ReferenceUpdated June 2026

The numerical reasoning formula sheet

Every formula the tests actually use — there are fewer than thirty. Each one comes with its classic trap, because in numerical reasoning the formula is rarely the problem; applying it to the wrong base is. Bookmark this page; revising it the night before is a legitimate strategy.

Percentages

Percentage of a numberpart = rate% × whole

Trap: "% of" multiplies — it never divides.

Percentage changechange% = (new − old) ÷ old × 100

Trap: The base is always the OLD value — using the new value is the most common mistake in the entire test.

Reverse percentage (find the original)original = final ÷ (1 ± rate)

Trap: After a 12% rise, divide by 1.12 — never subtract 12% from the final.

Successive changestotal = (1 ± r₁)(1 ± r₂) − 1

Trap: +10% then −10% is −1%, not 0 — percentages never simply add.

Percentage points vs percentpp = simple difference of two rates

Trap: From 20% to 25% is +5pp but +25% relative.

Ratios & proportion

Share in a ratioshare = total × part ÷ (sum of parts)

Trap: Divide by the sum of ALL parts (1:2:5 → ÷8), not by the largest part.

Solve a proportiona/b = c/x → x = b·c ÷ a

Trap: Cross-multiply once; don't chain operations.

Scaling (direct proportion)new = old × (new basis ÷ old basis)

Trap: Check whether the quantity scales up or down before multiplying.

Averages

Meanmean = sum ÷ count
Weighted averagemean = Σ(value × weight) ÷ Σ(weights)

Trap: Group sizes are the weights — averaging two group means equally is wrong unless the groups are equal.

Missing value from a meanmissing = mean × count − sum(known)
Combined mean of two groups(n₁m₁ + n₂m₂) ÷ (n₁ + n₂)

Rates, speed & conversion

Speed–distance–timespeed = distance ÷ time (cover the one you want)

Trap: Minutes must become hours before mixing with mph — 90 min = 1.5 h, not 0.9.

Rate per unitrate = output ÷ input
Currency conversionamount × rate (to foreign) · amount ÷ rate (back)

Trap: Check which direction the quoted rate goes before multiplying.

Per-capitaper head = total ÷ population

Money & finance

Simple interestinterest = principal × rate × years
Compound interestfinal = principal × (1 + rate)ʸᵉᵃʳˢ

Trap: Compound multiplies each year — simple-interest shortcuts undershoot.

VAT / sales taxgross = net × (1 + rate); net = gross ÷ (1 + rate)
Break-evenunits = fixed costs ÷ (price − variable cost)
Margin vs markupmargin = profit ÷ price · markup = profit ÷ cost

Trap: Same profit, different base — a 25% markup is a 20% margin.

Index numbersvalue = base value × index ÷ 100

Probability

Single eventP = favourable ÷ total
ComplementP(not A) = 1 − P(A)
Independent events both occurringP(A and B) = P(A) × P(B)

Trap: Multiply for AND (both), add for OR (either, minus overlap).

Two-set overlap (Venn)either = A + B − both

Knowing the formula isn’t the test

The tests measure whether you apply these fast, on unfamiliar data, under a clock. Ten minutes of practice tells you which of these you only think you know.

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