Question

The number of votes not cast for the Praja Party increased by 25% in the National General Election over those not cast for it in the previous Assembly Polls, and the Praja Party lost by a majority twice as large as that by which it had won the Assembly Polls. If a total 2,60,000 people voted each time, how many voted for the Praja Party in the Assembly Elections?

• A. 1,10,000
• B. 1,50,000
• C. 1,40,000
• D. 1,20,000

Hint:

Always set variables for unknowns, form equations step-by-step from the given percentage changes and margin relationships, and carefully match the total vote constraints for each election before solving.

Answer 1

The Correct Option is B

Let the number of votes received by the Praja Party in the Assembly Elections be $x$.
The total number of votes cast in the Assembly Elections = $2,60,000$.
Therefore, the number of votes not cast for the Praja Party in the Assembly Elections = $2,60,000 - x$.
In the Assembly Elections, the Praja Party won by a certain margin.
Winning margin in Assembly Elections = (Votes for Praja Party) $-$ (Votes against Praja Party)
= $x - (2,60,000 - x) = x - 2,60,000 + x = 2x - 2,60,000$.
In the National General Election:
The votes not cast for the Praja Party increased by 25%.
Votes not cast for Praja Party in Assembly = $2,60,000 - x$.
Votes not cast for Praja Party in National Election = $(2,60,000 - x) \times 1.25$.
Since total votes = $2,60,000$, votes cast for Praja Party in the National Election = $2,60,000 - 1.25(2,60,000 - x)$.
In the National Election, the Praja Party lost by a margin twice as large as the margin by which it had won the Assembly Elections.
Loss margin = $2 \times$ (Assembly win margin).
Loss margin in National = $(\text{Votes against Praja}) - (\text{Votes for Praja})$.
Votes against Praja in National = $1.25(2,60,000 - x)$.
Votes for Praja in National = $2,60,000 - 1.25(2,60,000 - x)$.
Loss margin in National = $1.25(2,60,000 - x) - [2,60,000 - 1.25(2,60,000 - x)]$.
Simplify: Loss margin in National = $1.25(2,60,000 - x) - 2,60,000 + 1.25(2,60,000 - x)$.
= $2.5(2,60,000 - x) - 2,60,000$.
Given: Loss margin in National = $2 \times$ Win margin in Assembly.
Win margin in Assembly = $2x - 2,60,000$.
So, $2.5(2,60,000 - x) - 2,60,000 = 2 \times (2x - 2,60,000)$.
Equation: $6,50,000 - 2.5x - 2,60,000 = 4x - 5,20,000$.
$3,90,000 - 2.5x = 4x - 5,20,000$.
$3,90,000 + 5,20,000 = 4x + 2.5x$.
$9,10,000 = 6.5x$.
$x = 1,40,000$. Wait — this doesn’t match our correct answer, so let's recheck calculations carefully.
Recalculation:
Assembly win margin = $x - (2,60,000 - x) = 2x - 2,60,000$.
National loss margin = $(1.25(2,60,000 - x)) - (2,60,000 - 1.25(2,60,000 - x))$.
Simplify the second term: $2,60,000 - 1.25(2,60,000 - x) = 2,60,000 - 3,25,000 + 1.25x = -65,000 + 1.25x$.
Loss margin in National = $3,25,000 - 1.25x - (-65,000 + 1.25x)$.
= $3,25,000 - 1.25x + 65,000 - 1.25x$.
= $3,90,000 - 2.5x$.
Given: $3,90,000 - 2.5x = 2(2x - 2,60,000)$.
$3,90,000 - 2.5x = 4x - 5,20,000$.
$3,90,000 + 5,20,000 = 4x + 2.5x$.
$9,10,000 = 6.5x$.
$x = 1,40,000$.
Thus, the number of votes for Praja Party in Assembly Elections is $\boxed{1,40,000}$.