Programming logic: study guide

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

A matrix where the rule is arithmetic on side counts: A three-by-three grid of polygons. The top row runs triangle, square, pentagon. The middle row runs square, pentagon, hexagon. The bottom row runs pentagon, hexagon, and an empty cell. Count sides rather than naming shapes and the structure appears immediately: 3, 4, 5 across the top; 4, 5, 6 across the middle; 5, 6, and the answer across the bottom. Sides increase by one along each row AND by one down each column, so the missing figure has seven sides: a heptagon. Writing the numbers into the cells is the whole technique, and it generalises: whenever a matrix contains countable things, replace every cell with its count before you look for anything else, because a numeric grid makes a progression visible that shapes hide. The distractor set on an item like this is instructive. It will contain a hexagon (correct rule, one step short), an octagon (right idea, overshot), a heptagon with the wrong shading, and a heptagon rotated. Three of those four are only wrong on a second attribute, which is why the answer must be checked on every attribute before you commit rather than on the one that solved the puzzle.

Union, intersection and exclusive-or look identical until you test them: Many matrices build the third cell of each row by combining the first two. Take a row where cell one contains a dot in the top-left corner and a cross in the centre, and cell two contains a cross in the centre and a dot in the bottom-right corner. Cell three contains a dot in the top-left and a dot in the bottom-right, with no cross. That is exclusive-or superposition: elements present in exactly one of the two inputs survive, and elements present in both cancel. If cell three had contained both dots AND the cross, the rule would be union. Everything from both inputs is kept. If it had contained only the cross, the rule would be intersection: only what appears in both survives. All three rules produce plausible-looking figures, so guessing from one row is unreliable. The reliable procedure is to identify a row where the two inputs share at least one element and differ in at least one other, because that is the only configuration where union, intersection and exclusive-or give three different answers. Confirm the rule there, then apply it to the row with the missing cell. Items in this family also hide a fourth variant, where shared elements survive but change colour; catching that one requires checking shading as a separate attribute rather than treating a black cross and a white cross as the same object.

Three attributes on three different clocks: A six-frame sequence. Frame 1: an arrow pointing up, unshaded, with one internal bar. Frame 2: arrow pointing right, shaded, two bars. Frame 3: arrow pointing down, unshaded, three bars. Frame 4: arrow pointing left, shaded, one bar. Frame 5: arrow pointing up, unshaded, two bars. What is frame 6? Take the attributes one at a time. Direction rotates ninety degrees clockwise every frame, a cycle of four, so after up comes right. Shading alternates, a cycle of two, so after unshaded comes shaded. The bar count runs 1, 2, 3, 1, 2, a cycle of three, so after two comes three. Frame 6 is a right-pointing shaded arrow with three bars. The reason this item defeats people is that the whole figure does not repeat until frame 13: the individual attributes come back round every four, two and three frames, but they only come back into phase together after twelve, so the sequence as printed looks like it has no period at all. Treat each attribute as an independent counter with its own cycle length and the difficulty evaporates. The practical habit is to rule three columns on your paper (direction, shading, count) fill them in for every given frame, and extend each column separately before you look at a single answer option.

Odd one out, where the obvious difference is the decoy: Five figures. (a) A square with one diagonal drawn. (b) A triangle with a line from the apex to the base. (c) A pentagon with a line joining two non-adjacent vertices. (d) A hexagon with two lines crossing inside it. (e) A circle with a chord. Which is the odd one out? The difference that jumps out is that (e) is curved and the others are straight-edged, and that is the decoy. It is a real difference, but it is not the one the item is built on. Count enclosed regions instead. Figures (a), (b), (c) and (e) are each divided into exactly two regions by a single line. Figure (d) has two crossing lines and is divided into more than two. The rule is region count, and (d) is the answer. Odd-one-out items reliably plant a salient irrelevant difference (one figure is the only curved one, or the only shaded one, or the only one with a right angle), and hide the operative rule in something countable. The defence is procedural: before choosing, write the value of at least three attributes for all five figures, for example number of enclosed regions, number of straight edges, number of line intersections. The correct answer is the figure that stands alone on exactly one row of that table while the others agree, and if two rows both isolate a figure, the item is ambiguous and you should prefer the countable rule over the categorical one.

