $12$ dollars plus $1.50$ dollars for each premium movie rented. A premium monthly plan costs $20$ dollars and includes $3$ free premium movie rentals, with each additional rental costing $1.00$ dollar. If a subscriber rents $x$ premium movies in a month, where $x > 3$ , which of the following expressions represents the savings, in dollars, of the premium plan compared to the basic plan?

0.50x - 8

5 - 0.50x

0.50x - 5

2.50x - 11

0.50x - 11

Show answer & explanation

Answer: $0.50x - 5$

Answer

0.50x - 5

The cost of the basic plan for

x

rentals is

12 + 1.50x

dollars. Under the premium plan, the first

3

rentals are free, so the subscriber only pays for

x - 3

rentals. The cost of the premium plan is

20 + 1.00(x - 3)

dollars, which simplifies to

17 + x

dollars. The savings of the premium plan compared to the basic plan is the difference between the basic cost and the premium cost:

(12 + 1.50x) - (17 + x) = 0.50x - 5

dollars.

Step-by-Step Solution

1

Write the expression for the cost of the basic monthly plan.

12 + 1.50x

The basic plan has a flat fee of

12

dollars and a rate of

1.50

dollars per movie for all

x

movies rented.

2

Write the expression for the cost of the premium monthly plan.

20 + 1.00(x - 3)

, which simplifies to

17 + x

.

The premium plan has a flat fee of

20

dollars. Since

3

rentals are free, the subscriber only pays

1.00

dollar each for the remaining

x - 3

rentals.

3

Subtract the premium plan cost from the basic plan cost to find the savings.

(12 + 1.50x) - (17 + x) = 0.50x - 5

Savings represents the difference in cost: Basic Cost minus Premium Cost.

Key Concept

Translating verbal descriptions of multi-step costs into linear algebraic expressions and simplifying them.

Estimated Time:1m 30s

Question 31Question →

Two cyclists start at opposite ends of a $90$ -mile trail at the same time and ride toward each other. One cyclist rides at a constant speed that is $3$ miles per hour faster than the other cyclist. If the two cyclists meet after exactly $3$ hours, what is the constant speed, in miles per hour, of the faster cyclist?

Show answer & explanation

Answer: 16.5

Answer

The speed of the faster cyclist is

16.5

miles per hour.

The correct answer of

16.5

is found by setting the speed of the slower cyclist to

s

and the faster cyclist to

s + 3

. Since both cyclists ride toward each other for

3

hours, their combined distance is

3s + 3(s + 3) = 90

. Solving for

s

yields

6s + 9 = 90

, which simplifies to

6s = 81

, or

s = 13.5

. Adding

3

to

13.5

gives the speed of the faster cyclist, which is

16.5

miles per hour.

Step-by-Step Solution

1

Define variables for the speeds of both cyclists in terms of a single variable.

Let

s

be the speed of the slower cyclist. The speed of the faster cyclist is

s + 3

.

Using a single variable simplifies the setup of a solvable linear equation.

2

Write a linear equation using the relationship

\text{Distance} = \text{Speed} \times \text{Time}

.

The equation is

3s + 3(s + 3) = 90

.

The sum of the distances traveled by both cyclists when they meet must equal the total length of the trail, which is

90

miles.

3

Solve the equation for

s

.

6s + 9 = 90 \implies 6s = 81 \implies s = 13.5

.

This yields the speed of the slower cyclist.

4

Calculate the speed of the faster cyclist.

13.5 + 3 = 16.5

.

The question specifically asks for the speed of the faster cyclist, which is represented by

s + 3

.

Key Concept

Translating distance-rate-time relationships from word problems into solvable linear equations.

Alternative Method

An alternative approach is to use the concept of relative speed. Since the two cyclists are moving directly toward each other, their relative speed of approach is the sum of their individual speeds. They cover a total of

90

miles in

3

hours, which means their combined speed is

\frac{90}{3} = 30

miles per hour. If the speed of the faster cyclist is

f

and the slower is

s

, then

f + s = 30

and

f - s = 3

. Adding these two equations gives

2f = 33

, which yields

f = 16.5

miles per hour.

