Troubleshooting and fault finding: study guide

Worked examples and pitfalls

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.

How to practise this skill

• Write the question you are actually being asked at the top of your working, in your own words. Problem-solving items usually supply more information than the question needs, and the commonest failure is a correct calculation of something nobody asked for.
• Before any comparison of two options, ask where they cross. One quoted quantity never settles which is cheaper or faster, and the break-even is usually a single line of algebra away.
• State the worst-case number of steps before starting a search or a fault trace. If bisection applies, say so and say why; if the fault is not monotone or a test only works at the ends, say that instead and pick a different method.
• Estimate before you compute, then decide explicitly which regime you are in. Missing the target by an order of magnitude ends the question; landing within a factor of two means the estimate proved nothing and the exact numbers are now compulsory.
• Finish every allocation or capacity problem by listing how much of each constraint your answer consumes. The constraint you exhaust is the one that explains the answer, and the one you do not is where a per-unit heuristic will mislead you.
• Always verify a backwards solution by running it forwards. Attempts stay on this device, so the extra minute costs nothing and catches the inverted-fraction error that makes this family's most attractive wrong answer.

Glossary

Break-even point

The value at which two options cost or take the same. Below it one option wins and above it the other does, which is why a single quoted case never answers a comparison question.

Binding constraint

The limit that is fully consumed by the chosen solution. Identifying it explains the answer and reveals which per-unit ratios are relevant and which are noise.

Slack

The unused portion of a constraint at a proposed solution. Spare capacity in one resource is what lets a mix include items that look inefficient on the binding resource.

Bisection

Halving a search space with each test. It reduces a search of n candidates to about log base two of n tests, but only where the fault is monotone along the chain.

Working backwards

Solving from a known end state by inverting each operation in reverse order. The reliable method whenever the final value is given and every step is individually invertible.

Order-of-magnitude estimate

A rough calculation intended to settle feasibility rather than produce a figure. It is decisive when the answer is off by a factor of ten and useless when the margin is within a factor of two.

Root cause

The condition whose removal stops the fault recurring, as opposed to a symptom whose treatment suppresses it temporarily. A candidate cause that cannot be tested by changing it remains a hypothesis.

Monotone fault

A fault where everything on one side of a break behaves differently from everything on the other. The precondition that makes bisection valid, and the assumption most often left unstated.

Study this next

• Problem solving skill hub
• Problem solving practice bank
• Diagrammatic reasoning lesson
• Maintenance technician suite
• Supply chain planner suite
• Problem solving lesson on Novus Learn

Where this material comes from

• Every figure above was computed and checked for Novus Learn: the courier break-even at about 53.3 km, the six-test bisection bound for 40 candidates, the 4,400 starting budget verified forwards, and the four-X-two-Y loading enumerated across all feasible whole-pallet mixes.
• Terminology checked against the public Wikipedia articles 'Binary search algorithm', 'Break-even', 'Shadow price' and 'Root cause analysis'. Definitions only; every scenario above is original.
• Novus Learn aptitude construct registry (catalog seed) for the construct scope and related suite mapping.
• Public educational framing only: not affiliated with any official exam board, publisher or employer, and no copyrighted test item is reproduced.

Worked examples and pitfalls

Half-splitting beats swapping parts: A conveyor will not stop when its photo-eye is blocked. The chain is: sensor, field cable, junction box, PLC input card, PLC program, output card, interposing relay, contactor. Eight places the signal can die. Swapping parts one at a time means four or five attempts on average, since the culprit is equally likely to sit anywhere in the eight, and every attempt costs a part. Half-splitting starts in the middle instead: watch the PLC input LED while a colleague blocks the beam. If it toggles, the sensor, field cable, junction box and input card are all proven good in a single observation, and eight candidates become four. Force the output in the PLC and watch the contactor: if it pulls in, the output card, interposing relay and contactor are good as well, so the fault is in the program logic rather than the hardware. Three checks resolve eight stages, because each check halves the remaining suspects. The trap is starting at whichever end is easiest to reach, which resolves one stage per check instead of half of them. The second trap is the sentence 'I replaced the sensor and it still fails', that proves only that the sensor was not the fault, at the price of a part and an hour.

Tolerance: in spec, or scrap?: A drawing calls a shaft 25.00 mm with a tolerance of plus 0.05 and minus 0.10. That is an asymmetric tolerance, so the acceptance window runs from 24.90 mm to 25.05 mm and the nominal is not at its centre. A part measuring 24.92 mm is inside the window and passes, even though it is below nominal: the most common wrong rejection on this style of item, made by anyone who silently reads the tolerance as plus or minus 0.05. A part at 25.06 mm fails by 0.01 mm, but it fails oversize, so material can still be removed and it is rework rather than scrap. A part at 24.85 mm fails undersize and there is no recovering it. One more layer the better items include: if you took that 25.06 reading on a caliper with 0.02 mm resolution, the reading is at the very limit of what the instrument can resolve, and the honest next step is to re-measure with a micrometer before anyone scraps or reworks anything.

