SHL-style ability-test familiarization: study guide

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

The three-way rule on one short passage: Passage: 'The Riverton depot handled 4,200 parcels in March, a 12 percent increase on February. A night shift was introduced in January; in March it processed 40 percent of the parcels the depot handled.' Now take four statements. (1) 'The depot handled 3,750 parcels in February.' February is March divided by 1.12: 4,200 / 1.12 = 3,750 exactly, so this is True - the passage determines it even though the number is not printed. (2) 'The night shift caused the March increase.' Cannot Say. The passage puts the night shift and the increase in the same depot in the same period; it never claims one produced the other, and it never denies it either. Most candidates mark this False, reasoning correctly that correlation is not causation but then choosing the wrong label: False is reserved for statements the passage contradicts. (3) 'The night shift processed more than 1,700 parcels in March.' Forty percent of 4,200 is 1,680, which is not more than 1,700, so this is False - contradicted by arithmetic on the passage's own figures. (4) 'The depot handled more parcels in March than in January.' Cannot Say: January is mentioned only as the month the shift started, and no January volume is given.

Quantifier drift and illicit conversion: Passage: 'All accredited suppliers submit quarterly audits. Some suppliers in the northern region are accredited.' Statement A: 'Some suppliers in the northern region submit quarterly audits.' True, and it is worth seeing why the chain holds: at least one northern supplier is accredited, and every accredited supplier submits, so at least one northern supplier submits. Statement B: 'Every supplier that submits quarterly audits is accredited.' This reverses the first sentence. 'All accredited suppliers submit' leaves the door wide open for unaccredited suppliers to submit as well - perhaps voluntarily, perhaps under another rule - so the answer is Cannot Say. Turning 'all A are B' into 'all B are A' is called illicit conversion and it is the single most productive trap in this format, because the reversed sentence sounds like a paraphrase. Statement C: 'No unaccredited supplier submits quarterly audits' is the same reversal wearing a negative coat, and it is Cannot Say for the same reason. Statement D: 'All northern suppliers submit quarterly audits' is also Cannot Say: 'some are accredited' says nothing about the rest.

Asserted versus reported: Passage: 'The committee's report claims that the scheme cut waiting times by a third. Two of the four regional boards have disputed that figure.' Statement A: 'The scheme cut waiting times by a third.' Cannot Say. What the passage asserts is that a report claims this; the claim's truth is never settled, and the dispute does not settle it either. Statement B: 'At least two regional boards disagree with the report's waiting-time figure.' True, and note 'at least' - the passage says two disputed it, and two of four disputing is consistent with 'at least two'. Statement C: 'All four regional boards accept the figure.' False, directly contradicted. Statement D: 'The report is wrong.' Cannot Say. This item type appears constantly in policy and diplomacy passages, where a paragraph stacks a claim, a source, and a reaction, and the reader has to keep three different truth-conditions apart. The reliable habit is to underline the reporting verb - claims, argues, estimates, alleges, projects - and remember that everything downstream of it belongs to the source, not to the passage.

Percentages are not symmetric: Passage: 'Membership fell from 8,000 to 6,000 over two years, then recovered by 25 percent in the third year.' Statement: 'Membership at the end of year three was higher than at the start.' The fall is 2,000 on a base of 8,000, which is 25 percent. The recovery is 25 percent of the new base: 6,000 x 1.25 = 7,500. That is below 8,000, so the statement is False. The trap is elegant, because a 25 percent fall followed by a 25 percent rise feels like it should cancel. It does not: to get back from 6,000 to 8,000 you need a 33.3 percent rise, since 2,000 / 6,000 = 0.333. Whenever a verbal item states a change as a percentage, write down what the percentage is a percentage of before you touch it. A related version supplies the recovery as an absolute number instead - 'recovered by 2,000 members' - which does return the total to 8,000, and candidates who answered the first version from memory get the second one wrong.

Verbal analogies: name the relation as a sentence: Item: 'ANTISEPTIC is to INFECTION as ___ is to ___.' Options: (a) vaccine : immunity, (b) insulation : heat loss, (c) medicine : illness, (d) bandage : wound. Write the stem relation as a full sentence first: 'An antiseptic is applied in order to prevent an infection from occurring.' Now test each option against that exact sentence. Option (a) runs the other way - a vaccine is given to produce immunity, not to prevent it. Option (c) is the right family but the wrong specificity: medicine typically treats an illness that has already started. Option (d) is applied after the wound exists. Option (b) fits precisely: insulation is applied in order to prevent heat loss from occurring. The general method is the same on the simpler item types. 'BOOK is to LIBRARY as PAINTING is to ___' has canvas (the material), artist (the maker), and gallery (the place a collection is kept and shown) among its options; all three are genuine relations to 'painting', and only the third matches the stem sentence. Options in analogy items are chosen to be true statements about the words, just not the stated relation.

