Manufacturing, production, maintenance, and quality · suite apt-164-lean-and-continuous-improvement · generated 2026-09-15T15:26:20.427Z
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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.
Selecting methods, combining information, troubleshooting, and reaching practical solutions.
Problem solving is the construct that asks you to choose a method, not just execute one: to combine a table with a rule, decide whether an estimate settles the question or an exact calculation is needed, narrow a fault by halving the search space, and check the answer against the constraint that actually binds. It shows up across operations, logistics, manufacturing, technician and analyst selection, and in the situational sections of public-safety batteries. It is also the construct where the marking rewards a defensible route as much as a number. Everything you practise stays on this device unless you export it.
What you should be able to do after this lesson:
Two price structures and the distance where they cross: A delivery of 55 km. Courier A charges 8.00 base plus 0.45 per km. Courier B charges 20.00 flat for the first 40 km, then 0.90 per km beyond that. Which is cheaper? Courier A costs 8 plus 0.45 times 55, which is 8 plus 24.75, giving 32.75. Courier B costs 20 plus 0.90 times 15, which is 20 plus 13.50, giving 33.50. Courier A wins by 0.75. Now the question the item is really testing: is A always cheaper? At 45 km, A costs 8 plus 20.25 which is 28.25, while B costs 20 plus 4.50 which is 24.50. B wins comfortably. So the answer flips somewhere between, and finding where is one line of algebra. Above 40 km, B costs 20 plus 0.9 times distance minus 40, which simplifies to 0.9d minus 16. Set that equal to A's 8 plus 0.45d: 8 plus 0.45d equals 0.9d minus 16, so 24 equals 0.45d, so d is about 53.3 km. Below 53.3 km B is cheaper; above it A is. Check at the crossover: A is 8 plus 24 which is 32.00, and B is 48 minus 16 which is also 32.00. The general lesson is that whichever option has the lower per-unit rate always wins eventually, regardless of the base charges, and the base charges only decide where eventually starts. A single quoted distance never answers a which-is-cheaper question for a fleet.
Halving the search space beats walking the chain: Forty sensors sit on a single daisy-chained cable and exactly one connection is broken, cutting off everything downstream. How many tests do you need in the worst case? Walking the chain from one end and testing each sensor in turn takes up to 40 tests and 20 on average. Test the midpoint instead. If sensor 20 responds, the break is in the upper half and you have eliminated 20 candidates in one measurement; if it does not, the break is below and you have eliminated the other 20. Repeat on the surviving half: 40 becomes 20, then 10, then 5, then 3, then 2, then 1. Six tests, worst case, because each test halves what is left and two to the power six is 64, comfortably more than 40. The saving grows as the problem grows: 1,000 candidates need only ten tests. Two conditions have to hold for bisection to be valid, and stating them is part of the answer. The fault must be monotone, meaning everything on one side of the break behaves differently from everything on the other; and a single test at any point must tell you which side you are on. When there are two independent breaks, or when a test is only meaningful at the ends, bisection does not apply and a different strategy is needed. Candidates who reach for bisection reflexively on a problem that fails those conditions lose more than they save.
Working backwards from the number you were given: A team started a quarter with an unknown budget. In week one it spent half of it. In week two it spent 400 of what remained. In week three it spent a third of what remained after that. It finished with 1,200. How much did it start with? Attempting this forwards means carrying an unknown through three operations. Backwards it is arithmetic. After week three, 1,200 is what is left having spent a third, so 1,200 represents two thirds of the week-three opening balance, which was therefore 1,800. Before week two's spend of 400, the balance was 1,800 plus 400, which is 2,200. That 2,200 is what remained after spending half, so the starting budget was 4,400. Verify forwards, always: 4,400 less half is 2,200; less 400 is 1,800; less a third of 1,800, which is 600, leaves 1,200. Correct. The mechanical rule is to invert each operation and apply the inversions in reverse order. The inverse of spending a third is dividing by two thirds, not multiplying by three, and that specific slip is the most common wrong answer in this family. Working backwards is the right tool whenever the end state is known exactly and the operations are individually invertible, which covers most budget, mixture and journey problems phrased as how much did it start with.
