CMA Foundation Paper 3 cheat sheet

Ratio a:b means a/b; to divide amount N in ratio a:b:c, shares are N·a/(a+b+c), N·b/(a+b+c), N·c/(a+b+c).

Proportion: a:b = c:d implies ad = bc; for continued proportion a:b = b:c, b² = ac and b is the mean proportional.

Percentage change = (new − old)/old × 100; successive changes of x% and y% give net change x + y + xy/100 percent.

Simple interest SI = PRT/100; amount A = P(1 + RT/100); interest is always computed on the original principal only.

Compound interest A = P(1 + R/100)^n for annual compounding; for half-yearly use rate R/2 with 2n periods.

Effective annual rate for nominal rate R compounded m times a year = (1 + R/m)^m − 1, expressed as a percentage.

Discount: true discount = simple interest on present value; banker's discount = simple interest on face value, so banker's discount exceeds true discount.

Average due date uses base date plus sum of (amount × days from base) divided by sum of amounts, giving a weighted mean of dates.

Laws of indices: a^m·a^n = a^(m+n); a^m/a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1 for a ≠ 0; a^(−n) = 1/a^n.

Laws of logarithms: log(xy) = log x + log y; log(x/y) = log x − log y; log x^n = n log x; change of base log_b x = log x / log b.

Quadratic ax² + bx + c = 0 has roots with sum −b/a and product c/a; discriminant D = b² − 4ac: D > 0 real distinct, D = 0 equal, D < 0 complex.

Arithmetic progression: nth term a + (n−1)d; sum of n terms = n/2[2a + (n−1)d] = n/2(first + last).

Geometric progression: nth term ar^(n−1); sum of n terms a(r^n − 1)/(r − 1) for r ≠ 1; infinite sum a/(1 − r) only when |r| < 1.

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

Permutations: nPr = n!/(n−r)!; combinations nCr = n!/[r!(n−r)!]; nCr = nC(n−r) and nC0 = nCn = 1.

Solve simultaneous linear equations by elimination or substitution; consistency requires the determinant of coefficients a1b2 − a2b1 ≠ 0 for a unique solution.

Triangle area = ½ × base × height; also Heron's formula Area = √[s(s−a)(s−b)(s−c)] with s = (a+b+c)/2.

Circle: circumference = 2πr, area = πr²; sector area = (θ/360)πr² and arc length = (θ/360)2πr for angle θ in degrees.

Rectangle area = l × b, perimeter = 2(l + b); parallelogram area = base × height; trapezium area = ½(a + b) × h.

Cuboid: volume = lbh, total surface area = 2(lb + bh + hl); cube with side a has volume a³ and surface area 6a².

Cylinder: volume = πr²h, curved surface = 2πrh, total surface = 2πr(r + h); cone: volume = ⅓πr²h, curved surface = πrl with l = √(r² + h²).

Sphere: volume = (4/3)πr³, surface area = 4πr²; hemisphere total surface area = 3πr².

Pythagoras theorem: in a right triangle, hypotenuse² = sum of squares of the other two sides; 3-4-5 and 5-12-13 are common triples.

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, interior angles of an n-sided polygon sum to (n − 2) × 180°.

Derivative from first principles: f′(x) = limit as h→0 of [f(x + h) − f(x)]/h; it measures instantaneous rate of change.

Power rule d/dx(x^n) = n·x^(n−1); derivative of a constant is 0; d/dx(k·f) = k·f′.

Product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Exponential and logarithmic: d/dx(e^x) = e^x; d/dx(a^x) = a^x ln a; d/dx(ln x) = 1/x.

Stationary points occur where f′(x) = 0; use f″(x) > 0 for a minimum, f″(x) < 0 for a maximum, f″(x) = 0 needs further test.

Integration is the reverse of differentiation: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C.

Definite integral ∫ from a to b of f(x) dx equals F(b) − F(a), giving the area under the curve between a and b.

Marginal cost is d(TC)/dQ and marginal revenue is d(TR)/dQ; profit is maximised where marginal revenue equals marginal cost.

