Every financial decision in business reduces, at some point, to a ratio. When Tata Consultancy Services announces a 3:2 bonus issue, or Reliance Industries compares its debt-to-equity at 0.4 versus an industry average of 1.2, they are speaking the language of ratio and proportion. CA Foundation Paper 3 — Quantitative Aptitude — opens with these foundational tools because they underlie compound interest, logarithmic scales, population-growth models, and exponential decay curves studied later in the same paper. Indices (powers) are the compact notation behind compound growth: ₹1,00,000 growing at 10% p.a. for 20 years requires you to compute 1.1²⁰ — an index expression. Logarithms are the inverse: when a bank's analyst asks "how many years to double at 7% compounded monthly?", she reaches for log rules to unlock the exponent. Together, these three topics — ratio/proportion, indices, and logarithms — form the arithmetic toolkit that makes every later quantitative topic tractable. This chapter follows the ICAI CA Foundation syllabus and is examined with 60 MCQs carrying 100 marks (negative marking 0.25 per wrong answer). Mastery here saves you minutes on every downstream calculation, and minutes translate directly to marks.
Before You Start
🎯 Why learn this?
Ratio and proportion appear in partnership profit-sharing, balance sheet analysis, and mixing/alligation problems. Indices drive compound interest and growth models. Logarithms give you a direct route to solving "how many periods?" problems without trial and error.
📚What you'll learn
- •Defining ratio, proportion, and their properties
- •Componendo-dividendo and other rules for simplifying proportions
- •Laws of indices and surds (irrational numbers)
- •Common and natural logarithms and their properties
- •Antilogarithm and characteristic-mantissa notation
- •Solving equations using logarithms
✅ After this chapter, you can…
- ✓Apply ratio and proportion to financial and mixing problems
- ✓Simplify index expressions using the six laws
- ✓Convert between exponential and logarithmic form
- ✓Solve for unknown exponents using logarithm tables or a calculator
- ✓Identify the characteristic and mantissa of a logarithm
💼 Real problems this solves at work
- →Bonus and rights-issue calculations (ratio of new to existing )
- →Comparing financial ratios across Sensex companies
- →Compound interest and EMI calculations (index/log methods)
- →Population and sales-growth projections for business plans
01Ratio
A ratio expresses how many times one quantity contains another of the same kind. The ratio of a to b (written a : b or a/b) is a pure number — it has no units. Ratios are used in accounting for profit-sharing between partners, diluting chemical solutions, and mixing ingredients in manufacturing.
- Ratio a : b requires both quantities to be in the SAME UNIT before comparison.
- A ratio is already in its simplest form when HCF of the terms is 1.
- Duplicate ratio of a : b is a² : b²; sub-duplicate ratio is √a : √b.
- Triplicate ratio of a : b is a³ : b³; sub-triplicate is ∛a : ∛b.
- Compound ratio: (a : b) and (c : d) → ac : bd.
- Inverse ratio of a : b is b : a.
Profit sharing in a CA firm
Partners A, B, C invest ₹2,00,000; ₹3,00,000; ₹5,00,000. Net profit = ₹6,00,000. Share in profit-sharing ratio of their investments.
- →PSR = 2 : 3 : 5 (already simplified, HCF of 2,3,5 is 1).
- →A's share = (2/10) × 6,00,000 = ₹1,20,000.
- →B's share = (3/10) × 6,00,000 = ₹1,80,000.
- →C's share = (5/10) × 6,00,000 = ₹3,00,000.
Exam tip: ICAI frequently tests ratios in mixture/alligation questions: "Two grades of coffee mixed in ratio 3:2 sell at ₹400/kg; one grade costs ₹350, other ₹450. Find cost per kg of mix." Always set up ratio before solving.
02Proportion
When two ratios are equal, the four quantities are said to be in proportion: a/b = c/d, written a : b :: c : d. Here a and d are EXTREMES; b and c are MEANS. The fundamental property: product of extremes = product of means → ad = bc.
- Continued proportion: a : b :: b : c → b² = ac; b is the GEOMETRIC MEAN of a and c.
- Mean proportional between a and c = √(ac).
- