Question

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?

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

When several pairwise sums are given, convert each statement into a linear equation and reduce everything to one variable. Often the totals then automatically sum to 100, so the values double as percentages for a pie chart.

Answer 1

The Correct Option is A

Step 1: Translate statements into equations.
Let the counts be \(V, Y, R, G\). Then \[ \begin{aligned} \text{(i)}&:\ Y+2V=50,\\ \text{(ii)}&:\ V+G=50, \\ \text{(iii)}&:\ Y+R=50,\\ \text{(iv)}&:\ V+2R=50. \end{aligned} \]

Step 2: Express \(Y\) and \(R\) in terms of \(V\).
From (i): \(Y=50-2V\).
From (iii): \(R=50-Y=50-(50-2V)=2V\).

Step 3: Use (iv) to determine \(V\).
Substitute \(R=2V\) into (iv): \[ V+2(2V)=50 \Rightarrow 5V=50 \Rightarrow V=10. \]

Step 4: Find \(Y, R, G\).
\[ Y=50-2V=50-20=30,\qquad R=2V=20. \] From (ii): \[ G=50-V=50-10=40. \]



Step 5: Check total and map to percentages.
\[ V+Y+R+G=10+30+20+40=100. \] Hence the composition (and therefore the pie chart) is \[ V:10%, R:20%, Y:30%, G:40%. \] Final Answer:
\[ \boxed{\text{Option (A)}} \]