Thomas GIA-style familiarization: study guide

Worked examples and pitfalls

Chunk the string, stop at the first mismatch: Compare 4820 6193 5507 with 4820 6913 5507. Read them as three blocks, not as twelve digits. Block 1 matches, 4820 against 4820. Block 2 does not: 6193 against 6913, where the 1 and the 9 have swapped. You are done. On a same-or-different item there is no reason to check block 3, and checking it anyway costs you about a second on every item, which over eighty items is more than a minute of pure waste. The reason chunking works is capacity: a twelve-digit run exceeds what most people can hold for a comparison, so reading it whole forces you back to the source repeatedly, while a four-digit block fits in one go. The instruction wording decides whether early stopping is allowed. 'Are these identical?' permits it. 'How many of the five fields differ?' does not, and stopping early there converts a correct three into an incorrect one. Read the instruction once, classify the section, then run the matching procedure without re-deciding it per item.

Homoglyphs: 0 and O are where the time goes: Three pairs to judge. Nguyen, Thi-Hoa against Nguyen, Thi-Hoa: identical. Muller, Jorg against Mueller, Jorg: different, and the difference is in the surname spelling rather than anywhere in the given name, which is where most candidates look first. RN0O1I5 against RNO01I5: different, because positions three and four hold zero-then-letter-O in the first string and letter-O-then-zero in the second. That last pair is the killer, and it is deliberate. In most screen and print faces, a zero and a capital O differ by a few pixels, and the digit one, capital I and lower-case L can be visually identical. Production systems avoid these characters in generated references for exactly this reason; test writers include them for exactly the opposite one. The practical rule: whenever a string mixes letters and digits, drop your rate for that item, and use context to disambiguate. If a field is described as numeric, an O in it is a zero, and if it is a postcode you can often tell from the format which positions are letters. Guessing on a homoglyph is not a speed decision, it is a coin flip you can avoid by spending half a second.

Memorise the key, then never look at it again: A routing section supplies two small tables. Region: NW = 12, NE = 14, SW = 21, SE = 23, CE = 31. Priority: A = 1, B = 4, C = 7. Each item names a region and a priority and asks for the combined code, so 'SE, priority B' is 23-4 and 'NE, priority C' is 14-7. There are only eight mappings. If the section holds forty items in three minutes, you have 180 / 40 = 4.5 seconds per item, and a candidate who glances back at the key twice per item (once for the region, once for the priority) spends around two seconds of that on lookup alone, which is roughly eighty seconds of the section, or nearly half of it. Twenty seconds spent memorising eight pairs before the first item pays for itself about ten items in and then keeps paying. The general principle for speeded sections: any fixed information that appears in every item should be moved into memory as early as the format allows, because per-item costs multiply by the item count and one-off costs do not.

Cancellation drills and mirror-image distractors: Build your own cancellation drill rather than reusing a published one. Write twenty rows of forty characters drawn only from b, d, p and q, each carrying zero, one or two tick marks above or below. Define the target as 'b with exactly two marks' and cross out every occurrence, timed. What the drill teaches is feature overlap: a p with two marks is a vertical mirror of the target and a d with two marks is a horizontal mirror, and both take measurably longer to reject than an obviously different character would, because they share every feature except orientation. The same effect is why 6 and 9, and M and W, are expensive in real checking work. Two things improve with practice on this drill and neither is eyesight. The first is that you stop re-deriving the target on each row and start rejecting non-targets pre-attentively. The second is that you learn your own decay point, the row at which misses start appearing, which is the information you actually need in order to pace a real section.

Raster order on a scanning grid: A ten-by-ten grid of two-digit numbers, and the task is to mark every value ending in 7. Scan strictly left to right, top to bottom, one row at a time, keeping a finger or the cursor on the current row. The reason is not neatness. Free-roaming the eye over a hundred cells means some cells get visited twice and others not at all, and there is no way to know which is which, so you finish with no idea whether you are done. If you re-visit even fifteen percent of the grid, that is fifteen wasted fixations, and at roughly a quarter of a second each that is close to four seconds on one grid: often the difference between completing the section and leaving items blank. On a two-column layout, treat each column as its own raster and finish it before moving; jumping between columns doubles the re-visit rate. The same discipline transfers directly to real work: scanning a picking list, a passenger manifest or a stock count row by row is slower per glance and faster per document.

