CA Foundation Jan 2027 Maths formula sheet: chapter-wise
This CA Foundation Maths formula sheet lists every formula Paper 3 (Quantitative Aptitude) needs for January 2027, chapter by chapter, in the notation of ICAI's own study material. Each table says where the formula is used, and the most-tested ones carry a one-line example or the trap that turns them into a wrong option. Bookmark it or print it: it is the formula sheet PDF most students search for, checked against ICAI's chapters.
Paper 3 is 100 MCQs in 2 hours with no reading time, and each wrong answer costs a quarter mark (Guidance Notes, PDF (opens in a new tab); January 2027 schedule, PDF (opens in a new tab)). That is about 72 seconds a question, so a formula you have to derive in the hall is a formula you don't have. The full paper format is in CA Foundation exam pattern and marking.
Does ICAI give a CA Foundation Maths formula sheet?
No. ICAI publishes no separate formula sheet, and none is printed in the question paper. The formulas sit inside the 18 chapters of the Paper 3 study material, which for January 2027 is the August 2024 edition (Reprint August 2025 onwards), with no RTP named for this paper (ICAI notice on boslive.icai.org, PDF (opens in a new tab)). The chapter PDFs are free on ICAI's website, icai.org, and linked from our CA Foundation ICAI study material page. Every table below links the chapter it comes from.
Two scope notes before you trust any other sheet:
- No trigonometry. ICAI's syllabus covers calculus "Excluding the trigonometric applications" (Paper 3 syllabus, PDF (opens in a new tab)). Sheets full of sin and cos identities, complex numbers or conic sections are school sheets, not Foundation sheets.
- Logical Reasoning has no formulas. Part B (Chapters 9 to 12: number series and coding, directions, seating, blood relations) is 20 marks of method. Practise it as puzzles. Part A Business Mathematics and Part C Statistics are 40 marks each.
Most-tested CA Foundation Maths formulas
If you have one evening, these do most of the work. For how many questions each chapter actually gets, see the counts in the CA Foundation Jan 2027 Maths mock test guide; this table doesn't repeat them.
| Formula | Chapter | One-line check |
|---|---|---|
| A = P(1 + i)ⁿ | 4 | ₹10,000 at 12% compounded quarterly for 1 year: i = 0.03, n = 4, A = ₹11,255.09 |
| E = (1 + i)ⁿ − 1 | 4 | Same deposit: E = 12.55%, never the nominal 12% |
| Annuity FV = A[(1 + i)ⁿ − 1] ÷ i | 4 | ₹10,000 a year, 10%, 5 years: ₹61,051 |
| Annuity PV = A[(1 + i)ⁿ − 1] ÷ [i(1 + i)ⁿ] | 4 | Same annuity: ₹37,907.87 |
| EMI = V ÷ P(n, i) | 4 | ₹1,00,000 loan, 12% a year, 12 monthly EMIs: ₹8,884.88 |
| ⁿPᵣ = n! ÷ (n − r)!; ⁿCᵣ = n! ÷ [r!(n − r)!] | 5 | ⁵P₂ = 20, ⁵C₂ = 10 |
| Sₙ = n/2 × [2a + (n − 1)d] | 6 | 1 + 2 + … + 100 = 5,050 |
| Mode = 3 Median − 2 Mean | 14 | Mean 50, median 48: mode 44 |
| CV = σ ÷ x̄ × 100 | 14 | Compare spread of two series with different means |
| r = ±√(b_yx × b_xy) | 17 | b_yx = 0.8, b_xy = 0.45: r = 0.6 |
| Fisher = √(Laspeyres × Paasche) | 18 | L = 125, P = 120: Fisher = 122.47 |
| Binomial mean np, variance npq | 16 | n = 6, p = ½: mean 3, variance 1.5 |
Ratio, indices and logarithms formulas (Chapter 1)
From Chapter 1 (PDF) (opens in a new tab).
| Topic | Formula |
|---|---|
| Laws of indices | aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1 ÷ aⁿ |
| Proportion | If a : b = c : d, then ad = bc; mean proportional of a and b = √(ab) |
| Duplicate, triplicate ratio | a² : b² and a³ : b³; sub-duplicate √a : √b |
| Log rules | log(mn) = log m + log n; log(m/n) = log m − log n; log mⁿ = n log m |
| Change of base | logₐm = log_b m ÷ log_b a, so log_b a × logₐb = 1; logₐa = 1; logₐ1 = 0 |
Trap: log(m + n) is not log m + log n. If an option comes from that step, it is the wrong one.
