Question

diego owns a small ice cream shop called sweet scoops. to make his ice cream, he needs to buy milk. his supplier sells milk in two sizes: gallons and pints. there is a proportional relationship between any volume measured in gallons and its same volume measured in pints.

1. if you know the volume of something in gallons, you can calculate its volume in pints. complete the table below to show this relationship, then answer the questions.

gallons, g	1/8	1/2	2	4	1
pints, p			16	32

a. does this represent a proportional relationship? how do you know?
b. what is the constant of proportionality for this relationship?
c. write an equation to represent the relationship between gallons, g, and pints, p.
p = ______

Explanation:

Part 1: Completing the Table

Step 1: Find the conversion factor

From the given values, when \( g = 2 \), \( p = 16 \); when \( g = 4 \), \( p = 32 \). So the conversion factor (pints per gallon) is \( \frac{16}{2}=8 \) or \( \frac{32}{4} = 8 \). So \( p = 8g \).

Step 2: Calculate for \( g=\frac{1}{8} \)

Substitute \( g=\frac{1}{8} \) into \( p = 8g \).
\( p=8\times\frac{1}{8}=1 \)

Step 3: Calculate for \( g=\frac{1}{2} \)

Substitute \( g = \frac{1}{2} \) into \( p = 8g \).
\( p=8\times\frac{1}{2}=4 \)

Step 4: Calculate for \( g = 1 \)

Substitute \( g = 1 \) into \( p = 8g \).
\( p=8\times1 = 8 \)

Part 2: a. Proportional Relationship Check

A proportional relationship has the form \( y = kx \) (or \( p=kg \) here), where \( k \) is constant. We check the ratio \( \frac{p}{g} \) for each pair:

• For \( g = 2, p = 16 \): \( \frac{16}{2}=8 \)
• For \( g = 4, p = 32 \): \( \frac{32}{4}=8 \)
• For \( g=\frac{1}{8}, p = 1 \): \( \frac{1}{\frac{1}{8}}=8 \)
• For \( g=\frac{1}{2}, p = 4 \): \( \frac{4}{\frac{1}{2}}=8 \)
• For \( g = 1, p = 8 \): \( \frac{8}{1}=8 \)

Since the ratio \( \frac{p}{g} \) is constant (\( k = 8 \)), it is a proportional relationship.

Part 2: b. Constant of Proportionality

The constant of proportionality \( k \) in \( p=kg \) is the ratio \( \frac{p}{g} \). From the calculations above, \( k = 8 \) (pints per gallon).

Part 2: c. Equation for the Relationship

Answer:

s:
a. Yes, because \( \frac{p}{g} \) is constant (\( 8 \)).
b. The constant of proportionality is \( 8 \).
c. The equation is \( \boldsymbol{p = 8g} \).