Correct per minute, not percent correct: Two candidates, four minutes each. A attempts 62, gets 60 right: 15.0 correct per minute at 96.8 percent accuracy. B attempts 96, gets 84 right: 21.0 correct per minute at 87.5 percent accuracy. Under raw scoring, count the correct answers, B wins, 84 to 60. Under correct minus incorrect, A scores 60 - 2 = 58 and B scores 84 - 12 = 72, so B still wins. Under correct minus three times incorrect, A scores 60 - 6 = 54 and B scores 84 - 36 = 48, and A wins. The same two performances rank three different ways depending on a single line in the instructions. This is why percent-correct is close to useless as a self-assessment on a speeded test: it ignores the rate, and the rate is the whole construct. Log both numbers after every practice session (attempts and correct, from which rate and accuracy both fall out), and compare sessions on rate. A session where accuracy improved but the rate fell is not necessarily progress.

How to practise this skill

• Practise in short, intense blocks of two to four minutes rather than long sessions. The construct is a rate under pressure, and a relaxed twenty-minute session trains a different behaviour from the one being measured.
• Never sub-vocalise a string you are comparing. Saying '4820' in your head takes about as long as saying it aloud and caps your rate; comparing the shapes of the blocks does not.
• Deliberately vary the drill format between name checking, number checking, symbol cancellation and table lookup. Speed built on one format transfers less than candidates expect, and real sections mix them.
• Slow down on purpose for mixed alphanumeric strings and speed back up on pure numeric ones. A uniform pace is the wrong strategy when the items have unequal difficulty.
• Fix your scan path before starting (chunk size, direction, whether you finish a column before moving), and do not change it mid-section. Changing procedure under time pressure costs more than the procedure was ever going to save.
• Record attempts and correct answers separately for every session so you can plot rate over time. Scores stay on this device unless you export them, so the log is yours and is worth keeping across weeks rather than days.

Glossary

Speeded test

A test whose items are easy enough that nearly everyone could answer them given unlimited time, so the score reflects how many were completed. Contrasted with a power test, where difficulty rather than the clock is the limiting factor.

Cancellation task

A format in which a target character or symbol must be marked wherever it appears in a dense field of similar non-targets, scored on hits, misses and false marks within a fixed time.

Distractor (perceptual)

A non-target that shares visual features with the target. The more features shared, the longer the rejection takes, which is why mirror-image pairs such as b and d are the most expensive distractors of all.

Homoglyph

A character that looks nearly identical to another in a given typeface: 0 and O, 1 and I and l, 5 and S. The commonest single cause of unavoidable errors in alphanumeric checking.

Chunking

Grouping a long string into units of three or four characters so each comparison fits within working memory. It is the difference between one pass over a reference number and several.

Raster scan

Covering a field in a fixed order, row by row in one direction, so that every cell is visited exactly once. Its value is not tidiness but the guarantee that the search is complete.

Rate measure

Performance expressed per unit of time, such as correct answers per minute. It is the only fair basis for comparing two speeded sessions of different lengths.

Ceiling effect

What happens when a test is easy enough that many candidates reach the maximum score, destroying its ability to discriminate. Speeded sections are deliberately long enough that almost nobody finishes, which is why an unfinished section is normal rather than a failure.

