Finance, accounting, banking, and insurance · suite apt-239-financial-analysis · generated 2026-09-15T21:49:44.786Z
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This file contains the whole study outline for this suite: every skill it draws on, the full lesson for each of those skills, worked examples, practice tips, a glossary, and where each piece of material comes from. Nothing here is a summary of a page you still have to visit.
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| Mode | Duration | What it is for |
|---|---|---|
| Guided practice | 15 minutes | Untimed, with feedback after every item. |
| Mini-test | 18 minutes | A short timed set for checking pace. |
| Full simulation | 45 minutes | Full length and full time, in one sitting. |
This suite draws on 5 skill constructs. Each one below carries its complete lesson.
Arithmetic, fractions, percentages, ratios, rates, estimation, word problems, and number relationships.
Numerical reasoning is the ability to take a small set of supplied numbers (a price list, a staffing table, a fuel figure), and reach a defensible answer in around ninety seconds without a formula sheet. It is the most widely used quantitative section in graduate, management and public-sector screening, and it appears in banking, retail, armed forces and healthcare entry batteries alike. The arithmetic itself is deliberately ordinary: percentages, ratios, rates and averages. What is actually being measured is whether you pick the right operation quickly, keep the units straight, and answer the question that was asked rather than the one you started computing. Practice here is device-local: no account, and nothing leaves this device unless you export it.
What you should be able to do after this lesson:
Reverse percentages: the item most candidates get backwards: A subscription price rose by 15 percent and now stands at 57.50 pounds. What was it before? The correct move is to divide, not subtract: the new price is 115 percent of the old, so the old price is 57.50 / 1.15 = 50.00. Check it forwards: 15 percent of 50 is 7.50, and 50 + 7.50 = 57.50. The tempting wrong answer takes 15 percent of the NEW figure, 0.15 x 57.50 = 8.625, and reports 48.88. That distractor is always on the option list because it is what most people do under time pressure, and it is wrong by 2.25 percent, small enough to look plausible. The same asymmetry drives the successive-change item: a price that rises 20 percent and then falls 20 percent does not return to where it started. Multiply the factors: 1.20 x 0.80 = 0.96, a net fall of 4 percent. Starting at 500 pounds you get 600 then 480, not 500. Percentages compose by multiplication; they never add.
Ratio splits: count the parts before you divide: A 4,830 pound budget is split between three teams in the ratio 3:5:6. Add the parts first: 3 + 5 + 6 = 14. One part is 4,830 / 14 = 345. The shares are 3 x 345 = 1,035, 5 x 345 = 1,725 and 6 x 345 = 2,070, and they sum back to 4,830, which is the check you should always run. Two classic failures. The first is dividing by 3 because there are three teams. That gives 1,610 each and ignores the ratio entirely. The second is treating 5 as a fraction and computing five sixths or five fourteenths of something other than the total. The more interesting version of this item gives you a difference instead of the total: 'the second team receives 690 pounds more than the first, what is the whole budget?' The difference between the shares is 5 - 3 = 2 parts, so one part is 690 / 2 = 345, and the budget is 14 x 345 = 4,830. Same number, reached from the other end, and the arithmetic is trivial once you have made the parts explicit.
Rates and units: 2 h 15 min is 2.25, not 2.15: A delivery van covers 174 km in 2 hours 15 minutes. Average speed is distance over time, and the time must be in hours: 15 minutes is 15/60 = 0.25 h, so 174 / 2.25 = 77.3 km/h. Type 2.15 into the calculator instead and you get 80.9 km/h: an error of roughly 4.7 percent that sits comfortably inside the plausible range and will match one of the options. Now extend it. The van consumes 8.6 litres per 100 km, so the trip needs 174 x 8.6 / 100 = 14.964 litres, and at 1.48 pounds per litre that is 14.964 x 1.48 = 22.15 pounds. Notice the units doing the work: (km) x (L / 100 km) leaves litres; (L) x (pounds / L) leaves pounds. If your intermediate line has km still attached at the end, you have divided when you should have multiplied. Write the unit next to every number and the method checks itself.
