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.
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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.

Answer 1

To analyze Jerome's energy drink mixture, we'll start by determining the part-to-part and part-to-whole ratios.

Part-to-Part Ratios

The part-to-part ratios compare one component of the mixture to another:

1. Water to Drink Mix Ratio: \[ \text{Water : Drink Mix} = 8 : 3 \]

Part-to-Whole Ratios

The part-to-whole ratio compares one component to the total mixture:

2. Total Amount of Mixture: \[ \text{Total} = \text{Water} + \text{Drink Mix} = 8 + 3 = 11 \text{ cups} \]

3. Water to Total Mixture Ratio: \[ \text{Water : Total} = 8 : 11 \]

4. Drink Mix to Total Mixture Ratio: \[ \text{Drink Mix : Total} = 3 : 11 \]

Now we have established the ratios needed to understand the mixture before we explore the adjustments proposed by Jerome’s siblings.

Step 2: Analysis of Jerome's Brother's and Sister's Suggestions

Jerome’s Brother’s Suggestion: Jerome’s brother suggests adding one cup of water for every one cup of drink mix added. This means if Jerome adds \( x \) cups of drink mix, he must add \( x \) cups of water.

Let's set \( x \) as the amount of drink mix added:

• New water amount = original water (8 cups) + \( x \)
• New drink mix amount = original drink mix (3 cups) + \( x \)

The new ratio of water to drink mix will be: \[ \text{New Water : New Drink Mix} = (8 + x) : (3 + x) \]

For the ratio to remain at \( 8 : 3 \), we need: \[ \frac{8 + x}{3 + x} = \frac{8}{3} \]

Cross-multiplying, we have: \[ 3(8 + x) = 8(3 + x) \] This simplifies to: \[ 24 + 3x = 24 + 8x \implies 3x = 8x \implies 0 = 5x \implies x = 0 \] This means that while adding equal amounts of water and drink mix does not maintain the original ratio.

Jerome’s Sister’s Suggestion: Jerome’s sister suggests multiplying both the amounts by the same number \( k \). If we let \( k \) be the multiplier:

• New water amount = \( 8k \)
• New drink mix amount = \( 3k \)

The new ratio of water to drink mix will be: \[ \text{New Water : New Drink Mix} = 8k : 3k \] This simplifies directly to: \[ \frac{8k}{3k} = \frac{8}{3} \] Thus, the ratio remains consistent.

Conclusion

Based on our analysis, Jerome’s sister is correct. To maintain the original ratio of 8 cups of water to 3 cups of drink mix, Jerome should multiply both amounts by the same number (k) instead of adding one cup of water for every cup of drink mix added as suggested by his brother.

Summary of Findings

• Part-to-part ratio of water to drink mix: \( 8 : 3 \)
• Part-to-whole ratios were established.
• Jerome's brother's method alters the original ratio, while his sister's method maintains it.