Study this next

• Perceptual speed skill hub
• Perceptual speed practice in the Cognitive Skills Lab
• Processing speed lesson
• Clerical checking suite
• Numeric keypad suite
• Perceptual speed lesson on Novus Learn

Where this material comes from

• All strings, code tables, grids and drills above were written for this lesson. The rate and score comparisons were computed and checked, including the three different orderings produced by the three scoring rules.
• Timing figures given for fixations, lookups and per-item costs are illustrative arithmetic showing the shape of the trade-off, not measured research findings; use your own logged times in their place.
• Terminology (speeded versus power tests, cancellation tasks, chunking, homoglyphs) cross-checked against standard public references such as the Wikipedia articles 'Speed test (psychometrics)', 'Chunking (psychology)' and 'Homoglyph'.
• 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

Substitution: turning a lookup into a recall: A key maps six letters to digits: Q = 1, W = 2, E = 3, R = 4, T = 5, Y = 6. The sequence W E Y Q R T E W becomes 2 3 6 1 4 5 3 2. Trivial, and that is the point. The item content is not the difficulty. Break the per-item cost into three parts: a glance up to the key and back, the match itself, and writing or keying the digit. Say those cost roughly 0.3, 0.4 and 0.5 seconds while you are still reading the key, giving 1.2 seconds per item and 72 seconds for sixty items. Once the six pairs are held in memory, the glance disappears and each item costs about 0.9 seconds, so sixty items take 54 seconds: a saving of about 25 percent. The investment arithmetic matters: at 0.3 seconds saved per item, ten seconds spent deliberately rehearsing six pairs is repaid after about 34 items and everything after that is profit, whereas thirty seconds of rehearsal would need 100 items to break even. So on a short section, learn the key quickly and imperfectly; on a long one, learn it properly. Use your own measured times rather than the illustrative ones above. The structure of the decision is what transfers, not the numbers.

The optimum error rate is not zero: A 90-second section scored one point per correct answer, nothing deducted for a wrong one. Three paces. Careful at 1.8 seconds an item: 90 / 1.8 = 50 attempted, 96 percent right, 48 correct and 2 wrong. Brisk at 1.2 seconds: 75 attempted, 92 percent right, 69 correct and 6 wrong. Reckless at 0.75 seconds: 120 attempted, 70 percent right, 84 correct and 36 wrong. Under raw scoring the totals are 48, 69 and 84, so faster keeps winning right up to the point where accuracy collapses entirely, and most candidates sit nowhere near that point, which is why 'slow down and be careful' is such common and such bad advice on a no-penalty section. Now change the rule to correct minus incorrect: 48 - 2 = 46, 69 - 6 = 63, 84 - 36 = 48. The brisk pace wins and the reckless pace has fallen back to roughly where the careful one sits. Change it again to correct minus twice incorrect: 44, 57 and 12. Brisk wins by more, and reckless is now catastrophic. Three scoring rules, three different optimal paces, and under none of them is the answer either extreme. The instruction screen gives you the penalty; only your own practice log gives you the accuracy you actually hold at each pace.

Switch cost: the same items, slower, for free: Take forty items, twenty additions and twenty alphabetical comparisons. Present them in two blocks of twenty and people complete them faster than when the same forty items are interleaved one after the other, even though not a single item has changed. The extra time is switch cost: reconfiguring the mental task set on every alternation. If a blocked pace is 1.0 second an item and a mixed pace is 1.25, eighty items cost 80 seconds blocked and 100 seconds mixed, and the 20-second difference bought you nothing. The practical consequences are concrete. On a paper form or a scrollable list where you control the order, do all items of one type first and then the other. On a randomised screen where you cannot, expect the mixed rate and do not interpret it as having got worse. And in real work, batching the same kind of task (all the invoices, then all the emails) is the same effect, which is why an interruption costs far more than the seconds it occupies.

Rate, not raw score: comparing two practice sessions: Session 1: six minutes, 148 correct, 9 wrong. Session 2: four minutes, 112 correct, 4 wrong. On raw correct, session 1 looks better by 36. Convert to rates: 148 / 6 = 24.7 correct per minute against 112 / 4 = 28.0. Convert the errors to rates too: 9 out of 157 attempted is 5.7 percent, against 4 out of 116 which is 3.4 percent. Session 2 is better on both dimensions, faster and more accurate, and the raw comparison had it backwards purely because it ran for two minutes less. This is the most common self-assessment error in speed practice, and it matters because it drives the wrong training decision: the candidate concludes that longer sessions suit them and keeps practising in a way that inflates the number they are watching. Log four figures per session: duration, attempted, correct, wrong. Everything else is derivable, and no comparison should ever be made on a figure that has duration baked into it.