Reading a system diagram: what can actually cause this?: A tank fill line is drawn as supply, isolation valve V1, strainer, pump P1, check valve, control valve CV1, tank. A high-level switch LSH-1 is wired to close CV1. The reported symptom is that the tank overfilled. Work the path between the measurement and the element that stops flow: a CV1 that has stuck open, an LSH-1 that never actuated, and a broken wire in the LSH-1 loop are all consistent with the symptom. A blocked strainer is not: restricting the inlet reduces flow, and no amount of restriction causes an overfill. Nor is the check valve, whose function is to prevent reverse flow, not forward flow. Candidates pick the strainer because it is the component they know fouls in service, which is a memory of maintenance history rather than a reading of the diagram. The discipline that earns the mark is directional: a component can only be responsible if it lies on the causal path AND its failure mode pushes the system in the direction of the symptom.

Instrument choice: resolution is not accuracy: A steel rule resolves to roughly 0.5 mm. A vernier caliper marked 0.02 mm resolves to 0.02 mm. A 0 to 25 mm micrometer resolves 0.01 mm on the thimble, or 0.001 mm if it carries a vernier. A dial indicator reads 0.01 mm of relative movement but tells you nothing about absolute size without a reference. Asked to verify a 25.00 mm shaft with a tolerance of plus or minus 0.02 mm, the tolerance band is 0.04 mm wide: two divisions on that caliper, which is not enough to judge anything reliably. The workshop convention is that the instrument should resolve to about a tenth of the tolerance band, here 0.004 mm, so even the micrometer is marginal and comparison against gauge blocks is the defensible answer. The trap the item is built around is a digital display: showing four decimal places is a statement about resolution, not accuracy. An uncalibrated digital caliper will report 25.0000 mm with total confidence and be 0.03 mm out.

Closed loop: which element failed?: A room is meant to hold 21 degrees Celsius. A thermostat containing the sensor and the controller drives a valve on a radiator. The symptom: the room reaches 28 degrees Celsius and the valve stays open. Three explanations survive first inspection. The sensor reads low so the controller still believes the room is cold, the valve is mechanically jammed open, or the controller output has failed in the on state. One measurement separates them. Put an independent thermometer beside the thermostat. If the thermostat displays 17 degrees while the thermometer reads 28, the sensor is lying and everything downstream is behaving correctly. If the thermostat displays 28 and is still calling for heat, the sensor is fine and the fault is in the controller or the valve, which you then split by checking whether the valve actuator is being energised. The tempting non-answer is 'the room is too hot, so lower the setpoint'. That treats the symptom, and if the sensor reads seven degrees low the loop will simply settle seven degrees high again at the new setpoint.

Nameplates: read the qualifier, not just the number: A welding machine is rated 200 A at 40 percent duty cycle over a ten-minute period. That means four minutes of arc time and six minutes of cooling in every ten, at the full 200 A. It does not mean 40 percent of 200 A, and it does not mean 40 percent of an hour. Both wrong readings feel entirely natural, which is why they make good distractors. Run the machine continuously at 200 A and the thermal cut-out will open. The same discipline transfers across the whole trade: a hoist's working load limit is not its breaking load, a hose's working pressure is not its burst pressure, a motor's service factor describes a short-term overload allowance and not a continuous rating, and a pump curve's flow figure is quoted at a stated head. Whenever an item hands you a number in a table or on a plate, underline the qualifier printed next to it before you calculate anything; the wrong options are usually built by dropping exactly one qualifier.

How to practise this skill

• Trace every diagram with a pencil from input to output and name each block as you pass it. Technical items punish skimming far harder than they punish slow arithmetic.
• Rehearse half-splitting on systems you already know (a home network that has dropped out, a car that will not crank), and count the checks. The habit transfers to the test intact.
• Underline the qualifier beside every number a question supplies: per hour, at 20 degrees Celsius, at 40 percent duty, nominal, maximum. That single mark-up defuses most distractors.
• Convert the whole question to one unit system in one pass before calculating, rather than converting each intermediate result and accumulating rounding.
• Train yourself to state what a measurement PROVES rather than what it reads. '12 V present at the coil' proves the supply and the whole upstream path; it says nothing about whether the coil itself is good.
• Keep a running list of the schematic symbols and component names you personally keep getting wrong, and drill only those. Five focused minutes beats another full untargeted set, and your attempt history stays local to this device so the list is yours alone.

Glossary

Half-split fault finding

Testing at the midpoint of a chain of stages so that each measurement eliminates half the remaining suspects. A chain of eight stages resolves in about three checks rather than four swaps.