Two premises with 'some' prove nothing: Argument: 'Some depot managers are qualified engineers. Some qualified engineers hold a rigging certificate. Therefore some depot managers hold a rigging certificate.' This is invalid, and the way to prove it is a counter-model rather than an argument. Let the depot managers be Ana and Ben; let the qualified engineers be Ben and Cara; let the certificate holders be Cara alone. Premise one holds: Ben is a manager and an engineer. Premise two holds: Cara is an engineer with the certificate. The conclusion fails: neither Ana nor Ben holds the certificate. Since a single consistent world makes both premises true and the conclusion false, the argument cannot be valid, so in a True / False / Cannot Say framing the conclusion is Cannot Say. Two premises that both begin 'some' never combine into a conclusion, because 'some' gives you an overlap without telling you where the overlap sits. Contrast a valid pair: 'All accredited labs are inspected annually. No inspected facility is exempt from the fee.' Every accredited lab is inspected, and no inspected facility is exempt, so no accredited lab is exempt - True.

How to practise this skill

• Answer from the passage even when you know the subject. Candidates with a background in the topic score worse on verbal items than they expect, because their own knowledge quietly supplies the missing premise that turns a Cannot Say into a True.
• Run the two-worlds test on every candidate 'Cannot Say': can you imagine a world where all the passage's sentences hold and the statement is true, and another where they hold and it is false? If both, it is Cannot Say. If only the false world exists, it is False.
• Underline the quantifiers and hedges in the statement (all, some, only, most, may, must, likely) and find their counterparts in the passage. Roughly half of the wrong answers in this construct come from a single word swapped between the two.
• Budget about 25 to 30 seconds per statement rather than per passage, and read the passage once for structure before touching the statements. Re-reading the whole passage for each statement is what causes candidates to run out of time on the last set.
• Keep a wrong-answer log with one label per error: quantifier, causation, reported-versus-asserted, arithmetic, outside knowledge. A clear pattern almost always emerges within about forty items, and it is usually a single label.
• Work untimed until the three-way rule is automatic, then add the clock. Attempts are stored on this device only, so a slow first pass through a passage set costs you nothing but the time you spend on it.

Glossary

Entailment

A statement is entailed by a passage when it cannot be false while every sentence of the passage is true. Entailment is the only thing that earns a 'True' in this format; being plausible, likely, or well known does not.

Cannot Say

The verdict for a statement the passage neither entails nor contradicts. It is a claim about the passage, not about the world, which is why a statement you know to be true in real life can still be Cannot Say.

Quantifier

A word fixing how much of a group a claim covers: all, most, some, few, none, only. Swapping one quantifier for another changes the logical content completely while barely changing how the sentence reads.

Illicit conversion

The invalid move from 'all A are B' to 'all B are A', or from 'if P then Q' to 'if Q then P'. It preserves the words and destroys the logic, which is why converted sentences make such effective wrong answers.

Counter-model

A concrete, consistent scenario in which the premises hold and the conclusion fails. Producing one is the fastest possible proof that an argument is invalid, and it takes two or three named individuals.

Hedge

A qualifier such as may, could, is expected to, or is associated with. A hedged sentence in a passage cannot support an unhedged statement, and an unhedged passage sentence is not weakened by a hedged statement.

Reporting verb

A verb such as claims, argues, estimates, or alleges that attributes the following content to a source. Everything downstream of it is the source's assertion, and the passage takes no position on it.

Analogy relation

The specific link between the two stem words in an analogy item - tool and user, item and container, action and purpose. Naming it as a full sentence before reading the options removes almost all of the guesswork.

Study this next

• Verbal reasoning skill hub
• Practise the verbal reasoning bank
• Critical thinking lesson
• Reading comprehension lesson
• General civil-service aptitude suite
• Verbal reasoning lesson on Novus Learn

Where this material comes from

• Worked items written for Novus Learn. Every passage, statement set and figure above is original and invented for this lesson; no published or copyrighted test item is reproduced.
• Terminology follows standard, widely published usage in introductory logic and assessment writing - entailment, quantifier, illicit conversion, counter-model.
• Novus Learn aptitude construct registry (catalog seed) for the construct scope and the suite mapping shown in the related links.
• Public educational framing only - not affiliated with any official exam board, publisher or employer.

