CA Foundation Paper 4 flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Simple interest formula

SI = P × r × t, where P is principal, r the annual rate as a decimal, and t the time in years. Amount = P + SI = P(1 + rt).

Compound interest formula

A = P(1 + r)^n for annual compounding, where n is the number of years. Interest = A − P. For m conversions per year use A = P(1 + r/m)^(mn).

Effective annual rate

The true annual growth rate when compounding occurs more than once a year: (1 + r/m)^m − 1. It lets you compare nominal rates that have different compounding frequencies.

Annuity present value

PV = A × [1 − (1 + i)^(−n)]/i, where A is the regular payment, i the rate per period and n the number of periods. Multiply by (1 + i) for an annuity-due.

Permutation nPr

The number of ordered arrangements of r items chosen from n distinct items: n!/(n − r)!. Order matters, so AB and BA count as different arrangements.

Combination nCr

The number of unordered selections of r items from n distinct items: n!/[r!(n − r)!]. Order does not matter, so AB and BA count as one selection.

Addition rule of probability

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events the intersection term is zero, so the probabilities simply add.

Conditional probability

P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0. For independent events P(A|B) = P(A), which is equivalent to P(A ∩ B) = P(A)P(B).

Arithmetic mean

The sum of all observations divided by their number: Σx/n for raw data, or Σfx/Σf for grouped data where f is frequency and x the class midpoint.

Median of grouped data

The value at cumulative frequency N/2, found as L + [(N/2 − cf)/f] × h, where L is the lower boundary, cf the cumulative frequency before the median class.

Mode of grouped data

The most frequent value, computed as L + [(f1 − f0)/(2f1 − f0 − f2)] × h, where f1 is the modal class frequency and f0, f2 the adjacent frequencies.

Standard deviation

The positive square root of variance: √[Σ(x − x̄)²/n]. It is expressed in the same units as the data, unlike variance, which is in squared units.

Coefficient of variation

CV = (standard deviation / mean) × 100, expressed as a percentage. It compares variability across data sets with different units or very different means.

Quartile deviation

QD = (Q3 − Q1)/2, also called semi-interquartile range. Its coefficient is (Q3 − Q1)/(Q3 + Q1), a unit-free measure of spread around the median.

Karl Pearson correlation coefficient

r = Σ(x − x̄)(y − ȳ)/[√Σ(x − x̄)² × √Σ(y − ȳ)²]. It ranges from −1 to +1, is unit-free, and is unchanged by shifting or rescaling either variable.

Regression coefficients product

For two variables, b_yx × b_xy = r², where b_yx is the slope of y on x. The two regression lines coincide only when r² = 1, that is perfect correlation.

Laspeyres price index

Σ(p1q0)/Σ(p0q0) × 100, using base-year quantities as weights. It tends to overstate price rises because it ignores substitution toward cheaper goods.

Fisher ideal index

The geometric mean of the Laspeyres and Paasche indices: √(L × P). It satisfies both the time reversal and factor reversal tests, unlike either component index.

Syllogism validity test

A conclusion is valid only if it is true in every Venn diagram consistent with the premises. One counterexample diagram is enough to reject the conclusion.

Clock hand angle

The angle between hour and minute hands is |30H − 5.5M| degrees, where H is the hour and M the minutes past the hour. Take the smaller angle if it exceeds 180.

Odd days in a century

100 ordinary years contain 5 odd days and 400 years contain 0. An ordinary year shifts the weekday by 1, a leap year by 2, because 365 = 52 weeks + 1 day.

Painted cube formula

For an n × n × n cube painted on all faces: 3 painted faces = 8; 2 faces = 12(n − 2); 1 face = 6(n − 2)²; 0 faces = (n − 2)³.

Successive percentage change

Two successive changes of a% and b% produce a net change of a + b + ab/100 per cent. A rise then an equal fall of x% always leaves a net loss of x²/100 per cent.

Average speed for equal distances

When equal distances are covered at speeds v1 and v2, average speed = 2v1v2/(v1 + v2), the harmonic mean. It is always less than the arithmetic mean of the speeds.

Time and work together

If A alone takes a days and B alone takes b days, working together they finish in ab/(a + b) days. Work rates, not times, are the quantities that add.

Alligation rule

For mixing two components at prices c and d to obtain a mean price m, the quantity ratio is (m − c) : (d − m), with c < m < d.

CI minus SI shortcut

For two years, compound interest exceeds simple interest by P(r/100)². For three years the excess is P(r/100)²(3 + r/100).

Circular permutation

The number of ways to arrange n distinct objects around a circle is (n − 1)!. If clockwise and anticlockwise orders are treated as identical, as in a necklace, divide by 2.

Combined mean of groups

For two groups of sizes n1 and n2 with means x̄1 and x̄2, the combined mean is (n1x̄1 + n2x̄2)/(n1 + n2). It is a size-weighted average, not a simple average of means.

Scaling effect on dispersion

Multiplying every observation by k multiplies the mean and standard deviation by |k| and the variance by k². Adding a constant shifts the mean but leaves dispersion unchanged.

Empirical skewness relation

For moderately skewed distributions, mean − mode ≈ 3(mean − median). Karl Pearson's coefficient of skewness is (mean − mode)/standard deviation.

Normal distribution rule

A normal curve is symmetric with mean = median = mode. Roughly 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.