Automating the easy arithmetic: Processing speed sections use arithmetic that is deliberately easy, which means what is being measured is whether it has become automatic. Compensation is the main technique: 38 + 47 becomes 40 + 45 = 85, moving two across so there is no carry to track. Rounding and correcting handles multiplication: 6 x 24 is 6 x 25 - 6 = 150 - 6 = 144, and 49 x 8 is 50 x 8 - 8 = 392. Percentages decompose: 15 percent of 320 is 10 percent, 32, plus half of that, 16, giving 48. Division by four is halving twice, so 96 becomes 48 becomes 24. None of these is clever, and that is exactly why they work under time pressure. Each replaces a multi-step procedure with one you can run without holding intermediate state. Drill them until you stop noticing which method you used. The distinction that matters is between knowing a shortcut and having automated it; on a two-second-per-item section, a shortcut you have to consciously select is barely faster than the long way.

What actually moves a speeded score on the day: Four things reliably cost points and all four are controllable. First, an unfamiliar input device: practising on a laptop and sitting the test with an external mouse, or drilling on a full keyboard with a dedicated numeric pad and meeting a compact one where the digits live on the top row, changes a motor skill you had automated. Second, interruptions: a notification mid-block costs the item you were on plus the next one or two while the task set is rebuilt, which is switch cost arriving uninvited. Third, no warm-up: the first thirty seconds of a cold speeded section are measurably slower, so two minutes of low-stakes practice before a timed attempt is not superstition. Fourth, unrecorded conditions: a bad session with no note of why looks in your log like a decline in ability. Write down the device, the time of day, the sleep and whether you warmed up. A speeded score is sensitive to all of these in a way a reasoning score simply is not, and half of an apparent plateau usually turns out to be uncontrolled conditions rather than a real ceiling.

How to practise this skill

• Warm up for two minutes before any timed attempt. Cold starts under-report by enough to make an untimed comparison meaningless, and the warm-up costs less time than the items it saves.
• Practise on the device and layout you will actually use. If the test is on a desktop with a mouse, do not train on a trackpad; if it involves numeric entry, do not train on the keyboard top row.
• Read the penalty rule first and pick a pace from arithmetic, not nerve. Write your own accuracy at two different speeds in a log so the arithmetic has real inputs.
• Where you control the order of items, batch by type to avoid switch cost. Where you do not, accept the mixed rate rather than trying to force the blocked one and making errors instead.
• Drill the small arithmetic until you stop choosing a method. Automaticity, not knowledge of shortcuts, is what a two-second-per-item section rewards.
• Record duration, attempted, correct and wrong for every session, plus the conditions. Attempts stay on this device unless you export them, and four numbers per session is enough to see a genuine trend within a fortnight.

Glossary

Processing speed

The rate at which simple cognitive operations requiring a decision can be completed accurately. Distinguished from perceptual speed by the presence of a rule to apply rather than only a match to make.

Substitution (coding) task

A format supplying a key that maps symbols to digits or letters, with the candidate transcribing a long sequence through the key. Performance improves sharply once the key is recalled from memory rather than read.

Speed-accuracy trade-off

The relationship in which going faster raises the error rate. It has an optimum for any given scoring rule, and the optimum sits at neither extreme.

Switch cost

The extra time taken per item when task types alternate rather than being grouped, caused by reconfiguring the mental task set. It applies even when the items themselves are unchanged.

Automaticity

The state in which an operation runs without deliberate selection or monitoring. It is the practical goal of drilling small arithmetic, because a consciously chosen shortcut is barely faster than the long method.

