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.
s Jerome’s sister correct? Explain why or why not, using examples from the ratio table to support your argument. (4 points)

Answer 1

To determine who is correct between Jerome's brother and sister, we first need to analyze the ratios presented in Jerome's original mixture and how they change based on their suggestions.

Original Mixture:

Jerome uses:

• Water = 8 cups
• Drink Mix = 3 cups

Finding the Ratios:

1. Part-to-Part Ratio (Water to Drink Mix): \[ \text{Water : Drink Mix} = 8 : 3 \] This means for every 8 cups of water, there are 3 cups of drink mix.

2. Part-to-Whole Ratio (Water to Total Mixture): \[ \text{Total Mixture} = \text{Water} + \text{Drink Mix} = 8 + 3 = 11 \text{ cups} \] So, \[ \text{Water : Total Mixture} = 8 : 11 \]

Analyzing Suggestions:

1. Jerome's Brother's Suggestion:

Jerome's brother suggests to add one cup of water for every one cup of drink mix added.

If Jerome adds 1 cup of drink mix:

• New amount of Drink Mix = \(3 + 1 = 4\) cups
• New amount of Water = \(8 + 1 = 9\) cups

This results in:

• New Ratio (Water : Drink Mix) = \(9 : 4\)

Analysis:

The new ratio \(9 : 4\) is not equivalent to the original ratio \(8 : 3\) (since \( \frac{9}{4} \neq \frac{8}{3} \)). Therefore, Jerome's brother's suggestion does not keep the ratio the same.

2. Jerome's Sister's Suggestion:

Jerome's sister suggests to multiply both the amount of water and the drink mix by the same number.

Let's say Jerome wants to double the amounts:

• New Water = \(8 \times 2 = 16\) cups
• New Drink Mix = \(3 \times 2 = 6\) cups

This results in:

• New Ratio (Water : Drink Mix) = \(16 : 6\)
• Simplified Ratio = \(8 : 3\)

Analysis:

The new ratio \(16 : 6\) simplifies back to the original ratio \(8 : 3\). Therefore, Jerome's sister's suggestion keeps the ratio the same.

Conclusion:

Jerome's sister is correct. By multiplying both the amount of water and the drink mix by the same number, the ratio remains consistent with the original 8:3 ratio (water to drink mix). Jerome’s brother’s method, on the other hand, changes the ratio and does not maintain the original proportions necessary for his energy drink.

Final Judgement:

Jerome should follow Jerome's sister's advice if he wants to increase the total amount of energy drink while preserving the same flavor consistency and strength of the original blend.