Percentages Model 5 — The Election Model

Percentages · 20 min

Aptitude · Percentages · Model 5

The majority is the gap, not the winner's share

A candidate takes 57% of the votes and wins by 42,000. That 42,000 is not 57% of anything — it is the 14-point gap between winner and loser. Getting that one reading right is the entire model.

Set the winner’s share and the majority →
Two candidates: winner w%, loser (100 − w)%, and the majority is (2w − 100)% — the difference, not the winner's share.

01 The idea

Why the gap is double the lead

Two candidates contest an election. The winner takes 57% of the votes, so the loser takes 43%, and the winner’s majority — the margin of victory — is the difference between them: 57 − 43 = 14%. If that majority is 42,000 votes, then 14% is 42,000, so 1% is 3,000 and the total is 300,000.

The number to be careful with is 14, not 57 and not 7. Students reach for the winner’s 57%, or for the 7 percentage points above half. The majority counts the votes the winner has more than the loser, and every vote above 50% is worth two in that gap — one gained and one denied.

That doubling is worth stating as a rule: for two candidates, a winner on w% has a majority of (2w − 100)%. At 57% the majority is 14%; at 60% it is 20%; at 72% it is 44%. It is always twice the lead over half.

The harder variant adds invalid votes, and it adds a second base. If 10% of the votes cast are invalid, then the candidates’ percentages are shares of the valid votes while the 10% is a share of the votes cast. Two different denominators in one question, and mixing them is what the variant is set to catch.

Every vote above half counts twice in the majority — once gained, once denied. So the majority percentage is twice the winner’s lead over 50%.
MajorityThe margin of victory: winner’s votes minus loser’s votes. As a percentage of the valid votes it is 2w − 100.
Valid votesThe votes actually counted. The candidates’ percentages are shares of these, not of the total cast.
Votes castEverything including the invalid ones. If i% are invalid then valid votes are (100 − i)% of the votes cast — a different base.

02 Worked example

57% of the votes, a majority of 42,000

The standard form of the model. Two candidates contested an election. The winning candidate secured 57% of the total votes and won by a majority of 42,000. Find the total number of votes polled.

1
Give the loser the restWith two candidates the shares must total a hundred.winner 57%, loser 100 − 57 = 43%
2
The majority is the differenceThe margin is how many more votes the winner got, so subtract the two shares.57% − 43% = 14% of the total
3
Match it to the 42,000The stated majority is that 14%, not the winner’s 57%.14% → 42,000
4
Find one per centOne short division.1% = 42,000 / 14 = 3,000
5
Scale to a hundredThe total is a hundred of those.100% = 3,000 × 100 = 3,00,000 votes

Check it by counting both candidates: 57% of 300,000 is 171,000 and 43% is 129,000, and the difference is exactly 42,000. Note the trap answers this question generates: matching 42,000 to the winner’s 57% gives about 73,684, and matching it to the 7-point lead over half gives 600,000. Both are in the options, and only the 14% reading is right.

03 The method

The variants, including the second base

The two-candidate case is one reading. The last two rows are where a second denominator appears.

Two candidates, winner on w%: majority = (2w − 100)% of the valid votes. With i% invalid, valid = (100 − i)% of the votes cast.
With invalid votes, scale twice and keep the bases apart. The gap gives you the valid votes; the invalid percentage then gives you the votes cast. If 10% are invalid, the winner takes 70% of the valid and beats the loser by 1,800: the gap is 40% of the valid, so valid = 4,500, and votes cast = 4,500 × 100/90 = 5,000. Two unitary steps on two different bases, in that order.
Winner’s shareLoserMajority as % of valid
57%43%14%
60%40%20%
70%30%40%
72%28%44%
Three candidates—third = 100 − the other two
With i% invalid—valid = (100−i)% of cast
Majority = winner’s sharewrongit is the difference

05 Cheat sheet

Model 5 on one page

One reading, one relation, and the second base that the harder version adds.

CaseRouteWorked
Loser’s share100 − w57% → 43%
Majority %2w − 10057% → 14%
Total from the majoritymaj × 100/gap42,000 → 3,00,000
Winner’s votesw% of the total57% → 1,71,000
Third candidate100 − the other two40 + 36 → 24%
Valid from cast(100 − i)% of cast10% invalid → 90%
Majority = w%the standard trapgives 73,684 not 3,00,000
Every vote above half counts twiceOne gained by the winner and one denied to the loser. So the majority percentage is twice the lead over 50%, which is why 57% gives 14%.
Two candidates onlyThe 2w − 100 relation needs exactly two candidates. With three, work out each share separately — the third is 100 minus the other two.
Invalid votes are a second baseCandidate shares are of the VALID votes; the invalid percentage is of the votes CAST. Scale the gap first, then convert to votes cast.

06 Where & why

Where Model 5 shows up

A small, reliably set model whose entire difficulty is one reading.

SSC CGL · RRB · CPO
Winner’s share and a majority

The standard form. Both the correct total and the winner’s-share misreading are among the options every time.

Bank PO
Invalid votes added

The harder variant, and the two-base structure is what is being tested rather than the arithmetic.

Three-candidate versions
Two shares given, the third wanted

Simpler than it looks — the third share is a hundred minus the other two, and then it is one multiplication.

