Irish Numerical Reasoning cheat sheet

Arithmetic sequence: common difference d between consecutive terms; nth term = a + (n-1)d; sum = n/2 × (first + last).

Geometric sequence: common ratio r; nth term = a × r^(n-1); sum of n terms = a(r^n - 1)/(r - 1) for r ≠ 1.

Check differences first, then second differences: constant second difference means a quadratic pattern of form an² + bn + c.

Alternating or interleaved series: split odd-positioned and even-positioned terms and treat each as its own sequence.

Ratio test: if successive terms give a constant quotient, the series is geometric; if differences are constant, it is arithmetic.

Fibonacci-type rule: each term equals the sum of the two preceding terms; verify by checking at least three consecutive terms.

Square and cube patterns: compare terms to n², n³, n² ± k, or n³ ± k before assuming a linear rule.

Prime-based series: gaps follow the prime gaps 1, 2, 2, 4, 2, 4, 2, 4, 6; test primes when no simple arithmetic rule fits.

Sum of first n integers = n(n+1)/2; sum of first n squares = n(n+1)(2n+1)/6; sum of first n cubes = [n(n+1)/2]².

Translate 'of' to multiplication, 'per' to division, and 'more than'/'less than' to addition/subtraction before computing.

Average (mean) = total sum ÷ count; total sum = mean × count, which lets you recover a missing value.

Work backwards from the answer choices by substituting them into the word problem when the algebra is messy.

Combined rate for two workers: 1/T = 1/a + 1/b, so T = ab/(a + b) for two workers completing a job together.

Simple interest = Principal × Rate × Time ÷ 100; the rate is a decimal fraction of the principal per year.

Profit or loss percentage is always calculated on cost price: % = (selling price − cost price)/cost price × 100.

Consecutive integer problems: let the middle term be n, so terms are n−1, n, n+1; the sum is 3n.

Age problems: set up equations using present ages and add or subtract the same number of years to both people.

Unit consistency check: convert all quantities to the same unit (minutes, cent, kg) before applying any formula.

Percentage change = (new − old)/old × 100; a rise then an equal-percent fall does not return to the original value.

Successive percentage changes multiply: net factor = (1 + a)(1 + b), not 1 + a + b.

A 20% increase followed by a 20% decrease gives a net 4% decrease, since 1.2 × 0.8 = 0.96.

To increase by p%, multiply by (1 + p/100); to decrease by p%, multiply by (1 − p/100).

Reverse percentage: if a value after a p% increase is X, the original is X ÷ (1 + p/100).

Ratio a : b : c splits a total T into shares aT/(a+b+c), bT/(a+b+c), cT/(a+b+c).

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

To compare ratios, convert each to a decimal or cross-multiply: a/b vs c/d compares ad vs bc.

Percentage points vs percent: a change from 10% to 15% is 5 percentage points but a 50% relative increase.

Read the axis scale and units first; a chart may start at a non-zero baseline, exaggerating apparent differences.

For pie charts, each sector angle = (value/total) × 360°, and each percentage = (value/total) × 100.

For bar charts, compare bar heights only after checking whether the vertical axis is linear and starts at zero.

For line graphs, the slope represents rate of change, not the absolute value at any single point.

When a table has a 'total' row or column, use it to cross-check your sums before answering.

Percentage of a total from a table: divide the cell by the row or column total, depending on which base the question names.

Estimate before computing: round table values to one significant figure to eliminate obviously wrong answer choices.

Watch for mixed units across columns (thousands vs millions, euro vs cent) and convert before comparing.

If a question asks for a ratio between two table entries, cancel common units so the ratio is dimensionless.

Core relation: distance = speed × time; speed = distance ÷ time; time = distance ÷ speed.

Unit conversion: 1 km/h = 5/18 m/s; 1 m/s = 18/5 km/h; multiply km/h by 5/18 to get m/s.

Average speed for equal distances = 2ab/(a + b), the harmonic mean, not the arithmetic mean (a + b)/2.

Average speed = total distance ÷ total time, so add distances and times separately before dividing.

Relative speed: same direction = difference of speeds; opposite directions = sum of speeds.

Work rate adds: if A finishes in a days and B in b days, together they take ab/(a + b) days.

If a pipe fills in x hours and a drain empties in y hours, net rate = 1/x − 1/y when both are open.

Train crossing a pole uses its own length as distance; crossing a platform uses train length plus platform length.

Work done = rate × time; express each worker's rate as a fraction of the job per unit time before combining.

Round each number to one or two significant figures before multiplying or dividing to get a fast estimate.

For addition and subtraction, round to the same place value (nearest ten, hundred) rather than to significant figures.

Estimate by front-end rounding: 497 × 21 ≈ 500 × 20 = 10,000; compare with choices to pick the closest.

Use benchmark fractions: 1/3 ≈ 0.333, 1/6 ≈ 0.167, 1/7 ≈ 0.143, 1/9 ≈ 0.111, 1/11 ≈ 0.091.

Square root bounds: if a² < n < b², then a < √n < b; use perfect squares to bracket the answer.

Rule of 70 or 72: doubling time ≈ 70 ÷ percentage growth rate per period.

Percentage estimate: 1% of a number moves the decimal two places left; 10% moves it one place left.

Check reasonableness with order of magnitude: 0.48 × 0.52 is about 0.25, not 2.5 or 0.025.

When dividing, round the divisor to a friendly number (e.g., 19.8 ≈ 20) and adjust the estimate slightly upward.

Eliminate answer choices by estimating the magnitude of the result before doing exact arithmetic.

For questions with units in the answer choices, check units first; mismatched units eliminate options immediately.

If a calculation is long, look for cancellation between numerator and denominator rather than multiplying fully.

Verify a computed answer against a rough estimate; a discrepancy of a factor of 10 signals a decimal-place error.

For multi-step problems, write the intermediate result with its unit to prevent mixing operations.

When two answer choices differ only in sign or decimal place, the question is testing place value and sign handling.

Time-per-question budgeting: if a question exceeds about 90 seconds, estimate and move on rather than computing exactly.

To find a ratio from a table, read both values from the same row or column and cancel common units.

Part-to-whole ratio from a table: divide the cell by the column total, not the row total, unless stated.

Percentage share in a table = cell ÷ grand total × 100 when the question asks for share of the whole.

If two tables have different totals, convert each to a percentage before comparing across tables.

Cross-tabulation: the marginal totals equal the sum of the interior cells in their row or column.

When a table gives rates, multiply rate by the relevant base to recover the absolute count.

Scaling ratios: if a : b = 3 : 5 and a = 12, the multiplier is 4, so b = 20.

Check whether a table's 'total' is a sum or an average; averaging rows then summing gives a different result.