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.
Your teacher will give you further directions on submitting your work. You may be asked to upload the document, e-mail it to your teacher, or hand in a hard copy.
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.
How much water and how much drink mix should Jerome use to make enough energy drink for all the runners? Create a table to show your work and explain your results in writing. (4 points)

Answer 1

Step 1: Finding the Part-to-Part and Part-to-Whole Ratios

Given Data:

• Water: 8 cups
• Drink Mix: 3 cups

Part-to-Part Ratio

The part-to-part ratio of water to drink mix can be calculated as follows:

\[ \text{Part-to-Part Ratio} = \frac{\text{Water}}{\text{Drink Mix}} = \frac{8}{3} \]

This means for every 8 cups of water, there are 3 cups of drink mix.

Part-to-Whole Ratio

The part-to-whole ratio can be calculated by first finding the total amount of the mixture, which is the sum of the water and the drink mix:

\[ \text{Total Mixture} = \text{Water} + \text{Drink Mix} = 8 + 3 = 11 \text{ cups} \]

Now, we can calculate the part-to-whole ratios:

• Part-to-whole ratio for Water: \[ \text{Water Ratio} = \frac{\text{Water}}{\text{Total Mixture}} = \frac{8}{11} \]

• Part-to-whole ratio for Drink Mix: \[ \text{Drink Mix Ratio} = \frac{\text{Drink Mix}}{\text{Total Mixture}} = \frac{3}{11} \]

Creating a Table

| Component | Amount (cups) | Ratio (Part-to-Part) | Ratio (Part-to-Whole) | |-------------|----------------|-----------------------|------------------------| | Water | 8 | 8:3 | 8/11 | | Drink Mix | 3 | 8:3 | 3/11 | | Total | 11 | - | - |

Explanation of Results

The part-to-part ratio of 8:3 indicates that for every 8 cups of water, there should be 3 cups of drink mix to maintain the desired flavor and concentration. The part-to-whole ratios of 8/11 for water and 3/11 for drink mix show the proportions of each component in the total mixture.

Conclusion

To meet the needs of the increased demand for energy drinks, Jerome can either follow his brother or sister’s suggestion for adjusting the quantities while maintaining this ratio.

• If Jerome's brother's suggestion is followed (adding 1 cup of water for every 1 cup of drink mix added), the new amounts would be 9 cups of water for 4 cups of drink mix (which changes the ratio to 9:4).

• If Jerome's sister's suggestion is followed (multiplying both amounts by the same number), the mixture would remain in the original ratio of 8:3.

Thus, Jerome's sister is correct because multiplying both components by the same factor preserves the established ratio of 8 cups of water to 3 cups of drink mix, ensuring the same taste consistency while producing a larger quantity of the energy drink for the race.