Estimated Time:1m 15s

Question 32Question →

A local courier service offers two delivery options inside the city limits: Standard and Express. A Standard delivery costs a flat fee of $15.00 plus$ 0.75 per mile. An Express delivery costs a flat fee of $25.00 plus$ 0.50 per mile. If the cost of a Standard delivery is exactly $5.00 less than the cost of an Express delivery, what is the total cost of the Express delivery?

$30.00

$35.00

$45.00

$55.00

$60.00

Show answer & explanation

Answer: $35.00

Answer

The total cost of the Express delivery is $35.00.

The correct answer is

35.00. First, represent the costs as algebraic functions of the distance

m

:

15 + 0.75m

for Standard and

25 + 0.50m

for Express. Set up the equation reflecting that the Standard cost is

5.00 less than the Express cost:

15 + 0.75m = (25 + 0.50m) - 5

. Simplifying this equation gives

15 + 0.75m = 20 + 0.50m

. Subtracting

0.50m

and

15

from both sides results in

0.25m = 5

, which yields

m = 20

. Finally, plug

m = 20

into the Express cost function:

25 + 0.50(20) = 35.00

dollars.

Step-by-Step Solution

1

Define the variables and write the cost functions for both options.

Let

m

represent the number of miles. The Standard delivery cost is

S = 15 + 0.75m

, and the Express delivery cost is

E = 25 + 0.50m

.

Translating the verbal descriptions into algebraic expressions establishes the basis for solving the problem.

2

Set up the equation using the given relationship that Standard is 5.00 dollars less than Express.

15 + 0.75m = (25 + 0.50m) - 5

The phrase 'Standard cost is 5.00 dollars less than Express cost' translates to

S = E - 5

.

3

Solve the equation for

m

.

15 + 0.75m = 20 + 0.50m \implies 0.25m = 5 \implies m = 20

miles.

Solving for

m

determines the delivery distance that satisfies the cost relationship.

4

Calculate the total cost of the Express delivery by substituting the value of

m

into the Express cost function.

E = 25 + 0.50(20) = 25 + 10 = 35

dollars.

The question specifically asks for the total cost of the Express delivery, not the distance or the Standard delivery cost.

Key Concept

Translating verbal descriptions of costs and differences into algebraic equations and solving them.

Alternative Method

You can test the choices. For example, if the Express cost is

35.00, then the distance

m

is found by solving

25 + 0.50m = 35 \implies 0.50m = 10 \implies m = 20

. At

20

miles, the Standard cost is

15 + 0.75(20) = 30

. Since

30

is indeed exactly

5

less than

35$, this choice is correct.

Estimated Time:1m 30s

Question 33Question →

A botanist is monitoring the heights of two bamboo plants. Plant A is $20$ inches tall and grows at a constant rate of $1.5$ inches per day. Plant B is $12$ inches tall and grows at a constant rate of $2.5$ inches per day. After how many days will Plant B be exactly $6$ inches taller than Plant A?

Show answer & explanation

Answer: 14

Answer

The correct answer is 14 days.

The correct answer is

14

days because when we translate the relationship, we get the equation

12 + 2.5d = 20 + 1.5d + 6

. Solving this equation yields

d = 14

.

Step-by-Step Solution

1

Define the variable

d

as the number of days and write the expressions for the heights of both plants.

Plant A's height is

20 + 1.5d

inches, and Plant B's height is

12 + 2.5d

inches.

To represent the growth of each plant algebraically over time.

2

Set up an equation representing that Plant B's height is

6

inches more than Plant A's height.

12 + 2.5d = (20 + 1.5d) + 6

To translate the verbal relationship into a mathematical equation.

3

Simplify the equation and solve for

d

.

d = 14

To find the number of days that satisfies the given condition.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 15s

Question 34Question →

A company's net profit is calculated by subtracting its monthly operating costs from its monthly revenue. In a certain month, the company's revenue is $260,000$ dollars minus twice its monthly operating costs. If the company wants its monthly net profit to be at least $110,000$ dollars, which of the following inequalities represents all possible values of its monthly operating costs, $C$ , in dollars?

