How does superintelligence work when an AI sees three impressive percentages—75% interested, 40% eligible and 25% attending—and must work out how many pupils actually arrived? Adding the figures gives 140%, which is meaningless here. Each percentage describes a smaller group within the preceding one. Out of 240 original invitations, only eighteen pupils attend, or 7.5% of all those invited.
The skill is sequential percentage reasoning. At each stage, the denominator changes from all invitees to those interested, then to those eligible. A trustworthy AI assistant keeps the stage definitions, exact counts and conditional proportions together. It multiplies nested percentages where appropriate and checks whether the remaining people are accounted for rather than applying every percentage to the same starting total.
This guide uses a fictional school workshop and a full four-stage register. It explains why a percentage of a percentage is not simply their sum, how a conversion funnel is reconciled, and how to distinguish eligibility from attendance. All numbers are invented teaching data, not actual student records, real reservations or measured AI performance. The series title “How Superintelligence Works” does not imply that currently deployed AI has achieved artificial superintelligence.
The complete answer at a glance
- 240 distinct invitations.
- 180 pupils expressed interest: 75% of invitees.
- 72 of those interested were eligible: 40% of the interested group.
- 18 of those eligible attended: 25% of eligible pupils.
- Overall attendance: 18/240 = 7.5%.
- Nonattending groups: 60 uninterested, 108 ineligible after interest, 54 eligible but absent.
- Full reconciliation: 60 + 108 + 54 + 18 = 240.
1. Understand the population before the percentage
The initial fictional population consists of 240 different pupils, not 240 emails that may have been sent repeatedly to fewer children. Its membership is assumed verified for the purpose of this exercise. Whenever a rate is described as a share of all invitees, 240 is the relevant denominator.
A real campaign could contain duplicate messages, changes of address or unknown recipients. An AI should identify whether it is counting invitations, unique individuals or responses before computing rates. This teaching fixture removes that ambiguity deliberately.
2. A stage is a subset of the previous stage
In a simple conversion funnel, each successive stage contains people from the stage before it. Here someone must first express interest to be assessed for eligibility, and must be eligible to attend. This arrangement makes the group sizes nested rather than independent.
A model cannot validly add the counts 240, 180, 72 and 18 to estimate different pupils, because the same people appear repeatedly as they advance. Each stage is a new status for members of the original population, not a wholly new group.
3. The first conversion rate is 75%
Out of 240 invited pupils, 180 express interest. Divide 180 by 240 to obtain 0.75, or seventy-five percent. This accurately describes the proportion of invitees who were interested.
It would be misleading to call seventy-five percent the attendance rate. Interest is only the first response and does not confirm eligibility, registration or participation. An assistant should preserve the name of the measured event.
4. Count the pupils without interest
The first stage excludes 240 − 180 = 60 pupils who did not express interest. The rate is 60/240 = 25% of all invitees. Together, 180 interested and sixty not interested reconcile to 240.
This is the complement of the initial 75% interest rate. A proper audit of a funnel looks at both those who advanced and those who did not. Otherwise some people may disappear from the report.
5. The second conversion has a new denominator
Of the 180 interested pupils, only 72 satisfy the fictional workshop’s eligibility conditions. The conditional eligibility rate is 72/180 = 0.40 = 40%.
The denominator is not 240, because the forty-percent statement is explicitly about interested pupils. An AI that calculates 40% of all invitees obtains 96 and changes the meaning of the second-stage rule.
6. Calculate eligibility among all invitees
Although 40% is measured within the interested group, the overall eligible share is 72/240 = 30%. This value answers a different but valid question: what percentage of everyone invited was eventually verified as eligible?
The same result follows by multiplying the first two rates: 0.75 × 0.40 = 0.30. The product works because the eligible group is nested inside those who expressed interest.
7. Count interested but ineligible pupils
Subtract eligible pupils from the interested group: 180 − 72 = 108. Those 108 expressed interest but did not qualify under the fictional rules.
They represent 108/180 = 60% of interested pupils and 108/240 = 45% of all invitees. Both proportions are correct but have different denominators. A report needs to name which group it uses.
8. The third percentage describes eligible pupils
Of the 72 eligible pupils, eighteen attend. Therefore the conditional attendance rate is 18/72 = 0.25, or 25% of the eligible group.
This does not mean that a quarter of the entire 240-person invitation list attended. That different calculation would produce sixty and contradict the recorded attendee count. The phrase of eligible pupils is part of the numerical instruction.