Tolerance band

The distance between the upper and lower acceptance limits of a dimension. A tolerance written as plus 0.05 and minus 0.10 has a 0.15 mm band that is not centred on the nominal.

Resolution

The smallest change an instrument can display. A four-decimal readout has fine resolution and may still be inaccurate if the instrument is out of calibration.

Accuracy

How close a reading is to the true value, established by calibration against a traceable standard. Independent of resolution, and the property that decides whether a part passes.

Interlock

A condition wired or programmed to inhibit an action until it is satisfied, such as a guard-door switch that prevents a motor start while the door is open.

Closed-loop control

A control arrangement where a sensor measures the process, a controller compares that measurement to a setpoint, and an actuator drives the process until the difference closes.

Duty cycle

The proportion of a stated period during which equipment may operate at a stated output, for example 40 percent of a ten-minute period, after which it must cool.

Schematic versus pictorial diagram

A schematic shows function and connection logic with no regard to physical layout; a pictorial or exploded view shows physical arrangement and assembly order but hides the logic.

Root cause

The condition whose removal stops a failure recurring, as opposed to the symptom, which is only what became visible.

Study this next

• Technical reasoning skill hub
• HVAC and refrigeration suite
• Instrumentation and control suite
• Engineering and technical design careers
• Encyclopedia search: troubleshooting
• Technical reasoning lesson on Novus Learn

Where this material comes from

• Diagnostic and metrology examples written for Novus Learn from general maintenance practice: binary-search fault isolation, asymmetric dimensional tolerance, and the ten-to-one instrument selection convention.
• Terminology checked against the public Wikipedia articles 'Troubleshooting', 'Engineering tolerance', 'Accuracy and precision' and 'Control loop'. Definitions only; every scenario above is original.
• Novus Learn aptitude construct registry (catalog seed) for the construct scope and related suite mapping.
• Public educational framing only: not affiliated with any official exam board, publisher or employer. Ratings and limits described here are illustrative and never override the equipment documentation in front of you.

Worked examples and pitfalls

One conditional, four moves, two of them wrong: Take a warehouse rule: if a pallet is flagged by the scanner, then it is inspected before dispatch. Write it as F implies I. Four things can now happen. You learn a pallet was flagged: F is true, so I is true: inspected. That is modus ponens and it is valid. You learn a pallet was not inspected: not-I, so not-F. It was not flagged. That is modus tollens, equally valid, and it is the move most candidates never reach for. Now the two traps. You learn a pallet WAS inspected and conclude it must have been flagged. Invalid: the rule says nothing about why else a pallet might be inspected: random audit, a customer complaint, a damaged corner. That is affirming the consequent. You learn a pallet was NOT flagged and conclude it was not inspected. Invalid for the same reason; that is denying the antecedent. Both feel right because in ordinary conversation people say if when they mean if and only if. Assessment items exploit exactly that slip, so the fix is mechanical: the instant you meet a conditional, write F implies I on one line and its contrapositive not-I implies not-F underneath. Those two lines are everything the statement gives you. Anything else on the page is a distractor.

Chaining conditionals, then running the chain backwards: Three statements from a release policy. If the build fails, the deploy is blocked. If the deploy is blocked, the release date slips. If the release date slips, the client is notified. Write them: B implies D, D implies S, S implies N. Chaining forwards is easy: a failed build guarantees a notified client. The item almost never asks that. It says: the client was not notified. What follows? Take contrapositives and chain them the other way: not-N implies not-S, not-S implies not-D, not-D implies not-B. So the build did not fail, the deploy was not blocked, and the date did not slip. All three follow. Now the near-miss version, and the one candidates get wrong: the client WAS notified. What follows? Nothing about the build. The chain only runs one way, and a client can be notified for reasons the three statements never mention. Answer: none of the above must be true. A useful habit for chain items is to draw the arrows on one line, B implies D implies S implies N, and remember that you can travel left-to-right when you are given a truth and right-to-left when you are given a falsehood, never the reverse.

The four phrases that flip the arrow: Most lost marks in conditional items come from translation, not from reasoning. Only if introduces the consequent: you may board only if you hold a boarding pass means board implies pass. It does NOT mean a pass gets you on the plane. The pass is necessary, not sufficient, and you still need the gate to be open and your name off the no-fly list. Unless is read as if not: unless you check in you cannot board is not-check-in implies not-board, whose contrapositive is board implies check-in, again the check-in is necessary. None but staff may enter is enter implies staff. No temporary badge grants server-room access is temporary badge implies not server-room access. Compare all four with a plain sufficient condition: swiping a manager card opens the door is card implies open, and here the card really is enough. The discipline is to ask one question of every clause. Is this thing the guarantee or the requirement? A guarantee sits at the tail of the arrow, a requirement sits at the head. Get that one decision right and the rest of the item is bookkeeping.

