Concession Confections

Cedric, the manager at the pool, decided to sell bags of chips for $0.40. He made a table to show the cost of the chips.

Number of Candy Bars / Cost
1 / $0.40
2 / $0.80
3 / $1.20
4 / $1.60

He realized making a table of the prices would be too large, so he decided to make a graph. The points for one, two, and three bags of chips were graphed. Since the points were going in a line, he decided to draw a line to find the cost for any number of bags.

a.Use the graph to find the price of 8 bags of chips. ______

b.Sandra has $2.40 to spend. How many bags of chips can she buy? ______

For every bag of chips added to the table, the cost goes up ______. Each time a bag of chips is added to the graph, you move one to the right and the line goes up another ______. That is because the cost of one bag of chips is 40 cents.

  1. A ratio is a way to compare two numbers or quantities.
  1. What was the ratio of cost to bags of chips in the first problem? ______
  2. Write two other equal ratios for the chip problem? ______

Ratios can also be written as fractions. To compare bags of chips and cost, we can write the ratio

Price $0.40 $1.20

Bags of Chips 1 3

When the fractions are equal, the ratios are equal and a proportion is made.

  1. Complete the following proportions:

Price $0.40 $2.80

Bags of Chips 1 x

Price $0.40 .y

Bags of Chips 1 9

  1. Cedric is thinking about adding some new candy to the concession stand. Below are parts of tables about the prices of two candy bars – Chocolate Delight and Peanut Butter Crème.

Chocolate Delight Peanut Butter Crème

Number of Candy Bars / Price
2 / $1.40
$2.80
5
Number of Candy Bars / Price
3 / $1.80
4
$3.00
6
  1. What is the cost of 6 Chocolate Delight candy bars? How did you determine the price?
  1. How much would he need to pay for 5 Peanut Butter Crème candy bars? How did you determine the price?

Complete the table for Chocolate Delight and Peanut Better Crème. The unit rate is the cost for one candy bar.

  1. What is the unit rate for Chocolate Delight? How did you determine the answer?
  2. What is the unit rate for Peanut Butter Crème? How did you determine your answer?
  1. In the table for Chocolate Delight, what happens to the cost as the number of candy bars is increased by one?
  1. In the table for Peanut Butter Crème, what happens to the cost as the number of candy bars is increased by one?
  1. According to the table, which candy bar is the most expensive? ______
  1. Graph the information (number, cost) for the two candy bars on the graph below.

  1. According to the graph, which candy bar is the most expensive? ______How did you determine your answer?
  1. Each time a Chocolate Delight candy bar is added to the graph, you move one to the right and the line goes up another ______.
  1. Each time a Peanut Butter Crème candy bar is added to the graph, you move one to the right and the line goes up another ______.
  1. What is the relationship between the unit rate and the line?
  1. The graph below shows the prices of two sodas, Thirst Quencher and Cherry Delight.

  1. Which soda is the least expensive? How can you tell?
  1. What is the unit rate for one ounce of Cherry Delight? ______
  1. What is the unit rate for one ounce of Thirst Quencher? ______
  1. Use the Thirst Quencher soda graph to determine the change in price when the number of ounces increases from:
  1. 2 ounces to 3 ounces______
  2. 2 ounces to 4 ounces______
  3. 3 ounces to 6 ounces______
  1. The rate of change is the change that occurs in the price for each ounce. In 4b, the price change from 2 ounces to 4 ounces is 40 cents; however, this change occurred over 2 ounces The rate of change is 20 cents (.40 divided by 2).

Is the rate of change from 3 ounces to 6 ounces (4c above) also 20 cents? Explain how you determined your answer?

  1. What would be the rate of change for Cherry Delight? ______How did you determine your answer?
  2. What is the relationship between unit rate and rate of change?
  1. How are the graphs for the costs of bags of chips, candy bars and sodas the same?
  1. Equations

In each situation above, the manager needed to find the cost for any number of bags of chips, candy bars or ounces of soda. Although tables and graphs are useful, equations are another tool to show mathematical relationships.

The cost of one bag of chips was $0.40. Every time a bag was added the cost increased $0.40. The two variables in this situation are Cost (C) and Number of bags (N). An equation relating these variables would be

C = $0.40n

  1. Use the equation to find the cost be for 20 bags of chips? ______
  2. Josh collected $4.80 for chips. How many bags did he sell? ______

Write equations to represent the following scenarios:

  1. Cost of any number of Chocolate Delight candy bars: ______
  2. Cost of any number of Peanut Butter Crème candy bars: ______
  3. Cost of any number of ounces of Cherry Delight soda: ______
  4. Cost of any number of ounces of Thirst Quencher soda: ______

In each of the equations above, what does the coefficient represent? ______

______

Each of the variables in the above scenarios is related in a way that the ratio of their values always remains the same. This relationship is called direct variation. Notice that all the graphs are in a straight line that passes through (0, 0). All points on the line represent this ratio or unit rate. Direct variation equations are in the form y = kx, where k represents the unit rate or rate of change. In direct variation, k is called the constant of proportionality.

Concession Confections Answer Key

Cedric, the manager at the pool, decided to sell bags of chips for $0.40. He made a table to show the cost of the chips.

Number of Candy Bars / Cost
1 / $0.40
2 / $0.80
3 / $1.20
4 / $1.60

He realized making a table of the prices would be too large, so he decided to make a graph. The points for one, two, and three bags of chips were graphed. Since the points were going in a line, he decided to draw a line to find the cost for any number of bags.

  1. Use the graph to find the price of 8 bags of chips. $3.20
  2. Sandra has $2.40 to spend. How many bags of chips can she buy? 6 bags

For every bag of chips added to the table, the cost goes up .40. Each time a bag of chips is added to the graph, you move one to the right and the line goes up another .40. That is because the cost of one bag of chips is 40 cents.

