CA Foundation Paper 3 cheat sheet

Simple interest: SI = P × r × t / 100, where r is annual rate in percent and t is time in years; amount A = P + SI.

Compound interest: A = P(1 + r/100)^n for annual compounding; for m conversions per year use A = P(1 + r/(100m))^(mn).

Effective annual rate from nominal rate r compounded m times: EAR = (1 + r/(100m))^m − 1, expressed as a decimal.

Annuity present value (ordinary): PV = A × [1 − (1 + i)^(−n)] / i, where i is the rate per period and n the number of payments.

Annuity future value (ordinary): FV = A × [(1 + i)^n − 1] / i; for annuity-due multiply by (1 + i).

Permutations: nPr = n! / (n − r)!; combinations: nCr = n! / [r!(n − r)!]; nCr = nC(n−r).

Logarithm rules: log(ab) = log a + log b; log(a/b) = log a − log b; log(a^k) = k log a; log_b a = log a / log b.

Ratio and proportion: if a:b = c:d then ad = bc; in a mixture, component amount = (ratio part / sum of parts) × total quantity.

Equation of a straight line: y = mx + c with slope m = (y2 − y1)/(x2 − x1); two lines are perpendicular when m1 × m2 = −1.

Number series: check successive differences first; if differences are constant the series is arithmetic; if ratios are constant it is geometric.

Coding–decoding: map each letter to its position (A=1 … Z=26) and test shifts, reversals, or position-sum rules before assuming a pattern.

Syllogism: a conclusion follows only if it is true in every possible Venn diagram consistent with the premises; a possibility conclusion fails if one counter-diagram exists.

Blood relations: draw a family tree using vertical lines for generations and horizontal lines for siblings/spouses; never assume gender from a name.

Direction sense: track net displacement with signed coordinates; final distance = √(x² + y²) and final direction uses the signs of x and y.

Seating arrangement: fix the person with the most constraints first; for circular arrangements of n distinct people there are (n − 1)! distinct clockwise orders.

Calendar: a normal year shifts the weekday by 1 and a leap year by 2; 100 years give 5 odd days, 200 give 3, 300 give 1, and 400 give 0.

Clock: angle between hands = |30H − 5.5M| degrees; hands coincide 11 times in 12 hours and are opposite 11 times in 12 hours.

Analogy and classification: identify the relation (part–whole, cause–effect, worker–tool) and apply the same relation; in classification, find the one item lacking the common property.

Arithmetic mean: x̄ = Σx / n for raw data; for grouped data x̄ = Σfx / Σf using class midpoints.

Median for grouped data: M = L + [(N/2 − cf) / f] × h, where L is lower class boundary, cf cumulative frequency before the median class, f its frequency, h class width.

Mode for grouped data: Mo = L + [(f1 − f0) / (2f1 − f0 − f2)] × h, where f1 is modal frequency and f0, f2 adjacent frequencies.

Quartiles: Q1 = L + [(N/4 − cf)/f] × h and Q3 = L + [(3N/4 − cf)/f] × h; quartile deviation = (Q3 − Q1)/2.

Mean deviation about mean = Σ|x − x̄| / n for raw data; coefficient = MD / mean.

Standard deviation: σ = √(Σ(x − x̄)² / n) for population; coefficient of variation = (σ / x̄) × 100.

Combined mean of two groups: x̄ = (n1x̄1 + n2x̄2) / (n1 + n2); combined variance uses within-group and between-group deviations.

Correlation coefficient r lies between −1 and 1; r = 0 means no linear correlation; regression line of y on x is y − ȳ = b_yx (x − x̄) with b_yx = r σy/σx.

Probability: P(A or B) = P(A) + P(B) − P(A and B); for independent events P(A and B) = P(A) × P(B); conditional P(A|B) = P(A and B)/P(B).

Present value of a single sum: PV = FV / (1 + i)^n; discount factor decreases as i or n increases.

Sinking fund: periodic deposit A = FV × i / [(1 + i)^n − 1], where i is rate per period and n number of deposits.

Perpetuity present value: PV = A / i for a level payment A received at the end of every period forever.

Bills of exchange: true discount = Amount × r × t / (100 + r × t); banker's discount = Amount × r × t / 100; banker's gain = BD − TD.

Average due date: base date plus weighted average of days from base, weights being amounts; use for settling multiple debts on one date.

Matrix multiplication is defined only when columns of the first equal rows of the second; AB ≠ BA in general, and AB = 0 does not imply A = 0 or B = 0.

Determinant of a 2×2 matrix [[a,b],[c,d]] is ad − bc; a matrix is singular when its determinant is zero and has no inverse.

Set operations: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for three sets subtract pairwise intersections and add back the triple intersection.

Statement–assumption: an assumption is something the speaker takes for granted; reject options that restate the statement or add unrelated conditions.

Statement–argument: a strong argument is relevant, substantial, and follows from the statement; weak arguments are trivial, emotional, or ambiguous.

Cause–effect: the stated cause must precede and plausibly produce the effect; a common cause may explain both events without direct causation.

Course of action: a valid action must be practical, address the problem directly, and be within the authority implied by the statement.

Input–output and machine logic: track each step's transformation rule (sorting, shifting, swapping) and apply it to the new input, not the old one.

Puzzle with multiple attributes: build a grid with one row per attribute value and mark eliminations; a confirmed match eliminates all other options in that row and column.

Ranking: if A is 7th from the left and 12th from the right in a row, total people = 7 + 12 − 1 = 18.

Data sufficiency: decide whether the data is sufficient, not what the answer is; each statement is evaluated independently unless the question says otherwise.

Measures of dispersion: range = maximum − minimum; coefficient of range = (max − min)/(max + min).

For a symmetrical distribution, mean = median = mode; for a moderately skewed distribution, mode ≈ 3 median − 2 mean.

Karl Pearson's skewness = (mean − mode)/σ or 3(mean − median)/σ; positive skew means a longer right tail.

Index numbers: Laspeyres uses base-year quantities, Paasche uses current-year quantities; Fisher's ideal index is the geometric mean of the two.

Time series: four components are trend, seasonal, cyclical, and irregular; a moving average smooths short-term fluctuations to isolate trend.

Probability distribution: for a discrete random variable, ΣP(x) = 1 and expected value E(X) = Σx·P(x).

Binomial distribution: P(X = r) = nCr p^r q^(n−r) with mean np and variance npq; Poisson mean and variance both equal λ.

Normal distribution is symmetric about its mean; approximately 68% of values lie within 1σ, 95% within 2σ, and 99.7% within 3σ.

Sampling: in simple random sampling every unit has an equal chance; a statistic is a function of sample values, a parameter describes the population.