Discussion
A bag contains Violet (V), Yellow (Y), Red (R), and Green (G) balls. On counting them, the following results are obtained:
(i) The sum of Yellow balls and twice the number of Violet balls is 50.
(ii) The sum of Violet and Green balls is 50.
(iii) The sum of Yellow and Red balls is 50.
(iv) The sum of Violet and twice the number of Red balls is 50.
Which one of the following Pie charts correctly represents the balls in the bag?
Answer: A
The correct chart is V 10%, R 20%, Y 30%, G 40%.
Write the four statements as equations: \(Y + 2V = 50\), \(V + G = 50\), \(Y + R = 50\) and \(V + 2R = 50\). The first and last both equal 50, so
\[ \begin{aligned} Y + 2V &= V + 2R \Rightarrow Y + V = 2R \\ (50 - R) + V &= 2R \Rightarrow V = 3R - 50 \\ (3R - 50) + 2R &= 50 \Rightarrow R = 20 \end{aligned} \]
Then \(V = 10\), \(Y = 50 - 20 = 30\) and \(G = 50 - 10 = 40\). The total is 100 balls, so the counts are also the percentages: V 10%, R 20%, Y 30%, G 40%.
The other charts fail a statement at once. V 40% would make \(Y + 2V\) at least 80, not 50; V 30% with Y 10% gives \(10 + 60 = 70\); V 20% with Y 40% gives \(40 + 40 = 80\).
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