Algebraic Expressions Portfolio

Question

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Question 1

A ticket to a movie costs t dollars, and there is a 15% amusement tax on each ticket.

Conor buys a ticket to see a movie. Write two expressions in terms of t showing his total cost.
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.
Suppose that Conor’s total cost to see the movie is greater than $17.50 and less than $20. What could be a value of t to make that possible?

Answer 1

Question 1

Part a: Writing Expressions

Let \( t \) represent the cost of a movie ticket in dollars. The amusement tax is 15% of the ticket price. Therefore, the total cost \( C \) of the ticket can be expressed in two ways:

1. Expression including tax directly: \[ C = t + 0.15t = 1.15t \]

2. Expression using the tax as a separate amount: \[ C = t + (0.15t) = t + \text{(15% of } t \text{)} \] which can also be written as: \[ C = t + 0.15t \]

Both expressions represent the total cost Conor incurs after including the tax.

Part b: Finding the Total Cost

Choose a value for \( t \) between $12 and $13. Let's use:

\[ t = 12.50 \]

Now, using the first expression to find the total cost:

\[ C = 1.15t = 1.15(12.50) \]

Calculating the total cost:

\[ C = 1.15 \times 12.50 = 14.375 \]

Rounding to the nearest penny:

\[ C \approx 14.38 \]

So, Conor's total cost is $14.38.

Part c: Finding a Value of \( t \)

We know that Conor's total cost is greater than $17.50 and less than $20. From the expression \( C = 1.15t \), we set up the following inequality:

\[ 17.50 < 1.15t < 20 \]

To solve for \( t \), we divide the entire inequality by 1.15:

1. Solving the left side: \[ \frac{17.50}{1.15} < t \]

   Calculating: \[ t > 15.2174 \quad (\text{approximately}) \]

2. Solving the right side: \[ t < \frac{20}{1.15} \]

   Calculating: \[ t < 17.3913 \quad (\text{approximately}) \]

Now we can combine the results:

\[ 15.2174 < t < 17.3913 \]

A possible value for \( t \) that falls within this range could be $16.00.

Thus, if Conor buys a ticket for $16.00, his total cost with the tax included would be:

\[ C = 1.15 \times 16.00 = 18.40, \]

which lies within the acceptable total cost range between $17.50 and $20.

Answer 2

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 Population Percentage Change
Pleasantville x –2%
Amity Center y +5%
Harmony Corners z +7%

Choose one of these towns. Use the information from the second and third columns to write two expressions showing the current estimated population of the town in terms of x, y, or z.
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.
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) above, 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.

Answer 3

Question 3

Part a: Writing Expressions for a Chosen Town

Let's choose Pleasantville for our calculations.

1. The last census population is represented by \( x \), and the population has decreased by 2%. Therefore, the current estimated population \( P \) can be expressed in two ways:
   • First Expression (accounting for the decrease): \[ P = x - 0.02x = (1 - 0.02)x = 0.98x \]
   • Second Expression (in terms of the percentage change): \[ P = x \cdot (1 - 0.02) = x \cdot 0.98 \]

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Adding Concord to the Table

Here’s the updated table with the inclusion of Concord:

| Town | Population | Percentage Change | |--------------------|------------|-------------------| | Pleasantville | \( x \) | -2% | | Amity Center | \( y \) | +5% | | Harmony Corners | \( z \) | +7% | | Concord | \( p \) | -3% |

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Part b: Writing Expressions for Concord

For Concord, which has a last census population of \( p \) and a decrease of 3%, the current estimated population can be expressed as:

1. First Expression: \[ P_{Concord} = p - 0.03p = (1 - 0.03)p = 0.97p \]

2. Second Expression: \[ P_{Concord} = p \cdot (1 - 0.03) = p \cdot 0.97 \]

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Choosing a Value for \( p \)

Let’s choose a value for \( p \) that is greater than 5,000 and less than 10,000. We’ll use:

\[ p = 7,500 \]

Part c: Calculating the Current Population of Concord

Now we can calculate the estimated current population of Concord using the expression:

\[ P_{Concord} = 0.97p \]

Substituting our value of \( p \):

\[ P_{Concord} = 0.97 \times 7,500 \]

Calculating the estimated current population:

\[ P_{Concord} = 7,275 \]

So, the estimated current population of Concord is 7,275 people.

Explanation

I used the expression \( P_{Concord} = 0.97p \) for my calculation because it directly incorporates the percentage change of -3% into the linear relationship with the original population \( p \). This approach simplifies the calculations by using the decimal equivalent of the percentage directly, yielding the final population efficiently. By rounding to the nearest whole number, we maintain a realistic population representation, as populations are typically whole numbers.

Answer 4

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=(13)b .

Write two different expressions giving the perimeter of the donkey’s pen. 
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. 
Explain why you used that expression to evaluate the perimeter of the donkey’s pen.  

Answer 5

Question 4

Part a: Writing Expressions for the Perimeter

The perimeter \( P \) of a rectangle can be calculated using the formula:

\[ P = 2 \times (\text{length} + \text{width}) \]

In this case, the length is \( a \) and the width is \( b \). Since we know that \( a = 13b \), we can write two expressions for the perimeter.

1. First Expression (using \( a \) and \( b \)): \[ P = 2(a + b) \]

2. Second Expression (substituting \( a = 13b \)): \[ P = 2(13b + b) = 2(14b) = 28b \]

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Part b: Choosing a Value for \( a \)

Let's choose a value for \( a \) that is greater than 5 and less than 15. We will use:

\[ a = 12 \]

Find \( b \) for the chosen value of \( a \)

We know from the equation \( a = 13b \):

\[ 12 = 13b \]

To find \( b \):

\[ b = \frac{12}{13} \approx 0.9231 \text{ yards} \]

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Evaluating the Perimeter

Now, we can calculate the perimeter using one of the expressions. Let's use the first expression:

\[ P = 2(a + b) \]

Substituting \( a = 12 \) and \( b \approx 0.9231 \):

\[ P = 2(12 + 0.9231) = 2(12.9231) = 25.8462 \]

Rounding to the nearest hundredth gives us:

\[ P \approx 25.85 \text{ yards} \]

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Explanation

I used the expression \( P = 2(a + b) \) to evaluate the perimeter because it clearly incorporates both dimensions of the rectangle directly (length and width). Using this expression allows for straightforward calculation without needing intermediate steps to find \( b \) separately first. Furthermore, it simplifies the understanding of how the values of \( a \) and \( b \) contribute to the overall perimeter.