Response time

The full interval from stimulus to completed response, including the physical act of keying or writing. Broader than reaction time, which refers only to the interval before the response begins.

Practice effect

The improvement that comes from familiarity with a format rather than from any change in underlying ability. It is largest on the first few attempts, which is why an early practice score is a poor baseline.

Error rate

Wrong answers as a proportion of attempted items. It is the only error figure comparable across sessions of different lengths, and it must be logged alongside rate for either to be interpretable.

Study this next

• Processing speed skill hub
• Processing speed practice in the Cognitive Skills Lab
• Perceptual speed lesson
• Short cognitive assessment familiarization suite
• Game-based assessment familiarization suite
• Processing speed lesson on Novus Learn

Where this material comes from

• Every score, rate and break-even calculation above was worked out and re-checked for this lesson, including the three scoring rules applied to the same three paces.
• Per-item timings, the blocked-versus-mixed pace figures and the 0.3-second lookup saving are illustrative arithmetic chosen to show the structure of each trade-off. They are not measured research results, and your own logged times should replace them.
• Concepts (speed-accuracy trade-off, task-switching cost, automaticity, practice effect) cross-checked against standard public references such as the Wikipedia articles 'Speed-accuracy tradeoff', 'Task switching (psychology)' and 'Mental chronometry'.
• 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

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

Rotation preserves handedness; reflection reverses it: A figure carries three distinguishable marks (say a red band, a blue band and a yellow band), which read red, blue, yellow going clockwise around its centre. Rotating that figure in the plane, by any angle, never changes that clockwise order: it still reads red, blue, yellow clockwise from whatever mark you start at. Mirroring it reverses the order, so a reflected copy reads red, yellow, blue clockwise. That single check answers the whole is-this-a-rotation-or-a-reflection family without imagining any motion. It works on letters, which is why no in-plane rotation of the letter R produces a backwards R, and no rotation of a lower-case b produces a d: b and d are mirror images across a vertical line, and turning one will never give you the other. Contrast b and q, which are related by a half turn and therefore genuinely are rotations of each other. The distinction is exactly what the clockwise-order test measures. In three dimensions the equivalent is handedness at a corner: pick three edges meeting at a vertex, and a rotation preserves whether they form a right-handed or left-handed set while a reflection swaps it. This is exactly why most Western dice, viewed at the corner where the 1, 2 and 3 faces meet, show those numbers running anticlockwise, and a die showing them clockwise is a mirror-image die rather than a differently rotated one. The one honest limitation: a figure with its own mirror line looks identical after reflection, so the test decides nothing for symmetric shapes, which is precisely why item writers use asymmetric ones.

Cube nets: two apart in a strip means opposite: A net made of six squares. Four of them, call them A, B, C and D, sit in a horizontal strip left to right. Square E sits directly above B, and square F sits directly below B. Fold it. The strip of four wraps around the cube as a band, so squares two apart in that strip end up on opposite faces: A is opposite C, and B is opposite D. E and F are the remaining pair, and they become the top and bottom, opposite each other. The two rules that solve almost every net item are exactly these. Squares separated by one square in a straight line end up opposite. Squares sharing an edge on the net end up adjacent on the cube, and can never be opposite. Everything else is applying them carefully. A die adds one more constraint worth carrying: opposite faces of a standard die sum to seven, so 1 faces 6, 2 faces 5 and 3 faces 4, and a net showing 1 and 6 two apart in a strip is consistent while a net showing 1 and 2 two apart is not a standard die. The classic error on these items is trying to fold the net in your head all at once. Do not. Pick one square as the base, identify its opposite by the two-apart rule, then repeat for a second pair, and you have determined the cube in two cheap steps rather than one expensive act of visualisation.

