CA Foundation (New Scheme)

Ratio, Indices and Logarithms

• Compounding ratiosa:b and b:c → make b common: if a:b = 2:3 and b:c = 4:5 then a:b:c = 8:12:15
• Proportiona:b = c:d ⇔ ad = bc; mean proportional of a, b = √(ab)
• Indicesaᵐ·aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ
• Log ruleslog(mn) = log m + log n, log(m/n) = log m − log n, log mⁿ = n log m
• Change of baselogₐ b = log b / log a, logₐ b × log_b a = 1

💡 For logₓ k = n questions rewrite as xⁿ = k first; it solves most of them in one line.

Mathematics of Finance

• Compound amountA = P(1 + r/k)ⁿᵏ for k compoundings a year
• Effective rateE = (1 + r/k)ᵏ − 1
• PV of annuityPV = A × [1 − (1 + i)⁻ⁿ] / i
• FV of annuityFV = A × [(1 + i)ⁿ − 1] / i
• Annuity duemultiply the ordinary annuity value by (1 + i)
• PerpetuityPV = A / i; growing: PV = A / (i − g)

💡 Always divide the annual rate and multiply the years by the number of compoundings per year before using any formula.

Permutations and Combinations

• PermutationsⁿPᵣ = n! / (n − r)!
• CombinationsⁿCᵣ = n! / [r!(n − r)!], ⁿCᵣ = ⁿCₙ₋ᵣ
• Circular(n − 1)! arrangements around a table
• Repeated itemsn! / (p! q! r!) for p, q, r alike objects

💡 Order matters → P; only selection matters → C. Fill restricted places first.

Sequences and Series

• APtₙ = a + (n − 1)d, Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
• GPtₙ = arⁿ⁻¹, Sₙ = a(rⁿ − 1)/(r − 1), S∞ = a/(1 − r) for |r| < 1
• Standard sumsΣn = n(n + 1)/2, Σn² = n(n + 1)(2n + 1)/6, Σn³ = [n(n + 1)/2]²

💡 Count terms with n = (last − first)/d + 1 before summing; off-by-one is the usual error.

Calculus Basics

• Derivativesd/dx xⁿ = nxⁿ⁻¹, d/dx eˣ = eˣ, d/dx log x = 1/x
• Product and quotient(uv)′ = u′v + uv′, (u/v)′ = (u′v − uv′)/v²
• Marginal costMC = dC/dx; marginal revenue MR = dR/dx
• Integrals∫xⁿ dx = xⁿ⁺¹/(n + 1) + c (n ≠ −1), ∫(1/x) dx = log x + c

💡 Maxima: set f′(x) = 0 and check f″(x) < 0.

Central Tendency and Dispersion

• Combined meanx̄ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)
• Empirical relationMode = 3 Median − 2 Mean
• MeansAM ≥ GM ≥ HM for positive numbers; for two numbers GM² = AM × HM
• Change of origin and scaleif y = a + bx then ȳ = a + bx̄ and SD(y) = |b| × SD(x)
• Coefficient of variationCV = (SD / Mean) × 100
• Quartile deviationQD = (Q₃ − Q₁)/2

💡 SD and other dispersion measures ignore change of origin but not change of scale, and they are never negative.

Probability and Distributions

• AdditionP(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• ConditionalP(A | B) = P(A ∩ B) / P(B)
• BayesP(Aᵢ | B) = P(Aᵢ)P(B | Aᵢ) / Σ P(Aⱼ)P(B | Aⱼ)
• Binomialmean = np, variance = npq (variance < mean)
• Poissonmean = variance = m
• Normalmean = median = mode; QD : MD : SD ≈ 10 : 12 : 15

💡 If a question gives binomial mean and variance, q = variance/mean; q must be below 1 or the data is impossible.

Correlation and Regression

• Correlationr = cov(x, y) / (σₓσᵧ), −1 ≤ r ≤ 1
• From regression coefficientsr² = b_yx × b_xy; r takes the common sign of both
• Regression linesboth lines pass through (x̄, ȳ); solve them together to get the means
• Spearman rankR = 1 − 6Σd² / [n(n² − 1)]

💡 If your assumed coefficients give b_yx × b_xy > 1, you assumed the lines the wrong way round: swap them.

Index Numbers

• LaspeyresΣp₁q₀ / Σp₀q₀ × 100 (base-year quantities)
• PaascheΣp₁q₁ / Σp₀q₁ × 100 (current-year quantities)
• Fisher√(Laspeyres × Paasche), the 'ideal' index
• TestsFisher satisfies both time reversal and factor reversal tests

💡 Remember 'L uses base, P uses present' to avoid mixing up Laspeyres and Paasche.

Logical Reasoning Shortcuts

• Directionsfacing North: right = East, left = West; draw every turn on paper
• Distancenet displacement = √(net east-west² + net north-south²)
• Blood relationsdraw a family tree: + for male, − for female, horizontal line for siblings
• Seriescheck differences, then ratios, then squares/cubes ± 1, then alternate terms

💡 Logical Reasoning is 20 quick marks; never leave it for the last 10 minutes.