Civil Service Numerical Test cheat sheet

Order of operations: brackets first, then powers and roots, then multiplication and division left to right, then addition and subtraction left to right.

Divisibility: by 3 or 9 when the digit sum is a multiple of 3 or 9; by 4 when the last two digits form a multiple of 4; by 8 when the last three digits do.

Dividing by a fraction equals multiplying by its reciprocal: a ÷ (p/q) = a × (q/p).

Difference of squares: a² − b² = (a + b)(a − b); for example 61² − 59² = 120 × 2 = 240.

Squaring a number ending in 5: multiply the leading part by the next integer and append 25, so 75² → 7 × 8 = 56 → 5625.

Multiplying a two-digit number by 11: insert the digit sum between the digits and carry when the sum exceeds 9, so 68 × 11 = 748.

Sign rules: negative × negative = positive, negative × positive = negative; an odd number of negative factors gives a negative product.

Subtraction by complement: 1000 − 764 = 236, found by subtracting each digit from 9 and the final digit from 10.

Check the axis before reading values: units, scale step, and whether the vertical axis starts at zero; a truncated axis exaggerates differences between bars.

Pie chart: share = sector angle ÷ 360; sector angle = share × 360°.

Two-way table: row percentage = cell ÷ row total; column percentage = cell ÷ column total; the grand total equals both the sum of row totals and the sum of column totals.

Percentage change between two figures = (later value − earlier value) ÷ earlier value × 100.

Line graph slope = vertical change ÷ horizontal change; equal slopes indicate equal rates of change.

Mean of plotted values = sum of the values ÷ number of points; it is not the midpoint between the highest and lowest plotted values.

In stacked bars, segment height = share × total, so segments can be compared only when the totals are the same.

Chart labels may be in thousands or millions; convert to the units used in the question before calculating.

x% of N = (x ÷ 100) × N; increasing by x% multiplies by (1 + x/100); decreasing by x% multiplies by (1 − x/100).

Successive changes multiply: a 20% rise then a 20% fall gives 1.2 × 0.8 = 0.96, a net fall of 4%.

Percentage points measure an absolute gap between two percentages; percent measures a relative change from the starting percentage.

Reverse a percentage change: original = new ÷ (1 ± r/100).

A ratio a:b contains a + b parts; the value of one part = total ÷ (a + b).

To compare ratios, convert each to a decimal or to a common second term.

Direct proportion y = kx: doubling one quantity doubles the other. Inverse proportion y = k/x: doubling one halves the other.

Percent ↔ fraction: fraction × 100 gives percent; percent ÷ 100 gives fraction; simplify before multiplying.

Mean = sum ÷ count; median = middle value of an ordered list; mode = most frequent value.

Weighted mean = Σ(weight × value) ÷ Σ(weights).

Combined mean of two groups = (n₁m₁ + n₂m₂) ÷ (n₁ + n₂).

Speed = distance ÷ time; distance = speed × time; time = distance ÷ speed.

Average speed for a whole journey = total distance ÷ total time; it equals the simple mean of the leg speeds only when each leg takes equal time.

1 m/s = 3.6 km/h; 1 km = 1000 m; 1 hour = 3600 seconds.

Metric conversions: 1 kg = 1000 g; 1 tonne = 1000 kg; 1 litre = 1000 ml; 1 m² = 10,000 cm².

Adding a value above the current mean raises the mean, adding one below it lowers the mean, and the deviations of all values from the mean sum to zero.

Assign a letter to the unknown, then write one equation for each stated relationship before doing any arithmetic.

Work rates add: combined rate = 1/a + 1/b, so the time together = ab ÷ (a + b).

Mixtures: pure amount = concentration × volume; on mixing, pure amounts add and volumes add, giving the new concentration.

Catch-up problems: time = starting gap ÷ (faster speed − slower speed); for head-on approach: time = gap ÷ (sum of speeds).

When only the final amount is given after a chain of changes, work backwards by inverting each operation in reverse order.

Translation rules: 'of' → ×, 'per' → ÷, 'is' → =, 'more than' → +, 'less than' → −.

Age problems: the difference between two people's ages stays constant however many years pass.

Substitute the chosen answer back into the original wording and confirm the units and size are plausible.

Simple interest = P × r × t; total amount = P(1 + rt), with r as a decimal per year and t in years.

Compound amount = P(1 + r)^n, where r is the decimal rate per period and n is the number of periods.

Profit = selling price − cost price; profit % = profit ÷ cost price × 100, measured against cost rather than selling price.

Discount: price after discount = marked price × (1 − d/100); two successive discounts multiply rather than add.

Tax added to a net price: gross = net × (1 + rate/100).

Budget variance = actual − budgeted; a positive figure is an overspend, and variance % = variance ÷ budgeted × 100.

Total cost = fixed cost + (variable cost per unit × units); break-even units = fixed cost ÷ (price − variable cost per unit).

Share of budget = line item ÷ total budget × 100; all line items must sum to the total budget.

Round every number to one or two significant figures, compute, then select the closest option; refine only when two options sit near the estimate.

Percentage anchors: 10% shifts the decimal one place left, 5% is half of that, 1% shifts two places, and sums give 15%, 25% or 35%.

Benchmark fractions: 1/6 ≈ 16.7%, 1/7 ≈ 14.3%, 1/8 = 12.5%, 1/9 ≈ 11.1%, 1/11 ≈ 9.1%.

Round one factor up and the other down so errors partly cancel: 48 × 52 ≈ 50 × 50 = 2500.

√n lies between the roots of the two neighbouring perfect squares; interpolate between them for a decimal estimate.

π ≈ 3.14 or 22/7; use 3 when only a rough area or circumference is needed.

Magnitude check: if the estimate is out by a factor of ten relative to the options, a decimal point has been misplaced.

Round every number to one or two significant figures, compute, then select the closest option; refine only when two options sit near the estimate.

Percentage anchors: 10% shifts the decimal one place left, 5% is half of that, 1% shifts two places, and sums give 15%, 25% or 35%.

Benchmark fractions: 1/6 ≈ 16.7%, 1/7 ≈ 14.3%, 1/8 = 12.5%, 1/9 ≈ 11.1%, 1/11 ≈ 9.1%.

Round one factor up and the other down so errors partly cancel: 48 × 52 ≈ 50 × 50 = 2500.

√n lies between the roots of the two neighbouring perfect squares; interpolate between them for a decimal estimate.

π ≈ 3.14 or 22/7; use 3 when only a rough area or circumference is needed.

Magnitude check: if the estimate is out by a factor of ten relative to the options, a decimal point has been misplaced.