Worked examples and pitfalls

Differences first, ratios second: Four series, each solved by the same opening move. Series one: 2, 6, 12, 20, 30. First differences are 4, 6, 8, 10: arithmetic, rising by 2, so the next difference is 12 and the next term is 42. The closed form is n times n-plus-one: 1x2, 2x3, 3x4, 4x5, 5x6, 6x7 = 42, which confirms it. Series two: 7, 10, 16, 28, 52. First differences are 3, 6, 12, 24. Those are doubling, so the next difference is 48 and the next term is 100. Equivalently each term is double the previous minus 4: 7x2-4 = 10, 10x2-4 = 16, 16x2-4 = 28, 28x2-4 = 52, 52x2-4 = 100. Series three: 3, 4, 7, 11, 18, 29. Nothing simple in the differences, so try summing neighbours: 3+4 = 7, 4+7 = 11, 7+11 = 18, 11+18 = 29, and the next term is 18+29 = 47. Series four: 1, 4, 9, 61, 52. That looks broken until you notice 61 and 52 are 16 and 25 with their digits reversed; the underlying series is the squares 1, 4, 9, 16, 25, 36, written back-to-front where reversing changes anything, so the next term is 63. The order of attack that solves most series is fixed: first differences, then second differences, then ratios, then sums of neighbours, then digit manipulation. Run it in that order and you stop staring.

Two rules fit; the fourth term decides: You are shown 2, 4, 8 and asked for the next term. Doubling gives 16. But first differences are 2 and 4, and if those differences are themselves rising by 2 the next difference is 6 and the next term is 14. Both rules fit every term you were given, so the sequence is genuinely under-determined and neither answer is more correct than the other from the data alone. This is not a trick; it is the central limitation of induction, and assessment writers manage it by giving you enough terms to separate the candidates. Add one term. If the series is 2, 4, 8, 14 then the differences are 2, 4, 6 and the next difference is 8, giving 22, doubling is dead, because doubling would have produced 16 at position four. If the series is 2, 4, 8, 16 the difference rule is dead instead, because it predicted 14. So the practical rule is: never commit on two terms, always test your candidate rule against the LAST given term as well as the first, and if two rules survive all given terms, look at the answer options, exactly one of them will normally correspond to a rule that fits, and that is legitimate evidence about what the writer intended.

Letters are numbers wearing a costume: The sequence C, F, J, O, U. Convert to alphabet positions: C is 3, F is 6, J is 10, O is 15, U is 21. First differences are 3, 4, 5, 6, rising by one, so the next gap is 7 and the next position is 28. The alphabet has only 26 letters, so wrap: 28 minus 26 is 2, which is B. Answer B. Three things go wrong on letter items. Candidates count gaps by reciting the alphabet on their fingers and drop a letter, which is why writing the numbers down beats counting in your head every time. Candidates forget to wrap, and answer with a position number rather than a letter. And candidates mishandle a series that runs backwards past A. Position 1 minus 3 is minus 2, which wraps to 26 minus 2, that is 24, the letter X. A related family uses letter pairs where the two letters move at different rates, for example AZ, CX, EV, GT: the first letters are 1, 3, 5, 7 going up by two and the second are 26, 24, 22, 20 going down by two, so the next pair is I and R, that is IR. Split the pair, treat each stream separately, and the item becomes two easy arithmetic series instead of one impossible one.

Interleaving: two clocks in one series: The series 5, 8, 6, 11, 7, 14, 8. First differences are 3, minus 2, 5, minus 4, 7, minus 6: alternating sign, no obvious progression, and this is the point where candidates guess. Split it instead. Terms in the odd positions are 5, 6, 7, 8, rising by one. Terms in the even positions are 8, 11, 14, rising by three. The next term sits in an even position, so it continues the second stream: 14 plus 3 is 17. The diagnostic that should trigger the split is alternating signs in the first differences, or a series that oscillates while drifting. Interleaving also appears with three streams, and with one stream constant: 4, 9, 4, 16, 4, 25 hides the squares 9, 16, 25 among repeated 4s, so the next term is 4 and the one after is 36. One caution worth carrying: a series that has been split should be checked back against the original by writing out your predicted term in place and reading the whole thing again. If the reassembled series looks stranger than the one you started with, the split was probably wrong.

