Making Energy Drinks

Question

Equivalent Ratios
Jerome makes and sells his own energy drinks. He mixes a liquid drink mix with water to create his own special blend.
Jerome usually uses 8 cups of water and 3 cups of drink mix. Tomorrow, there is a road race in town, however, and Jerome thinks he’ll be able to sell a lot more of his energy drinks than usual.
Jerome’s brother told him to add one cup of water for every one cup of drink mix he added to keep the ratio of water to drink mix the same and still increase the total amount of energy drink.
Jerome’s sister told him to multiply both the amount of water and the amount of drink mix by the same number to keep the ratio the same and still increase the total amount of energy drink.
Who is correct: Jerome’s brother or Jerome’s sister?
As you complete the task, keep this question in mind: Based on what you know about ratios, what do you think the answer will be?
Directions:
Complete each task, reading the directions carefully as you do.
You will be graded on the work you show, or your solution process, in addition to your answers. Make sure to show all your work and to answer each question as you complete the task. Type all of your work into this document so you can submit it to your teacher for a grade. You will be given partial credit based on the work you show and on the completeness and accuracy of your explanations.
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Now let’s get started!
Step 1: Finding the part-to-part and part-to-whole ratios
To make his special energy drink, Jerome uses 8 cups of water and 3 cups of drink mix.
b) Is Jerome’s brother correct? Explain why or why not, using examples from the ratio table to support your argument. (4 points)

Answer 1

To determine whether Jerome's brother is correct, we need to analyze the ratios involved when increasing the amount of water and drink mix.

Original Ratios

Jerome's original mixture contains:

• Water: 8 cups
• Drink mix: 3 cups

Part-to-Part Ratio of Water to Drink Mix: The ratio of water to drink mix can be represented as: \[ \text{Water:Drink Mix} = 8:3 \]

Part-to-Whole Ratio of the Mixture: The total mixture consists of both water and drink mix: \[ \text{Total} = 8 \text{ cups (water)} + 3 \text{ cups (drink mix)} = 11 \text{ cups (total)} \] The part-to-whole ratio (water to total) is: \[ \text{Water:Total} = 8:11 \] And for drink mix to total: \[ \text{Drink Mix:Total} = 3:11 \]

Analyzing Jerome's Brother's Method

Jerome's brother suggests adding one cup of water for every one cup of drink mix added. This means that if Jerome adds \( x \) cups of drink mix, he will also add \( x \) cups of water. Let's analyze:

Assuming Jerome adds \( y \) cups of drink mix:

• Water: \( 8 + y \)
• Drink Mix: \( 3 + y \)

The new ratio of water to drink mix would be: \[ \text{New ratio} = \frac{8 + y}{3 + y} \] To analyze whether this maintains the original ratio of \( \frac{8}{3} \), let's set up a proportion: \[ \frac{8 + y}{3 + y} = \frac{8}{3} \]

By cross-multiplying: \[ 3(8 + y) = 8(3 + y) \] Expanding both sides: \[ 24 + 3y = 24 + 8y \] Subtracting \( 3y \) from both sides gives: \[ 24 = 24 + 5y \] Then, \[ 0 = 5y \] This means \( y = 0 \) is the only solution. Therefore, the ratios do not remain equivalent or the same if we increase the amounts as Jerome's brother suggests.

Conclusion

Jerome’s brother is not correct. When he suggests adding 1 cup of water for every 1 cup of drink mix, while he tries to keep the amount of drink mix proportional, he disrupts the original ratio of water to drink mix \( (8:3) \). The correct approach to keep the ratio the same when increasing the amounts is to multiply both parts of the mixture by the same factor, which is what Jerome’s sister recommends.

Thus, the analysis demonstrates that the only way to maintain the \( 8:3 \) ratio is to multiply the initial amounts by some common factor (for example, multiplying both by 2 or 3, etc.), rather than changing both amounts in a fixed addition form.