C \ge 50,000

C \le 50,000

C \le 75,000

C \ge 75,000

C \le 150,000

Show answer & explanation

Answer: $C \le 50,000$

Answer

The inequality

C \le 50,000

The correct inequality is obtained by setting up the net profit as the revenue (

260,000 - 2C

) minus the operating costs (

C

), which simplifies to

260,000 - 3C

. Setting this profit to be at least

110,000

gives the inequality

260,000 - 3C \ge 110,000

. Subtracting

260,000

from both sides results in

-3C \ge -150,000

. Finally, dividing both sides by

-3

and reversing the inequality sign because of the division by a negative number yields

C \le 50,000

.

Step-by-Step Solution

1

Define the expression for net profit by subtracting the operating costs,

C

, from the revenue, which is

260,000 - 2C

.

Net profit =

(260,000 - 2C) - C = 260,000 - 3C

Net profit is defined as revenue minus operating costs.

2

Set up the inequality where the net profit is at least

110,000

.

260,000 - 3C \ge 110,000

The phrase 'at least' translates to the 'greater than or equal to' inequality sign (

\ge

).

3

Subtract

260,000

from both sides of the inequality.

-3C \ge -150,000

Isolate the term containing the variable

C

.

4

Divide both sides by

-3

and reverse the inequality sign.

C \le 50,000

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 35Question →

A community theater group sells tickets for a play. Student tickets cost $12$ each, and adult tickets cost $18$ each. For a Friday night performance, the theater sold a total of $150$ tickets. The revenue from the adult tickets sold was $360$ more than the revenue from the student tickets sold. What is the absolute difference between the number of student tickets and the number of adult tickets sold for this performance?

78

72

54

30

6

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. By setting the number of student tickets to

s

, the number of adult tickets is

150 - s

. Setting up the revenue relationship gives

18(150 - s) = 12s + 360

. Solving this equation gives

s = 78

student tickets and

150 - 78 = 72

adult tickets. The absolute difference between the two values is

78 - 72 = 6

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

s

be the number of student tickets sold. Since the total number of tickets sold was

150

, the number of adult tickets sold is

150 - s

.

Establishing single-variable representations allows us to model the entire situation using one equation.

2

Write an equation representing the revenue relationship.

The revenue from student tickets is

12s

and the revenue from adult tickets is

18(150 - s)

. The equation is:

18(150 - s) = 12s + 360

.

The problem states that the adult ticket revenue was

360

more than the student ticket revenue.

3

Solve the linear equation for

s

.

2700 - 18s = 12s + 360 \implies 2340 = 30s \implies s = 78

.

Solving the equation gives the exact number of student tickets sold.

4

Calculate the number of adult tickets sold.

Adult tickets sold =

150 - 78 = 72

.

Subtracting the number of student tickets from the total ticket count yields the number of adult tickets.

5

Find the absolute difference between the two quantities.

|78 - 72| = 6

.

The question asks for the absolute difference between the number of student tickets and the number of adult tickets sold.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 36Question →

Two volunteer teams, Team A and Team B, are packing food boxes for a local shelter. Team A packs at a constant rate of $15$ boxes per hour and begins packing at 8:00 a.m. Team B packs at a constant rate of $20$ boxes per hour and begins packing at 9:30 a.m. If both teams pack continuously at their respective rates, how many hours after Team A begins will both teams have packed the same total number of boxes?

Show answer & explanation

Answer: 6

Answer

Both teams will have packed the same total number of boxes

6

hours after Team A begins.

The correct answer is

6

. Let

x

be the number of hours Team A packs. Team B begins 1.5 hours later, so Team B packs for

x - 1.5

hours. Setting their total boxes packed equal gives

15x = 20(x - 1.5)

. Distributing yields

15x = 20x - 30

. Subtracting

20x

from both sides gives

-5x = -30

, which simplifies to

x = 6

.