Rotation or reflection, settled by clockwise order: A flag-like figure carries three distinguishable marks: a red band, a blue band and a yellow band, which read red, blue, yellow going clockwise around its centre. You are shown a candidate answer figure that also has three bands. Is it a rotation of the original or a reflection of it? Rotating a flat figure in the plane never changes the clockwise order of its features, so any genuine rotation still reads red, blue, yellow clockwise, from whatever starting point you choose. A mirror reflection reverses that order, so a reflected figure reads red, yellow, blue clockwise. Read the cyclic order and the question is answered without imagining any motion at all. This matters because reflections are the standard trap in rotation items: the reflected option looks exactly as plausible as the rotated one, and no amount of mental turning will produce it. The same test works on letters, no rotation of R in the plane produces a backwards R, and on three-dimensional figures, where the equivalent check is whether a right-handed set of three edges at a corner stays right-handed. Note the one honest caveat: a figure with a mirror line of its own is unchanged by reflection, so this test only decides the question for asymmetric figures, which is exactly why item writers use asymmetric ones.

Two dots on the perimeter, moving at different rates: A three-by-three grid where only the eight perimeter cells are used. Number them clockwise starting at the top-left corner: 0 top-left, 1 top-middle, 2 top-right, 3 right-middle, 4 bottom-right, 5 bottom-middle, 6 bottom-left, 7 left-middle. Frame 1 has both dots at cell 0. Frame 2 has one dot at cell 2 and the other at cell 7. Frame 3 has them at cells 4 and 6. Frame 4 has them at cells 6 and 5. What is frame 5? Track them separately. The first dot goes 0, 2, 4, 6, two cells clockwise each frame, so it lands on 8, which wraps to cell 0, the top-left corner. The second goes 0, 7, 6, 5, one cell anticlockwise each frame, so it lands on cell 4, the bottom-right corner. Frame 5 has one dot top-left and one bottom-right. Two counting errors dominate this family. The first is treating the perimeter as nine positions because the grid has nine cells; the centre is not on the path and the perimeter is a loop of eight. The second is assuming both markers share a direction or a speed, which is precisely the assumption the item is testing. When markers move at different rates, the frames where they coincide or sit adjacent are coincidences of the arithmetic, not part of the rule, and reading meaning into them is how a solvable item becomes impossible.

How to practise this skill

• Keep a written attribute checklist beside you until it is automatic: count, shape, size, shading, orientation, position, line style, symmetry. Scanning a fixed list is faster than free-associating, and it is what separates candidates who finish the section from those who stall on item four.
• Replace every countable cell with a number before looking for a rule. A matrix of shapes is hard to read; the same matrix written as 3, 4, 5 over 4, 5, 6 solves itself.
• Read matrices along rows and down columns as two separate passes. A large share of items encode one progression in each direction, and a rule that only works across rows is usually half the answer.
• Use the options as data. Sort them by one attribute, discard the group that cannot be right, then sort the survivors by the next. Eliminating on single attributes is far more reliable under time pressure than judging whole figures.
• Check the cyclic order of three features before accepting any rotation option, and remember that a reflected asymmetric figure can never be produced by turning the original in the plane.
• Practise untimed until you can state the rule in one sentence for every item you answer, including the ones you got right by feel. Attempts are stored on this device only, so an unhurried pass where you narrate each rule aloud costs nothing and is what actually builds the scanning habit.

Glossary

Attribute

A single independently varying property of a figure, such as shading, count or orientation. Most difficult items vary three or more attributes at once, each on its own cycle.

Matrix item

A grid of figures, usually three by three, with one cell missing. Rules typically run along rows, down columns, or both at once.

Superposition

A rule that builds one figure by combining two others. The three common variants keep the union of elements, the intersection, or the elements appearing in exactly one input.

