To evaluate whether Jerome’s brother or sister is correct regarding maintaining the ratio of water to drink mix while increasing the total amount of energy drink, let's examine each suggestion.
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Jerome’s Brother's Suggestion:
- He suggests adding one cup of water for every cup of drink mix added.
- In terms of maintaining the ratio: If the original ratio of water to drink mix is \( W:M \) (where \( W \) is the amount of water and \( M \) is the amount of drink mix), and Jerome adds \( x \) cups of drink mix, he would also add \( x \) cups of water.
- The new amounts would be:
- Water: \( W + x \)
- Drink mix: \( M + x \)
- The new ratio would be: \[ \text{New ratio} = \frac{W + x}{M + x} \]
- This alters the ratio because the increase in both components by the same amount does not necessarily maintain the initial ratio unless \( W = M \). Therefore, Jerome's brother's approach is not correct because it does not guarantee that the original water-to-drink mix ratio is preserved unless both quantities are equal.
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Jerome’s Sister's Suggestion:
- She suggests multiplying both the amount of water and the amount of drink mix by the same number (let's say \( k \)).
- If the original amounts are \( W \) (water) and \( M \) (drink mix), after multiplication, the new amounts would be:
- Water: \( k \cdot W \)
- Drink mix: \( k \cdot M \)
- The new ratio would be: \[ \text{New ratio} = \frac{k \cdot W}{k \cdot M} = \frac{W}{M} \]
- This keeps the ratio of water to drink mix the same as the original ratio because multiplying both quantities by the same number does not change the relative proportions. Thus, Jerome's sister's approach is correct because it successfully maintains the original ratio of water to drink mix.
In summary:
- Jerome’s brother is not correct because his method does not maintain the original ratio.
- Jerome’s sister is correct because multiplying both components by the same factor preserves the ratio.
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