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05 Aug 2026, 12:13
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At a home decor shop, 12 lamps were on display on Sunday evening, and each lamp had a different price. If, on Monday morning, the price of each of the 12 lamps was changed, did the average (arithmetic mean) price of the 12 lamps increase?
(1) The 6 most expensive lamps each increased in price by 8%.
(2) The 6 least expensive lamps each decreased in price by 8%.
(1) The 6 most expensive lamps each increased in price by 8%.
(2) The 6 least expensive lamps each decreased in price by 8%.
05 Aug 2026, 12:13
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Official Solution:
At a home decor shop, 12 lamps were on display on Sunday evening, and each lamp had a different price. If, on Monday morning, the price of each of the 12 lamps was changed, did the average (arithmetic mean) price of the 12 lamps increase?
The question asks whether the total price of the 12 lamps increased after the price changes.
(1) The 6 most expensive lamps each increased in price by \(8\%\).
This is not sufficient, since we do not know what happened to the 6 least expensive lamps.
Not sufficient.
(2) The 6 least expensive lamps each decreased in price by \(8\%\).
This is not sufficient, since we do not know what happened to the 6 most expensive lamps.
Not sufficient.
(1)+(2) Since all 12 prices are different, the total price of the 6 most expensive lamps must be greater than the total price of the 6 least expensive lamps. So \(8\%\) of the total price of the 6 most expensive lamps is greater than \(8\%\) of the total price of the 6 least expensive lamps. Therefore, the total increase is greater than the total decrease, so the overall total price increased. Hence, the average price also increased. Sufficient.
Answer: C
At a home decor shop, 12 lamps were on display on Sunday evening, and each lamp had a different price. If, on Monday morning, the price of each of the 12 lamps was changed, did the average (arithmetic mean) price of the 12 lamps increase?
The question asks whether the total price of the 12 lamps increased after the price changes.
(1) The 6 most expensive lamps each increased in price by \(8\%\).
This is not sufficient, since we do not know what happened to the 6 least expensive lamps.
Not sufficient.
(2) The 6 least expensive lamps each decreased in price by \(8\%\).
This is not sufficient, since we do not know what happened to the 6 most expensive lamps.
Not sufficient.
(1)+(2) Since all 12 prices are different, the total price of the 6 most expensive lamps must be greater than the total price of the 6 least expensive lamps. So \(8\%\) of the total price of the 6 most expensive lamps is greater than \(8\%\) of the total price of the 6 least expensive lamps. Therefore, the total increase is greater than the total decrease, so the overall total price increased. Hence, the average price also increased. Sufficient.
Answer: C