Exclusive-or rule

The superposition variant in which elements common to both inputs cancel and only the unshared elements survive. Distinguished from union and intersection by testing a row whose inputs both share and differ in elements.

Distractor

An incorrect answer option built to match the correct one on the attribute that solved the item while differing on another. It is why a figure must be checked on every attribute before selection.

Chirality

The handedness of an asymmetric figure. Rotation preserves it and reflection reverses it, so the clockwise order of three marked features tells the two transformations apart.

Cycle length

The number of frames after which an attribute repeats. When several attributes have different cycle lengths, the whole figure only repeats after their least common multiple.

Enclosed region

An area fully bounded by lines within a figure. It is one of the most common hidden rules in odd-one-out items because it is countable and visually unobtrusive.

Study this next

• Abstract reasoning skill hub
• Abstract reasoning practice bank
• Inductive reasoning lesson
• Diagrammatic reasoning lesson
• General academic aptitude suite
• Abstract reasoning lesson on Novus Learn

Where this material comes from

• Every figure, matrix and sequence described above was constructed and verified for Novus Learn. Side counts, superposition outcomes, attribute cycle lengths and perimeter positions were each checked by hand.
• Matrix-style non-verbal items are a long-established public assessment format; this lesson describes the format in general terms and reproduces no item from any published test.
• Terminology checked against the public Wikipedia articles 'Chirality', 'Exclusive or' and 'Least common multiple'. Definitions only.
• 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

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.

How to practise this skill

• Memorise the fraction-to-percentage table up to eighths and twelfths. Almost every 'nasty' percentage on these tests is a clean fraction in disguise, and the fraction route is two or three times faster than decimal multiplication.
• Write the unit beside every intermediate number, including the ones you keep in your head. Most numerical errors on timed sections are unit errors, not arithmetic errors, and units are the only cheap way to catch them.
• Rehearse reverse percentages as their own drill until dividing by 1.15 feels more natural than subtracting 15 percent. It is a single, identifiable technique that accounts for a disproportionate share of lost marks.
• Before selecting, re-read the last clause of the stem out loud. 'By how much did it fall' and 'what was it after the fall' have different answers, and both are always on the option list.
• Keep an error log with four columns (misread the question, wrong operation, unit slip, arithmetic slip), and tally which one you actually commit. Three sessions is usually enough to show that one column dominates, and that is the column to train.
• Work untimed until your method is stable, then add the clock in stages: generous, then realistic, then 20 percent tighter than the real test. Attempts stay on this device, so a slow first pass costs nothing.

Glossary

Percentage point

The arithmetic difference between two percentages. A rate moving from 4 percent to 5 percent rises by one percentage point but by 25 percent in relative terms; questions specify which they want and the two answers are both on the option list.

Reverse percentage

Recovering an original amount from a figure that already includes a percentage change, by dividing by the multiplier (1.15 for a 15 percent rise, 0.88 for a 12 percent fall) rather than subtracting the percentage from the new figure.

Multiplier (decimal factor)

The single number a quantity is multiplied by to apply a percentage change: 1.20 for plus 20 percent, 0.80 for minus 20 percent. Successive changes are handled by multiplying the factors, which is why plus 20 then minus 20 gives 0.96, not 1.

Ratio part

One unit of a ratio split. Dividing the total by the sum of the ratio terms gives the value of one part; every share and every difference between shares is then a whole number of parts.

Unit rate

A quantity expressed per one of something: price per litre, pages per minute, litres per 100 km. Comparisons are only valid once every option has been converted to the same unit rate.

Combined rate

The sum of two or more individual rates working simultaneously. Times cannot be added or averaged; only rates add, and the combined time is the total work divided by the combined rate.

Significant figure estimate

Rounding every input to one meaningful digit to get an approximate answer in a few seconds. Its purpose is not accuracy but elimination: it removes any option that is off by a factor or has the sign of the change wrong.

