Question

The following pie diagram represents the number of valid votes obtained by four students who ran for the post of school captain .
The total valid votes polled was 720. Answer the following questions :
(a) Who won the elections?
(b) What is the maximum number of votes and who got them?
(c) By how many votes did the winner defeat the nearest contestant?

Fig 6.27

Answer 1

(a)
$$\begin{gathered} \mathrm{Total}\mathrm{valid}\mathrm{votes}\mathrm{polled}=720 \\ \mathrm{Central}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{valid}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Raj}=80° \\ \mathrm{Central}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{valid}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Raj}=\frac{\mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Raj}}{\mathrm{Total}\mathrm{valid}\mathrm{votes}\mathrm{polled}}\times 360° \\ 80°=\frac{\mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Raj}}{720}\times 360° \\ \therefore \mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Raj}=\frac{80\times 720}{360}=160 \end{gathered}$$

$$\begin{gathered} \mathrm{Central}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{valid}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Albert}=100° \\ \mathrm{Central}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{valid}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Albert}=\frac{\mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Albert}}{\mathrm{Total}\mathrm{valid}\mathrm{votes}\mathrm{polled}}\times 360° \\ 100°=\frac{\mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Albert}}{720}\times 360° \\ \therefore \mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Albert}=\frac{100\times 720}{360}=200 \end{gathered}$$


$$\begin{gathered} \mathrm{Central}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{valid}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Suja}=60° \\ \mathrm{Central}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{valid}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Suja}=\frac{\mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Suja}}{\mathrm{Total}\mathrm{valid}\mathrm{votes}\mathrm{polled}}\times 360° \\ 60°=\frac{\mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Suja}}{720}\times 360° \\ \therefore \mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Suja}=\frac{60\times 720}{360}=120 \end{gathered}$$

$$\begin{gathered} \mathrm{Central}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{valid}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Nashima}=120° \\ \mathrm{Central}\mathrm{angle}\mathrm{of}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{valid}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Nashima}=\frac{\mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Nashima}}{\mathrm{Total}\mathrm{valid}\mathrm{votes}\mathrm{polled}}\times 360° \\ 120°=\frac{\mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Nashima}}{720}\times 360° \\ \therefore \mathrm{Number}\mathrm{of}\mathrm{votes}\mathrm{obtained}\mathrm{by}\mathrm{Nashima}=\frac{120\times 720}{360}=240 \end{gathered}$$

We can see that Nashima obtained the maximum number of votes, i.e. 240.
Therefore, Nashima won the elections.

(b) The minimum number of votes is 120 and these votes were obtained by Suja .

(c) The nearest contestant to the winner is Albert and he obtained 200 votes.
Therefore, Nashima defeated Albert by (240-200), i.e. 40 votes.