How to Use These
Nine worked items across the five families that appear in almost every numerical battery. Each one carries the full working and, more usefully, the error it was built to catch.
Attempt each before reading on. A worked solution you have not first struggled with teaches very little.
Family 1: Percentages of a Total
Item 1
Regional revenue for the quarter, in £m: North 42, South 58, East 31, West 29.
What percentage of total revenue came from South? (a) 29% (b) 36.25% (c) 40% (d) 58%
Answer: (b). Total = 42 + 58 + 31 + 29 = 160. South = 58 ÷ 160 = 0.3625, so 36.25%.
The trap is (d). Under time pressure the eye lands on 58, and 58 also appears as a percentage option. Any figure that appears in both the table and the answer list should be treated as suspicious.
Item 2
A company's market share rose from 12% to 15% over one year.
By what percentage did its market share increase? (a) 3% (b) 15% (c) 20% (d) 25%
Answer: (d). The change is 3 points on a base of 12: 3 ÷ 12 = 0.25, so a 25% increase.
The trap is (a). It rose by three percentage points, which is not a 3% rise. This single distinction is responsible for more lost marks than any other in numerical testing.
Family 2: Reverse Percentages
Item 3
After a 20% discount, a piece of equipment costs £96.
What was the price before the discount? (a) £76.80 (b) £115.20 (c) £120 (d) £124
Answer: (c). £96 represents 80% of the original, so the original is 96 ÷ 0.8 = £120. Check: 120 × 0.8 = 96. ✓
The trap is (b). Adding 20% to £96 gives £115.20, which feels right and is not. Percentages are not reversible by adding back what you subtracted, because the base has changed.
Family 3: Rates and Proportion
Item 4
A team of 4 people processes 1,200 claims in 5 working days.
At the same rate per person per day, how many claims would 6 people process in 10 days? (a) 1,800 (b) 2,400 (c) 3,000 (d) 3,600
Answer: (d). Reduce to a unit rate first: 1,200 ÷ (4 × 5) = 60 claims per person per day. Then 6 × 10 = 60 person-days, and 60 × 60 = 3,600.
Why the unit rate matters. Scaling both quantities at once invites doubling one and forgetting the other — (a) is what you get by scaling the days and not the people.
Family 4: Data Extraction
Item 5
A table headed Units sold (thousands) shows: 2024 — 425; 2025 — 340.
By what percentage did units sold fall? (a) 20% (b) 25% (c) 85% (d) Cannot be determined
Answer: (a). The fall is 425 − 340 = 85, on a base of 425: 85 ÷ 425 = 0.2, so 20%.
Two things to notice. The "thousands" heading is irrelevant here because it applies to both figures and cancels in a ratio — but it would matter instantly if the question asked for an absolute number.
The trap is (b). Dividing by the smaller figure, 85 ÷ 340, gives 25%. Percentage change is always measured against the starting value.
Item 6
Department A has 20 staff with an average score of 60. Department B has 80 staff with an average of 80.
What is the average score across both departments? (a) 70 (b) 74 (c) 76 (d) 78
Answer: (c). Weight by headcount: (20 × 60) + (80 × 80) = 1,200 + 6,400 = 7,600, over 100 staff = 76.
The trap is (a). Averaging the two averages gives 70, which is only correct when the groups are the same size. Here B is four times larger, so the combined figure must sit much closer to 80.
Family 5: Sequences
Item 7
3, 6, 11, 18, 27, ? (a) 34 (b) 36 (c) 38 (d) 40
Answer: (c). The differences are 3, 5, 7, 9 — consecutive odd numbers. The next difference is 11, giving 27 + 11 = 38.
The method that always works. Write the differences underneath before looking for anything cleverer. If the differences are not obvious, take the differences of the differences.
The Two Items That Separate Candidates
Item 8: when the data does not support an answer
A chart shows that 34% of survey respondents in Region A rated the service positively, and 41% in Region B did.
How many more people rated the service positively in Region B than in Region A? (a) 7 (b) 70 (c) 700 (d) Cannot be determined
Answer: (d). Percentages without base sizes cannot produce a count. If Region A surveyed 5,000 people and Region B surveyed 200, more people rated it positively in A despite the lower percentage.
Why this item matters. Candidates who never select "cannot be determined" are guessing on every item of this type — and candidates who select it whenever they feel unsure lose the items where the base was given three lines up.
Item 9: estimation as a weapon
What is 17% of 2,480? (a) 42.2 (b) 148.8 (c) 421.6 (d) 4,216
Answer: (c). 10% is 248, and 7% is 173.6, so the total is 421.6.
The point is not the arithmetic. A single glance at 10% = 248 eliminates three of the four options before any real calculation begins. On a paper you cannot finish, that is worth more than accuracy.
Which Family Costs the Most Marks?
Percentage change, by a wide margin — and not because it is hard. It is the only family where a candidate can execute the arithmetic perfectly and still be wrong, because the error is in choosing the base.
Data extraction runs second, and its errors are the most dangerous because they are silent. A figure taken from the wrong row is then processed flawlessly, and nothing about the result feels wrong.
Are Real Tests Harder Than This?
Not in mathematical content. Real papers use the same operations, and they stop at the same place.
They are harder in three specific ways: the tables carry considerably more data than any one question uses, the time per item is tighter than feels reasonable, and several items chain off a single dataset — so one misread figure can cost three marks rather than one.
That last property is the argument for re-checking your extraction rather than your arithmetic. The arithmetic on these papers is rarely where the marks go.