An ordering item, solved to a single arrangement: Five people present in five consecutive slots, one each: Aisha, Ben, Chen, Dana, Eli. Constraints: (1) Ben presents in the slot immediately before Dana. (2) Aisha is neither first nor last. (3) Chen presents at some point before Ben. (4) Eli is not immediately before or immediately after Dana. (5) Dana is not last. Start with the most restrictive item, the Ben-Dana block, and test each placement. Slots 1-2 is impossible: Chen must come before Ben and there is no slot before 1. Slots 4-5 is impossible because Dana would be last, breaking (5). Slots 3-4: Chen takes 1 or 2, and Aisha must take 2 because she cannot be first or last, so Chen is 1 and Eli is forced into 5, but 5 is immediately after Dana in slot 4, breaking (4). Dead. Slots 2-3: Chen must be 1. Aisha and Eli take 4 and 5, and since Aisha cannot be last, Aisha is 4 and Eli is 5. Check (4): Dana is in 3, the neighbouring slots are 2 and 4, and Eli is in 5: fine. The order is Chen, Ben, Dana, Aisha, Eli, and it is the only one. Two habits made that quick. First, place the block before the singletons; a two-slot block only has four possible positions in a five-slot line, so it does most of the elimination for you. Second, when a case dies, write down which constraint killed it. Later questions in the same set often ask what happens if one rule is dropped, and your dead-case notes answer them instantly.

A grouping item where the real answer is who is excluded: Exactly three of six people are picked: Farah, Gus, Hana, Ivo, Jo, Kit. Rules: (1) If Farah is picked, Gus is not. (2) Hana is picked only if Ivo is. (3) At least one of Gus and Jo is picked. (4) Kit and Ivo are never both picked. The question: if Hana is picked, which of the following must be true? Work it. Rule (2) is Hana implies Ivo, so Ivo is in. That is two of the three seats. Rule (4) then puts Kit out. The third seat goes to Farah, Gus or Jo. Suppose it went to Farah: then neither Gus nor Jo is picked, breaking (3). So the third seat is Gus or Jo, and both of those complete a legal team. Check Gus: rule (1) is vacuous because Farah is out, (2) holds, (3) holds, (4) holds. Check Jo: same. So the answer is not who joins Hana; that is genuinely open. The answer is that Farah is NOT picked, and it must be true in every legal arrangement. Grouping items are built around that distinction. Could be true means one arrangement exists; must be true means no counter-arrangement exists. The fastest way to kill a must-be-true option is to build a single legal team that violates it, which is why sketching two complete valid teams before reading the options is usually faster than reasoning about the options one at a time.

Some, all and none: the overlap nobody gave you: Two premises: all compliance officers have completed the ethics module, and some people who completed the ethics module work in the Leeds office. Does it follow that some compliance officers work in Leeds? No, and the diagram shows why. Draw a big circle for ethics-module completers. Compliance officers sit entirely inside it. Leeds staff overlap it somewhere, but nothing places that overlap on top of the compliance officers, so a world where every Leeds completer is a lab technician satisfies both premises and refutes the conclusion. Two overlapping particular statements never chain. Now three rules worth memorising. First, some means at least one and is silent about the rest, so the premise some invoices are overdue does not rule out all of them being overdue, though it does not establish that either. An option reading all invoices are overdue is therefore could-be-true rather than must-be-true, and a candidate who takes some to mean some but not all will wrongly mark it impossible. Second, some does convert: if some completers are Leeds staff then some Leeds staff are completers, and that is valid. All does not convert. All compliance officers are completers tells you nothing about most completers. Third, a valid chain needs a universal: all A are B plus no B are C gives no A are C, and that one is airtight. When an item mixes quantifiers, sketch three circles before you read a single answer option; the arrangement that satisfies the premises while breaking the conclusion is usually visible in about ten seconds.

How to practise this skill

• Notate before you reason. Every conditional gets two lines on your paper, the statement and its contrapositive, written in symbols. Items that look like paragraphs collapse to four or five arrows, and the arrows are what the question is actually about.
• Underline only if, unless, none but and no as you read. Those four phrasings cause more errors than every inference rule combined, and they are the cheapest thing on the page to get right.
• On ordering sets, draw a numbered row of slots and place the largest fixed block first. Keep a note of which constraint killed each dead case; follow-up questions in the same set reuse them.
• On must-be-true options, attack rather than verify. One complete legal arrangement that breaks the option kills it outright, and building one is usually faster than proving the option holds everywhere.
• Train the invalid moves deliberately. Write out an affirming-the-consequent item and a denying-the-antecedent item of your own each session until the shape of them is instantly recognisable under time pressure.
• Work untimed until you can state which rule licensed each step, then add the clock. Attempts are stored on this device only, so a slow, fully-narrated first pass through a constraint set costs nothing but your own time.

Glossary

Antecedent and consequent

In a conditional if P then Q, P is the antecedent and Q is the consequent. Getting the two the right way round during translation is where most conditional items are won or lost.

