In an election between two candidates, the one who gets 88% of the votes is elected by a majority of 684 votes. What is the total number of votes polled? (all polled votes are valid)
- (a)850
- (b)900
- (c)800
- (d)750
Answer
Why
Correct — B. Two candidates and every polled vote valid, so the two shares add to 100%.
Loser's share = 100 − 88 = 12%
Majority = winner − loser = 88 − 12 = 76% of the total
So 76% of the total = 684
Total = 684 × 100 ÷ 76 = 900 → option (b)
Check: 88% of 900 = 792 and 12% of 900 = 108, and 792 − 108 = 684.
Why the others are wrong
- (a)850 — 76% of 850 is 646, not 684. A poll of 850 would split 748 to 102, a lead 38 votes short of the one the stem states.
- (c)800 — 76% of 800 is 608. Rounding the total to a convenient 800 breaks the arithmetic — that poll splits 704 to 96.
- (d)750 — 76% of 750 is 570, the furthest of the four from 684. A poll of 750 splits 660 to 90.
Concept
In a two-candidate contest with nothing else to absorb votes, the loser's percentage is 100 minus the winner's.
The majority is then the difference between the two shares, not the winner's own share: 88% against 12% is a margin of 76 percentage points.
Once you know what percentage the majority is, one division recovers the base. The only question left is which base — here it is the total polled, because the stem rules out invalid votes.
The bracket (all polled votes are valid) is doing real work. It collapses electorate, votes polled and valid votes into a single number, so 76% can be taken on the total directly.
Where a paper separates them, the margin percentage bites only on the valid votes, and the total has to be recovered through the turnout and invalid-vote steps afterwards.
17 Sep 2024, 12:30, Quant Q.22 is that harder version: 80% of voters cast a vote and 5% of those polled votes are invalid.
Key facts
- With two candidates and no invalid votes, the loser's share is 100% minus the winner's share.
- The majority is the difference of the two shares, so 88% against 12% is a majority of 76% of the total.
- 684 is 76% of 900, and a 900-vote poll splits 792 to 108.
Study next
Common traps
- Treating 684 as 88% of the total, which returns about 777 and matches no option.
- Measuring the majority against half the votes rather than against the loser's share.
- Reading the bracket as a warning about invalid votes and inventing a turnout step the stem has removed.
Across the rows below the majority idea stays the same, and what changes is how many layers sit between the electorate and the valid votes.
17 Sep 2024, 12:30, Quant Q.22 puts 80% turnout and 5% invalid votes ahead of a winner on 70% of valid votes with a majority of 36,480. 23 Sep 2024, 12:30, Quant Q.25 stacks 20% non-voters and 10% invalid votes ahead of a majority of 45,360.
A ratio-flavoured version at 13 Sep 2024, 09:00, Quant Q.12 gives R 110% of S's votes and a lead of 770.
Related PYQs
No directly related past PYQ was found.