EPSO Numerical Reasoning cheat sheet

Percentage change = (new − old) ÷ old × 100; the denominator is always the original/base value, never the new one.

Percentage points vs percent: a rise from 20% to 25% is +5 percentage points but +25% relative change.

Increase by p% means multiply by (1 + p/100); decrease by p% means multiply by (1 − p/100).

Successive changes multiply: +20% then −20% gives 1.2 × 0.8 = 0.96, a net −4%, not 0%.

To reverse a change, divide rather than subtract: undoing +25% requires ÷1.25 (= −20%), not −25%.

'x is what percent of y' = x ÷ y × 100; 'x is p% of y' = p/100 × y; 'x is p% more than y' = x = y(1 + p/100).

A value that falls by p% needs a rise of p/(100 − p) × 100 to recover: a 20% fall needs +25%.

Multiplier form: a change from a to b corresponds to multiplier b/a; percentage change = (b/a − 1) × 100.

Ratio a : b means parts a and b of a total of a + b; each part = total ÷ (a + b).

Scaling all terms of a ratio by the same nonzero factor leaves the ratio unchanged; add terms only after matching units.

To split an amount A in ratio a : b : c, the shares are A × a/(a+b+c), A × b/(a+b+c), A × c/(a+b+c).

Direct proportion: y = kx, so y1/x1 = y2/x2. Inverse proportion: xy = k, so x1y1 = x2y2.

Combining two ratios: if a : b = 2 : 3 and b : c = 4 : 5, make b common (12) to get a : b : c = 8 : 12 : 15.

Map/scale problems: actual = scale length × scale factor; areas scale by the square of the linear factor, volumes by the cube.

To compare ratios as fractions: a : b > c : d exactly when ad > bc (cross-multiplication with positive terms).

If a quantity is shared so that one share is known, find the multiplier first: other share = known share × (other ratio term ÷ known ratio term).

Arithmetic mean = sum ÷ count; total = mean × count is the workhorse for reconstructing sums.

Weighted mean = Σ(w_i x_i) ÷ Σw_i; weights may be counts, populations, hours or amounts, not percentages unless they sum to 100.

Combined mean of two groups = (n1m1 + n2m2) ÷ (n1 + n2); it lies between the two means, closer to the larger group's mean.

Equal-sized groups: combined mean is the simple midpoint of the two means.

If the combined mean is known, the group-size ratio is found from n1/n2 = (m2 − M)/(M − m1).

Sum of deviations from the mean is zero; adding a constant c to every value raises the mean by c, multiplying by k multiplies the mean by k.

Median is the middle value of ordered data (mean of the two middle values if n is even); it is unaffected by extreme outliers, the mean is not.

Removing one item from a set: new total = old mean × n − removed value, then divide by n − 1.

Core relation: distance = speed × time; speed = distance ÷ time; time = distance ÷ speed. Keep units consistent (km/h with hours, m/s with seconds).

Unit conversion: 1 m/s = 3.6 km/h; to convert km/h to m/s divide by 3.6.

Average speed for a whole journey = total distance ÷ total time, not the mean of the segment speeds.

Equal distances at speeds a and b give average speed 2ab/(a + b), the harmonic mean.

Relative speed: same direction = |a − b|; opposite directions = a + b.

Meeting/catching problems: time to meet = initial gap ÷ relative speed.

Trains and bridges: a train fully clears a point in length ÷ speed and clears a bridge in (train length + bridge length) ÷ speed.

Upstream/downstream: downstream speed = boat + current, upstream = boat − current; boat speed = (down + up)/2.

Common fraction–decimal anchors: 1/8 = 0.125, 1/6 ≈ 0.1667, 1/3 ≈ 0.3333, 3/8 = 0.375, 5/8 = 0.625, 1/7 ≈ 0.1429, 1/9 ≈ 0.1111.

To compare fractions a/b and c/d with positive denominators, cross-multiply: a/b > c/d when ad > bc.

Dividing by a fraction is multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Percent ↔ fraction: p% = p/100; e.g. 12.5% = 1/8, 37.5% = 3/8, 62.5% = 5/8, 75% = 3/4.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiply mantissas and add exponents, divide mantissas and subtract exponents.

Rounding to one significant figure first gives a quick bound: if the estimate and the answer options differ by more than the rounding error, the estimate is decisive.