Distractor

A wrong option written deliberately to match a specific predictable error: subtracting instead of dividing, reporting the change instead of the result, averaging instead of combining rates. Recognising the pattern is often faster than recomputing.

Study this next

• Numerical reasoning skill hub
• Numerical reasoning practice in the Cognitive Skills Lab
• Data interpretation lesson
• Office numeracy suite
• Finance, accounting and insurance careers
• Numerical reasoning lesson on Novus Learn

Where this material comes from

• All worked figures above were written and checked for Novus Learn. Every division, ratio split and rate calculation in this lesson was verified by recomputing it in the reverse direction.
• Conventions only (percentage point, unit rate, significant figures) cross-checked against standard public references such as the Wikipedia articles 'Percentage', 'Ratio' and 'Rate (mathematics)'. No item text is drawn from any published test.
• Novus Learn aptitude construct registry (catalog seed) for construct scope and 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

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.

How to practise this skill

• Fix your scan path before you start, and never re-scan backwards to double-check. Backward re-scanning is what produces both double-counts and lost places; if you need to verify a count, run it again from the other end and compare.
• Read the scoring rule before the first item. Whether false alarms are penalised changes the correct strategy completely, and a criterion set for the wrong rule costs more marks than slow reading does.
• Check numeric strings in fixed groups of two or three and in a deliberately shuffled group order. Transpositions are the errors that survive whole-string comparison, and they are the ones planted on purpose.
• Time your practice in segments and log accuracy per segment, not per session. A flat error rate means you need speed; a rising one means you need endurance and breaks, and the two problems have opposite fixes.
• Name the target feature out loud before each conflict item. Pre-loading the relevant dimension is the cheapest available defence against the salient-but-wrong answer.
• Practise with an actual distractor present rather than in silence. A test hall has movement, coughing and page-turning, and attention practice in perfect quiet trains a condition you will not meet.

Glossary

Selective attention

Prioritising one source or feature while suppressing competing ones. It is what fails when the visually salient element of a display captures the response instead of the requested one.

Sustained attention (vigilance)

Maintaining detection performance over a long, low-event period. It is the capacity that checking, monitoring and inspection tasks are really sampling.

Vigilance decrement

The decline in detection performance over a prolonged watch, typically steepest early in the block. Mackworth's 1948 clock test is the standard public reference.

Divided attention

Handling two streams that compete for control. Performance is best modelled as fast alternation with a switch cost, not as genuine parallelism.

Hit, miss, false alarm, correct rejection

The four outcomes of any detection decision: flagging a real fault, missing one, flagging a clean record, and correctly leaving a clean record alone. Any honest accuracy claim needs at least the first three.

Response criterion

How much evidence you require before flagging. Shifting it trades misses against false alarms without changing your underlying sensitivity, which is why it must be set from the scoring rule.

Stroop interference

The slowing that occurs when an automatic response, such as reading a word, conflicts with the requested response, such as naming its ink colour. Named for Stroop's 1935 experiments.

Inattentional blindness

Failing to notice a fully visible, unexpected event because attention is committed elsewhere - the effect Simons and Chabris demonstrated in 1999 with a counting task.

Study this next

• Attention and concentration skill hub
• Cognitive Skills Lab: attention practice
• Emergency dispatch and 911 communications suite
• Error detection lesson
• Perceptual speed lesson
• Attention and concentration lesson on Novus Learn

Where this material comes from

• Character strings, invoice comparisons, batch counts and monitoring scenarios above are written for Novus Learn. All figures are original and no published test item is reproduced.
• Terminology and directions of effect follow widely published work: Mackworth (1948) on the vigilance decrement, Stroop (1935) on response conflict, Simons and Chabris (1999) on inattentional blindness, and the standard hit/miss/false-alarm framing from signal detection theory. Concepts only; no result figures are attributed to those studies.
• Novus Learn aptitude construct registry (catalog seed) for construct scope and suite mapping.
• Public educational framing only - not affiliated with any official exam board, publisher or employer, and not a clinical, diagnostic or attention-disorder screening measure.

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.