Three views, one solid, and the cubes you cannot see: A solid is built from unit cubes on a base one cube deep and three cubes wide. The column heights, left to right, are 3, 1 and 2. The front view is therefore a staircase profile of heights 3, 1, 2. The side view, looking along the row from the right, is a single column three cubes tall, because the tallest column hides the others behind it. The plan view is a three-by-one rectangle. Total cubes: 3 plus 1 plus 2, which is 6, and here the three views pin it down exactly because the plan view tells you the solid is only one cube deep. Now change one thing. Suppose the plan view is three by two, so the solid is two cubes deep. The front view can still read 3, 1, 2 while the count is anything from 6, one row of cubes with a hollow back, up to 12, a solid block matching the front profile in both rows. The views constrain the shape but do not determine it, and a well-written item will either supply the count or ask for the minimum and maximum rather than a single figure. That is the trap in this family: candidates who assume every hidden position is filled will over-count, and candidates who assume nothing is hidden will under-count. Read the plan view first to establish the footprint, then use the elevations to cap each column, then ask explicitly whether the item wants the minimum, the maximum, or a determined value.

Punched and unfolded, with coordinates: Take a square sheet 8 units across, with the origin at its centre so the corners are at plus or minus 4 in each direction. Fold the right half onto the left half across the vertical centre line, then fold the bottom half up onto the top half across the horizontal centre line. What is left visible is one quarter, the region where x runs from minus 4 to 0 and y runs from 0 to 4. Punch a single hole at the point minus 3, 3. Now unfold, one step at a time, and treat every unfold as a reflection. Undoing the horizontal fold reflects across the line y equals 0 and adds a hole at minus 3, minus 3. Undoing the vertical fold reflects both existing holes across x equals 0 and adds holes at 3, 3 and 3, minus 3. The finished sheet has four holes, at all four combinations of plus or minus 3. Two rules generalise from this. The number of holes is one punch multiplied by two for each fold the punch passed through, so two folds give four holes and three folds give eight, provided the punch went through every layer, which it does when the folds are through the middle. And the positions are mirror images across each fold line, never rotations. That second point is where the marks go: candidates who place the fourth hole by rotational symmetry get three holes right and one wrong, which is exactly the answer option the item includes.

Bearings, and the difference between the map and the walker: A walker leaves a gate, goes 300 m due north, then 400 m due east. The straight-line distance back to the gate is the hypotenuse of a 300 by 400 right triangle, which is 500 m: the 3-4-5 triangle, so no calculator is needed. The bearing from the gate to the walker is measured clockwise from north, and it is the angle whose tangent is 400 over 300, which is about 53 degrees, so the bearing is 053. The return bearing is that plus 180, which is 233. Bearings are always three digits and always clockwise from north, so 53 degrees is written 053 and a bearing of 350 is very slightly west of north, not south. Now the other half of this construct, which trips up more people than the trigonometry. You are facing south and you turn left. Which way are you now facing? East. Your left hand points east when you face south, and the standard wrong answer, west, comes from reading left as left-on-the-map while forgetting that the map is drawn north-up and you are not. The general fix is to state your heading before every turn and rotate your frame with the traveller: facing north, left is west; facing south, left is east; facing east, left is north; facing west, left is south. On route items, physically rotate the page so the direction of travel points up before working out a turn, and rotate it back only to read off the final compass direction.

What a cut actually reveals: Cross-section items ask what shape a plane makes as it slices a solid, and the reflex answer of circle is wrong most of the time. A cylinder cut perpendicular to its axis gives a circle. Cut parallel to the axis, it gives a rectangle. Cut obliquely so the plane crosses only the curved surface, it gives an ellipse; cut obliquely so the plane also crosses one flat end, it gives an ellipse with one side sliced flat, a D-shape. A cone is richer, because its sections are the conic sections that give the family its name: perpendicular to the axis gives a circle; an oblique cut that crosses every side of the cone gives an ellipse; a cut parallel to one slanted side gives a parabola; a cut parallel to the axis gives a hyperbola branch; and a cut through the apex gives a triangle. A cube is the one that surprises people. A cut parallel to a face gives a square, obviously. A cut through three vertices that are mutually adjacent to one corner gives an equilateral triangle. And a cut through the midpoints of six edges, perpendicular to a long diagonal of the cube, gives a regular hexagon: a six-sided section from a six-faced solid with no curves anywhere. For a flat-faced solid, work these by counting faces rather than by visualising: the section of a convex polyhedron gains exactly one straight edge for every face the plane passes through, so a cut crossing six faces of a cube must close into a six-sided figure and a cut touching only three cannot have more than three sides. For curved solids the question becomes whether the plane meets the curved surface all the way round, which gives a closed circle or ellipse, or only partway, which leaves an open arc closed off by a straight chord.

