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, let's first look at the original ratio of water to drink mix that Jerome uses. He typically uses:

• 8 cups of water
• 3 cups of drink mix

The ratio of water to drink mix can be expressed as \( \frac{8}{3} \).

Analyzing Jerome's Brother's Suggestion

Jerome's brother suggests adding one cup of water for every cup of drink mix that he adds. This means that for each cup of drink mix added, the amount of water increases by an equal amount (1 cup).

If we increase the drink mix by \( x \) cups, then the amount of water would also increase by \( x \) cups. Therefore, the new amounts would be:

• Water: \( 8 + x \)
• Drink mix: \( 3 + x \)

The new ratio will then be:

\[ \text{New Ratio} = \frac{8 + x}{3 + x} \]

To keep the ratio the same as the original \( \frac{8}{3} \), the following should hold true:

\[ \frac{8 + x}{3 + x} = \frac{8}{3} \]

Cross-multiplying gives:

\[ 3(8 + x) = 8(3 + x) \] \[ 24 + 3x = 24 + 8x \] \[ 3x = 8x \] \[ 0 = 5x \]

This implies that \( x \) must be 0 for the original ratio to be maintained. This means that if he applies his brother's suggestion, he will disrupt the original ratio unless he doesn't add anything at all.

Analyzing Jerome's Sister's Suggestion

Jerome's sister suggests multiplying both the amount of water and drink mix by the same number \( k \).

So the new amounts would be:

• Water: \( 8k \)
• Drink mix: \( 3k \)

The new ratio will be:

\[ \text{New Ratio} = \frac{8k}{3k} = \frac{8}{3} \]

In this case, the proportions remain the same because both quantities are scaled by factor \( k \).

Conclusion

Jerome's sister is correct. Multiplying both the amount of water and the amount of drink mix by the same number \( k \) keeps the ratio the same, allowing Jerome to increase the total amount of energy drink without changing the original proportion of water to drink mix. On the other hand, Jerome’s brother’s method would disrupt the original ratio unless no additional drink mix is added.