Statistics is classified into descriptive statistics (collecting, organising, summarising) and inferential statistics (estimating and testing from samples).

Primary data are collected first-hand by the investigator; secondary data already exist and were collected by someone else for another purpose.

A population is the complete set of units under study; a sample is a subset selected to represent it, with sampling error decreasing as sample size grows.

Sampling methods: simple random, stratified (strata internally homogeneous), systematic (every kth item), cluster, and quota/convenience as non-random types.

An ogive is the cumulative frequency curve; the less-than ogive rises and the more-than ogive falls, intersecting at the median.

Classification is arranging data into classes; the class mark is the midpoint (lower + upper)/2 and class width is upper limit minus lower limit.

A frequency polygon joins midpoints of class frequencies; a histogram uses adjacent bars whose areas represent class frequencies for unequal widths.

Time series has four components: secular trend, seasonal variation, cyclical variation and irregular (random) variation.

Arithmetic mean of raw data = Σx/n; for grouped data x̄ = Σfx/Σf where x is the class mark; it uses every observation and is affected by outliers.

Median is the middle value of ordered data; for grouped data median = L + [(n/2 − cf)/f] × h, where L is the lower limit of the median class.

Mode is the most frequent value; for grouped data mode = L + [(f1 − f0)/(2f1 − f0 − f2)] × h using the modal class.

Empirical relation for moderately skewed data: Mode ≈ 3 Median − 2 Mean; geometric mean of n values = (product of values)^(1/n).

Range = largest − smallest; quartile deviation = (Q3 − Q1)/2; mean deviation = Σ|x − A|/n taken about the mean or median.

Standard deviation σ = √[Σ(x − x̄)²/n] for a population; variance is σ²; for grouped data use Σfx²/Σf − (Σfx/Σf)².

Coefficient of variation = (σ / x̄) × 100 percent; it compares variability between series with different units or means.

A distribution with mean > median > mode is positively skewed; mean < median < mode indicates negative skewness.

Karl Pearson's coefficient r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² · Σ(y − ȳ)²]; r lies between −1 and +1 and is unit-free.

r = +1 is perfect positive correlation, r = −1 perfect negative, r = 0 indicates no linear relationship; correlation does not imply causation.

Spearman's rank correlation ρ = 1 − 6Σd²/[n(n² − 1)], where d is the difference of ranks; apply this when data are ordinal.

Regression line of y on x: y − ȳ = b_yx(x − x̄), where b_yx = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² = r·σy/σx.

Regression line of x on y: x − x̄ = b_xy(y − ȳ), where b_xy = r·σx/σy; generally b_yx ≠ b_xy and their geometric mean equals r.

The two regression lines intersect at the point (x̄, ȳ); r² = b_yx × b_xy and is the coefficient of determination.

If b_yx > 1 then b_xy < 1; both regression coefficients always carry the same sign, which is the sign of r.

The regression line of y on x minimises the sum of squared vertical deviations and is used to estimate y for a given x.

Classical probability P(A) = number of favourable outcomes / total equally likely outcomes; 0 ≤ P(A) ≤ 1 and P(A′) = 1 − P(A).

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the intersection term is zero.

Multiplication rule: P(A ∩ B) = P(A) × P(B|A); for independent events P(A ∩ B) = P(A) × P(B).

Conditional probability P(B|A) = P(A ∩ B)/P(A) for P(A) > 0; Bayes' theorem updates P(A|B) using prior probabilities and likelihoods.

Expected value E(X) = Σ x·P(x) for a discrete random variable; it is the probability-weighted mean of outcomes.

Variance of a random variable Var(X) = E(X²) − [E(X)]², and standard deviation is its positive square root.

For a binomial distribution with n trials and success probability p: P(X = r) = nCr·p^r·(1 − p)^(n−r), mean = np, variance = np(1 − p).

For a Poisson distribution P(X = r) = e^(−λ)·λ^r/r!, where mean = variance = λ; it models counts of rare events in a fixed interval.

In a fair game the expected net gain is zero; a positive expected value means the bettor gains on average over many repetitions.