Step-by-Step Solution

1

Define the variable for time and identify the time difference.

Let

x

be the number of hours Team A packs. Since Team B starts 1 hour and 30 minutes (which is

1.5

hours) later, Team B's time is represented as

x - 1.5

hours.

Establishing correct algebraic representations for time is necessary to set up the equation.

2

Set up an equation equating the total boxes packed by both teams.

The equation is

15x = 20(x - 1.5)

.

Since both teams pack a constant number of boxes per hour, multiplying their rate by their active time gives the total boxes packed.

3

Solve the linear equation for

x

.

Distribute the 20:

15x = 20x - 30

. Subtract

20x

from both sides:

-5x = -30

. Divide by

-5

:

x = 6

.

Solving the equation gives the number of hours after Team A starts when their packed boxes are equal.

Key Concept

Translating real-world rates and time shifts into linear equations and solving them.

Question 37Question →

A pet care service, Canine Care, charges a flat monthly registration fee of $25 plus$ 15 per dog walk. A competing service, Paws & Claws, charges a flat monthly registration fee of $45 plus$ 10 per dog walk. If a client's total monthly cost with Canine Care is $30 more than their total monthly cost with Paws & Claws for the same number of walks, how many dog walks did the client's dog receive during that month?

Show answer & explanation

Answer: 10

Answer

The client's dog received 10 dog walks during the month.

The correct answer is 10. By translating the verbal descriptions into algebraic expressions, we find that Canine Care's monthly cost is

15w + 25

and Paws & Claws' monthly cost is

10w + 45

. Setting up the equation where Canine Care is

30

more than Paws & Claws gives

15w + 25 = 10w + 45 + 30

. Simplifying the right side results in

15w + 25 = 10w + 75

. Subtracting

10w

from both sides gives

5w + 25 = 75

. Subtracting

25

from both sides gives

5w = 50

, and dividing by

5

yields

w = 10

.

Step-by-Step Solution

1

Define the variable and write the cost expression for Canine Care.

15w + 25

, where

w

is the number of walks.

To represent the total monthly cost of Canine Care algebraically based on the flat fee and per-walk rate.

2

Write the cost expression for Paws & Claws.

10w + 45

, where

w

is the number of walks.

To represent the total monthly cost of Paws & Claws algebraically based on the flat fee and per-walk rate.

3

Set up the equation relating the two costs.

15w + 25 = 10w + 45 + 30

The problem states the cost of Canine Care is $30 more than the cost of Paws & Claws.

4

Simplify the equation and solve for

w

.

5w = 50 \Rightarrow w = 10

Combine like terms and isolate the variable

w

to find the number of dog walks.

Key Concept

Translating and Solving Algebraic Word Problems

Question 38Question →

An online retail store charges a flat shipping fee of $10.00$ for any order, plus $5.00$ per pound of the package's weight. A customer places an order and uses a coupon that gives them a $20\%$ discount on the total cost (the flat shipping fee plus the weight cost). If the customer's total bill after the discount is $36.00$ , what would their total bill have been, in dollars, if they had not used the coupon?

30.00

38.00

42.50

43.20

45.00

Show answer & explanation

Answer: $45.00$

Answer

The total bill without the coupon would have been

45.00

.

The correct answer represents the total cost before the coupon discount. Since a

20\%

discount was applied to the entire bill, the final bill of

36.00

represents

80\%

of the original total bill. Setting up the equation

0.80T = 36.00

and dividing

36.00

by

0.80

yields the original total bill of

45.00

.

Step-by-Step Solution

1

Define the variable for the original total bill and set up the equation using the discount percentage.

Let

T

represent the total bill before the coupon discount. A

20\%

discount means the customer pays

100\% - 20\% = 80\%

of the original total bill. This gives the linear equation

0.80T = 36.00

.

Establishing a direct relationship between the original price and the discounted price allows for a direct solution without needing to calculate the weight of the package first.

