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WAEC 2019 · Paper 2 · Q9

  1. (a)

    A selection interview was conducted for 110 students into various subjects in a Science department. 25 were selected for Physics, 45 for Biology and 48 for Mathematics. 10 were selected for Physics and Mathematics, 8 for Biology and Mathematics and 6 for Physics and Biology, while 5 were selected for all the three subjects. (i) Illustrate the information on a Venn diagram. (ii) How many students were selected for Biology but neither Physics nor Mathematics? (iii) How many students were not selected for any of the three subjects?

    Separate values with commas, e.g. 3, −2

  2. (b)

    The mean of 2222, 1818, (2x+1)(2x + 1), 1010 and 2020 is 1515. Find the median.

Worked solution (try it first)

(a)(i)

  1. Draw three overlapping circles P, B and M in a rectangle of 110 students.
  2. Put 5 in the centre.
  3. Take it away from each pair to get the "two subjects only" regions: P and M only 10−5=510 - 5 = 5, B and M only 8−5=38 - 5 = 3, P and B only 6−5=16 - 5 = 1.
  4. Then the "one subject only" regions: Physics only =25−5−1−5=14= 25 - 5 - 1 - 5 = 14, Biology only =45−1−3−5=36= 45 - 1 - 3 - 5 = 36, Mathematics only =48−5−3−5=35= 48 - 5 - 3 - 5 = 35.

(ii)

  1. Biology but neither Physics nor Mathematics is Biology only: 36.

(iii)

  1. Inside the circles: 14+36+35+5+3+1+5=9914 + 36 + 35 + 5 + 3 + 1 + 5 = 99.
  2. So 110−99=11110 - 99 = 11 were not selected for any subject.

(b)

  1. The five numbers add up to 5×15=755 \times 15 = 75: 22+18+(2x+1)+10+20=7522 + 18 + (2x + 1) + 10 + 20 = 75, so 71+2x=7571 + 2x = 75 and x=2x = 2.
  2. The numbers are 22,18,5,10,2022, 18, 5, 10, 20.
  3. In order: 5,10,18,20,225, 10, 18, 20, 22.
  4. The median is 18.

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