When an estimate is enough, and when it is not: A maintenance window is three hours. Two technicians must service 1,850 units, each taking about 4.5 minutes, plus a shared 20-minute setup and 15-minute teardown. Feasible? Estimate first: 1,850 units at 4.5 minutes is 8,325 technician-minutes; split between two people that is about 4,163 minutes, which is roughly 69 hours. The window is 3 hours. The answer is no by a factor of more than twenty, and no refinement of the setup and teardown figures could change it. Precision here would be wasted effort. Now the same problem with 90 units. Ninety at 4.5 minutes is 405 technician-minutes, halved to 202.5 minutes, plus 35 minutes of setup and teardown, giving 237.5 minutes. That is 3 hours and 57 and a half minutes against a 3-hour window. Still no, but only just, and now every assumption matters: whether the technicians can genuinely work in parallel, whether setup is shared or duplicated, whether 4.5 minutes is a mean or a best case. The judgement being assessed is knowing which regime you are in. If your rough figure misses the target by an order of magnitude, stop and answer. If it lands within a factor of two, the estimate has not settled anything and you must do the exact arithmetic and name your assumptions.
The binding constraint decides, and it is rarely the obvious one: A van carries at most 900 kg and at most 6 cubic metres. Pallet type X weighs 150 kg, occupies 0.8 cubic metres and is worth 200. Pallet type Y weighs 90 kg, occupies 1.4 cubic metres and is worth 260. What mix maximises value? The instinctive heuristic is value per kilogram: X gives 200 over 150, about 1.33, while Y gives 260 over 90, about 2.89, so load Y. That is wrong, and the reason is that weight is not what runs out. Value per cubic metre tells the opposite story: X gives 250 and Y gives about 186. Enumerate the feasible whole-pallet mixes. Six X uses 900 kg and 4.8 cubic metres, worth 1,200. Five X and one Y uses 840 kg and 5.4 cubic metres, worth 1,260. Four X and two Y uses 780 kg and exactly 6.0 cubic metres, worth 1,320. Three X and three Y needs 6.6 cubic metres and does not fit. Two X and three Y uses 570 kg and 5.8 cubic metres, worth 1,180. Four Y alone uses 5.6 cubic metres and is worth 1,040. The best mix is four X and two Y at 1,320. Look at what binds it: volume is used to the last cubic metre while 120 kg of payload goes unused. Once volume is identified as the binding constraint, X's superior value per cubic metre explains the X-heavy answer, and the spare weight explains why two Y still get on board. The transferable move is to compute the usage of every constraint at your proposed answer and see which one is exhausted; a per-unit ratio computed against a non-binding resource is worse than no heuristic at all.
Cause, symptom, and the test that tells them apart: A pump trips out twice a shift. The obvious fix is to reset it, which works for about four hours. Ask why once and you find the thermal overload relay is tripping. Ask again and the motor is running hot. Again and the bearing is running hot. Again and the grease has dried out. Again and the lubrication interval was set for a duty cycle far lighter than the pump has actually been running since the line was rebalanced last year. Each level supports a different fix: resetting the trip costs nothing and lasts four hours; replacing the relay costs a part and lasts until the bearing seizes; regreasing lasts weeks; changing the lubrication schedule to match the real duty cycle is the one that stops the fault recurring. Stop asking why when the next answer stops being something you can act on, or when it leaves the boundary of the system you can change. There is one discipline that separates this from storytelling. A correlation is a lead, not a cause: the fact that the trips started after the line was rebalanced is suggestive, not proof. The confirming test is to intervene and observe: restore the original duty cycle, or apply the corrected greasing interval to this pump and not to the identical one on the parallel line, and see which one trips. If a proposed cause cannot be tested by changing it, treat it as a hypothesis and say so rather than closing the investigation.
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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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Resources, dependencies, deadlines, constraints, and contingency planning.
Learn to build a feasible schedule from dependencies, resource limits, deadlines, and uncertainty. A strong plan makes the critical sequence visible and includes triggers for replanning.
What you should be able to do after this lesson:
Worked scenario: a four-hour shutdown: Inspection takes 30 minutes before either repair can start. Two repairs take 90 and 120 minutes, but both need the same technician; testing takes 45 minutes after both finish. The minimum sequence is 30 + 90 + 120 + 45 = 285 minutes, longer than the shutdown. The correct response is to surface the conflict and change scope, staffing, or window, not to place overlapping bars on a chart and call the plan feasible.
Dependency and contingency check: Draw arrows between tasks, then mark scarce resources and approval gates. For the highest-risk dependency, define an early warning and a response: for example, if a permit is not approved by noon, defer nonessential work and notify the owner rather than discovering the conflict at start time.
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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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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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