9. Calculate attendance from the first population
The actual fictional attendance count is eighteen out of the 240 initially invited. The overall share is 18/240 = 0.075 = 7.5%.
This is the right answer when the user asks what proportion of the initial group reached the final stage. It is a different metric from 25% of those eligible. Neither should be presented without its reference population.
10. Multiply the conditional rates
An alternative calculation is 0.75 × 0.40 × 0.25 = 0.075. Multiply by one hundred to express the final fraction as 7.5%.
Multiplication is appropriate because each conditional proportion selects part of the previous group. Adding the three percentages describes no valid overall funnel conversion in this case.
11. Reconcile the eligible nonattenders
Seventy-two are eligible, but only eighteen attend. The eligible nonattender count is 72 − 18 = 54 pupils.
Within the eligible group, the nonattendance share is 54/72 = 75%. Relative to the original 240, the same fifty-four pupils make up 22.5%. A percentage is inseparable from the base used for division.
12. Reconcile all four final status groups
The complete population splits into sixty uninterested, 108 interested but ineligible, fifty-four eligible but not attending, and eighteen attending. These categories are mutually exclusive under the fictional process.
Add them: 60 + 108 + 54 + 18 = 240. The equality proves the accounting is internally consistent. It does not tell us why particular pupils stopped or whether the programme was successful in an educational sense.
13. Count everyone who did not attend
Only eighteen out of 240 attended. Therefore 240 − 18 = 222 were nonattenders from the initial invitation group.
The same result follows from 60 + 108 + 54 = 222. Reaching the same number through two different calculations is a useful quality check before producing an AI-generated summary.
14. The checked numerical ledger
- Invited 240; interested 180.
- Interest rate: 180/240 = 75%.
- Uninterested count: 60.
- Eligible count: 72.
- Eligible among interested: 72/180 = 40%.
- Eligible among all invited: 72/240 = 30%.
- Interested but ineligible: 108.
- Attended: 18.
- Attended among eligible: 18/72 = 25%.
- Attended among all invited: 18/240 = 7.5%.
- Eligible nonattenders: 54.
- All nonattenders: 222.
- Disjoint final categories: 60 + 108 + 54 + 18 = 240.
15. Why 75 + 40 + 25 is not the answer
The arithmetic sum of those percentages is 140, but the three percentages use different denominators. A mathematical operation is meaningful only when the quantities being combined answer compatible questions.
An AI that simply adds visible percentages may produce a confident-looking result while ignoring the grammar that defines them. The correct overall calculation uses nested conditional proportions and their multiplication.
16. Why 25% of 240 is also the wrong answer
A quarter of 240 equals sixty. That is correct arithmetic if a question genuinely asks for twenty-five percent of all invitees.
But our third-stage statement says twenty-five percent of eligible pupils, and the eligible group contains only seventy-two. The exact result is eighteen. The denominator must come from the text, not the assistant’s preferred round number.
17. Learn to locate the word ‘of’
In word problems, the words following of often name the population used as the base. For example, forty percent of interested pupils means the eligible group is a share of the interested group.
A language model should not extract forty percent while dropping the phrase that determines its meaning. Accurate reading and numerical calculation are parts of the same task.
18. Nested counts should not be summed
The interested 180 already includes the eligible seventy-two, and the eligible seventy-two includes the attendees eighteen. Summing 180 + 72 + 18 = 270 counts some pupils several times.
That figure represents stage membership appearances rather than distinct people. The number of original people remains 240, and the count of final attendees remains eighteen.
19. Percentages can be expressed as fractions
Seventy-five percent is three quarters, forty percent is two fifths and twenty-five percent is one quarter. Their product is 3/4 × 2/5 × 1/4 = 6/80 = 3/40.
Three fortieths equals 0.075, or 7.5%. This fraction method independently confirms the decimal multiplication and gives pupils a useful way to check the result.
20. Eligibility does not establish attendance
A pupil can qualify under the workshop rules and still not arrive. Eligibility is a condition; attendance is an event.
An assistant should not translate a verified eligible record into a confirmed visit or booking. The fictional source records seventy-two eligible pupils but only eighteen attendees, showing why the distinction is essential.
21. Interest is not a verified booking
Expressing interest may be as simple as ticking a survey box. It does not necessarily reserve a seat, satisfy a privacy requirement or create a completed transaction.
