In an election between two candidates, y% of the voters did not vote. 10% of the votes cast were declared invalid, while all the valid votes were cast in favour of either of the two candidates. The candidate who got 59.375% of the valid votes cast was declared elected by 2484 votes. If the number of people eligible to vote in that election was 16,000, what is the value of y?

Correct — A. Work backwards from the margin, because each percentage sits on a different base.

Winner 59.375%, loser 40.625%, so the margin is 18.75% of the valid votes.
That margin is 2484 votes: valid = 2484 ⁄ 0.1875 = 13,248

Valid votes are 90% of the votes cast, since 10% were invalid.
Votes cast = 13,248 ⁄ 0.9 = 14,720

Non-voters = 16,000 − 14,720 = 1,280
y = (1,280 ⁄ 16,000) × 100 = 8 → option (a)

Why the others are wrong

• (b)
  8.4 — y = 8.4 leaves 14,656 votes cast and 13,190.4 valid — not a whole number of votes, and the 18.75% margin would be 2,473, short of the stated 2,484.
• (c)
  7.5 — y = 7.5 gives 14,800 cast and 13,320 valid, so the 18.75% margin comes to 2,497.5 — above the stated 2,484, and not a whole number of votes.
• (d)
  7.2 — y = 7.2 gives 14,848 cast and 13,363.2 valid, a margin of 2,505.6 — the further y drops below 8, the larger the margin grows past 2,484.

Three percentages act on three different bases, and mixing them is the whole difficulty.

y% is taken on the 16,000 eligible voters. The 10% invalid is taken on the votes cast. The 59.375% share is taken on the valid votes.

In a straight two-candidate contest with every valid vote going to one of them, a winner's share p turns into a margin of (2p − 100)% of the valid votes — here 2(59.375) − 100 = 18.75%.

Key facts

• In a two-candidate contest, the margin is (2p − 100)% of the valid votes, where p is the winner's share.
• 18.75% = 3⁄16, so valid votes = 2484 × 16 ⁄ 3 = 13,248.
• Invalid votes are counted on votes cast, not on the electorate: cast = 13,248 ⁄ 0.9 = 14,720.
• Turnout here is 14,720 of 16,000, which is 92% — so y = 8.

Common traps

• Taking the 10% invalid votes off the 16,000 eligible voters instead of off the votes cast
• Using 59.375% − 50% = 9.375% as the margin instead of the full gap between the two candidates
• Reading 59.375% as a share of the votes cast rather than of the valid votes

No directly related past PYQ was found.