Unit pricing: normalise before you compare: Three pack sizes of the same fluid. Pack A: 750 ml for 3.60 pounds. Pack B: 2 litres for 9.40. Pack C: 1.5 litres for 7.20. Convert all three to price per litre. A: 3.60 / 0.75 = 4.80 per litre. B: 9.40 / 2 = 4.70. C: 7.20 / 1.5 = 4.80. So B is cheapest by 10p a litre, and A and C are identical despite looking like different deals. The 'bigger pack is cheaper' heuristic happens to hold here but is not a rule, and test writers know it. Half of these items are built specifically so the largest pack loses. Now add the multibuy that these questions love: pack A is on three-for-two. Three packs give 2.25 litres for the price of two, 7.20 pounds, which is 7.20 / 2.25 = 3.20 per litre and beats everything. The trap in the multibuy version is dividing by the number of packs paid for rather than the volume received.
Combined work rates: add rates, never times: Printer A completes a 3,000-page run in 50 minutes; printer B completes the same run in 75 minutes. Running together, how long? Convert to rates: A prints 3,000 / 50 = 60 pages per minute, B prints 3,000 / 75 = 40 pages per minute, and together they print 100 pages per minute, so the job takes 3,000 / 100 = 30 minutes. Verify by counting output: in 30 minutes A produces 1,800 pages and B produces 1,200, which is 3,000 exactly. The two wrong answers you will see on the option list are 62.5 minutes (the average of 50 and 75) and 125 minutes (the sum). Both are impossible on inspection: two machines working together must finish faster than the faster machine alone, so any answer above 50 minutes is wrong before you compute anything. The general form is 1 / (1/50 + 1/75), and it is worth being able to write that line directly, but the sanity bound catches the error faster than the algebra does.
Estimate first, then compute: killing distractors in ten seconds: 'A department spent 847,300 pounds in 2024. In 2025 the budget fell by 12.5 percent. What was the 2025 spend?' Options: 741,387.50 / 953,212.50 / 105,912.50 / 762,570. Estimate before anything else: 12.5 percent is exactly one eighth, one eighth of roughly 850,000 is roughly 106,000, so the answer is roughly 744,000. That single line eliminates three options. 953,212.50 applied the change upwards. 105,912.50 is the size of the reduction, not the resulting spend: the 'answered the wrong question' distractor, and the most commonly selected wrong option on items of this shape. 762,570 is a 10 percent cut, planted for anyone who misread the rate. Exact working: 847,300 x 7/8 = 5,931,100 / 8 = 741,387.50. Doing it as a fraction avoids the decimal multiplication altogether. Learn the fraction equivalents cold (12.5 percent is 1/8, 16.7 percent is 1/6, 37.5 percent is 3/8, 62.5 percent is 5/8) because a fraction turns most percentage items into one division.
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Evaluating workplace responses against role-relevant principles.
Learn a repeatable way to compare workplace responses: establish the facts, identify duties and risks, respect role boundaries, then choose a proportionate first action. This is educational preparation, not an official scoring guide.
What you should be able to do after this lesson:
Worked scenario: an unverified safety concern: A colleague reports a possible equipment fault while a deadline is approaching. First distinguish the known fact, the report, from the unverified cause. A strong response protects people and affected work, checks the concern through the right channel, tells the relevant lead, and records what was done. Ignoring the report underreacts; shutting down unrelated work or accusing someone before checking the facts overreacts.
Method: facts, duties, risks, response: Write four short notes before ranking options: what is known, who may be affected, which duty or boundary applies, and what safe next step is available now. Prefer an action that addresses the immediate issue and creates useful follow-through. Do not reward an option merely because it sounds decisive.
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Evaluating evidence, assumptions, arguments, credibility, and alternative explanations.
Critical thinking assessments ask you to take an argument apart: what is being concluded, what it rests on, what evidence would settle it, and whether a proposed conclusion actually follows. Published critical-thinking batteries typically run five task types - inference, recognition of assumptions, deduction, interpretation, and evaluation of arguments - and they are deliberately built so that agreeing with a conclusion and judging the argument as strong come apart. The construct is heavily weighted in policy analysis, audit, legal, investigative and graduate selection, and it is the same skill that stops a plausible chart from turning into a bad decision. Practice stays on this device; there is no account and nothing is uploaded unless you export it.