  1. A ratio is a way to compare two numbers or quantities.

1. What was the ratio of cost to bags of chips in the first problem? .40:1

2.Write two other equal ratios for the chip problem? .80:21.60:4

Ratios can also be written as fractions. To compare bags of chips and cost, we can write the ratio

Price $0.40 $1.20

Bags of Chips 1 3

When the fractions are equal, the ratios are equal and a proportion is made.

  1. Complete the following proportions:

Price $0.40 $2.80x = 7

Bags of Chips 1 xMultiply $0.40 and 1 by 7

Price $0.40 .yy = $3.60

Bags of Chips 1 9Multiply the 1 and $0.40 by 9

  1. Cedric is thinking about adding some new candy to the concession stand. Below are parts of tables about the prices of two candy bars – Chocolate Delight and Peanut Butter Crème.

Chocolate Delight Peanut Butter Crème

Number of Candy Bars / Price
1 / $0.70
2 / $1.40
3 / $2.10
4 / $2.80
5 / $3.50
6 / $4.20
Number of Candy Bars / Price
1 / $0.60
2 / $1.20
3 / $1.80
4 / $2.40
5 / $3.00
6 / $3.60
  1. What is the cost of 6 Chocolate Delight candy bars? How did you determine the price?

Six Chocolate Delight bars would cost $3.60. The price can be determined by using the table or by multiplying 6 times $0.60.

  1. How much would he need to pay for 5 Peanut Butter Crème candy bars? How did you determine the price?

Five Peanut Butter bars would cost $3.50. The price can be determined by using the table or by multiplying 5 times $0.70.

Complete the table for Chocolate Delight and Peanut Better Crème. The unit rate is the cost for one candy bar.

  1. What is the unit rate for Chocolate Delight? How did you determine the answer?

$0.60 is the unit rate since that is the cost of one bar. One possible answer to find the answer for 1 bar is to divide both the 3 and the $1.80 by 3.

  1. What is the unit rate for Peanut Butter Crème? How did you determine your answer?

$0.70 is the unit rate since that is the cost of one bar. One possible answer to find the answer for 1 bar is to divide both the 2 and the $1.40 by 2.

  1. In the table for Chocolate Delight, what happens to the cost as the number of candy bars is increased by one?

The price increases by 60 cents.

  1. In the table for Peanut Butter Crème, what happens to the cost as the number of candy bars is increased by one?

The price increases by 70 cents.

  1. According to the table, which candy bar is the most expensive? Peanut Butter Crème
  1. Graph the information (number, cost) for the two candy bars on the graph below.
  1. According to the graph, which candy bar is the most expensive? Peanut Butter Crème. How did you determine your answer? This is the steepest line.
  1. Each time a Chocolate Delight candy bar is added to the graph, you move one to the right and the line goes up another .60.
  1. Each time a Peanut Butter Crème candy bar is added to the graph, you move one to the right and the line goes up another .70.
  1. What is the relationship between the unit rate and the line? The steepness of the line represents the unit rate.
  1. The graph below shows the prices of two sodas, Thirst Quencher and Cherry Delight.
  1. Which soda is the least expensive? How can you tell?

Thirst Quencher because the line is less steep.

  1. What is the unit rate for one ounce of Cherry Delight? 0.25
  1. What is the unit rate for one ounce of Thirst Quencher? 0.20
  1. Use the Thirst Quencher soda graph to determine the change in price when the number of ounces increases from:
  1. 2 ounces to 3 ounces0.20 (0.40 – 0.60)
  2. 2 ounces to 4 ounces0.40 (0.40 – 0.80)
  3. 3 ounces to 6 ounces0.60 (0.60 – 1.20)
  1. The rate of change is the change that occurs in the price for each ounce. In 4b, the price change from 2 ounces to 4 ounces is 40 cents; however, this change occurs over 2 ounces The rate of change for one ounce is 20 cents (.40 divided by 2).

Is the rate of change from 3 ounces to 6 ounces (4c above) also 20 cents? Explain how you determined your answer?

The rate of change is still 20 cents because the 60 cents change occurs over 3 ounces; 60 cents divided by 3 ounces is 20 cents per ounce.

  1. What would be the rate of change for Cherry Delight? 0.25 How did you determine your answer?
  1. What is the relationship between unit rate and rate of change?

The unit rate and the rate of change are the same.

  1. How are the graphs for the costs of bags of chips, candy bars and sodas the same?

All are straight lines and all go through zero.

IV. Equations

In each situation above, the manager needed to find the cost for any number of bags of chips, candy bars or ounces of soda. Although tables and graphs are useful, equations are another tool to show mathematical relationships.

The cost of one bag of chips was $0.40. Every time a bag was added the cost increased $0.40. The two variables in this situation are Cost (C) and Number of bags (N). An equation relating these variables would be

C = $0.40n

  1. Use the equation to find the cost be for 20 bags of chips? $8.00
  2. Josh collected $4.80 for chips. How many bags did he sell? 12

Write equations to represent the following scenarios:

  1. Cost of any number of Chocolate Delight candy bars: C = .60n
  2. Cost of any number of Peanut Butter Crème candy bars: C = .70n
  3. Cost of any number of ounces of Cherry Delight soda: C = .25 n
  4. Cost of any number of ounces of Thirst Quencher soda: ______

In each of the equations above, what does the coefficient represent? The coefficient represents the unit rate.

Each of the variables in the above scenarios is related in a way that the ratio of their values always remains the same. This relationship is called direct variation. Notice that all the graphs are in a straight line that passes through (0, 0). All points on the line represent this ratio or unit rate. Direct variation equations are in the form y = kx, where k represents the unit rate or rate of change. In direct variation, k is called the constant of proportionality.

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