Inducing a rule from labelled cases: Five support tickets, each with a customer tier, a first-response time, and whether it was escalated. Ticket 1: Enterprise, 6 hours, escalated. Ticket 2: Enterprise, 2 hours, not escalated. Ticket 3: Standard, 9 hours, not escalated. Ticket 4: Enterprise, 5 hours, escalated. Ticket 5: Standard, 1 hour, not escalated. Three hypotheses are worth writing down. Hypothesis A: escalation happens when the response is slow. Ticket 3 kills it. A nine-hour Standard ticket was not escalated. Hypothesis B: escalation happens for Enterprise customers. Ticket 2 kills it. A fast Enterprise ticket was not escalated. Hypothesis C: escalation requires BOTH Enterprise tier and a response over four hours. It fits all five rows, and it is the only conjunction that does. Notice which rows did the work: the two that fit one hypothesis but not the label are worth more than the three that agree with everything. Now the honest limit. Does an Enterprise ticket answered in exactly four hours escalate? Hypothesis C as written says no, but the data contains no case between two and five hours, so the true threshold could be anywhere in that window. Induction from five rows gives you a rule and a region of ignorance, and naming the region is part of the answer.

Look for the case that would break you: Four cards lie on a table showing A, K, 4 and 7. Each card has a letter on one side and a number on the other. The rule to test: if a card has a vowel on one face, it has an even number on the other. Which cards must you turn? Most people say A, and many add 4. A is right: a vowel with an odd number on the back refutes the rule directly. The 4 is useless: the rule says nothing about what must be behind an even number, so a consonant there breaks nothing and a vowel there merely agrees. The card people miss is the 7. Turn it, find a vowel, and the rule is dead. So the answer is A and 7, the two cards that could produce a violation. This is the Wason selection task, a well-known public result in the psychology of reasoning, and the reason it belongs in an inductive lesson is that it isolates the habit the construct actually rewards: people search for confirming evidence by default and have to be trained to search for disconfirming evidence. Carry it into series items as a check. Once you have a candidate rule, do not run it forward on the terms it was built from; run it on the term you have not used yet, and treat a single mismatch as fatal rather than as noise.

How to practise this skill

• Write the first differences under every numeric series before you think about it at all. The mechanical step surfaces arithmetic and quadratic rules for free and tells you within seconds whether to move on to ratios.
• Fix an order of attack and follow it: first differences, second differences, ratios, sums of adjacent terms, then digit tricks. Most lost time in this construct is spent re-staring at a series rather than working through the list.
• Convert letters to numbers on paper the moment you see a letter series, and write the alphabet with positions at the top of your rough sheet once at the start of a session rather than counting on your fingers per item.
• Treat alternating signs in the first differences as a direct instruction to split the series into odd and even positions. The interleaved family is large and is where candidates most often decide, wrongly, that there is no pattern.
• Validate a candidate rule against the last given term, not the first two. A rule that explains the opening of a series and misses its final term is the standard wrong answer, and the answer options are usually built to reward it.
• When you induce a rule from a table of cases, name the row that eliminated the simpler hypothesis and name the gap the data leaves open. Both are recorded on this device only, so build the habit here where over-claiming is free.

Glossary

Inductive inference

Reasoning from particular observations to a general rule. The conclusion is supported but never guaranteed, which is the essential difference from deduction.

First difference

The gap between consecutive terms of a series. A constant first difference means an arithmetic progression; a constant second difference means a quadratic rule.

Geometric progression

A series where each term is the previous term multiplied by a fixed ratio. Detected by dividing neighbours rather than subtracting them.

Interleaved series

A single printed sequence containing two or more independent series in alternate positions. Signalled by first differences that alternate in sign.

Under-determination

The condition in which several different rules fit every term you were given, so the data alone cannot single one out. It is the reason short series need extra terms or answer options to be decidable.

Disconfirming instance

A case that a hypothesis predicts should not exist. One of these refutes a rule outright, whereas any number of agreeing cases only fails to refute it.

Confirmation bias

The tendency to seek evidence that agrees with a hypothesis already held. In series and rule-induction items it shows up as checking a rule against the terms that suggested it.

Wrap-around

Continuing a letter sequence past Z back to A, or before A back to Z, by adding or subtracting 26 from the alphabet position. The routine source of near-miss answers on letter items.

