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06 Aug 2026, 11:09
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At the beginning of a charity event, Lena had only red wristbands and blue wristbands, with at least one wristband of each color. During the morning shift, she handed out \(\frac{1}{9}\) of the red wristbands. If Lena did not receive any additional wristbands that day, did she have more than 27 wristbands at the beginning of the event?
(1) The number of red wristbands Lena handed out during the morning shift was 50% as many as the number of blue wristbands she had at the beginning of the event.
(2) The number of red wristbands Lena handed out during the morning shift was 75% as many as the number of red wristbands she handed out during the afternoon shift.
(1) The number of red wristbands Lena handed out during the morning shift was 50% as many as the number of blue wristbands she had at the beginning of the event.
(2) The number of red wristbands Lena handed out during the morning shift was 75% as many as the number of red wristbands she handed out during the afternoon shift.
06 Aug 2026, 11:09
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Official Solution:
At the beginning of a charity event, Lena had only red wristbands and blue wristbands, with at least one wristband of each color. During the morning shift, she handed out \(\frac{1}{9}\) of the red wristbands. If Lena did not receive any additional wristbands that day, did she have more than 27 wristbands at the beginning of the event?
(1) The number of red wristbands Lena handed out during the morning shift was 50% as many as the number of blue wristbands she had at the beginning of the event.
The number handed out in the morning was \(\frac{1}{9}\) of the red wristbands.
So:
\(\frac{1}{9}\) of red \(= \frac{1}{2}\) of blue
red : blue \(= 9:2\)
This gives only a ratio, not the actual numbers.
For example, 9 red and 2 blue gives 11 total, so the answer is NO.
But 27 red and 6 blue gives 33 total, so the answer is YES.
Not sufficient.
(2) The number of red wristbands Lena handed out during the morning shift was 75% as many as the number of red wristbands she handed out during the afternoon shift.
The number handed out in the morning was \(\frac{1}{9}\) of the red wristbands.
Statement (2) says:
\(\frac{1}{9}\) of red \(= \frac{3}{4}\) of afternoon
So:
afternoon \(= \frac{4}{27}\) of red
Thus, the number of red wristbands must be a multiple of 27. Since there was at least one red wristband, Lena had at least 27 red wristbands and at least 1 blue wristband.
Therefore, she had more than 27 wristbands in total.
Sufficient.
Answer: B
At the beginning of a charity event, Lena had only red wristbands and blue wristbands, with at least one wristband of each color. During the morning shift, she handed out \(\frac{1}{9}\) of the red wristbands. If Lena did not receive any additional wristbands that day, did she have more than 27 wristbands at the beginning of the event?
(1) The number of red wristbands Lena handed out during the morning shift was 50% as many as the number of blue wristbands she had at the beginning of the event.
The number handed out in the morning was \(\frac{1}{9}\) of the red wristbands.
So:
\(\frac{1}{9}\) of red \(= \frac{1}{2}\) of blue
red : blue \(= 9:2\)
This gives only a ratio, not the actual numbers.
For example, 9 red and 2 blue gives 11 total, so the answer is NO.
But 27 red and 6 blue gives 33 total, so the answer is YES.
Not sufficient.
(2) The number of red wristbands Lena handed out during the morning shift was 75% as many as the number of red wristbands she handed out during the afternoon shift.
The number handed out in the morning was \(\frac{1}{9}\) of the red wristbands.
Statement (2) says:
\(\frac{1}{9}\) of red \(= \frac{3}{4}\) of afternoon
So:
afternoon \(= \frac{4}{27}\) of red
Thus, the number of red wristbands must be a multiple of 27. Since there was at least one red wristband, Lena had at least 27 red wristbands and at least 1 blue wristband.
Therefore, she had more than 27 wristbands in total.
Sufficient.
Answer: B