What you should be able to do after this lesson:
The negation test finds assumptions; nothing else does: Argument: 'The council should fit bin sensors across the district. In the trial depot they cut collection trips by a fifth.' Candidate assumption A: 'The trial depot's waste pattern is broadly representative of the rest of the district.' Negate it - suppose the trial depot is nothing like the rest of the district - and the argument collapses, because a result that does not transfer supports nothing about the district. So A is a required assumption. Candidate assumption B: 'Bin sensors are the cheapest available technology for this purpose.' Negate it - suppose they are the most expensive - and the argument still stands as given, since it argued from a reduction in trips, not from cost. B is not required; it would strengthen a cost-based argument that was never made. Candidate C: 'Fewer collection trips reduce total operating cost.' Negate it and the recommendation loses its point, so C is required too, even though the argument never mentions cost. That last case is the one candidates miss, because the required assumption is what bridges the evidence to the recommendation, and bridging assumptions are by definition unstated. The failure mode across this whole item type is picking statements that support the conclusion instead of statements the argument cannot survive without.
Base rates: 90 percent accurate, 15 percent right: A production line runs an automated defect test. Defects occur in 1 percent of units. The test flags 90 percent of genuinely defective units, and wrongly flags 5 percent of good units. A unit has just been flagged - how likely is it to be defective? Work in whole units rather than probabilities. Take 10,000 units: 100 are defective and 9,900 are good. Of the 100 defective, the test flags 90. Of the 9,900 good, it wrongly flags 5 percent, which is 495. Total flags: 90 + 495 = 585. Of those, 90 are genuine, so the answer is 90 / 585 = 15.4 percent. Better than the 1 percent base rate, and nowhere near the 90 percent that intuition offers. The error has a name - confusing the probability of a flag given a defect with the probability of a defect given a flag - and it has a practical consequence: the rework queue has to be sized for 585 units a batch, not 100, and 495 of the 585 units sitting in it are perfectly good. Whenever a test, screening rule or model accuracy figure appears in an item, ask how many of the negatives there are, because with a rare condition the false positives from a large clean population swamp the true positives from a small affected one.
Three alternative explanations, and the comparison that settles it: Claim: 'Employees who attended the optional resilience workshop took 30 percent fewer sick days last year, so the workshop works.' Alternative one, selection: attendance was optional, so the people who signed up may have been the healthier and more engaged to begin with, and would have taken fewer sick days regardless. Alternative two, reverse direction: employees who were frequently unwell were the least able to attend a full-day workshop, so illness determined attendance rather than the other way round. Alternative three, a common third factor: the workshop ran on Thursday afternoons at head office, so attendance largely tracks being an office-based rather than shift-based worker, and shift workers take more sick leave for reasons that have nothing to do with resilience training. What would settle it: randomly assigning the offer to half the workforce and comparing the two groups. Failing that, the cheapest useful check already exists in the payroll data - compare attendees' and non-attendees' sick days in the year before the workshop. If attendees were already 30 percent lower, selection is doing the entire job. The habit worth building is to name the comparison group before you accept any before-and-after number.
Deduction: affirming the consequent versus modus tollens: Rule: 'If an invoice is over 10,000 euro, it requires two signatures.' Argument A: 'This invoice has two signatures, therefore it is over 10,000 euro.' Invalid - this is affirming the consequent. The rule makes two signatures necessary for large invoices; it says nothing that prevents a cautious manager from double-signing a 400 euro invoice, so a two-signature invoice of any size is consistent with the rule. Argument B: 'This invoice has only one signature, therefore it is not over 10,000 euro.' Valid - this is modus tollens, denying the consequent to deny the antecedent, and it holds assuming the rule was followed. Argument C: 'This invoice is not over 10,000 euro, therefore it does not have two signatures.' Invalid again, denying the antecedent. Two further points that deduction items test directly. First, validity is about form alone: 'All banks close on Sundays; Riverton Mutual is a bank; therefore Riverton Mutual closes on Sundays' is perfectly valid even if the first premise is false, and a valid argument with a false premise can deliver a false conclusion. Second, in these items you must accept the premises as given even when you know them to be untrue, because the question is whether the conclusion follows, not whether it is true.
Interpretation: what a survey figure does and does not license: Data: all 1,200 employees of a firm were surveyed. 62 percent said they would accept a four-day week at 90 percent pay. Of those who said yes, 71 percent were under 35. Conclusion one: 'A majority of this firm's employees would accept the trade.' This follows - 62 percent of a complete census of the firm is a majority, and the conclusion is properly limited to this firm. Conclusion two: 'A majority of the firm's under-35 employees would accept the trade.' This does not follow, and the arithmetic shows why. Sixty-two percent of 1,200 is 744 accepters, and 71 percent of 744 is about 528 accepters aged under 35. That is the composition of the accepters, not the acceptance rate within the under-35 group, and without knowing how many under-35s the firm employs the rate is undetermined: if there are 900, the rate is 528 / 900 = 59 percent; if there are 600, it is 88 percent; if there were 1,100 the rate would be below half. Conclusion three: 'The firm should move to a four-day week.' This does not follow either - a stated preference about pay says nothing about output, shift cover, customer hours or cost. Notice that conclusion two is the base-rate error from the screening example, wearing survey clothing.
