CA Foundation Paper 4 cheat sheet

Simple interest: SI = P × r × t; amount A = P(1 + rt). Compound interest: A = P(1 + r)^n for annual compounding; for m conversions per year use A = P(1 + r/m)^(mn).

Effective annual rate = (1 + r/m)^m − 1; continuous limit gives e^r − 1. Use effective rates to compare nominal rates with different compounding frequencies.

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

Permutations vs combinations: nPr = n!/(n − r)! orders selections; nCr = n!/[r!(n − r)!] ignores order. Identities: nCr = nC(n − r); nCr + nC(r − 1) = (n + 1)Cr.

Probability rules: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A′) = 1 − P(A). Independent events: P(A ∩ B) = P(A)P(B). Conditional: P(A|B) = P(A ∩ B)/P(B).

Measures of central tendency: mean = Σx/n (grouped: Σfx/Σf); median position = (n + 1)/2 for raw data; mode = value with highest frequency, or mode = L + [(f1 − f0)/(2f1 − f0 − f2)] × h for grouped data.

Dispersion: range = max − min; quartile deviation = (Q3 − Q1)/2; mean deviation = Σ|x − x̄|/n; variance = Σ(x − x̄)²/n (population) or /(n − 1) (sample); SD = √variance. Coefficient of variation = (SD/mean) × 100.

Correlation coefficient r lies in [−1, 1]; r = Σ(x − x̄)(y − ȳ)/[√Σ(x − x̄)² × √Σ(y − ȳ)²]. Regression y on x: y − ȳ = b_yx(x − x̄) with b_yx = r(SD_y/SD_x); b_xy × b_yx = r².

Index numbers: Laspeyres = Σ(p1q0)/Σ(p0q0) × 100; Paasche = Σ(p1q1)/Σ(p0q1) × 100; Fisher ideal = √(Laspeyres × Paasche). Time reversal and factor reversal tests are satisfied by Fisher.

Syllogism validity: a conclusion follows only if it holds in every Venn-model consistent with the premises. Universal affirmative (All A are B) does not convert to All B are A; it converts to Some B are A.

Number series and coding-decoding: identify the operation pattern (constant difference, ratio, alternating, squares/cubes ± k) before testing options; verify a rule against at least two consecutive pairs.

Blood relations: draw a family tree with gender markers; vertical lines for generations, horizontal for siblings/spouse. Never infer gender from a name alone.

Direction sense: track net displacement with a coordinate grid; final distance = √(x² + y²) for perpendicular moves; use Pythagoras for diagonal shortcuts.

Seating arrangement: fix one anchor person, then place constraints one at a time; when two arrangements satisfy all clues, the question is under-determined and both must be reported.

Odd-one-out and analogy: test the defining attribute (class, function, part-whole, degree). The correct answer shares the relation type, not surface similarity.

Calendar: ordinary year shifts the day by 1, leap year by 2. Century years are leap only if divisible by 400. Odd days in 100 years = 5, in 400 years = 0.

Clock: angle between hands = |30H − 5.5M| degrees. Hands coincide every 65 5/11 minutes; are opposite every 65 5/11 minutes offset by 32 8/11 minutes.

Cubes and dice: for an n × n × n painted cube, cubes with 3 painted faces = 8 corners; 2 faces = 12(n − 2); 1 face = 6(n − 2)²; 0 faces = (n − 2)³.

Class interval rules: equal width preferred; inclusive classes (e.g., 10–19) converted to exclusive (9.5–19.5) before computing midpoints. Midpoint = (lower + upper)/2.

Cumulative frequency (ogive) plots class boundaries against cumulative frequency; the median is read at N/2, quartiles at N/4 and 3N/4.

Weighted mean = Σwx/Σw. Combined mean of two groups = (n1x̄1 + n2x̄2)/(n1 + n2); combined variance requires the between-group term n1(x̄1 − x̄)² + n2(x̄2 − x̄)².

Skewness: for a moderately skewed distribution, mean − mode ≈ 3(mean − median). Karl Pearson's coefficient = (mean − mode)/SD.

Moments: μ1 = 0 for deviations about the mean; μ2 = variance; μ3 measures skewness; μ4 measures kurtosis. β1 = μ3²/μ2³ and β2 = μ4/μ2².

Normal distribution: symmetric, mean = median = mode; approximately 68% of observations within 1 SD, 95% within 2 SD, 99.7% within 3 SD.

