Algebraic Expressions Portfolio

Question

Worksheet
Directions: Use this worksheet to record your answers to the questions or
problems for the Algebraic Expressions Portfolio. When finished, save this
worksheet with your answers and submit it for a portfolio grade.
Question 1
Conor is going to the movie theater. A ticket to a movie costs two dollars, and there is a
15% amusement tax on each ticket.
a. Conor buys a ticket to see a movie. Write two expressions in terms of t
showing his total cost.
b. Choose a value for t greater than $12 and less than $13. Use one of the
expressions you wrote in part a to find Conor’s total cost, rounded to the
nearest penny. Show your work.
c. Suppose that Conor’s total cost to see the movie is greater than $17.50 and
less than $20. What could be the value of making that possible?
Question 2
Complete the following questions.
Winona and Takoda are shopping for a present for their mother. The chart shows
several possible presents. There is no sales tax in the state where they live.
Present Cost ($) Discount (%) Total Paid
Power tools d 15%
Water skis d + 15 25%
Jewelry d – 5 10%
a. Write two expressions in terms of d to show how much Winona and Takoda
would pay in total if they buy their mother the set of power tools.
b. What does the expression 0.90(d – 5) refer to? How do you know? How else
could this expression be written?
c. Choose a value for d that is greater than $25 and less than $100. Calculate the
total cost of each of the possible presents for the value you chose. Then write a
short paragraph telling Winona and Takoda which present they should buy and
explaining why.
Question 3
A county planner prepares the following table showing population trends in three local
towns. The first column gives the name of the town. The second column gives the
population as of the last census. The third column gives the estimated increase or
decrease in population since that census, expressed as a percentage of the previous
population.
Town Census Population Percentage Change
Pleasantville x –2%
Amity Center y +5%
Harmony Corners z +7%
a. Choose one of these towns. Use the second and third columns to write two
expressions showing the current estimated population of the town in terms of
x, y, or z.
b. Add a fourth row to the table showing the population information for the town
of Concord. Concord’s population as of the last census was p people; for its
percentage change, choose a negative integer not divisible by two. Write two
expressions showing the current estimated population of Concord in terms of p.
c. Choose a number greater than 5,000 and less than 10,000 for the value of p.
Based on the percentage change you chose in part b, calculate the estimated
current population of Concord, rounding to the nearest whole number if
needed. Explain why you used the expression you did to make your calculation.
Question 4
A’Leila is building a pen for her pet donkey. The pen is a rectangle with one side
measuring b yards and the adjacent side measuring a yards. A’Leila knows that
.
a. Write two different expressions giving the perimeter of the donkey’s pen.
b. Choose a value for a that is greater than 5 and less than 15. Evaluate the
perimeter of the pen for that value, using one of the expressions that you
wrote.
c. Explain why you used that expression to evaluate the perimeter of the
donkey’s pen

Answer 1

Algebraic Expressions Portfolio Worksheet

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Question 1

a.

The cost of one ticket is $2. The amusement tax is 15% of the ticket price. Thus, the total cost \( C \) in terms of \( t \) (the price of a ticket) can be represented as:

1. Total Cost including tax: \[ C_1 = t + 0.15t = 1.15t \]

2. Total Cost represented in terms of the ticket price, considering two factors: \[ C_2 = 2 + 0.15 \times 2 = 2 + 0.30 = 2.30 \]

b.

Choose \( t = 12.50 \): Using \( C_1 = 1.15t \): \[ C = 1.15 \times 12.50 = 14.375 \] Rounded to the nearest penny, the total cost is: \[ C \approx 14.38 \]

c.

We need to solve the expression \( 1.15t \) with \( 17.50 < C < 20 \): \[ 17.50 < 1.15t < 20 \] Dividing all parts by 1.15: \[ \frac{17.50}{1.15} < t < \frac{20}{1.15} \] Calculating: \[ 15.217 < t < 17.391 \] Thus, a possible value for \( t \) could be \( t = 16.00 \).

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Question 2

a.

For the power tools with \( d \):

1. Total cost with discount: \[ C_1 = d - 0.15d = 0.85d \]
2. Total paid: \[ C_2 = d - 0.15d = 0.85d \]

Since we need total payments with respect to discount, both expressions simplify to the same total cost formula.

b.

The expression \( 0.90(d - 5) \) refers to the total paid for a piece of jewelry after applying a 10% discount to the cost after subtracting 5 dollars from \( d \). It can also be expressed as: \[ 0.90d - 4.5 \]

c.

Choose \( d = 30 \):

• Power tools: \[ 0.85d = 0.85 \times 30 = 25.50 \]

• Water Skis: \[ C_W = (d + 15) \times (1 - 0.25) = (30 + 15)(0.75) = 45(0.75) = 33.75 \]

• Jewelry: \[ (d - 5) \times 0.90 = (30 - 5)(0.90) = 25(0.90) = 22.50 \]

Final costs:

• Power tools: $25.50
• Water skis: $33.75
• Jewelry: $22.50

Winona and Takoda should buy the power tools because they offer the best value with a total cost of $25.50, making it a great gift option for their mother.

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Question 3

a.

Let’s choose Pleasantville, where \( x \) represents the population:

1. Current population: \[ P_1 = x - 0.02x = 0.98x \]
2. In terms of growth: \[ P_2 = x \times (1 - 0.02) = 0.98x \]

b.

For Concord, let \( p \) be its population and I will choose a negative change of -3%:

1. Current estimated population: \[ P_C_1 = p - 0.03p = 0.97p \] \[ P_C_2 = p \times (1 - 0.03) = 0.97p \]

c.

Choose \( p = 7000 \): Using \( P_C = 0.97p \): \[ P_C = 0.97 \times 7000 = 6790 \] Thus, the estimated current population of Concord is approximately 6790. I used this expression as it represents a direct calculation of the population decrease at the specified percentage change.

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Question 4

a.

The perimeter \( P \) of A’Leila's rectangular pen can be given by:

1. Perimeter: \[ P_1 = 2a + 2b \]
2. Simplified perimeter: \[ P_2 = 2(a + b) \]

b.

Choose \( a = 10 \): Using \( P_1 = 2a + 2b \): \[ P = 2(10) + 2b = 20 + 2b \] The value of \( b \) was not defined in the instructions; let's assume \( b = 5 \): \[ P = 20 + 10 = 30 \]

c.

I used this expression because it fully accounts for both lengths of the pen sides combined to find the overall total perimeter. Adjusting for \( b \) allows greater flexibility in outcome calculations depending on chosen side lengths.

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This completed portfolio reflects the calculations and logic behind the algebraic expression problems as directed in the instructions.