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WAEC 2022 · Paper 2 · Q1

In a school, 100 students indicated their interest in the following sports: football, volleyball and hockey. 48 liked football, 42 liked volleyball and 36 liked hockey. 18 liked football only, 6 volleyball only and 4 hockey only. 10 liked all three sports while 12 liked football and volleyball only.

  1. (a)

    Illustrate the information on a Venn diagram.

    Model answer
    U = 100FVH1864128141028

    Football and hockey only is 48−18−12−10=848 - 18 - 12 - 10 = 8; volleyball and hockey only is 42−6−12−10=1442 - 6 - 12 - 10 = 14. The circles hold 18+6+4+12+8+14+10=7218 + 6 + 4 + 12 + 8 + 14 + 10 = 72 students, so 100−72=28100 - 72 = 28 go outside.

  2. (b)

    Find the number of students who liked: (i) two types of sports only; (ii) none of the three sports.

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

Worked solution (try it first)

(a)

  1. Draw three overlapping circles F, V and H in a rectangle of 100.
  2. Put 10 in the centre, 12 in F and V only, and 18, 6 and 4 in the "only" regions.
  3. Football holds 48: 18+12+10+(F and H only)=4818 + 12 + 10 + (\text{F and H only}) = 48, so F and H only =8= 8.
  4. Volleyball holds 42: 6+12+10+(V and H only)=426 + 12 + 10 + (\text{V and H only}) = 42, so V and H only =14= 14.
  5. Check with hockey: 4+8+14+10=364 + 8 + 14 + 10 = 36, which matches.

(b)(i)

  1. Two sports only: 12+8+14=3412 + 8 + 14 = 34.

(ii)

  1. Inside the circles: 18+6+4+12+8+14+10=7218 + 6 + 4 + 12 + 8 + 14 + 10 = 72.
  2. So 100−72=28100 - 72 = 28 liked none of the three.

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