Our exercise supplies no actual booking provider or confirmation receipts. The 180 interested pupils are a mathematical stage, not proof that 180 physical workshop places were booked.
22. Reasons for nonattendance remain unknown
The numbers say fifty-four eligible pupils did not attend. They do not reveal whether the pupils were unavailable, chose another activity or faced a transport problem.
A responsible AI can suggest what information might help investigate, but should not invent personal reasons or portray hypothetical causes as measured facts.
23. The funnel has no demonstrated learning outcome
The end point is recorded attendance, not academic improvement. Eighteen attendees do not by themselves establish the effectiveness of a lesson or whether the pupils learned the intended vocabulary.
An evaluation of educational quality requires appropriate assessments, participant context and a meaningful comparison. The arithmetic funnel cannot answer those questions alone.
24. What if the second-stage rate changes?
Suppose the 180 interested pupils remain the same but a future hypothetical process verifies ninety eligible rather than seventy-two. The eligible share among interested would be 90/180 = 50%.
This is a scenario, not a corrected historical record. Any attendance calculation must then specify a suitable new attendee count or rate. A model should label what-if results separately.
25. Why an expected count can be fractional
If ninety were eligible and a model assumed an independent 25% chance of attendance for each, the expected number attending would be 90 × 0.25 = 22.5.
An expected count can be fractional even though the observed number of pupils must be a whole number. The model should not call 22.5 an actual observed attendance without a defined rounding and interpretation.
26. Another hypothetical with whole counts
Imagine ninety eligible and thirty actual attendees among the original 240 invitations. The attendance rate among eligible would be 30/90 = one third, or about 33.3%.
The overall attendance conversion would be 30/240 = 12.5%. Both figures are valid for the new hypothetical case, but neither should replace the original eighteen-attendee record.
27. Compare the two overall rates
The original overall attendance share is 7.5%, while the separate thirty-attendee hypothetical produces 12.5%. The difference is five percentage points.
As a relative increase from the original 7.5%, it is 5/7.5 ≈ 66.7%. Percentage points and relative percentage increases describe the same movement with different denominators.
28. A larger first stage is not always better
A campaign may attract more initial interest while converting a smaller fraction of those interested into eligible applicants or attendees. The final outcome depends on every stage.
A useful AI summary should report the complete chain when evaluating the programme. Selecting only the most flattering percentage can make a source look stronger than the outcome it actually supports.
29. Check that stage definitions stay comparable
One programme may count invitations delivered, another invitations opened, and a third unique pupils contacted. Their first-stage denominators are not interchangeable.
Before comparing conversion rates, verify that each stage measures the same event over comparable periods. A higher number may reflect a different definition rather than better performance.
30. Twenty-four claims with reference labels
- 01. The starting group contains 240 distinct pupils — SUPPORTED.
- 02. Exactly 180 expressed interest — SUPPORTED.
- 03. The interest rate among invitees is 75% — SUPPORTED.
- 04. Sixty did not express interest — SUPPORTED.
- 05. Seventy-two interested pupils were eligible — SUPPORTED.
- 06. The eligibility rate among interested pupils is 40% — SUPPORTED.
- 07. The eligible share of all invitees is 40% — CONTRADICTED.
- 08. Exactly eighteen eligible pupils attended — SUPPORTED.
- 09. Attendance among eligible pupils is 25% — SUPPORTED.
- 10. Overall attendance conversion is 7.5% — SUPPORTED.
- 11. Adding 75%, 40% and 25% gives the overall rate — CONTRADICTED.
- 12. Multiplying the three stage rates gives the overall rate — SUPPORTED.
- 13. There were 108 interested but ineligible pupils — SUPPORTED.
- 14. There were fifty-four eligible nonattenders — SUPPORTED.
- 15. A total of 222 invitees did not attend — SUPPORTED.
- 16. The disjoint exit counts add to 240 — SUPPORTED.
- 17. Everyone interested attended — CONTRADICTED.
- 18. Everyone invited was eligible — CONTRADICTED.
- 19. The source gives each nonattender’s reasons — NOT ESTABLISHED.
- 20. The record proves every attendee improved academically — NOT ESTABLISHED.
- 21. The source verifies real bookings on an external service — NOT ESTABLISHED.
- 22. The eligible share of all invited is 30% — SUPPORTED.
- 23. The attendee count exceeds the eligible count — CONTRADICTED.
- 24. The interested share of all invited is 75% — SUPPORTED.
