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manalq8
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12 Jan 2012, 07:51
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On a certain day, the first flight out of Phoenix airport had a delayed departure, while the next three flights departed on-time. What is the minimum number of subsequent flights that must depart from Phoenix on-time, to achieve an on-time departure rate of over 90% that day?
A. 6
B. 7
C. 9
D. 10
E. 11
A. 6
B. 7
C. 9
D. 10
E. 11
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24 Feb 2023, 12:37
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manalq8
Official Solution:
On a certain day, the first flight out of Phoenix airport had a delayed departure, while the next three flights departed on-time. What is the minimum number of subsequent flights that must depart from Phoenix on-time, to achieve an on-time departure rate of over 90% that day?
A. 6
B. 7
C. 9
D. 10
E. 11
We want an on-time departure rate of over 90%, which means a delayed departure rate of less than 10%. Since we need the minimum number of subsequent flights that will make (delayed)/(total) less than 10%, then the one that was already delayed should be the only one.
Let \(x\) be the minimum number of subsequent flights that must depart on-time to achieve this. Then, the delayed departure rate is \(\frac{1}{x + 4} \), since there are already 4 flights, including the first delayed one. We want this rate to be less than 10%, so we have the inequality \(\frac{1}{x + 4} < \frac{1}{10}\). Solving for \(x\), we get x > 6. Therefore, the minimum number of subsequent flights that must depart on-time is 7.
Answer: B
General Discussion
My approach:
If the first (one) flight had a late departure, then 1 flight out of 1 flight had a late departure (100% of the flights so far). Then, the stem says that the next 3 flights departed on-time, meaning that 1 out of 4 flights had a late departure (25% of the flights so far). Therefore, following this logic, if we add another 6 on-time planes, we will know that 1 out of 10 flights had a late departure (thus 10% was late and 90% was on-time).
The question asks how many subsequent flights need to depart from Phoenix on-time, for the airport's on-time departure rate to be HIGHER than 90%. As we have seen so far, if 6 subsequent planes were on-time, we would have a rate of EXACTLY 90% of late departure. Therefore, for the airport's on-time departure rate to be HIGHER than 90%, we need 7 subsequent planes to be on-time (1 out of 11 had a late departure or aprox. 9% of the flights -- the remaining 91% is the rate of on-time flights).
Ans: B
If the first (one) flight had a late departure, then 1 flight out of 1 flight had a late departure (100% of the flights so far). Then, the stem says that the next 3 flights departed on-time, meaning that 1 out of 4 flights had a late departure (25% of the flights so far). Therefore, following this logic, if we add another 6 on-time planes, we will know that 1 out of 10 flights had a late departure (thus 10% was late and 90% was on-time).
The question asks how many subsequent flights need to depart from Phoenix on-time, for the airport's on-time departure rate to be HIGHER than 90%. As we have seen so far, if 6 subsequent planes were on-time, we would have a rate of EXACTLY 90% of late departure. Therefore, for the airport's on-time departure rate to be HIGHER than 90%, we need 7 subsequent planes to be on-time (1 out of 11 had a late departure or aprox. 9% of the flights -- the remaining 91% is the rate of on-time flights).
Ans: B
