Jerome makes and sells his own energy drinks. He mixes a liquid drink mix with water to create his own special blend.

Question

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?

Answer 1

To determine who is correct between Jerome's brother and sister, we need to analyze the ratios involved in the energy drink mix.

Jerome's original recipe consists of:

• Water: 8 cups
• Drink mix: 3 cups

The ratio of water to drink mix is: \[ \text{Ratio} = \frac{\text{Water}}{\text{Drink Mix}} = \frac{8}{3} \]

Evaluating Jerome's Brother's Proposal: Jerome's brother suggests that for every cup of drink mix added, Jerome should add one cup of water. If Jerome decides to add \( x \) cups of drink mix, he should also add \( x \) cups of water.

If we start from the original recipe (8 cups water and 3 cups drink mix) and add \( x \) cups of each:

• New amount of water: \( 8 + x \)
• New amount of drink mix: \( 3 + x \)

The new ratio becomes: \[ \text{New Ratio} = \frac{8 + x}{3 + x} \]

For the ratio to remain the same, this new ratio should equal the original ratio \(\frac{8}{3}\): \[ \frac{8 + x}{3 + x} = \frac{8}{3} \]

Cross-multiplying gives: \[ 3(8 + x) = 8(3 + x) \] \[ 24 + 3x = 24 + 8x \] Subtracting \( 3x \) gives: \[ 24 = 24 + 5x \] This cannot hold true unless \( x = 0 \). Thus, the method of adding equal amounts of each component does not maintain the original ratio.

Evaluating Jerome's Sister's Proposal: Jerome's sister suggests multiplying both the amounts by the same number \( k \) (where \( k > 1 \)) to increase the total amount while keeping the ratio the same.

If we multiply both amounts by \( k \):

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

The new ratio is: \[ \text{New Ratio} = \frac{8k}{3k} = \frac{8}{3} \]

Since this new ratio is the same as the original ratio, Jerome's sister's method maintains the ratio of water to drink mix.

Conclusion: Jerome's sister is correct. Multiplying both the water and drink mix amounts by a same factor keeps the ratio the same, while still allowing Jerome to increase the total amount he produces.