31. Count the answer categories
Supported are 01–06, 08–10, 12–16, 22 and 24: sixteen. Contradicted are 07, 11, 17, 18 and 23: five. Not established are 19, 20 and 21: three.
Sixteen plus five plus three equals twenty-four. The unknown statements are not errors in the arithmetic; they ask about real-world facts that the fabricated ledger never supplied.
32. A second practice example
One hundred worksheet questions are assigned. Half require an explanation, and twenty percent of those explanation questions require a diagram. Exactly 100 × 0.50 × 0.20 = 10 questions meet both conditions.
Ten percent of all questions require diagrams in this simple nested model. Adding 50 and 20 to obtain 70% would ignore the changing denominator.
33. A reading-club transfer exercise
An invented group has eighty eligible pupils. Sixty percent begin a reading challenge, and twenty-five percent of those who begin finish all four books. The final count is 80 × 0.60 × 0.25 = 12.
The finishing share relative to the full eighty-person group is 12/80 = 15%. The counts illustrate nested percentages, not actual learning outcomes or an observed programme.
34. Design a useful funnel table
A reliable spreadsheet shows stage name, entrants, continuing count, stage conversion rate and overall conversion rate. The two rate columns have different denominators even when displayed in the same row.
In our example the overall rates are 100% invited, 75% interested, 30% eligible and 7.5% attending. Keeping both columns prevents the final-stage 25% from being mistaken for an overall rate.
35. Charts need denominators too
A funnel chart displays narrowing bars, but it may not show whether percentages refer to the preceding stage or the original group. An AI interpreting a chart should inspect the labels instead of relying on bar widths.
A clear graphic could display 18 attended next to both 25% of eligible and 7.5% of invited. That pair explains the two legitimate interpretations without creating a contradiction.
36. The fastest arithmetic check
Take the final count of eighteen and divide by the initial population of 240. The result is 7.5%. Then multiply the individual stage rates and verify that the product is also 7.5%.
If those figures disagree, the assistant should inspect the stage definitions, missing cases and source counts. The two independent paths make the calculation auditable.
Extended audit — Confirm every denominator
A reviewer can rebuild the funnel one stage at a time without trusting the summary. First divide 180 by 240, then 72 by 180, then 18 by 72. Those three results must be 75%, 40% and 25% respectively. Each stage’s counted entrants become the next stage’s denominator.
The cumulative rates are different: interested is 180/240 = 75%, eligible is 72/240 = 30%, and attending is 18/240 = 7.5%. An AI-produced table should show conditional and cumulative rates in separate columns, because identical-looking percentages may answer different questions.
Extended audit — Check impossible stage growth
In a strictly nested funnel with no new entrants, a later stage cannot contain more distinct pupils than its preceding stage. Here 240 ≥ 180 ≥ 72 ≥ 18, so the sequence respects that invariant. If an AI reported ninety attendees from only seventy-two eligible people, the model would contradict the established inclusion rule.
This check does not prove that every count is correct; a wrong but decreasing sequence might still pass. It is an inexpensive consistency test that should be paired with source verification, distinct-person checks and reconciliation of the pupils who did not progress.
Extended audit — Preserve the time window
A conversion rate is tied to the period during which its stages were measured. If the invitations were sent in one week but the attendance count only included the following morning, late attendees could change the final rate. Mixing different reporting windows can mislead even when the fractions use their correct denominators.
Our fictional fixture treats the four counts as one complete closed process. A real report should disclose invitation dates, when interest was measured, the eligibility cutoff and the attendance event. A model cannot infer those missing timestamps merely because the numerical funnel is internally consistent.
37. The central lesson
Sequential percentages represent percentages of progressively smaller groups. The denominator changes at each stage, so the complete conversion is found by multiplying the relevant conditional proportions, not by adding them.
A trustworthy AI keeps the population, stage names, exact counts and missing facts visible. It can explain why eighteen is the final answer without pretending the toy source proves anyone’s motivation or learning.
Frequently asked questions
- Why don’t 75%, 40% and 25% add up to overall conversion? — They use different denominators.
- How many fictional pupils attended? — Eighteen.
- Why isn’t 25% of 240 the correct last-stage calculation? — Attendance is 25% of the seventy-two eligible pupils.
- What is the overall attendance rate? — Eighteen divided by 240, or 7.5%.
- Why does the multiplication work? — Each stage filters the preceding one.
- Does eligibility prove that a place was booked? — No.
- Can the funnel establish why people did not attend? — No, not without separate evidence.