In an election between two candidates, P received 42% of the valid votes and Q won by 45,360 votes. 20% people did not cast their vote. If 10% of the votes cast were found invalid, what is the total number of votes registered in the poll booth?
- (a)3,59,000
- (b)3,53,790
- (c)3,93,750
- (d)3,93,600
Answer
Why
Correct — C. Let T be the registered voters and apply the filters in order.
20% did not vote, so votes cast = 0.80 T
10% of those cast were invalid, so valid = 0.90 × 0.80 T = 0.72 T
P took 42% of the valid votes, so Q took 58%:
margin = 58% − 42% = 16% of the valid votes
Set that equal to 45,360:
0.16 × 0.72 T = 0.1152 T = 45,360
T = 45,360 ⁄ 0.1152 = 3,93,750 → option (c)
Why the others are wrong
- (a)3,59,000 — 3,59,000 gives 2,58,480 valid votes, and 16% of that is 41,356.8 — not a whole number of votes, and well short of the 45,360 margin stated.
- (b)3,53,790 — 3,53,790 does not even survive the first filter cleanly: 72% of it is 2,54,728.8, so the valid-vote count would be fractional.
- (d)3,93,600 — 3,93,600 is 150 short of the true total and is the near-miss decoy. Its valid pool is 2,83,392, and a 16-point margin on that is 45,342.72, not 45,360.
Concept
This is a chain of successive percentage filters, and each one applies to the pool immediately above it, not to the original total.
Registered → cast is × 0.80, because 20% stayed home. Cast → valid is × 0.90, because 10% of the votes cast were rejected. So valid = 0.72 T, not 0.70 T.
Then the margin. With P on 42% of valid votes, Q holds the other 58%, and the winning margin is the difference, 16 points — never the winner's own share. One multiplier carries the whole question: 0.16 × 0.72 = 0.1152.
Working forwards from the answer confirms it: 3,93,750 registered gives 3,15,000 cast and 2,83,500 valid, split 1,19,070 to P and 1,64,430 to Q, a margin of exactly 45,360.
Key facts
- Votes cast are 80% of those registered, and valid votes are 90% of those cast, so valid = 72% of registered.
- With P on 42% of valid votes, Q is on 58% and the margin is 16% of the valid votes.
- 16% of 72% is 11.52%, so the margin is 0.1152 times the registered total.
- 45,360 ⁄ 0.1152 = 3,93,750 registered voters.
Study next
Common traps
- Taking the 10% invalid off the registered total instead of off the votes cast, which gives 70% valid.
- Using the winner's 58% as the margin instead of the 58 − 42 = 16 point difference.
- Answering with the valid votes, 2,83,500, when the question asks for the registered total.
The election frame lets SSC stack turnout, invalidity and vote share into one chain.
9 Sep 2024, 09:00, Quant Q.12 builds the same three-layer filter with the turnout itself unknown at y%.
11 Sep 2024, 09:00, Quant Q.13 gives the electorate and asks for the loser's valid votes.
18 Sep 2024, 12:30, Quant Q.17 strips the filters off entirely — all polled votes valid, a majority of 684 — so the same setup runs anywhere from one step to four.
Related PYQs
No directly related past PYQ was found.