Study this next

• Inductive reasoning skill hub
• Inductive reasoning practice bank
• Abstract reasoning lesson
• Ability-test style familiarization suite
• Crime and intelligence analysis suite
• Inductive reasoning lesson on Novus Learn

Where this material comes from

• Worked series written for Novus Learn from standard sequence analysis: first and second differences, geometric ratios, neighbour sums, alphabet-position arithmetic, and interleaved streams. Every series above was extended and checked term by term.
• The four-card selection problem is the Wason selection task, first published by P. C. Wason in 1968 and widely reproduced in the public psychology-of-reasoning literature; the description above is a plain restatement of the standard version, not an item from any commercial test.
• Terminology checked against the public Wikipedia articles 'Inductive reasoning', 'Arithmetic progression', 'Geometric progression' and 'Wason selection task'. 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

Read the header: (£000) changes every answer by a factor of a thousand: Table titled 'Regional revenue, year to March (£000)': North 1,240; South 986; East 1,455; West 719. Total = 1,240 + 986 + 1,455 + 719 = 4,400, so the business turned over 4,400 thousand pounds, that is 4.4 million. East's share is 1,455 / 4,400 = 33.1 percent. The whole set: North 28.2 percent, South 22.4, East 33.1, West 16.3, summing to 100. Two things go wrong here. The first is reading East's revenue as 1,455 pounds and then reporting a business with a total turnover of 4,400 pounds, which nobody notices because every option is scaled the same way, until the question asks for revenue in millions and only one option is right. The second is the comparison wording. East's share is (1,455 - 719) / 4,400 = 16.7 percentage points above West's, and East's revenue is (1,455 - 719) / 719 = 102 percent more than West's, that is slightly more than double. 'Sixteen point seven' and 'a hundred and two' both describe the same two cells honestly, and the question decides which one is correct. Note that the percentage-point figure must be computed from the unrounded shares rather than by subtracting the rounded ones, or the last digit will not survive.

One pair of rows, three correct increases: A complaints table: 2023, 120,000 orders, complaint rate 4.0 percent; 2024, 150,000 orders, complaint rate 5.0 percent. Three defensible answers to 'how much did complaints increase?'. The rate rose by 1.0 percentage point. The rate rose by (5.0 - 4.0) / 4.0 = 25 percent in relative terms. And the count of complaints rose from 0.04 x 120,000 = 4,800 to 0.05 x 150,000 = 7,500, which is (7,500 - 4,800) / 4,800 = 56.25 percent. All three are arithmetically right; only one answers the question in front of you. The pattern to internalise is that a rate and a count move together only when the denominator is fixed, and here it is not. Order volume grew 25 percent as well. If the question is about customer experience, the rate is the honest figure; if it is about how many complaint handlers to hire, the count is. Test items usually ask for the one you would not have chosen.

Index numbers: five points is not five percent: A cost index with 2020 = 100 reads 104 in 2021, 111 in 2022 and 109 in 2023. From 2021 to 2023 the index rose 5 points, but the percentage change is 5 / 104 = 4.8 percent, because the base for the comparison is 104, not 100. From 2022 to 2023 it fell 2 points, which is -2 / 111 = -1.8 percent, and note that costs fell even though the index remains 9 percent above the 2020 base. A level and a change are different claims. The only comparison where points and percent coincide is against the base year itself: 2020 to 2023 is 100 to 109, exactly plus 9 percent. Watch also for a rebased series, where a table switches to 2022 = 100 partway down; the two segments cannot be compared directly without converting one of them, and an item that quietly rebases is testing whether you read the column heading.

Joining two tables: totals and per-head figures disagree on purpose: Table 1, headcount by site: Leeds 84, Derby 47, Bristol 129. Table 2, absence days recorded in the same period: Leeds 630, Derby 300, Bristol 903. 'Which site has the worst absence problem?' On raw totals Bristol is worst at 903 days. Normalise per head and the ranking changes: Leeds 630 / 84 = 7.50 days per employee, Bristol 903 / 129 = 7.00, Derby 300 / 47 = 6.38. Leeds is worst, Bristol is merely biggest. The organisation-wide figure is 1,833 / 260 = 7.05 days per head, which is a useful reference line: Leeds is above it, the other two below. The general rule is that any comparison between units of different size demands a denominator, and the denominator has to come from the other table. Items are built so the raw-total answer and the per-head answer are both on the option list, and so the site with the biggest total is never the site with the highest rate.