Equations, sets and limits (Chapters 2 and 7)
From Chapter 2 (PDF) (opens in a new tab) and Chapter 7 (PDF) (opens in a new tab).
| Topic | Formula |
|---|---|
| Quadratic ax² + bx + c = 0 | x = [−b ± √(b² − 4ac)] ÷ 2a |
| Roots | Sum = −b/a; product = c/a; b² − 4ac > 0 real and unequal, = 0 equal, < 0 not real |
| Equation from roots α, β | x² − (α + β)x + αβ = 0 |
| Sets | n(A ∪ B) = n(A) + n(B) − n(A ∩ B); De Morgan: (A ∪ B)′ = A′ ∩ B′ |
| Subsets | A set of n elements has 2ⁿ subsets; n(A × B) = n(A) × n(B) |
| Standard limits | lim (xⁿ − aⁿ) ÷ (x − a) = naⁿ⁻¹ as x → a; lim (eˣ − 1) ÷ x = 1 and lim (aˣ − 1) ÷ x = logₑa as x → 0; lim (1 + 1/n)ⁿ = e as n → ∞ |
Fastest check for a quadratic MCQ: put each option's roots into "sum = −b/a, product = c/a" instead of solving. For x² − 5x + 6 = 0 the roots must add to 5 and multiply to 6, so 2 and 3.
Mathematics of finance formula sheet (Chapter 4)
From Chapter 4, Mathematics of Finance (PDF) (opens in a new tab). ICAI writes i for the rate per conversion period as a decimal and n for the number of periods.
| What you need | Formula |
|---|---|
| Simple interest | SI = Pit; amount A = P(1 + it) |
| Compound amount | A = P(1 + i)ⁿ; i = annual rate ÷ conversions a year, n = years × conversions a year |
| Compound interest | P[(1 + i)ⁿ − 1] |
| Effective rate | E = (1 + i)ⁿ − 1, with n = conversions in one year |
| Depreciation (reducing balance) | Value after n years = P(1 − r)ⁿ |
| FV, annuity regular | A × [(1 + i)ⁿ − 1] ÷ i |
| FV, annuity due | FV of annuity regular × (1 + i) |
| PV, annuity regular | V = A × P(n, i), where P(n, i) = [(1 + i)ⁿ − 1] ÷ [i(1 + i)ⁿ] |
| PV, annuity due | PV of annuity regular × (1 + i), same as first payment + annuity regular for n − 1 periods |
| EMI | A = V ÷ P(n, i) |
| Sinking fund | Payment = Target × i ÷ [(1 + i)ⁿ − 1] |
| Perpetuity | PV = R ÷ i |
| Growing perpetuity | PV = R ÷ (i − g), first payment R at the end of period 1 |
| Bond value | PV of each coupon + PV of the redemption amount, at the investor's required rate |
| CAGR | (Ending value ÷ Beginning value)^(1/n) − 1 |
| NPV | PV of inflows − PV of outflows; accept if NPV > 0 |
| Real rate (ICAI's form) | Nominal rate − Inflation = Real rate |
Worked example 1: annuity due or regular?
Question (written by us): you will deposit ₹10,000 at the start of each year for 5 years at 10% compounded annually. What is the amount at the end of 5 years?
- First payment is today, so it is an annuity due.
- FV of annuity regular = 10,000 × [(1.1)⁵ − 1] ÷ 0.1 = 10,000 × 6.1051 = ₹61,051.
- Annuity due = 61,051 × 1.1 = ₹67,156.10.
The trap option is ₹61,051, the answer for deposits made at the end of each year. Look for "beginning", "in advance" or "today" in the question before you choose the formula.
ICAI's own bond example is worth one read: a 3-year ₹1,000 bond paying 10% a year, bought to earn 14%, is worth 100 ÷ 1.14 + 100 ÷ 1.14² + 1,100 ÷ 1.14³ ≈ ₹907.13 (ICAI's rounded factors give ₹907.125). When the required return is above the coupon rate, the price is below par; check that before you calculate.
