WAEC 2017 · Paper 2 · Q7

(a)
If $\log_6 y + 2\log_6 x = 3$ , express $y$ in terms of $x$ .
(b)
A number of students were interviewed to find out which of the sporting activities they liked (football, boxing and volleyball). $70\%$ of those interviewed liked football, $60\%$ boxing and $45\%$ volleyball. $45\%$ liked football and boxing, $15\%$ boxing and volleyball, $25\%$ football and volleyball, $5\%$ liked all three sports and $x\%$ did not like any of the three sports. (i) Draw a Venn diagram to illustrate this information. (ii) Use the diagram to find the percentage of students who liked: (α) football and volleyball but not boxing; (β) exactly two sports; (γ) none of the three sports.
Separate values with commas, e.g. 3, −2
Model answer
Fill the Venn diagram from the middle outwards. Start with the $5\%$ who like all three. Then subtract it from each pair: football and boxing only $45 - 5 = 40$ , boxing and volleyball only $15 - 5 = 10$ , football and volleyball only $25 - 5 = 20$ . Then each sport on its own: football only $70 - 40 - 20 - 5 = 5$ , boxing only $60 - 40 - 10 - 5 = 5$ , volleyball only $45 - 20 - 10 - 5 = 10$ . The regions add up to $95\%$ , so $x = 5$ lie outside all three circles.

From the diagram: ( $\alpha$ ) $20\%$ ; ( $\beta$ ) exactly two sports $40 + 10 + 20 = 70\%$ ; ( $\gamma$ ) $5\%$ .

Worked solution (try it first)

(a)

$2\log_6 x = \log_6 x^2$ , so $\log_6 y + \log_6 x^2 = \log_6(yx^2) = 3$ .
In index form, $yx^2 = 6^3 = 216$ , so $y = \frac{216}{x^2}$ .

(b)(i)

Draw three overlapping circles F, B and V in a rectangle (100%).
Put 5 in the centre.
Two sports only: F and B $45 - 5 = 40$ , B and V $15 - 5 = 10$ , F and V $25 - 5 = 20$ .
One sport only: F $70 - 40 - 20 - 5 = 5$ , B $60 - 40 - 10 - 5 = 5$ , V $45 - 20 - 10 - 5 = 10$ .

(ii)

(α)** Football and volleyball but not boxing: $20\%$ .

(β)

Exactly two sports: $40 + 10 + 20 = 70\%$ .

(γ)

The regions add up to $5 + 5 + 10 + 40 + 10 + 20 + 5 = 95\%$ , so $x = 100 - 95 = 5\%$ liked none.

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