Contrapositive

For P implies Q, the statement not-Q implies not-P. It is always logically equivalent to the original, which is what lets you reason backwards from a denied consequent.

Modus ponens

The valid inference from P implies Q together with P to the conclusion Q. The most common inference in assessment items, and the safest.

Modus tollens

The valid inference from P implies Q together with not-Q to the conclusion not-P. Under-used by candidates, and the move most backwards-chaining items are built around.

Affirming the consequent

The invalid move from P implies Q together with Q to the conclusion P. It is tempting because ordinary speech often means if and only if when it says if.

Necessary condition

Something that must hold for an outcome to occur, but does not by itself bring it about. Signalled by only if, unless, none but, and required for.

Sufficient condition

Something whose presence guarantees the outcome. Signalled by a plain if, whenever, or any time that. A condition can be necessary, sufficient, both, or neither.

Must be true versus could be true

Must be true holds in every arrangement the constraints permit; could be true holds in at least one. Nearly every grouping and ordering answer option is one of these two, and mistaking which is being asked costs the mark.

Study this next

• Logical reasoning skill hub
• Logical reasoning practice bank
• Deductive reasoning lesson
• Programming logic suite
• Law, compliance and justice careers
• Logical reasoning lesson on Novus Learn

Where this material comes from

• Worked items written for Novus Learn from standard introductory logic: conditional translation, the contrapositive, categorical syllogisms, and linear ordering under constraints. Every puzzle above was solved and checked for a unique answer before publication.
• Terminology checked against the public Wikipedia articles 'Modus ponens', 'Modus tollens', 'Contraposition', 'Affirming the consequent' and 'Syllogism'. Definitions only; all items are original.
• Novus Learn aptitude construct registry (catalog seed) for the construct scope and related suite mapping.
• Public educational framing only: not affiliated with any official exam board, publisher or employer, and no copyrighted test item is reproduced.

Worked examples and pitfalls

Levers: balance the moments, then name the class: A wheelbarrow carries 60 kg whose centre of mass sits 0.4 m from the wheel axle, and you grip the handles 1.2 m from that same axle. Take moments about the axle: lift force x 1.2 m = 60 kgf x 0.4 m, so the lift force is 60 x 0.4 / 1.2 = 20 kgf, about 196 N. The mechanical advantage is simply the arm ratio, 1.2 / 0.4 = 3. Two traps live in this item. The first is measuring the load arm from your hands instead of from the fulcrum; the arm is always the perpendicular distance to the pivot, and the pivot here is the wheel contact, not the barrow body. The second is calling it a first-class lever because that is the one everyone pictures. A wheelbarrow is second class: fulcrum at one end, load in the middle, effort at the far end, which is why its mechanical advantage is always greater than one. Compare tweezers or a pair of tongs, where the effort sits between fulcrum and load: that is third class, mechanical advantage below one, and you are deliberately trading force away to buy speed and control at the tip. Your own forearm lifting a weight is the same arrangement, which is why a 5 kg dumbbell loads the biceps far more than 5 kg.

Gear trains: tooth counts set speed, meshes set direction: A 12-tooth driver turning at 300 rpm meshes directly with a 36-tooth gear. The ratio is driven teeth over driver teeth, 36/12 = 3:1, so the output turns at 300/3 = 100 rpm and, ignoring friction, carries about three times the torque. One external mesh reverses rotation, so the output turns opposite to the driver. Now drop a 20-tooth idler between them. Step it through: 300 x 12/20 = 180 rpm at the idler, then 180 x 20/36 = 100 rpm at the output. The overall ratio is unchanged at 3:1, the idler's tooth count cancels, but there are now two external meshes, two reversals, so the output turns the SAME way as the driver. That is the whole reason idlers are fitted. The tempting wrong answer treats the idler as another reduction stage and reports 180 rpm or some product of both ratios. The check that never fails: only the first and last gear in a simple train affect the ratio, and the direction depends on whether the number of external meshes is odd (reversed) or even (same). An internal or ring mesh, as in a planetary set, does not reverse at all.

Pulleys: count the rope parts that carry the load: A 200 kg load, about 1,962 N. Hung from a single pulley bolted to a beam, the pulley only changes the direction you pull; both rope parts still meet at a fixed axle, the mechanical advantage is 1, and you pull the full 200 kgf. Hang the pulley on the load instead, with one rope end anchored above and the other in your hands, and two rope parts now support the moving block: mechanical advantage 2, effort about 100 kgf or 981 N, but you must pull 2 m of rope for every 1 m the load rises. Build a tackle with four parts supporting the moving block and the effort falls to 200/4 = 50 kgf, about 490 N, at the cost of 4 m of rope pulled per metre of lift. The trap is counting sheaves instead of supporting parts. A three-sheave arrangement gives three parts if the dead end is made off to the fixed block and four if it is made off to the moving block, and the answer differs by 33 percent. The second trap is treating the figure as delivered force: real sheaves lose a few percent each to bearing and rope friction, so quoted mechanical advantage is the ideal velocity ratio, and the effort you actually feel is higher.