How to practise this skill

• Replace visualisation with a rule wherever one exists. Two-apart-means-opposite on nets, and clockwise-order-preserved for rotations, both give a checkable answer in seconds and are reliable when mental imagery gets tired late in a section.
• Physically rotate the paper or the device on route and orientation items so the direction of travel points up, work out the turn, then rotate back to read the compass direction. Fighting the north-up convention in your head is the single largest source of left-right errors.
• On view and cube-count items, read the plan view first to fix the footprint, then use the elevations to cap each column, then decide explicitly whether the question wants a minimum, a maximum, or a determined count.
• Treat every unfold as a reflection across the fold line. Say the word reflection out loud as you draw each new hole; the rotational version of the pattern is a deliberately included wrong answer in almost every folding item.
• Build a physical reference once. Fold a paper cube net, mark the faces, and hold it: ten minutes with a real cube fixes the opposite-face rule more firmly than an hour of diagrams, and the memory transfers to items you have never seen.
• Practise untimed first and check whether your errors cluster in rotation, folding, views or orientation, because the four subskills train differently. Attempts are stored on this device only, so an honest error log costs nothing and is what tells you which drill to repeat.

Glossary

Chirality (handedness)

The property that distinguishes a figure from its mirror image. Rotation preserves it and reflection reverses it, which is what makes the clockwise-order test decisive for asymmetric figures.

Net

A flat arrangement of faces that folds into a solid. On a cube net, squares two apart in a straight strip become opposite faces and squares sharing an edge become adjacent ones.

Plan view

The view looking straight down on an object. It fixes the footprint of a stacked-cube solid, which is why it is the first view to read.

Elevation

A view looking horizontally at an object, usually given as front and side. Each elevation caps the heights it shows but hides anything directly behind.

Orthographic projection

A drawing convention showing an object through separate parallel views with no perspective. Together the views constrain a solid but do not always determine it.

Bearing

A direction measured clockwise from north and written with three digits, so due east is 090 and a heading slightly west of north is 350. The reverse bearing differs by 180 degrees.

Egocentric versus allocentric frame

The difference between directions relative to a traveller who is turning and directions relative to a fixed map. Left means different compass headings in the two frames, which is why headings must be restated before every turn.

Cross-section

The shape revealed where a plane cuts a solid. It gains one straight edge for each flat face the plane crosses and one curve for each curved surface, which turns the question into a count.

Study this next

• Spatial reasoning skill hub
• Spatial reasoning practice bank
• Mechanical comprehension lesson
• Firefighter spatial reasoning suite
• Architecture and design aptitude suite
• Spatial reasoning lesson on Novus Learn

Where this material comes from

• Worked items written for Novus Learn from standard introductory geometry: cube-net opposite-face rules, orthographic view reconstruction, reflection across fold lines, right-triangle distances and bearings, and the conic and polyhedral cross-sections. The 3-4-5 distance, the 053 and 233 bearings, and the four hole coordinates were each computed and checked.
• Terminology checked against the public Wikipedia articles 'Net (polyhedron)', 'Multiview orthographic projection', 'Chirality', 'Bearing (navigation)' and 'Conic section'. Definitions only; every figure described above is original.
• Novus Learn aptitude construct registry (catalog seed) for the construct scope and related suite mapping.
• Public educational framing only: not affiliated with any official exam board, publisher or employer, and no copyrighted test item is reproduced.