Permutations, combinations and sequences (Chapters 5 and 6)
From Chapter 5 (PDF) (opens in a new tab) and Chapter 6 (PDF) (opens in a new tab).
| Topic | Formula |
|---|---|
| Permutations | ⁿPᵣ = n! ÷ (n − r)!; 0! = 1 |
| With repeated items | n! ÷ (p! q! r!) when p, q, r items are alike |
| Combinations | ⁿCᵣ = n! ÷ [r!(n − r)!]; ⁿCᵣ = ⁿCₙ₋ᵣ; ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ |
| Circular arrangement | (n − 1)! |
| Necklace or garland | ½ × (n − 1)!, since clockwise and anticlockwise look the same |
| AP | tₙ = a + (n − 1)d; Sₙ = n/2 × [2a + (n − 1)d] = n/2 × (first + last) |
| GP | tₙ = arⁿ⁻¹; Sₙ = a(rⁿ − 1) ÷ (r − 1) for r ≠ 1 (or a(1 − rⁿ) ÷ (1 − r), same value) |
| GP sum to infinity | a ÷ (1 − r), only when −1 < r < 1 |
| AM and GM of a, b | AM = (a + b) ÷ 2, GM = √(ab); AM ≥ GM for positive numbers |
| Special series | Σn = n(n + 1)/2; Σn² = n(n + 1)(2n + 1)/6; Σn³ = [n(n + 1)/2]² |
Five people at a round table: (5 − 1)! = 24 ways. Five beads on a necklace: 24 ÷ 2 = 12. And ¹⁰C₃ = ¹⁰C₇ = 120, so when r is large, work with n − r.
Calculus formulas (Chapter 8)
From Chapter 8 (PDF) (opens in a new tab).
| Topic | Formula |
|---|---|
| Derivatives | d/dx(xⁿ) = nxⁿ⁻¹; d/dx(eˣ) = eˣ; d/dx(aˣ) = aˣ logₑa; d/dx(log x) = 1/x; d/dx(c) = 0 |
| Product, quotient | (uv)′ = u′v + uv′; (u/v)′ = (u′v − uv′) ÷ v² |
| Maxima and minima | Set dy/dx = 0; d²y/dx² < 0 gives a maximum, > 0 a minimum |
| Marginal cost | MC = d(TC)/dx; marginal revenue = d(TR)/dx |
| Integrals | ∫xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + c (n ≠ −1); ∫(1/x) dx = log x + c; ∫eˣ dx = eˣ + c |
In calculus ICAI takes log x to mean logₑx.
CA Foundation Statistics formulas (Chapters 13 and 14)
Statistics is Part C, 40 marks. From Chapter 13 Unit II on sampling (PDF) (opens in a new tab), Chapter 14 Unit I on central tendency (PDF) (opens in a new tab) and Unit II on dispersion (PDF) (opens in a new tab).
| What you need | Formula |
|---|---|
| Median, grouped | l₁ + [(N/2 − Nₗ) ÷ (Nᵤ − Nₗ)] × C; Nₗ, Nᵤ = less-than cumulative frequencies at the median class's lower and upper limits, so Nᵤ − Nₗ is its frequency |
| Mode, grouped | l₁ + [(f₀ − f₋₁) ÷ (2f₀ − f₋₁ − f₁)] × C |
| Empirical relation | Mean − Mode = 3(Mean − Median), so Mode = 3 Median − 2 Mean |
| AM, GM, HM | AM ≥ GM ≥ HM for positive values; for two numbers, GM² = AM × HM |
| Combined mean | (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂) |
| Range and coefficient | L − S; (L − S) ÷ (L + S) × 100 |
| Quartile deviation | (Q₃ − Q₁) ÷ 2; coefficient (Q₃ − Q₁) ÷ (Q₃ + Q₁) × 100 |
| Standard deviation | σ = √[Σ(x − x̄)² ÷ n] = √[Σx² ÷ n − x̄²] |
| Coefficient of variation | σ ÷ x̄ × 100 |
| Combined SD | √{[n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²] ÷ (n₁ + n₂)}, d₁ = x̄₁ − x̄, d₂ = x̄₂ − x̄ |
| y = a + bx | ȳ = a + bx̄; σ_y = abs(b) × σ_x (adding a leaves range, QD, MD and SD unchanged; CV changes, since the mean moves) |
| Standard error of mean | σ ÷ √n with replacement; × √[(N − n) ÷ (N − 1)] without replacement |
| Standard error of proportion | √(PQ ÷ n), Q = 1 − P; same correction without replacement |
Worked example 2: combined standard deviation
Question (written by us): Group A has 20 students with mean 50 and SD 4. Group B has 30 students with mean 60 and SD 5. Find the combined SD.