Hydraulics: pressure is shared, force and travel are traded: A jack has a 2 square centimetre input piston and a 50 square centimetre output ram. Push the small piston with 100 N and the pressure in the fluid is 100 / 2 = 50 N per square centimetre, which is 500 kPa. Pascal's principle says every part of the confined fluid sees that same 500 kPa, so the large ram feels 50 N/cm2 x 50 cm2 = 2,500 N. Mechanical advantage 25. Nothing is free: fluid is effectively incompressible, so volume in equals volume out. A 25 cm stroke on the small piston moves 2 x 25 = 50 cubic centimetres, and 50 cubic centimetres spread over a 50 square centimetre ram raises it 1 cm. Twenty-five centimetres of pumping buys one centimetre of lift: exactly the force-times-distance bargain a lever strikes. Candidates lose this item by assuming both pistons travel the same distance, or by concluding the jack manufactures energy. A related item asks about a tank: the pressure at the bottom depends on the height of fluid above and its density, not on the width of the vessel, so a narrow 3 m standpipe reads the same bottom pressure as a 3 m deep swimming pool.

Belts and chains: same belt speed, different rpm: A motor pulley 100 mm in diameter runs at 1,450 rpm and drives a 250 mm pulley through an open V-belt. The quantity that is genuinely shared is belt speed, not rpm, so the driven pulley turns at 1,450 x 100/250 = 580 rpm. Note the direction of the ratio: with gears you divide by driven teeth, with belts you divide by driven diameter, and the arithmetic looks the same only because both are proportional to circumference. An open belt drives both pulleys the same way; a crossed belt reverses the driven pulley, which is the entire point of the classic crossed-belt diagram item. The trap is slip. A flat or V-belt under an overload slips, so the driven speed falls below 580 rpm while the motor holds 1,450: the symptom of a glazed belt or a weak tensioner. A chain or a toothed timing belt physically cannot slip a tooth without damage, which is why camshaft and indexing drives use them and why a belt-driven answer and a chain-driven answer to the same question can legitimately differ.

Springs in series and parallel: the answer most people reverse: Two identical coil springs, each 20 N/mm. Mount them side by side, both carrying the same load, and they act in parallel: stiffness adds to 40 N/mm, so a 400 N load compresses the pair 400/40 = 10 mm. Now stack them end to end, in series. Each spring carries the whole 400 N, force is not split down a chain, so each deflects 400/20 = 20 mm and the total is 40 mm. The combined stiffness is 400/40 = 10 N/mm, half of one spring. Most candidates reverse the two because 'series' sounds like the case where things add. Two reliable checks: in parallel the arrangement is always stiffer than one spring, and in series it is always softer than the softest spring. That is also why suspension designers add a helper spring in parallel to raise rate under load, and why a long slender bolt clamps more forgivingly than a short stubby one carrying the same preload.

How to practise this skill

• Sketch the free body and mark the pivot before writing a single number. Mechanical items are lost by choosing the wrong fulcrum far more often than by mis-multiplying.
• Fix the three lever classes to a tool you can picture (pliers and a see-saw for first class, a wheelbarrow and a nutcracker for second, tweezers and your own forearm for third), and say the class out loud before answering.
• Sanity-check every answer against the work rule: if a machine multiplies force it must divide distance. An answer where the effort both moves less and is smaller than the load is wrong somewhere.
• Convert units before you multiply, not after. A moment computed with millimetres on one side and metres on the other is off by a thousand and looks perfectly plausible.
• For any direction question, walk the mesh or the belt with a finger: each external gear pair reverses, an internal or ring mesh does not, an open belt keeps direction and a crossed belt flips it.
• Work untimed until you can explain the reasoning aloud, then add the clock. Attempts are recorded on this device only, so a messy first pass through gear trains costs you nothing but time.

Glossary

Moment (torque)

A force multiplied by the perpendicular distance from its line of action to the pivot, in newton-metres. A body is in rotational balance when clockwise moments equal anticlockwise moments about the same point.

Fulcrum

The pivot a lever turns about. Its position relative to the load and the effort defines the lever class and therefore whether the tool multiplies force or multiplies movement.

Mechanical advantage

Output force divided by input force. Ideal mechanical advantage is the geometric figure; actual mechanical advantage is what you get after friction, and it is always lower.

Velocity ratio

Distance moved by the effort divided by distance moved by the load. In a lossless machine it equals the ideal mechanical advantage, which is why force gain always costs travel.

Gear ratio

Driven teeth divided by driver teeth. A ratio above one reduces speed and multiplies torque; only the first and last gear of a simple train affect it.

