Two teams contest a fictional school council election. One team receives more votes altogether, while the other team fills more council seats. A student points to the totals and says, ‘The result must be a counting mistake.’ But the votes and seats answer different questions, and a correctly counted election can produce both figures.
Vote share is the proportion of valid votes received by a team or candidate among the votes being counted. Seat share is the proportion of available council seats won under the specified selection rule. They use different denominators: votes for the first, seats for the second. A higher overall vote share does not always produce more seats when separate groups each choose their own delegate.
Did you know? This distinction is a compact lesson in Mathematics, civic reasoning, clear data presentation and reading the rules before calculating. We will use entirely fictional pupils, teams and constituencies—not an account of a Singapore school, an actual election or an endorsement of any voting system.
The question comes before the percentage
Before calculating, check the unit of counting. Is the table counting ballot papers, people who voted, valid votes for each candidate, constituencies, seats or a percentage of one of those? In a simple one-vote election those counts may be related, but they are not interchangeable. A class can be right about every number and wrong about what the numbers represent.
- Who was eligible, and who actually voted?
- Did each voter cast one vote or choose multiple names?
- Are blank or invalid ballots reported separately from valid candidate votes?
- How many seats are available, and how are they allocated?
- Are results decided school-wide or separately by class, house or constituency?
- Do the figures being compared cover the same election and reporting period?
If an election’s rules are unknown, do not infer its seat allocation from vote totals. The examples below specify rules explicitly so the arithmetic can be checked.
Worked example: three imaginary houses elect three delegates
Assume a made-up school council election has three independent constituencies, A, B and C. Each constituency elects exactly one delegate; the team receiving more valid votes in that constituency wins its seat. Every fictional voter casts one valid vote for Red or Blue, with no abstentions, invalid ballots or ties in this example. These are teaching assumptions, not real school policy.
- Constituency A: Red 45 votes, Blue 35 votes; total 80. Red wins the A seat.
- Constituency B: Red 29 votes, Blue 31 votes; total 60. Blue wins the B seat.
- Constituency C: Red 49 votes, Blue 51 votes; total 100. Blue wins the C seat.
Step 1: count valid votes school-wide
Red receives 45 + 29 + 49 = 123 votes. Blue receives 35 + 31 + 51 = 117 votes. There are 123 + 117 = 240 valid votes, which also matches the constituency totals 80 + 60 + 100 = 240. This cross-check is essential: both methods reach the same denominator.
Step 2: calculate vote shares
Red’s vote share is 123 ÷ 240 × 100 = 51.25%. Blue’s vote share is 117 ÷ 240 × 100 = 48.75%. The total is 100%. Red has more votes across the whole imaginary school: a lead of six votes, or 2.5 percentage points of the valid vote.
Step 3: count seats, not people
The stipulated rule gives the A seat to Red, and the B and C seats to Blue. Red wins 1 of 3 seats; Blue wins 2 of 3 seats. Red’s seat share is 1 ÷ 3 × 100 = about 33.33%. Blue’s seat share is 2 ÷ 3 × 100 = about 66.67%. Rounding means the two displayed seat shares may add to 100.00%, but the underlying fractions are exact.
Both statements are true: Red earned 51.25% of all valid votes and Blue won two of the three seats. The explanation is the rule that awards a whole seat within each constituency, rather than combining all school votes to divide the three seats proportionally.
Why aggregating ballots is not the same as awarding seats
Imagine one extra Red supporter voting in A. Under the specified rule, Red already wins A, so that extra vote changes the total vote count but does not change the number of seats. By contrast, a small change in B could switch the B seat if it reversed the local winner. The seat outcome depends on where votes occur as well as how many there are in total.
This is not a universal judgement that one allocation rule is better. Different councils might use a school-wide election, ranked choices, multi-member constituencies or proportional allocations. Each requires its own instructions. Our example teaches the effects of one stated rule, not a verdict on real electoral institutions.
A denominator audit that catches wrong answers
- Wrong: ‘Blue won 66.67% of the votes.’ Blue won approximately 66.67% of seats; its share of valid votes was 48.75%.
- Wrong: ‘Red won 123 seats.’ The number 123 counts valid votes for Red, not available positions.
- Wrong: ‘The school had 240 council members.’ The illustrative model has 240 valid ballots and three seats.
- Wrong: ‘51.25% minus 33.33% shows votes were lost.’ These percentages describe different populations; subtracting them is a comparison of shares, not a count of missing ballots.