Strong and weak arguments on the same question: Question: should the agency publish raw inspection scores for every site? Strong argument for: 'In the neighbouring authority, publication was followed by a measurable fall in repeat violations, which is the stated aim of the inspection regime.' It is directly about this decision, it addresses the regime's own objective, and it offers evidence - and if true it would change your view. Weak argument for: 'Yes, because transparency is always good.' Sweeping, unsupported, and it would apply identically to publishing anything at all, which is the tell. Strong argument against: 'Raw scores omit the severity weighting, so a site with one critical failure can rank above a site with six minor ones, and a member of the public cannot see the difference.' Specific, mechanistic, directly about the proposal, and it would change your view. Weak argument against: 'No, because businesses will object.' It may be true and it may matter politically, but it does not bear on whether publication achieves the regime's aim, and no evidence is offered. Two tests separate strong from weak: is the argument about the exact question asked, and would accepting it change the decision? Note that you can hold a firm personal view on publication and still be required to mark the argument on the other side as the strong one - that split is what the item type is measuring.
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Sustained focus, selective attention, and accuracy under time pressure.
Attention and concentration is the skill of still noticing on minute forty of a checking block as reliably as you did on minute two, and of pointing the noticing at the right thing when two things compete. It has three distinguishable parts - selective attention, sustained attention or vigilance, and divided attention - and assessments load them differently: cancellation and checking tasks load vigilance, conflict tasks load selection, and dispatch-style monitoring loads division. It is the construct that decides whether a records clerk, a dispatcher, an air-side controller or a quality inspector catches the one wrong digit in a shift. Practice here is device-local, with no account and nothing uploaded.
What you should be able to do after this lesson:
A cancellation count you can actually check: Count every 7 in this row: 4 7 1 7 7 3 9 7 2 8 7 5 7 6 0 7. Working left to right and tapping once per hit: hits at the second, fourth, fifth, eighth, eleventh, thirteenth and sixteenth positions. That is seven sevens. Two error modes produce nearly all the wrong answers. The first is the adjacent pair - the 7 7 at positions four and five gets counted once, because the eye takes a repeated character as one perceptual object. The second is losing the place after the 9, where the visually similar 9 and 7 force a moment of re-checking and the scan restarts a character early, producing an over-count of eight. The defence for both is a fixed scan path with a physical anchor: a fingertip or cursor moving one character at a time, never jumping back. Re-scanning to check is the thing that creates the double-count, so if you must verify, verify by counting a second time from the RIGHT-hand end and comparing totals, never by re-reading part of the row.
Hits, misses and false alarms: the eager candidate loses: A checking batch contains 300 records, 40 of which are genuinely faulty. Candidate A flags 46 records and 36 of them are truly faulty. Hits 36, misses 40 minus 36 equals 4, false alarms 46 minus 36 equals 10. Candidate B flags 38 and 34 are truly faulty: hits 34, misses 6, false alarms 4. On hit rate alone A wins, 36 out of 40 which is 90 percent against B's 34 out of 40 which is 85 percent. Now apply the scoring rule that most checking tasks actually use, where a false alarm cancels a hit: A scores 36 minus 10 equals 26, B scores 34 minus 4 equals 30. B wins by four despite catching two fewer faults. This is why 'flag anything that looks odd' is bad advice on a scored checking task, and why the first thing to read on a checking item is whether wrong flags are penalised. Your response criterion - how much evidence you demand before flagging - is adjustable, and it should be set from the scoring rule, not from your temperament.
The transposition that passes every gist check: Compare these two lines and decide whether they match. Invoice 4820-7391-06. Invoice 4820-7931-06. They do not: the middle group reads 7391 in the first and 7931 in the second, with the 3 and the 9 swapped. Transpositions are the most-missed error class in record checking for a structural reason - the character SET is identical, the length is identical, the first and last characters of the group are identical, so every fast check the visual system runs comes back clean. Substitutions and omissions change the character inventory and get caught; transpositions do not. Two habits raise the catch rate. Read digits in fixed groups of two rather than as a whole number, so 73-91 against 79-31 becomes a mismatch at the first group instead of a subtle difference somewhere in a four-digit blur. And check groups in a deliberately non-natural order - last group, first group, middle group - because reading left to right lets the confirmation you built at the start carry you through the middle, which is exactly where the swap is usually planted.