Idler gear

A gear placed between driver and driven purely to add a mesh. It reverses the output direction and bridges a distance without altering the overall ratio.

Pascal's principle

Pressure applied to a confined fluid is transmitted undiminished to every part of it, which is what lets a small piston at high pressure produce a large force on a bigger ram.

Head

The height of a fluid column above a point. Pressure at depth is density x gravity x height and is independent of the vessel's width or shape.

Study this next

• Mechanical comprehension skill hub
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• Encyclopedia search: simple machine
• Mechanical comprehension lesson on Novus Learn

Where this material comes from

• Worked examples written for Novus Learn from standard introductory mechanics: moments about a pivot, gear ratios from tooth counts, rope-part counting for tackle, and Pascal's principle.
• Terminology checked against the public Wikipedia articles 'Simple machine', 'Lever', 'Gear train' and 'Pascal's law'. Definitions only; every number and item above is original.
• Novus Learn aptitude construct registry (catalog seed) for the construct scope and related suite mapping.
• Public educational framing only: not affiliated with any official exam board, publisher or employer, and no copyrighted test item is reproduced.

Worked examples and pitfalls

Ohm's law on real components: A 12 V battery feeds a 4 ohm heater element. Current is 12/4 = 3 A, and power is 12 x 3 = 36 W. Put the same element on 24 V and current becomes 6 A while power becomes 24 x 6 = 144 W: doubling the voltage quadruples the power, because both factors in P = VI doubled. Candidates who answer 72 W have doubled one factor and forgotten the other. The nastier version of this item uses lamps. On the same 230 V supply, which has the higher resistance, a 60 W lamp or a 100 W one? Since P = V x V / R, resistance is V x V / P: the 60 W lamp is 52,900/60 = 882 ohms and the 100 W lamp is 52,900/100 = 529 ohms. The more powerful lamp has the LOWER resistance and draws more current, which feels backwards to anyone who reads 'more powerful' as 'more of everything'. Fix the direction of the relationship once: at a fixed voltage, more power means less resistance.

Series and parallel: two rules, two consequences: Three 6 ohm resistors across an 18 V supply. In series the resistances add to 18 ohms, the current is 18/18 = 1 A and it is the same at every point in the loop, and each resistor drops 1 A x 6 ohms = 6 V. The three drops sum to the supply, 6 + 6 + 6 = 18 V, which is the check you should run every time. In parallel each resistor sees the full 18 V, so each branch carries 18/6 = 3 A, the supply delivers 9 A, and the equivalent resistance is 18/9 = 2 ohms. The same 6/3 the reciprocal rule gives. The consequences matter more than the arithmetic. Adding a parallel branch always lowers total resistance and raises total current, which is exactly why plugging a fourth appliance into one socket circuit trips the breaker. Two checks catch nearly every slip: parallel resistance is always less than the smallest branch, and the branch with the smallest resistance always carries the biggest current.

Fault-finding by what stays lit: Six identical lamps in series on 12 V drop 2 V each. Break one filament and the current has nowhere to go: all six go out, which is the old fairy-light string that fails completely for one dead bulb. Finding the break is the counter-intuitive part. Across every good lamp your meter now reads 0 V, because no current is flowing to develop a drop, while across the broken one it reads the full 12 V. Candidates who expect the faulty component to read zero walk straight past it. Wire the same six in parallel instead and each lamp sees the full 12 V on its own branch, so a broken filament kills that lamp alone and the supply current merely drops by one sixth. That contrast is what makes a half-dead string diagnostic. If lamps 1 to 3 are lit and 4 to 6 are dead, they cannot be in series at all, because one break in a series loop kills every lamp; they are parallel branches hanging off a shared feed, and the fault is a break in that feed between position 3 and position 4. Probe the feed there rather than testing lamps: the last lit branch and the first dead one bracket the break.

Voltage dividers, and where each meter belongs: A 2 kilohm and a 3 kilohm resistor in series across 10 V. Total 5 kilohms, current 10/5,000 = 2 mA, which is the same in both. The 3 kilohm resistor drops 0.002 x 3,000 = 6 V and the 2 kilohm drops 4 V, splitting the supply in the ratio of the resistances. The bigger resistor always takes the bigger share. Now the meters. An ammeter goes IN the current path, in series with the component, and is designed with a near-zero resistance so it barely disturbs the circuit. A voltmeter goes ACROSS the component, in parallel, and has a very high resistance for the same reason. Swap them and the consequences are asymmetric: a voltmeter placed in series reads almost the whole supply and the circuit stops working, while an ammeter placed straight across a battery is a near-zero resistance across a source: a deliberate short circuit, a blown fuse at best. That asymmetry is the point of the item.

