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 $20. What could be a value of t to make that possible? Question 2 Winona and Takoda are shopping for a present for their mother. The chart shows several possible presents. There is no sales tax in the state where they live. Present Cost ($) Discount (%) Total Paid ($) Power tools d dollars 15% Water skis d + 15 dollars 25% Jewelry d – 15 dollars 10% Write two expressions in terms of d to show how much Winona and Takoda would pay in total if they bought their mother the set of power tools. What does the expression 0.90(d – 5) refer to? How do you know? How else could this expression be written? 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 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. 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.
Question 1:
a) The expression for Conor's total cost after the amusement tax would be t + 0.15t, as the tax is 15% of the ticket price.
b) If Conor's total cost is greater than $17.50 and less than $20, we can say that t + 0.15t is greater than $17.50 and less than $20. Let's solve this inequality:
17.50 < t + 0.15t < 20
17.50 < 1.15t < 20
15.22 < t < 17.39
Therefore, a possible value for t could be $16.
Question 2:
a) The expression for the total cost if they bought the set of power tools would be 0.85d, as they have a 15% discount on the original cost.
b) The expression 0.90(d - 5) refers to the total paid after a 10% discount on an item that originally cost d - $5. We know this because it says in the chart that there is a 10% discount on jewelry, which has a cost of d - $15.
This expression could also be written as 0.9d - 4.5.
Let's choose d = $50 and calculate the total cost for each present:
Total cost for power tools = 0.85(50) = $42.50
Total cost for water skis = 0.75(50 + 15) = $48.75
Total cost for jewelry = 0.90(50 - 15) = $31.50
Based on the calculations, Winona and Takoda should buy the set of power tools because it has the lowest total cost of $42.50.
Question 3:
Let's choose the town of Pleasantville.
a) The expression for the current estimated population of Pleasantville in terms of x would be x - 0.02x = 0.98x, as there is a 2% decrease in population.
b) The expression for the current estimated population of Pleasantville in terms of x would be x + 0.05x = 1.05x, as there is a 5% increase in population.
Let's add a fourth row to the table for the town of Concord:
Town Population Percentage Change
Pleasantville x -2%
Amity Center y +5%
Harmony Corners z +7%
Concord p -3%
The expression for the current estimated population of Concord in terms of p would be p - 0.03p = 0.97p, as there is a 3% decrease in population.
Let's choose p = 7000 and calculate the estimated current population of Concord:
Estimated current population of Concord = 0.97(7000) = 6790 (rounded to the nearest whole number)
We used the expression 0.97p because there is a 3% decrease in population, as indicated by the given percentage change.
Question 4:
The expressions for the perimeter of the donkey's pen would be:
Perimeter = 2a + 2b
Perimeter = 2(13b) + 2b = 26b + 2b = 28b
Let's choose a = 8. Evaluate the perimeter of the pen using the expression 28b:
Perimeter = 28(8) = 224 yards
We used the expression 28b because it represents the sum of all sides of the rectangle (2b + 2a) when a is equal to (13)b.
Question 1a) To find Conor's total cost, we need to add the cost of the ticket to the amusement tax. The expression for the total cost can be written as t + 0.15t. Simplifying this expression, we get 1.15t which represents Conor's total cost.
b) If Conor's total cost is greater than $17.50 and less than $20, we can set up an inequality to find the possible value of t. The inequality is 17.50 < 1.15t < 20. To solve this inequality, we divide all the terms by 1.15, giving us 15.22 < t < 17.39. Therefore, a possible value of t to make Conor's total cost within that range is any value greater than $15.22 and less than $17.39.
Question 2a) To find the total cost of the set of power tools, we need to subtract the discount from the original cost d. The expression for the total cost can be written as d - 0.15d. Simplifying this expression, we get 0.85d which represents the total amount Winona and Takoda would pay.
b) The expression 0.90(d - 5) refers to the total paid for the jewelry present, which includes a 10% discount and a $5 deduction. We know this because we multiply the original cost d by 0.90 (representing the 10% discount), and then subtract an additional 5 dollars.
In terms of the value of d, let's choose d = $50.
For the power tools, the total cost would be 0.85(50) = $42.50.
For the water skis, the total cost would be 0.75(50 + 15) = $48.75.
For the jewelry, the total cost would be 0.90(50 - 15) = $31.50.
Based on the calculations, Winona and Takoda should buy the power tools as it has the lowest total cost of $42.50.
Question 3a) To express the current estimated population of a town in terms of x, y, or z, we simply add or subtract the percentage change to the population. For example, the estimated population of Pleasantville in terms of x would be x - 0.02(x), which simplifies to 0.98x.
b) Let's consider Concord, whose population as of the last census was p people. For the percentage change, we choose a negative integer not divisible by two, let's say -7%. Thus, the expressions for the current estimated population of Concord would be p + (-0.07)p or p(1 - 0.07).
Choosing a value for p between 5,000 and 10,000, let p = 7,500.
Using the expression p(1 - 0.07), we can calculate the estimated current population of Concord as 7,500(1 - 0.07) = 6,975 (rounded to the nearest whole number if needed).
We used the expression p(1 - 0.07) because it represents a decrease of 7% from the original population, which is consistent with the chosen percentage change. By multiplying p by (1 - 0.07), we find the percentage decrease and calculate the new estimated population accordingly.
Question 4a) We know that a = 13b, so we can substitute 13b for a in the expressions for the perimeter.
The two expressions for the perimeter can be written as 2(b + 13b) = 2 * 14b = 28b, and 2(a + b) = 2(13b + b) = 2 * 14b = 28b.
b) Let's choose a value for b between 5 and 15, for example, b = 10.
Using the expression 28b, we can calculate the perimeter as 28(10) = 280 yards.
We used the expression 28b to evaluate the perimeter of the donkey's pen because it represents the sum of all four sides of the rectangle, where two sides have length b and the other two sides have length 13b.
Question 1:
a) The two expressions in terms of t showing Conor's total cost are:
Expression 1: t + 0.15t
Expression 2: t(1 + 0.15) or 1.15t
b) Let's choose t = $12.50.
Using Expression 1: Total cost = 12.50 + 0.15(12.50) = 12.50 + 1.875 = $14.375
Rounded to the nearest penny, Conor's total cost is $14.38.
c) To find a value of t that makes Conor's total cost between $17.50 and $20, we can solve the inequality:
17.50 < t + 0.15t < 20
Subtracting t from all sides, we get:
17.50 < 0.15t < 20 - t
Dividing every term by 0.15 gives:
116.67 < t < 133.33
So, a possible value of t could be $117.
Question 2:
a) The two expressions in terms of d to show how much Winona and Takoda would pay for the set of power tools are:
Expression 1: d - 0.15d
Expression 2: d(1 - 0.15) or 0.85d
b) The expression 0.90(d - 5) refers to a 10% discount of (d - 5). We know this because the expression is multiplied by 0.90, which corresponds to a 10% discount (100% - 10% = 90%). In other words, it shows the total paid after a 10% discount is applied to the original cost (d - 5). This expression can also be written as 0.9d - 4.5.
c) Let's choose d = $50.
For the power tools: Total paid = 0.85(50) = $42.50
For the water skis: Total paid = (50 + 15)(1 - 0.25) = 64.50 - 16.125 = $48.375
For the jewelry: Total paid = (50 - 15)(1 - 0.10) = 35(0.90) = $31.50
Based on the calculations, Winona and Takoda should buy the jewelry as it will cost them the least, $31.50.