- Combined mean = (20 × 50 + 30 × 60) ÷ 50 = 56.
- d₁ = 50 − 56 = −6; d₂ = 60 − 56 = 4.
- σ² = (20 × 16 + 30 × 25 + 20 × 36 + 30 × 16) ÷ 50 = 2,270 ÷ 50 = 45.4.
- σ = √45.4 = 6.74 (about).
The trap is leaving out the d² terms: √(1,070 ÷ 50) = 4.63. When the group means differ, the combined SD is always bigger than that.
Probability and theoretical distributions (Chapters 15 and 16)
From Chapter 15 (PDF) (opens in a new tab) and Chapter 16 (PDF) (opens in a new tab).
| Rule | Formula |
|---|---|
| Addition | P(A ∪ B) = P(A) + P(B) − P(A ∩ B); drop the last term if mutually exclusive |
| Conditional | P(B/A) = P(A ∩ B) ÷ P(A) |
| Independent events | P(A ∩ B) = P(A) × P(B) |
| Complement, "at least one" | P(at least one) = 1 − P(none) |
| Expected value | E(x) = Σ x·p(x); variance = E(x²) − [E(x)]² |
| Binomial | P(x) = ⁿCₓ pˣ qⁿ⁻ˣ; mean np; variance npq, always less than the mean |
| Binomial mode | Largest integer in (n + 1)p; if (n + 1)p is a whole number, two modes: (n + 1)p and (n + 1)p − 1 |
| Poisson | P(x) = e⁻ᵐ mˣ ÷ x!; mean = variance = m |
| Normal | Mean = median = mode = μ; QD = 0.675σ; MD = 0.8σ; Q₁, Q₃ = μ ∓ 0.675σ |
ICAI's areas under the normal curve, which its answer keys use: μ ± σ covers 0.6828, μ ± 2σ covers 0.9546 and μ ± 3σ covers 0.9973. Learn ICAI's figures; other books round them to 0.6827 and 0.9545.
Binomial mode in one line: n = 10, p = 0.3 gives (n + 1)p = 3.3, so the mode is 3.
Correlation and regression formulas (Chapter 17)
From Chapter 17 (PDF) (opens in a new tab).
| What you need | Formula |
|---|---|
| Karl Pearson's r | r = Cov(x, y) ÷ (σₓ × σ_y), with Cov(x, y) = Σxy ÷ n − x̄ȳ |
| Spearman's rank correlation | r_R = 1 − 6Σd² ÷ [n(n² − 1)] |
| Concurrent deviations | r_c = ±√[±(2c − m) ÷ m]; m = pairs − 1; use the sign of (2c − m) inside and outside |
| Regression coefficients | b_yx = r × σ_y ÷ σₓ; b_xy = r × σₓ ÷ σ_y |
| Regression lines | y − ȳ = b_yx(x − x̄); x − x̄ = b_xy(y − ȳ); both pass through (x̄, ȳ) |
| r from the two coefficients | r = ±√(b_yx × b_xy), same sign as the coefficients |
| Coefficient of determination | r²; non-determination = 1 − r² |
| Origin and scale | r keeps its value (sign flips if the two scale factors differ in sign); regression coefficients ignore origin but change with scale |
Spearman in one line: 5 ranks with Σd² = 10 gives 1 − 60 ÷ 120 = 0.5. To decide which of two given lines is y on x, assume one, read off both coefficients and multiply: the product is r², so it can't be above 1. If it is, swap. A worked r-from-coefficients sign trap is in the Maths mock test guide.
Index number formulas (Chapter 18)
From Chapter 18 (PDF) (opens in a new tab). ICAI writes P for price, Q for quantity, subscript 0 for the base year and 1 for the current year.
