Question

4% of the votes cast were found to be invalid in an election. Raman got 75% of the total valid votes and won the election by 240 votes from ritesh who was the only opponent. Find the total number of people who cast the votes provided one person can cast one vote only

a.650

b. 600

c. 500

d. 550

Answer 1

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$\underline{\underline{\large{\textbf{Solution:}}}}$

Given,
4% of the Votes cast are invalid.

Raman Got 75% of the total valid Votes and won the election by 240 votes from ritesh who was the only opponent.

Total number of people who cast the votes = ?

$\underline{\large{\textbf{Step-1:}}}$
Assume total Number of Voters as a Variable.

Let Total number of people who cast the votes be x

$\underline{\large{\textbf{Step-2:}}}$
Find No. of Total Valid Votes

Valid Votes + Invalid Votes = 100% of (Total Votes)

Given,
Invalid Votes = 4% of Total votes casted

Valid Votes + 4% of x = 100% of x

No. of Total Valid Votes = $96\%\: of\: X = (\frac{96}{100} \times x) = 0.96x$

$\underline{\large{\textbf{Step-3:}}}$
Find the No. of Votes casted to Raman(Winner)

Votes casted to Raman = 75% of Total Valid Votes

Votes casted to Raman = 75% of (0.96x)

= $(\frac{75}{100}) \times 0.96x$

= $(\frac{3}{4}) \times (\frac{96}{100})x$

= $(\frac{18}{25})x$

Votes casted to Raman = $0.72x$

$\underline{\large{\textbf{Step-4:}}}$
Find the No. of Votes casted to Ritesh(Raman's only opponent)

Votes casted to Ritesh = No. of Total Valid Votes - Votes Casted to Raman.

Votes casted to Ritesh = 0.96x - 0.72x = 0.24x

$\underline{\large{\textbf{Step-5:}}}$
Find the Difference in Votes casted to Raman and Ritesh

Difference in Votes = Votes casted to Raman - Votes casted to Ritesh

Difference in Votes = 0.72x - 0.24x = 0.48x

$\underline{\large{\textbf{Step-5:}}}$
Equate the Difference in Votes casted to Raman and Ritesh to 240.

$0.48x = 240$

$\implies \frac{48}{100}\, x = 24$

$\implies 48x = 2400$

$\implies x = 50$

$\therefore$

$\bigstar$
If the Difference in No. of Votes = 240

Total No. of Votes casted by the people = $\underline{\large{\textbf{50}}}$

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If the Difference in No. of Votes = 2400
$0.48x = 2400$

$\implies \frac{48}{100}\, x = 2400$

$\implies 48x = 240000$

$\implies x = 500$

Total No. of Votes casted by the people = $\underline{\large{\textbf{500}}}$

Option (c) is Correct ✓
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