Scatter diagrams show direction and strength of association but not causation; a high |r| can arise from a lurking third variable.

Sampling: simple random, stratified, systematic and cluster are probability methods; quota and convenience sampling are non-probability and cannot support margin-of-error statements.

Ratio and proportion: if a : b = c : d then ad = bc. Compound ratio of a : b and c : d is ac : bd. Continued proportion a : b = b : c gives b² = ac.

Percentage change and successive changes: net change for +a% then +b% is a + b + ab/100. A rise of x% followed by a fall of x% leaves a net loss of x²/100 %.

Profit and loss: profit % = (SP − CP)/CP × 100 when based on cost; if based on selling price, convert to cost base first. Marked price with discount d: SP = MP(1 − d).

Time, speed and distance: distance = speed × time; average speed for equal distances at v1 and v2 = 2v1v2/(v1 + v2), not the arithmetic mean.

Time and work: if A finishes in a days and B in b days, together they take ab/(a + b) days. Work rates add; use LCM of days as total work units.

Simple and compound interest comparison: CI − SI for 2 years = P(r/100)²; for 3 years = P(r/100)²(3 + r/100).

Permutations with repetition: arrangements of n items where p are alike, q alike = n!/(p! q!). Circular arrangements of n distinct items = (n − 1)!; necklace (flip allowed) = (n − 1)!/2.

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

Logarithm use in growth problems: log(1 + i)^n = n log(1 + i); solve for n = log(A/P)/log(1 + i) when compounding annually.

Percentage-point vs percentage change: a move from 20% to 25% is 5 percentage points but a 25% relative increase. Read the question wording before dividing.

Table interpretation: check units and totals first; a row or column total that does not reconcile signals a misread or a 'others' category.

Ratio comparison across tables: convert to a common base (per unit or per 100) before ranking; raw totals mislead when bases differ.

Average of a series with a missing term: use the given mean to back-solve, then verify the term satisfies any stated ordering condition.

Approximation discipline: in DI, options are usually spaced enough that rounding to two significant figures selects the answer; exact arithmetic is needed only when options are close.

Letter series: convert letters to positions (A = 1 … Z = 26); alternating series often use two independent progressions on odd and even positions.

Alphanumeric series questions: answer by position counting from the stated end, and re-count after each rearrangement step rather than tracking mentally.

Mean is affected by every observation; median and mode are resistant to outliers. If a distribution has extreme values, the mean shifts while the median stays near the centre.

Changing the scale of every observation by multiplying by k multiplies the mean and SD by |k| and the variance by k²; adding a constant c shifts the mean but leaves SD and variance unchanged.

The sum of deviations from the arithmetic mean is always zero: Σ(x − x̄) = 0. This is the fastest check on a computed mean.

Geometric mean = (x1 × x2 × … × xn)^(1/n), used for rates of growth; harmonic mean = n/Σ(1/x), used for averaging speeds or ratios. GM ≤ AM always.

For grouped data, the assumed-mean or step-deviation method gives the same mean as the direct method; a mismatch indicates an arithmetic slip, not a different answer.

Correlation is unit-free and unaffected by change of origin or scale; regression coefficients are not unit-free and do change with scale.

A conclusion about a population from a sample requires a stated sampling method and sample size; descriptive statistics from a full population need no inference.

Always state units and the base year in index-number answers; a price index without a base year is not interpretable.

Equation framing: assign the unknown the quantity asked for, translate each sentence into one equation, and check the number of independent equations equals the number of unknowns.

Mixtures and alligation: ratio of quantities = (mean price − cheaper price) : (dearer price − mean price). Repeated replacement reduces pure quantity to P(1 − k/V)^n.

Simple interest installments and equated payments: treat each installment as a future value and equate to the principal plus interest at the settlement date.

Boats and streams: downstream speed = b + s, upstream = b − s; time ratio downstream : upstream = (b − s) : (b + s) for equal distances.

Partnership: profit shares follow capital × time. If capitals are in ratio a : b for periods in ratio m : n, shares are am : bn.

Permutation restriction rule: for 'not all together', compute total arrangements minus arrangements with the group treated as one block.

Data sufficiency: decide whether each statement alone is sufficient before combining; do not compute a numeric answer when only sufficiency is asked.

Verification habit: substitute the answer back into the original condition, and reject roots that violate domain constraints (negative time, non-integer people).