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Last week, Edmon had lunch in two cafés, Café A and Café B. In Café A, he left an additional \(x\%\) tip on the check, while in Café B, he left an additional \(y\%\) tip on the check. Did Edmon pay more in Café A than in Café B?
(1) The amount of the tip Edmon left in Café A was greater than the tip he left in Café B.
(2) The ratio of \(x\) to \(y\) is greater than 3 to 1.
(1) The amount of the tip Edmon left in Café A was greater than the tip he left in Café B.
(2) The ratio of \(x\) to \(y\) is greater than 3 to 1.
29 Jul 2024, 05:51
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Official Solution:
Last week, Edmon had lunch in two cafés, Café A and Café B. In Café A, he left an additional \(x\%\) tip on the check, while in Café B, he left an additional \(y\%\) tip on the check. Did Edmon pay more in Café A than in Café B?
Use as few variables and as little algebra as possible for this question. Plug in numbers, don't get lost in algebraic jungles.
(1) The amount of the tip Edmon left in Café A was greater than the tip he left in Café B.
This is clearly insufficient because, even though Edmon left a larger tip in Café A than in Café B, we know nothing about the initial checks in the cafés.
(2) The ratio of \(x\) to \(y\) is greater than 3 to 1.
This implies \(\frac{x}{y} > 3\), which is also clearly insufficient.
(1)+(2) When combining, we know that the tip in Café A was greater than in Café B and that \(\frac{x}{y} > 3\):
Case 1:
• If the check in Café A was $100, and the tip was $100 (making x equal to 100%), the total amount paid in Café A would have been $200.
• If the check in Café B was $200, and the tip was $20 (making y equal to 10%), the total amount paid in Café B would have been $220.
For this case, the answer would be NO.
Case 2:
• If the check in Café A was $100, and the tip was $100 (making x equal to 100%), the total amount paid in Café A would have been $200.
• If the check in Café B was $100, and the tip was $10 (making y equal to 10%), the total amount paid in Café B would have been $110.
For this case, the answer would be YES.
We have two different answers. Hence, even taken together, we cannot answer the question.
Answer: E
Last week, Edmon had lunch in two cafés, Café A and Café B. In Café A, he left an additional \(x\%\) tip on the check, while in Café B, he left an additional \(y\%\) tip on the check. Did Edmon pay more in Café A than in Café B?
Use as few variables and as little algebra as possible for this question. Plug in numbers, don't get lost in algebraic jungles.
(1) The amount of the tip Edmon left in Café A was greater than the tip he left in Café B.
This is clearly insufficient because, even though Edmon left a larger tip in Café A than in Café B, we know nothing about the initial checks in the cafés.
(2) The ratio of \(x\) to \(y\) is greater than 3 to 1.
This implies \(\frac{x}{y} > 3\), which is also clearly insufficient.
(1)+(2) When combining, we know that the tip in Café A was greater than in Café B and that \(\frac{x}{y} > 3\):
Case 1:
• If the check in Café A was $100, and the tip was $100 (making x equal to 100%), the total amount paid in Café A would have been $200.
• If the check in Café B was $200, and the tip was $20 (making y equal to 10%), the total amount paid in Café B would have been $220.
For this case, the answer would be NO.
Case 2:
• If the check in Café A was $100, and the tip was $100 (making x equal to 100%), the total amount paid in Café A would have been $200.
• If the check in Café B was $100, and the tip was $10 (making y equal to 10%), the total amount paid in Café B would have been $110.
For this case, the answer would be YES.
We have two different answers. Hence, even taken together, we cannot answer the question.
Answer: E
