Question

A cafeteria serves lemonade that is made from a powdered drink mix. There is a proportional relationship between the number of scoops of powdered drink mix and the amount of water needed to make it. For every 2 scoops of mix, one half gallon of water is needed, and for every 5 scoops of mix, one and one fourth gallons of water are needed.

Part A: Find the constant of proportionality. Show every step of your work.

Part B: Write an equation that represents the relationship. Show every step of your work.

Part C: Describe how you would graph the relationship. Use complete sentences.

Part D: How many gallons of water are needed for 12 scoops of drink mix?

Answer 1

A. 1/4 gal/scoop

B. w = 1/4s

C. line through the origin with slope 1/4; s-, w- axes labeled "scoops" and "gallons"

D. 3 gallons

Explanation:

Given 2 scoops of powdered drink mix needs 1/2 gallon of water, you want (A) the constant of proportionality, (B) the equation for the relationship, (C) a description of the graph, (D) gallons for 12 scoops.

Proportional relationship

A proportional relationship between gallons of water (w) and scoops of drink mix (s) can be written as ...

w = ks

where k is the constant of proportionality.

A Proportionality constant

Solving the equation for k, we get ...

k = w/s

Using the given values, we find k to be ...

k = (1/2 gal)/(2 scoops) = 1/4 gal/scoop

The constant of proportionality is 1/4 gallon per scoop.

B Equation

We show the equation in part A. Filling in the value of k gives ...

w = 1/4s . . . . . . . where s is number of scoops, and w is gallons of water

C Graph

The attached shows a graph. A line starts at the origin with slope 1/4; s-, w- axes are labeled "scoops" and "gallons". The practical domain is s ≥ 0, so no part of the graph is in the 3rd quadrant.

D 12 scoops

The equation tells you that 12 scoops will require ...

w = 1/4·12 = 3 . . . . gallons

3 gallons of water are needed for 12 scoops of drink mix.

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