In an election only two candidates contested. 30% of the registered votes did not cast their votes and 180 votes were declared invalid. The winner got 684 votes more than his opponent. The number of valid votes received by the winner is 42% of the number of registered voters. How many registered voters cast their votes?

Let total number of registered voters = $$100x$$

Number of voters who cast their votes = $$\frac{70}{100} \times 100x = 70x$$

Number of valid votes = $$70x - 180$$

Valid votes received by the winner = $$\frac{42}{100} \times 100x = 42x$$

=> Valid votes received by the loser = $$(70x - 180) - 42x$$

Acc. to ques, => $$42x - [(70x - 180) - 42x] = 684$$

=> $$84x - 70x = 684 - 180 = 504$$

=> $$x = \frac{504}{14} = 36$$

$$\therefore$$ Number of registered voters who cast their votes = $$70 \times 36 = 2520$$