ADF JOA Numerical cheat sheet

Check first differences before anything else; constant first differences mean an arithmetic sequence with rule a_n = a_1 + (n-1)d.

Constant ratios between consecutive terms indicate a geometric sequence; a_n = a_1 × r^(n-1).

If first differences form a simple pattern, take second differences; constant second differences mean a quadratic rule of the form an² + bn + c.

Alternating signs usually signal a multiplier with a negative ratio, or two interleaved series.

When terms grow fast, test factorials, squares, cubes, or doubling-plus-a-constant before assuming a polynomial.

Two interleaved series (odd positions, even positions) is the standard fix when a single series shows no clean rule.

Fibonacci-type rules: each term equals the sum (or sum plus constant) of the two preceding terms.

For series of fractions, treat numerator and denominator as separate series before comparing whole values.

Prime-number series and square/cube series are common distractors; test whether terms sit at n²±1 or n³±1.

Order of operations: brackets, indices, division/multiplication left to right, then addition/subtraction left to right.

Distributive shortcut: a × b + a × c = a(b + c); factor common terms to reduce work.

Difference of squares: a² − b² = (a − b)(a + b), useful for products like 63 × 57 = 60² − 3².

Multiplying by 5, 25, 50: halve and append zeros (×5 = ×10 ÷ 2; ×25 = ×100 ÷ 4; ×50 = ×100 ÷ 2).

Squaring numbers ending in 5: result ends in 25 and begins with n(n+1) where n is the leading part.

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

Estimate first by rounding to one significant figure to catch place-value errors before exact computation.

Subtraction check: add the answer back to the subtrahend; if it does not return the minuend, the result is wrong.

Percent change = (new − old) ÷ old × 100; a rise then equal fall of x% leaves a net loss of x²/100 percent.

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

To reverse a percentage change, divide by the multiplier rather than subtracting the stated percent.

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

Direct proportion: a/b = c/d; inverse proportion: a × b = c × d (one rises as the other falls).

Percentage points differ from percent: going from 20% to 25% is a 5-point rise but a 25% relative increase.

To compare fractions or ratios, cross-multiply or convert to decimals rather than eyeballing.

Successive percentage changes multiply: two 10% rises give ×1.1 ×1.1 = ×1.21, a 21% total increase.

Read the axis scale and units before extracting any number; a common trap is a scale that jumps by 5, 20, or 100.

For tables, confirm whether totals are given or must be computed; never assume a column sums correctly.

Pie charts: each sector's value equals its percentage of the stated total; angles sum to 360° and percentages to 100%.

Line graphs show trend and rate of change; bar charts compare magnitudes - choose the comparison the question asks for.

When asked for an average across categories, divide by the number of categories, not by the number of data points.

Compute ratios between two rows or years only after aligning the same units and the same category definitions.

Approximate to one or two significant figures when answer choices are widely spaced; exact arithmetic wastes time.

Watch for 'other' or 'not stated' categories: they are often needed to make a total reach 100%.

Translate words to an equation: 'is' means equals, 'of' means multiply, 'per' means divide, 'more than' means add.

Work-rate rule: combined rate = 1/a + 1/b; time together = ab ÷ (a + b) for two workers.

Mixture rule: amount of pure substance is conserved, so concentration × volume before equals concentration × volume after.

Average speed for equal distances = 2xy ÷ (x + y), not the arithmetic mean of the two speeds.

Set up a table of quantities (rate, time, distance or price, quantity, total) to keep unknowns aligned.

For age problems, express every person's age relative to one variable and one point in time.

Check the answer against the story: units must match, and quantities such as people or items must be whole numbers.

Metric ladder: km → m → cm → mm each step is ×10 (km to m is ×1000); mass kg → g → mg likewise.

1 litre = 1000 mL = 1000 cm³; 1 m³ = 1000 litres.

Time: 1 hour = 60 minutes = 3600 seconds; convert all times to one unit before calculating.

Perimeter of a rectangle = 2(l + w); area = l × w; area of a triangle = ½ × base × height.

Circle: circumference = 2πr; area = πr²; use π ≈ 3.14 or 22/7 as the answer choices suggest.

Volume of a rectangular prism = l × w × h; volume of a cylinder = πr²h.

Estimation: round each quantity to one significant figure, compute, then confirm the exact answer is within about 10%.

Unit conversion factor method: multiply by a fraction equal to 1 (e.g. 1000 m / 1 km) so unwanted units cancel.

Answer easy items first; mark hard items and return, since every question carries equal weight.

Eliminate answer choices by magnitude and last digit before doing full computation.

Use benchmark fractions: 1/8 = 12.5%, 1/6 ≈ 16.7%, 1/3 ≈ 33.3%, 3/8 = 37.5%, 5/8 = 62.5%.

Memorise squares to 25 and cubes to 10 to skip calculation on series and arithmetic items.

For data interpretation, read the question stem before the table or graph so you extract only the needed figures.

Check only the final step of a multi-step problem if time is short; intermediate errors usually show as implausible magnitudes.

Keep a consistent scratch layout (one problem per line) to reduce transcription errors under time pressure.