Conflict: why reading fights you: The classic demonstration is Stroop's, published in 1935: the word RED printed in blue ink, with the instruction to name the ink colour. Naming takes measurably longer, and errors go up, because reading a familiar word is automatic and cannot be switched off, so the automatic response has to be suppressed before the controlled one can be produced. The same conflict has a numeric version you can test on yourself in a second: how many characters are in the string 4 4 4? The answer is three, and the digit 4 pulls at you the entire way. In an assessment this appears wherever the salient feature and the asked-for feature come apart - a chart where the tallest bar is not the answer to the question, a form where the highlighted field is not the one being verified, a row where the bold total is not what the stem requested. The practical move is to name the target feature out loud before you look - 'ink colour', 'character count', 'the value for March' - because pre-loading the target biases selection before the automatic reading response gets a chance to win.
Where the errors actually appear in a 45-minute block: Errors in a long checking block are not spread evenly. Mackworth's 1948 clock-watching study established the pattern that gives the effect its name: detection declines over a prolonged watch, with the sharpest deterioration early rather than at the very end. Practically, on a 45-minute self-timed checking block, expect your per-minute error rate in minutes 20 to 45 to run visibly above minutes 1 to 20 even though nothing about the material changed, and expect the subjective sense of effort to lag the actual decline, so it will not feel like you are getting worse. Two things work against it. Break the block into three fifteen-minute segments with a ten-second reset between them - look away, unfocus, breathe out - which costs thirty seconds of a 45-minute block, about one percent of the time, and buys back more than that in caught errors. And score your practice by segment rather than as one number, because a single overall accuracy figure hides exactly the information you need: whether your problem is skill, which shows as flat error rate, or endurance, which shows as a rising one.
Two streams: alternate on a cadence, do not try to merge: A dispatch-style monitoring task: keep a running total of the numbers announced on channel one while watching channel two for the code word AMBER. Genuine simultaneity is not available - the two tasks compete for the same control resource - so the choice is not whether to alternate but whether to alternate deliberately or accidentally. Deliberate looks like this: fix the arithmetic to a rhythm, updating the total only at each announcement and holding a single number between updates, which frees the gaps for channel two. If the total is 34 and the next announcement is 7, you spend under a second reaching 41 and then you are free again. Accidental looks like re-deriving the running total from the beginning because you did not trust it, which locks up the whole window and is when AMBER goes past unheard. The measurable failure of divided attention is almost never a failure to hear the target; it is a failure to have any spare capacity at the moment it arrived. The related phenomenon worth knowing is inattentional blindness, illustrated by Simons and Chabris in 1999: an unexpected and perfectly visible event is missed entirely when attention is committed to a counting task.
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Tables, charts, graphs, dashboards, trends, comparisons, and evidence-based conclusions.
Data interpretation is the discipline of getting a correct number out of a table, chart or dashboard that was not built to make your question easy, and of saying so when the data cannot answer it at all. It dominates graduate and analyst screening, and it is the section where strong arithmetic still fails, because the marks are lost in the header row, the axis scale and the wording of the question. The same skill is the daily work of anyone who reports on a management pack, a clinical audit or a stock ledger. Practice is stored on this device only; there is no account and nothing is uploaded unless you export it.
What you should be able to do after this lesson:
Read the header: (£000) changes every answer by a factor of a thousand: Table titled 'Regional revenue, year to March (£000)': North 1,240; South 986; East 1,455; West 719. Total = 1,240 + 986 + 1,455 + 719 = 4,400, so the business turned over 4,400 thousand pounds, that is 4.4 million. East's share is 1,455 / 4,400 = 33.1 percent. The whole set: North 28.2 percent, South 22.4, East 33.1, West 16.3, summing to 100. Two things go wrong here. The first is reading East's revenue as 1,455 pounds and then reporting a business with a total turnover of 4,400 pounds, which nobody notices because every option is scaled the same way, until the question asks for revenue in millions and only one option is right. The second is the comparison wording. East's share is (1,455 - 719) / 4,400 = 16.7 percentage points above West's, and East's revenue is (1,455 - 719) / 719 = 102 percent more than West's, that is slightly more than double. 'Sixteen point seven' and 'a hundred and two' both describe the same two cells honestly, and the question decides which one is correct. Note that the percentage-point figure must be computed from the unrounded shares rather than by subtracting the rounded ones, or the last digit will not survive.