Any two-way split
Margin questions generally

Sales targets, poll leads, market share. Whenever a question gives a share and a margin, the margin is the doubled gap.

One sentence carries this model: the majority is the difference between the two candidates, so it is twice the winner’s lead over half. Everything else is a unitary step.

07 Interview questions

What gets asked

Nine, and the first two are the whole model.

A winner takes 57% of the votes and wins by 42,000. Find the total votes.
Three lakh. The loser took 43%, so the majority is 57 − 43 = 14% of the total. That 14% is 42,000, so 1% is 3,000 and the total is 3,00,000. Check: 1,71,000 against 1,29,000 is a gap of 42,000.
Why is the majority 14% rather than 57% or 7%?
Because the majority is the difference between the two candidates. The winner is 7 points above half, but each of those points is worth two in the gap — one vote gained and one denied — so the margin is 14%. That doubling is the model.
Give the general relation.
For two candidates with the winner on w%, the majority is (2w − 100)% of the valid votes. So 60% gives 20%, 70% gives 40% and 72% gives 44%. It needs exactly two candidates to hold.
A candidate gets 40% and loses by 298 votes. Find the total.
1,490. The winner took 60%, so the gap is 20%, and 20% is 298 votes. That makes 100% equal to 298 × 5 = 1,490. Note the loser’s share was given rather than the winner’s, which changes nothing about the method.
A candidate on 60% wins by 14,000. How many votes did the WINNER poll?
42,000. The gap is 20% and equals 14,000, so 1% is 700 and the total is 70,000. The winner took 60% of that, which is 42,000 — or note that 60% is three times the 20% gap, so it is 14,000 × 3 directly.
How do invalid votes change things?
They add a second base. The candidates’ percentages are of the valid votes, while the invalid percentage is of the votes cast. So scale the gap up to the valid votes first, then convert valid to cast by dividing by (100 − i)%.
Ten per cent of votes are invalid, the winner takes 70% of the valid and wins by 1,800. How many were cast?
Five thousand. The gap is 40% of the valid votes, so the valid votes are 1,800 × 100/40 = 4,500. Those are 90% of the votes cast, so the total cast is 4,500 × 100/90 = 5,000.
How do you handle three candidates?
The 2w − 100 relation no longer applies, so work with the shares directly. If the first got 40% and the second 36%, the third got 100 − 76 = 24%, and you apply that to the total. Majorities between specific pairs are just differences of their shares.
What is the fastest way to spot the trap in this model?
Ask whether the number you were given is a share or a margin. The words “majority”, “won by” and “defeated by” all mean a margin, and a margin is always the doubled gap rather than a candidate’s share.

08 Practice problems

Six elections

In each one, write down the gap percentage before touching the votes. Two of these add invalid votes.

The standard form

Easy
In an election between two candidates, the winner secured 65% of the votes and won by a majority of 45,000. Find the total number of votes polled.
Follow-up
The gap is not 65% and not 15%. Work out the loser’s share first and subtract, then check by counting both candidates.
Show the hint
The winner is 15 points above half — how much is that worth in the gap?

The loser’s share given

Easy
In an election, a candidate secured 45% of the votes and was defeated by a majority of 1,600 votes. Find the total number of votes.
Follow-up
The percentage given belongs to the loser this time. Nothing about the method changes, but reading it as the winner’s share gives a different answer.
Show the hint
If the loser has 45%, what does the winner have?

Find the winner’s votes

Medium
A candidate getting 60% of the votes is elected by a majority of 16,000 votes. How many votes did the winning candidate poll?
Follow-up
Two steps, and there is a one-step route: the winner’s share is a whole multiple of the gap, so you can scale directly without finding the total.
Show the hint
The gap is 20% and the winner has 60% — how many gaps is that?

Three candidates

Medium
In an election with three candidates, the first received 42% of the votes and the second 33%. If the total number of votes was 48,000, find how many votes the third candidate received and by how many votes the first beat the second.
Follow-up
The doubled-gap rule does not apply with three candidates, so work from the shares. The second part is a plain difference between two specific shares.
Show the hint
The third candidate’s share is what is left after the other two.

Invalid votes

Medium
In an election, 8% of the votes cast were declared invalid. The winner obtained 65% of the valid votes and won by 2,700 votes. Find the total number of votes cast.
Follow-up
Two bases, in a fixed order: the gap gives the valid votes, then the invalid share gives the votes cast. Applying the 8% to the wrong quantity is the intended error.
Show the hint
The gap is 30% of the VALID votes — find those first.

Derive the rule, and find its limit

Hard
(a) Show that for two candidates with the winner on w% of the valid votes, the majority is (2w − 100)% of those votes. (b) Hence find the smallest whole-number winning percentage that produces a majority of at least 30% of the votes. (c) A student applies the same 2w − 100 rule to a three-candidate election in which the leader has 45%, and reports a majority of −10%. Explain what has gone wrong and what the correct majority over the runner-up depends on.
Follow-up
Part (c) is the point: the rule silently assumes the two shares sum to a hundred, which is false with three candidates, and the negative answer is the formula reporting its own precondition failing. A formula that misbehaves exactly where its assumption breaks is one you can trust elsewhere.
Show the hint
For (a), write the loser’s share in terms of w and subtract. For (c), ask what the leader with 45% is actually being compared against.