| Index | Formula (× 100) | Weights |
|---|---|---|
| Simple aggregative | ΣP₁ ÷ ΣP₀ | None |
| Laspeyres | ΣP₁Q₀ ÷ ΣP₀Q₀ | Base-year quantities |
| Paasche | ΣP₁Q₁ ÷ ΣP₀Q₁ | Current-year quantities |
| Marshall-Edgeworth | ΣP₁(Q₀ + Q₁) ÷ ΣP₀(Q₀ + Q₁) | Base and current together |
| Fisher's ideal | √(Laspeyres × Paasche) | GM of the two |
| Value index | ΣP₁Q₁ ÷ ΣP₀Q₀ | — |
| Deflated value | Current value ÷ current price index (× 100) | — |
Which index passes which test, as ICAI states it:
| Test | What it checks | Laspeyres | Paasche | Fisher |
|---|---|---|---|---|
| Time reversal | P₀₁ × P₁₀ = 1 | No | No | Yes |
| Factor reversal | P₀₁ × Q₀₁ = value index | No | No | Yes |
| Circular | Base can be shifted | No | No | No |
The circular test is met by the simple geometric mean of price relatives and by the weighted aggregative index with fixed weights.
Common traps in CA Foundation Maths formulas
Each of these is a formula used one step wrong, which is exactly how wrong options are built:
- i and n not converted together. 12% compounded quarterly for 5 years is i = 0.03 and n = 20. Change one and not the other and you land on a printed wrong option.
- Annuity due read as regular. Payments at the start: multiply by (1 + i).
- Growing perpetuity with the wrong rate. ICAI's example: ₹50 a year growing at 5%, discounted at 7%, is 50 ÷ (0.07 − 0.05) = ₹2,500, not 50 ÷ 0.07.
- GP sum to infinity with r outside −1 to 1. The formula doesn't apply; check r first.
- Circular vs necklace. Divide by 2 only when the arrangement can be flipped over.
- Combined SD without d². Shown above.
- Binomial mode when (n + 1)p is whole. There are two modes.
- Sign of r. It follows the regression coefficients, which always share a sign.
- Index subscripts swapped. Laspeyres keeps Q₀ on both lines; Paasche keeps Q₁.
- Fisher and the circular test. Fisher passes both reversal tests and still fails the circular one.
More option traps and a calculator routine for powers and roots (no scientific calculator is allowed) are in the Maths mock test guide; the questions ICAI set twice in 2026 are in CA Foundation maths important questions.
How to make this formula sheet stick before January
- This week: read each table against its ICAI chapter and mark every formula you can't use without looking.
- Daily, 15 minutes: write the marked formulas from memory on a blank page, then check. Stop when a table comes out clean twice.
- After every mock: write the formula behind each wrong answer next to it. Repeats tell you what to drill.
- Last evening: one read of the most-tested table above, nothing new.
Our CA Foundation Jan 2027 chapter wise weightage shows where Paper 3 sits beside the other three papers.
Frequently asked questions
Is there an official CA Foundation Maths formula sheet PDF from ICAI?
No. ICAI does not publish a separate formula sheet, and the paper does not print one. The formulas are inside the chapters of the Paper 3 study material (August 2024 edition, Reprint August 2025 onwards), free on icai.org. This page collects them chapter by chapter with ICAI's links, and you can print it.
Which formulas are most important for CA Foundation Maths?
Compound interest, effective rate, annuities and EMI from mathematics of finance; mean, median, mode, SD and combined SD from Statistics; then probability, correlation and regression, and index numbers. The most-tested table above has all of them with a check value.
Which Statistics formulas are needed for CA Foundation?
Grouped median and mode, the empirical relation, quartile deviation, SD, CV and combined SD, standard error, probability rules, binomial, Poisson and normal values, Karl Pearson's and Spearman's r, regression coefficients, and the Laspeyres, Paasche and Fisher indices with their tests.
Is trigonometry in the CA Foundation Maths syllabus?
No. ICAI's syllabus covers calculus "Excluding the trigonometric applications". You need differentiation and integration of powers, eˣ and log x, and maxima and minima.
Do I need to memorise the normal distribution areas?
Yes, in ICAI's figures: 0.6828 within one SD of the mean, 0.9546 within two and 0.9973 within three, plus QD = 0.675σ and MD = 0.8σ.
Test the formula sheet under the clock
Knowing a formula and using it in 72 seconds are different skills. Sit ICAI's latest Paper 3 against a 2-hour timer first, then move to full CA mock tests. Our CA Foundation Quantitative Aptitude mock tests are new papers on ICAI's pattern, set by 15+ qualified CA faculty: you write by hand, upload one PDF and get your checked copy within 24 hours, with the reason for every mark cut and suggested answers that show the working. The CA Foundation test series covers all four papers, and the CA test series home page lists every level. To plan your mock dates, read the CA Foundation Jan 2027 mock test plan.