SPJAT cheat sheet

Percentage change: increase then decrease of a% and b% gives net change a + b + ab/100 percent; equal rise and fall of x% always yields a net loss of x²/100 percent.

Profit and loss: SP = CP × (100 + profit%)/100; when two articles sell at the same price with equal gain% and loss%, the seller always loses (loss% = x²/100).

Simple interest SI = PRT/100; compound interest CI = P(1 + R/100)^T − P. For 2 years, CI − SI = P(R/100)².

Time-speed-distance: average speed for equal distances at speeds u and v is 2uv/(u+v); for equal times it is the simple mean (u+v)/2.

Trains and boats: relative speed adds for opposite directions, subtracts for same direction; downstream speed = boat + stream, upstream = boat − stream.

Work: if A alone takes a days and B alone b days, together they take ab/(a+b) days; efficiency is inversely proportional to time taken.

Ratio and mixtures: in an alloy of ratio a:b, replacing quantity with the same component keeps total constant; use alligation (cheaper:dearer = (mean − cheap):(dearer − mean)).

Permutation and combination: nPr = n!/(n−r)!; nCr = n!/[r!(n−r)!]; number of diagonals of an n-sided polygon = n(n−3)/2.

Number properties: divisibility by 3 or 9 depends on digit sum; by 4 on last two digits; by 8 on last three digits; by 11 on alternating digit sum difference.

Syllogism: a conclusion follows only if true in every possible Venn diagram consistent with the premises; 'some A are B' does not imply 'some B are not A'.

Blood relations: draw a family tree with gender symbols; 'brother of my father's only son' resolves to self or brother, not uncle.

Direction sense: track net displacement with a coordinate grid; final distance uses Pythagoras when turns are at right angles.

Coding-decoding: check letter position shifts (A=1...Z=26), reversal, or alternating patterns before assuming a single rule.

Seating arrangement: fix the most constrained person first, then apply left/right and facing-direction rules; clockwise/anticlockwise reverses for people facing outward.

Series: test differences, then ratios, then squares/cubes or alternating sub-series; letter series often skip by a fixed step.

Clocks and calendars: angle between hands = |30H − 5.5M| degrees; in a century there are 24 leap years (or 25 if the century is divisible by 400).

Statement-assumption: an assumption is valid only if it is necessary for the statement to hold, not merely plausible.

Subject-verb agreement: 'each', 'every', 'either', 'neither' take singular verbs; 'neither...nor' agrees with the nearer subject.

Parallelism: items joined by and/or/but must share grammatical form; a list of verbs should not mix gerunds and infinitives.

Modifier placement: a participial phrase at the start modifies the subject of the main clause, so the subject must be the logical doer.

Pronoun reference: a pronoun must have one clear antecedent; ambiguous 'it/they/this' is a common error to fix.

Comparatives: use 'than' after comparative adjectives; 'as...as' for equality; avoid double comparatives like 'more better'.

Idioms and prepositions: 'comprise' takes no 'of'; 'superior to', 'prefer to', 'consist of', 'capable of' are fixed pairs.

Reading comprehension: main-idea answers must cover the whole passage, not one paragraph; extreme words (always, never) often signal wrong options.

Vocabulary in context: infer meaning from surrounding contrast or cause-effect cues rather than memorized definitions.

Read the axis units and scale first; a bar chart in thousands versus a table in units causes most calculation errors.

Percentage of total = (part/total) × 100; to compare shares across years, recompute totals rather than comparing raw parts.

Growth rate = (new − old)/old × 100; average annual growth over n years uses the geometric mean, not the arithmetic mean.

Ratio comparison across categories: convert each to a common base (per unit or per 100) before ranking.

Pie charts: each sector angle = (value/total) × 360°; a 1% share equals 3.6 degrees.

Approximation strategy: round to two significant figures when options are far apart, but keep exact values when options differ by less than 5%.

Mixed charts: extract only the series the question names; ignore unrelated legends to save time.

Table totals: verify row and column sums before answering; a mismatch signals a misread cell.