One pair of rows, three correct increases: A complaints table: 2023, 120,000 orders, complaint rate 4.0 percent; 2024, 150,000 orders, complaint rate 5.0 percent. Three defensible answers to 'how much did complaints increase?'. The rate rose by 1.0 percentage point. The rate rose by (5.0 - 4.0) / 4.0 = 25 percent in relative terms. And the count of complaints rose from 0.04 x 120,000 = 4,800 to 0.05 x 150,000 = 7,500, which is (7,500 - 4,800) / 4,800 = 56.25 percent. All three are arithmetically right; only one answers the question in front of you. The pattern to internalise is that a rate and a count move together only when the denominator is fixed, and here it is not. Order volume grew 25 percent as well. If the question is about customer experience, the rate is the honest figure; if it is about how many complaint handlers to hire, the count is. Test items usually ask for the one you would not have chosen.
Index numbers: five points is not five percent: A cost index with 2020 = 100 reads 104 in 2021, 111 in 2022 and 109 in 2023. From 2021 to 2023 the index rose 5 points, but the percentage change is 5 / 104 = 4.8 percent, because the base for the comparison is 104, not 100. From 2022 to 2023 it fell 2 points, which is -2 / 111 = -1.8 percent, and note that costs fell even though the index remains 9 percent above the 2020 base. A level and a change are different claims. The only comparison where points and percent coincide is against the base year itself: 2020 to 2023 is 100 to 109, exactly plus 9 percent. Watch also for a rebased series, where a table switches to 2022 = 100 partway down; the two segments cannot be compared directly without converting one of them, and an item that quietly rebases is testing whether you read the column heading.
Joining two tables: totals and per-head figures disagree on purpose: Table 1, headcount by site: Leeds 84, Derby 47, Bristol 129. Table 2, absence days recorded in the same period: Leeds 630, Derby 300, Bristol 903. 'Which site has the worst absence problem?' On raw totals Bristol is worst at 903 days. Normalise per head and the ranking changes: Leeds 630 / 84 = 7.50 days per employee, Bristol 903 / 129 = 7.00, Derby 300 / 47 = 6.38. Leeds is worst, Bristol is merely biggest. The organisation-wide figure is 1,833 / 260 = 7.05 days per head, which is a useful reference line: Leeds is above it, the other two below. The general rule is that any comparison between units of different size demands a denominator, and the denominator has to come from the other table. Items are built so the raw-total answer and the per-head answer are both on the option list, and so the site with the biggest total is never the site with the highest rate.
The truncated axis: measure the numbers, not the bars: A quarterly satisfaction chart with a y-axis running from 78 to 82 shows bars at 79.2, 79.8, 80.4 and 81.1. Visually the last bar looks several times taller than the first, because only the top 4 points of a 100-point scale are drawn. The actual movement is 81.1 - 79.2 = 1.9 points, which on the score's own scale is a relative rise of 1.9 / 79.2 = 2.4 percent. If the question asks 'by approximately what percentage did satisfaction improve', the answer is about 2 percent, and the distractor built from the bar heights will be something like 40 or 400 percent. Related presentation effects to check before answering: a dual-axis chart where two series use different scales and appear to cross meaningfully when they do not; a cumulative series, where a flattening line still means the total is growing, just more slowly; and a logarithmic axis, where equal vertical distances are equal ratios rather than equal amounts. In every case the defence is the same. Find the printed numbers, or read the gridline values, and compute.
Cannot say: revenue is not profit: A product table shows units sold and total revenue. Product P: 4,200 units, 71,400 pounds. Product Q: 1,800 units, 41,400 pounds. Average selling price is 71,400 / 4,200 = 17.00 for P and 41,400 / 1,800 = 23.00 for Q, so Q earns more per unit while P earns more in total. Now the statement to evaluate: 'P is more profitable than Q.' The correct response is cannot say. Profit needs cost, and the table has no cost column; a product with a 17 pound price and a 16 pound unit cost is less profitable than one priced at 23 with a cost of 9, and nothing here rules that out. Contrast with 'Q generated more revenue per unit than P', which the table fully supports and which is true. The habit worth building is to finish every cannot-say judgement with the missing input named out loud ('cannot say, because unit cost is not given') because that forces you to distinguish a genuinely unanswerable item from one you simply have not worked hard enough on.
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