- Wrong: ‘The winner of the total vote must win every constituency.’ A combined total does not determine each local comparison.
When students make any of these mistakes, ask them to fill in the phrase ‘___ out of ___’. For 51.25%, it is 123 votes out of 240 valid votes. For 66.67%, it is two seats out of three seats. The denominator tells you which question was answered.
What changes if the election is held under different rules?
Suppose the same imaginary school instead elects only one school-wide representative using the largest total valid vote. With the same 123 Red and 117 Blue votes, Red would win that one seat under that different rule. You cannot reuse the three-constituency seat result for a school-wide election.
Suppose there are three seats allocated proportionally rather than by one seat per constituency. You would need the actual rounding, threshold and allocation rules before computing an exact result. Merely multiplying 51.25% by three gives 1.5375 not an automatically valid seat count; people cannot normally occupy fractional seats. The rule for assigning whole seats matters.
Progressive practice: guide, share, release
Guided: compare two denominators
A fictional club gives two seats to Team A and one to Team B. What are the seat shares? A has 2/3, approximately 66.67%; B has 1/3, approximately 33.33%. This alone says nothing about how many votes either team received.
Shared: count valid votes
Another fictional ballot has 72 votes for Yellow, 48 for Green and no other valid votes. Yellow has 72 ÷ 120 × 100 = 60% vote share; Green 40%. Ask the learner to name what extra information is required to determine seats: the number of positions and the allocation rules.
Independent: use a fresh two-seat design
Two imaginary classes each elect one representative. In Class X, Orange gets 40 votes and Purple 20; in Class Y, Orange gets 19 and Purple 21. Overall, Orange has 59 of 100 votes, or 59%, yet each team wins one of the two seats. Seat shares are 50% each. The learner should show both totals and explain the difference without a hint.
Transfer: read a real published table responsibly
Find a trustworthy public report of a school club’s election or a clearly documented practice election. Identify votes, seats, district boundaries if any and the stated method. If those details are not available, report that limitation rather than applying the fictional model. Never invent absent vote totals or assume a real school’s rules match this article.
Diagnose learning, not political preference
For teachers, ask pupils to calculate two percentages, give the correct denominator for each and explain one possible reason the rankings differ. Score the reasoning, units and fidelity to the given rule. A student can disagree about whether a fictional system is desirable and still calculate the numbers correctly. Keep mathematical interpretation separate from personal preference.
Parents can invite the child to explain how the three seats are assigned by pointing at each constituency. Then hide the worked solution and present new numbers with a different number of valid voters. Genuine progress is visible when the child can work from the written rules and say why a total vote lead may not imply a seat lead.
What this example cannot prove
These invented figures do not establish a wider claim about the fairness, representativeness or consequences of any particular real election. They exclude turnout differences, invalid ballots, candidate-level rules, multi-member seats and several other complications. Real systems can contain safeguards or additional rules that change the result.
A school’s student council also has educational aims beyond a count: representation, participation, responsibility and respectful debate. The mathematical lesson is to label each measure accurately before using it in a broader discussion. No political party, candidate or real-world voting system is being recommended.
Frequently asked questions
Is vote share the same as turnout?
No. Vote share generally uses valid candidate votes as its denominator in this example. Turnout concerns how many eligible people participated. The two measures answer different questions.
Can one team win more votes but fewer seats?
Yes, as the checked three-constituency example demonstrates under the specified one-seat-per-constituency rule.
Should seat percentages always add to 100%?
When all available seats are allocated among the reported groups and shares use the same total, their exact fractions sum to 100%. Displayed rounded percentages may differ slightly from 100%.
Can we work out seat shares from total votes alone?
Not unless the allocation rules and necessary constituency details are also known. Avoid making up missing information.
Is a vote share of 51.25% a ‘51.25 percentage-point lead’?
No. In this two-team example, Red leads Blue by 51.25 − 48.75 = 2.5 percentage points, not 51.25 points.
Closing: keep the measure attached to its meaning
A thoughtful reader can say ‘more votes’ and ‘more seats’ without pretending those are the same thing. The core educational habit is to identify the counted object, read the allocation rule, calculate carefully and describe only the conclusion the evidence supports.
Related learning, verified on eduKateSG
Core Aim: distinguish a voting majority from the largest count
Core Aim: compare totals with per-person measures without confusing their denominators
Clementi Mathematics learning reference: diagnosis, explanation and transfer