Why cable size is an I x I x R question: A 2.5 kW heater on 230 V draws 2,500/230 = 10.9 A. Suppose the supply cable has 0.5 ohms of total loop resistance. The voltage lost in the cable is 10.9 x 0.5 = 5.4 V, and the heat dissipated in the cable itself is I x I x R = 10.9 x 10.9 x 0.5 = about 59 W: quietly warming a coiled extension lead. Fit a thicker conductor so the loop resistance halves to 0.25 ohms and both the drop and the heat halve. Double the current instead, to 21.7 A, and the drop doubles but the heat goes up FOURFOLD, to about 236 W, because the current appears twice. Candidates who treat cable loss as proportional to current under-estimate it badly. This squared term is the reason a partly uncoiled drum overheats, the reason long runs need a larger conductor for the same load, and the reason transmission networks push power at high voltage and low current.

Transformers: turns ratio, and why they ignore DC: An ideal transformer has 400 primary turns and 40 secondary turns, a 10:1 ratio, and sits on 230 V AC. The secondary voltage is 230 x 40/400 = 23 V. Power in equals power out in the ideal case, so if the secondary supplies 5 A into a load, the secondary is delivering 23 x 5 = 115 W and the primary must draw 115/230 = 0.5 A. Voltage steps down by the turns ratio, current steps up by the same ratio, and no power is created. Two traps. The first is multiplying rather than dividing, giving 2,300 V from a step-down winding. Always ask which winding has more turns before you touch the arithmetic. The second is the DC version of the question: connect that primary to a 24 V battery and the secondary produces nothing once the initial switch-on transient passes, because a steady current produces a steady flux and only a CHANGING flux induces a secondary voltage. Meanwhile the primary, with only its winding resistance to limit it, draws enough current to burn out. That is also why the mains figure quoted as 230 V is an RMS value, and why the waveform actually peaks near 230 x 1.414 = 325 V.

How to practise this skill

• Write the three Ohm's law rearrangements and the three power forms across the top of your rough paper before question one. Most electrical items are a single substitution away once the right form is already in front of you.
• Normalise units first: put everything into volts, amps and ohms, or knowingly pair milliamps with kilohms (which conveniently gives volts). Mixing milliamps with plain ohms produces an answer that is wrong by a factor of a thousand and looks fine.
• Test every parallel result against the smallest branch. If your combined resistance is bigger than the smallest resistor in the group, you have added instead of combined.
• Practise fault items by predicting the meter reading rather than naming the culprit: 'zero volts across each good lamp, full supply across the break' is what you will actually see, and it is what the answer options are built from.
• Drill the schematic symbols in short bursts until they are instant: cell versus battery, fixed versus variable resistor, fuse, normally-open versus normally-closed contacts, lamp, ammeter, voltmeter. These items carry full marks and cost seconds.
• Keep safe-working questions in a separate mental bucket from circuit questions. Isolate, lock off, prove dead, then re-prove the tester is the expected answer even when a cleverer electrical shortcut exists, and attempts you log stay on this device either way.

Glossary

Ohm's law

V = I x R. Voltage across a resistive component equals the current through it multiplied by its resistance; rearranged as I = V/R and R = V/I.

Potential difference

The energy transferred per unit charge between two points, measured in volts and always measured ACROSS a component rather than through it.

Current

The rate of charge flow, in amperes. It is identical at every point of a series path and divides between parallel branches in inverse proportion to their resistances.

Resistance

Opposition to current, in ohms. Series resistances add directly; a parallel combination is always smaller than its smallest member.

Power

The rate of energy conversion in watts, obtainable three ways: P = V x I, P = I x I x R when you know the current, and P = V x V / R when you know the voltage.

Short circuit

An unintended low-resistance path around a load. Current rises until something limits it, which is precisely the job a fuse or circuit breaker exists to do.

Open circuit

A break in the conducting path. Current is zero and, in a live circuit, the full supply voltage appears across the break, the reading that locates it.

Turns ratio

Primary turns divided by secondary turns in a transformer. Voltage divides by this ratio while current multiplies by it, so the power either side is nominally equal.

RMS value

The steady DC value that would deliver the same heating as a given AC waveform. For a sine wave the peak is about 1.414 times the RMS value.

Study this next

• Electrical reasoning skill hub
• Electrician apprenticeship suite
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• Utilities, energy, mining and resources careers
• Encyclopedia search: Ohm's law
• Electrical reasoning lesson on Novus Learn

Where this material comes from

• Circuit examples computed for Novus Learn from first principles using Ohm's law, the series and parallel resistance rules, and the ideal transformer relation.
• Terminology checked against the public Wikipedia articles 'Ohm's law', 'Series and parallel circuits', 'Electric power' and 'Transformer'. Definitions only; every value above is original.
• Novus Learn aptitude construct registry (catalog seed) for the construct scope and related suite mapping.
• Public educational framing only: not affiliated with any official exam board, publisher or employer. Safe-working guidance here is illustrative and never replaces a competent person or your local wiring rules.