A fraction's value is unchanged if numerator and denominator are multiplied or divided by the same nonzero number; simplify before multiplying large numbers.

Absolute change = new − old (same units as data); relative change = absolute change ÷ base × 100 (unit-free).

The largest absolute increase need not be the largest percentage increase: a small base produces a big percentage from a small absolute move.

Stacked/segmented charts show parts of a whole: read each segment's share as segment ÷ total, and remember the total may change between periods.

Pie charts encode proportions only: a slice's angle = share × 360°, and slice sizes cannot be compared across two pies with different totals.

Line charts can use a truncated y-axis, which exaggerates visual slopes; check the axis start and step before judging growth.

Index numbers rebase a series to 100 at a base period: value in period t = (actual_t ÷ actual_base) × 100.

When a table gives totals and one component, the remainder is total minus the component; percentages of a changing total are not additive across periods.

Rate per unit (e.g. per capita, per employee) = absolute figure ÷ population or count; compare rates, not raw totals, when group sizes differ.

Work rate adds: if A does 1/a of a job per hour and B does 1/b, together they do 1/a + 1/b, so combined time = ab/(a + b).

Time and rate are inversely proportional: doubling the number of identical machines at the same rate halves the time.

General work formula: work = rate × time; for n identical workers with individual rate r, total work = n × r × t.

If A is k times as fast as B, A does k/(k+1) of the work when they work together.

Worker-days: total work = workers × days (at constant daily output); use this to rescale when workers or days change.

Pipes and tanks: filling rates add, emptying rates subtract; net rate = Σ fill − Σ drain.

Resource allocation: allocate a fixed budget or headcount in the given ratio, then check that the sum of allocations equals the total.

Efficiency comparisons: output per worker = total output ÷ number of workers; output per hour = output ÷ hours worked.

Profit = revenue − costs; profit margin = profit ÷ revenue × 100; markup = profit ÷ cost × 100.

Revenue = price × quantity; a p% price cut with a q% volume rise gives new revenue = old × (1 − p/100)(1 + q/100).

Indexation: indexed value = base value × (index_new ÷ index_base); real change = nominal change adjusted by the same index ratio.

Compound growth: final = initial × (1 + r)^n; simple growth: final = initial × (1 + nr).

VAT/sales tax: price including tax = net × (1 + rate); to strip tax, divide by (1 + rate), do not subtract the rate.

Budget variance = actual − budgeted; a negative variance on costs is favourable, a negative variance on revenue is unfavourable.

Depreciation by a fixed annual percentage is compound: value after n years = cost × (1 − r)^n.

Break-even quantity = fixed costs ÷ (unit price − unit variable cost).

Probability = favourable outcomes ÷ total equally likely outcomes; 0 ≤ P ≤ 1 and P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B). Mutually exclusive events: P(A or B) = P(A) + P(B).

General addition rule: P(A or B) = P(A) + P(B) − P(A and B).

Permutations (order matters): nPr = n! ÷ (n − r)!. Combinations (order does not matter): nCr = n! ÷ [r!(n − r)!].

Arrangements of a multiset with repeated items: n! ÷ (a! b! …) for repeated counts a, b.

With replacement, probabilities stay constant; without replacement, the denominator and numerator both fall by one after each draw.

Expected value = Σ(value × probability); it is a long-run average, not a guaranteed outcome.

At least one occurrence: P(at least one) = 1 − P(none); for n independent trials with success p this is 1 − (1 − p)^n.

Sufficiency test: a statement is sufficient only if it forces a single numerical answer; two different valid answers mean insufficient.

Check each piece of information alone first, then combined; do not assume unstated constraints such as integers, positivity or ordering.

Statement types: 'must be true' needs a proof for all cases; 'could be true' needs one valid example; 'cannot be true' needs a contradiction.

Counterexample method: to reject a general claim, one valid numerical counterexample is enough.

Watch for hidden assumptions in percentages: 'a 10% rise then a 10% fall returns to the start' is false because the bases differ.

Validate totals: percentages of a whole must sum to 100; shares of a budget must sum to the budget; a component cannot exceed its total.

Rounding traps: an answer that matches the exact computation but not the rounded one may be a distractor; recompute with full precision before choosing.

Correlation is not causation: a table showing two series moving together does not establish that one causes the other.