The truncated axis: measure the numbers, not the bars: A quarterly satisfaction chart with a y-axis running from 78 to 82 shows bars at 79.2, 79.8, 80.4 and 81.1. Visually the last bar looks several times taller than the first, because only the top 4 points of a 100-point scale are drawn. The actual movement is 81.1 - 79.2 = 1.9 points, which on the score's own scale is a relative rise of 1.9 / 79.2 = 2.4 percent. If the question asks 'by approximately what percentage did satisfaction improve', the answer is about 2 percent, and the distractor built from the bar heights will be something like 40 or 400 percent. Related presentation effects to check before answering: a dual-axis chart where two series use different scales and appear to cross meaningfully when they do not; a cumulative series, where a flattening line still means the total is growing, just more slowly; and a logarithmic axis, where equal vertical distances are equal ratios rather than equal amounts. In every case the defence is the same. Find the printed numbers, or read the gridline values, and compute.

Cannot say: revenue is not profit: A product table shows units sold and total revenue. Product P: 4,200 units, 71,400 pounds. Product Q: 1,800 units, 41,400 pounds. Average selling price is 71,400 / 4,200 = 17.00 for P and 41,400 / 1,800 = 23.00 for Q, so Q earns more per unit while P earns more in total. Now the statement to evaluate: 'P is more profitable than Q.' The correct response is cannot say. Profit needs cost, and the table has no cost column; a product with a 17 pound price and a 16 pound unit cost is less profitable than one priced at 23 with a cost of 9, and nothing here rules that out. Contrast with 'Q generated more revenue per unit than P', which the table fully supports and which is true. The habit worth building is to finish every cannot-say judgement with the missing input named out loud ('cannot say, because unit cost is not given') because that forces you to distinguish a genuinely unanswerable item from one you simply have not worked hard enough on.

How to practise this skill

• Read the title, the units line, the row and column headers and any footnote before you look at a single value. Roughly the first fifteen seconds of an item should contain no arithmetic at all, and that fifteen seconds is what prevents the thousand-fold and percentage-point errors.
• For every item, write down which of the three quantities is wanted (share of total, relative change, or change in percentage points), before computing. Most wrong answers on this construct are correct arithmetic applied to the wrong quantity.
• Whenever two groups differ in size, ask what the denominator should be. If a question compares sites, teams, countries or periods of unequal length and you have not divided by something, you are almost certainly answering the wrong question.
• Practise the cannot-say items separately and force yourself to name the missing variable each time. Candidates who train only on computational items reliably over-answer inference statements under time pressure.
• Do not redraw or re-scale charts in your head. Locate the gridline values or the data labels, and if neither exists, interpolate between two labelled gridlines and state the bound rather than guessing a point value.
• Time yourself per item rather than per section. Data interpretation sets share a stimulus, so the first item costs the reading time and the rest should be fast; if item four takes as long as item one, you did not build a mental map of the table.

Glossary

Units multiplier

A scaling note in a table title or column header, such as (£000), (millions) or (per 1,000 population). It applies to every value in scope and is the single most common source of order-of-magnitude errors.

Index number

A series rescaled so a chosen base period equals 100. Changes between two non-base periods must be divided by the earlier value, so a movement in index points is not a percentage change except when measured from the base.

Rebasing

Restating an index against a new base period. Segments of a series with different bases cannot be compared directly, and a table that rebases partway down is testing whether you read the headings.

Truncated axis

A chart whose value axis does not start at zero, which exaggerates the apparent size of differences between bars or points. Legitimate for showing small movements in a large quantity, misleading if read as area or height.

Cumulative series

A line showing a running total rather than each period's value. It can only go up or stay flat, so a flattening cumulative line means the periodic figure is falling, not that the total is.

Weighted average

An average in which each value is multiplied by the size of the group it represents. Averaging two group percentages directly is only correct when the groups are the same size, which in these tables they rarely are.

Normalisation

Dividing a raw figure by an exposure measure (headcount, units sold, population, days open), so groups of different size can be compared. The denominator usually lives in a second table.

Cannot say

The verdict when a statement is neither supported nor contradicted by the data supplied. A correct cannot-say answer can always be defended by naming the specific missing variable.

Study this next

• Data interpretation skill hub
• Data interpretation practice in the Cognitive Skills Lab
• Numerical reasoning lesson
• Financial data interpretation suite
• Scientific data interpretation suite
• Data interpretation lesson on Novus Learn

Where this material comes from

• Every table, index series and chart described above was constructed for Novus Learn, and each figure was verified by recomputing the totals and the reverse calculation.
• Definitions of index numbers, rebasing and weighted averages cross-checked against standard public references such as the Wikipedia articles 'Index (economics)' and 'Weighted arithmetic mean'. Terminology only; no data or item text is taken from any source.
• 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 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.