The answer
(a)(i) \(72\) teachers
(ii) comparison below
(b)(i) \(\dfrac{5}{12}\)
(ii) \(\dfrac{22}{69}\)
(iii) \(\dfrac{56}{253}\)
O-Level E-Math 2023 Paper 2 Question 7 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 7 of the O-Level E-Math 2023 Paper 2. It tests box-and-whisker plots (quartiles, in the Dispersion / Probability area. It is worth 9 marks: (a) 1 + 3, (b) 1 + 2 + 2. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a)(i) The upper quartile of the teachers' times is 50 minutes, so a quarter of the teachers took more than 50 minutes. Hence \(18 = \tfrac{1}{4} \times N\), giving \(N = 72\) teachers.
(ii) Averages: the teachers' median (\(\approx 36\) min) is greater than the students' median (\(\approx 28\) min), so on average the teachers took longer to travel to school. Distributions: the teachers' times have a larger range (\(84 - 24 = 60\) min) than the students' (\(63 - 10 = 53\) min), so the teachers' times are more spread out, even though the interquartile ranges are similar (\(50 - 32 = 18\) min for teachers, \(34 - 16 = 18\) min for students).
(b) Bus travellers \(= 24 - 8 - 6 = 10\).
(i) \(P(\text{bus}) = \dfrac{10}{24} = \dfrac{5}{12}\).
(ii) Same type (without replacement): \(P = \dfrac{8}{24}\cdot\dfrac{7}{23} + \dfrac{6}{24}\cdot\dfrac{5}{23} + \dfrac{10}{24}\cdot\dfrac{9}{23} = \dfrac{56 + 30 + 90}{552} = \dfrac{176}{552} = \dfrac{22}{69}\).
(iii) Two walk (from 8) and one does not (from the other 16), in any of 3 orders: \(P = 3 \times \dfrac{8}{24}\cdot\dfrac{7}{23}\cdot\dfrac{16}{22} = \dfrac{56}{253}\). *(Equivalently \(\dfrac{\binom{8}{2}\binom{16}{1}}{\binom{24}{3}} = \dfrac{28 \times 16}{2024} = \dfrac{56}{253}\).)*
Answer: (a)(i) \(72\) teachers
(ii) comparison below
(b)(i) \(\dfrac{5}{12}\)
(ii) \(\dfrac{22}{69}\)
(iii) \(\dfrac{56}{253}\)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a box-and-whisker plots (quartiles question from Dispersion / Probability, worth 9 marks: (a) 1 + 3, (b) 1 + 2 + 2.
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