2

Solve the equation for

T

by dividing the discounted cost by the remaining percentage factor.

T = \frac{36.00}{0.80} = 45.00

Dividing both sides of the equation by

0.80

isolates the variable

T

, representing the original total bill in dollars.

Key Concept

Translating percentage changes and linear cost models into algebraic equations.

Alternative Method

You can also solve this by finding the weight of the package first. Setting up the equation with the weight

w

gives

0.80(10.00 + 5.00w) = 36.00

. Dividing by

0.80

simplifies the expression to

10.00 + 5.00w = 45.00

, which is the original bill. Solving further gives

5.00w = 35.00

and

w = 7

pounds. Substituting

7

back into the original cost formula yields

10.00 + 5.00(7) = 45.00

.

Estimated Time:1m 30s

Question 39Question →

A landscaping company charges a one-time equipment mobilization fee of $45$ plus an hourly labor rate of $32.50$ per worker. A homeowner hires a crew of $3$ workers to clear their yard. If the total bill for the job is $435$ dollars, for how many hours did the crew work?

Show answer & explanation

Answer: 4

Answer

4

The total cost of

435

dollars is the sum of the one-time

45

dollar mobilization fee and the hourly labor cost of

32.50

dollars per worker for

3

workers over

h

hours. This translates to the equation

45 + 3(32.50)h = 435

, which simplifies to

45 + 97.50h = 435

. Subtracting

45

from both sides gives

97.50h = 390

. Dividing by

97.50

yields

h = 4

hours.

Step-by-Step Solution

1

Set up the algebraic expression representing the total cost based on the number of hours worked,

h

. The total cost consists of a fixed mobilization fee of

45

dollars and a variable labor cost of

32.50

dollars per worker per hour.

45 + 3(32.50)h

To represent the relation between the hours worked and the total charge.

2

Equate the expression for the total cost to the actual total bill of

435

dollars and simplify the labor rate coefficient.

45 + 97.50h = 435

To form a solvable linear equation in terms of the unknown number of hours,

h

.

3

Subtract the fixed mobilization fee of

45

from both sides of the equation.

97.50h = 390

To isolate the term containing the variable

h

.

4

Divide both sides of the equation by

97.50

to solve for

h

.

h = 4

To determine the number of hours the crew worked.

Key Concept

Formulating and solving linear equations from real-world contexts

Estimated Time:1m 30s

Question 40Question →

A smartphone's battery is fully charged to $100\%$ . When the phone is in sleep mode, the battery drains at a constant rate of $1\%$ per hour. When the phone is in active use, the battery drains at a constant rate of $6\%$ per hour. Over a $20$ -hour period, the phone is either in sleep mode or in active use, and the battery level decreases to $45\%$ . For how many hours was the phone in active use during this period?

5

7

9

11

13

Show answer & explanation

Answer: 7

Answer

The phone was in active use for 7 hours.

To find the active hours, first calculate the total percentage of the battery consumed:

100\% - 45\% = 55\%

. Let

x

represent the number of hours in active use. The remaining time,

20 - x

, represents the hours in sleep mode. Set up the equation representing the total battery drain:

6x + 1(20 - x) = 55

. Simplifying this equation gives

5x + 20 = 55

, which reduces to

5x = 35

, resulting in

x = 7

hours.

Step-by-Step Solution

1

Calculate the total percentage of the battery that drained during the period.

100\% - 45\% = 55\%

To determine the exact battery percentage consumed by both active use and sleep mode combined.

2

Set up a system of equations using

x

for hours in active use and

y

for hours in sleep mode.

x + y = 20

and

6x + 1y = 55

To represent the constraints on the total duration (20 hours) and the total battery drain (55%).

3

Substitute

y = 20 - x

into the drain equation and solve for

x

.

6x + (20 - x) = 55 \implies 5x + 20 = 55 \implies 5x = 35 \implies x = 7

To find the number of hours the phone was actively used.

Key Concept

Translating and solving a system of linear equations from a real-world word problem.

Translating and Solving Algebraic Word Problems

Answer

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