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Question

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Question
The constant of proportionality is the amount a relationship will change for each unit.

It is represented by the value of this ratio:

k=yx


One of the ways a proportional relationship is identified is with a table showing the input and output values that change by the same unit rate.
(table)
p c
0 0
1 2.25
2 4.5
3 6.75
4 9
On this table, the pounds of peaches (p) is the input and the cost of the pounds (c) is the output. The constant of proportionality for the table may be determined:

k=yx = 2.251=2.25


All of the ratios on the table are equivalent. 4.52=2.25
6.753=2.25
94=2.25


k = $2.25 per pound of peaches



(4 points)
Determine the constant of proportionality for each table. If the table does not have an constant of proportionality, enter the letter "n" in box.

(table)
gallons of gas 2, 4, 6, 8, 10,
total cost 5.50, 11, 16.50, 22, 27.50

k = $
per gallon of gas

================================



Candy bars sold 10, 20, 30, 40, 50
profit 5, 35, 65, 95, 125
k = $
per candy bars sold

================================

batches of sugar cookies 5 10 15 20
number of eggs 5 10 15 20

k =
eggs per batch of sugar cookies

================================


pounds of bananas 2 3 5 7
cost 2.50 3.75 6.25 8.75

k = $
per pound of bananas

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Answer 1

Let’s determine the constant of proportionality (k) for each table provided.

1. Gallons of gas:

   • Table:
     • Gallons of gas: 2, 4, 6, 8, 10
     • Total cost: 5.50, 11, 16.50, 22, 27.50

   To find k, we will take the total cost divided by the gallons of gas:

   • For 2 gallons: \( k = \frac{5.50}{2} = 2.75 \)
   • For 4 gallons: \( k = \frac{11}{4} = 2.75 \)
   • For 6 gallons: \( k = \frac{16.50}{6} = 2.75 \)
   • For 8 gallons: \( k = \frac{22}{8} = 2.75 \)
   • For 10 gallons: \( k = \frac{27.50}{10} = 2.75 \)

   All ratios yield the same constant of proportionality:

   • k = $2.75 per gallon of gas

2. Candy bars sold:

   • Table:
     • Candy bars sold: 10, 20, 30, 40, 50
     • Profit: 5, 35, 65, 95, 125

   To find k, we calculate profit divided by candy bars sold:

   • For 10 candy bars: \( k = \frac{5}{10} = 0.5 \)
   • For 20 candy bars: \( k = \frac{35}{20} = 1.75 \)
   • For 30 candy bars: \( k = \frac{65}{30} \approx 2.17 \)
   • For 40 candy bars: \( k = \frac{95}{40} = 2.375 \)
   • For 50 candy bars: \( k = \frac{125}{50} = 2.5 \)

   The ratios are not equivalent, so there is no constant proportionality:

   • k = n

3. Batches of sugar cookies:

   • Table:
     • Batches of sugar cookies: 5, 10, 15, 20
     • Number of eggs: 5, 10, 15, 20

   For the number of eggs per batch of sugar cookies:

   • Each batch requires 1 egg (5 eggs for 5 batches, 10 eggs for 10 batches, and so on).
   • \( k = \frac{5}{5} = 1 \)

   This is constant throughout:

   • k = 1 egg per batch of sugar cookies

4. Pounds of bananas:

   • Table:
     • Pounds of bananas: 2, 3, 5, 7
     • Cost: 2.50, 3.75, 6.25, 8.75

   Calculate k as follows:

   • For 2 pounds: \( k = \frac{2.50}{2} = 1.25 \)
   • For 3 pounds: \( k = \frac{3.75}{3} = 1.25 \)
   • For 5 pounds: \( k = \frac{6.25}{5} = 1.25 \)
   • For 7 pounds: \( k = \frac{8.75}{7} \approx 1.25 \)

   Each ratio is equivalent:

   • k = $1.25 per pound of bananas

Summary:

• Gallons of gas: k = $2.75
• Candy bars sold: k = n
• Batches of sugar cookies: k = 1
• Pounds of bananas: k = $1.25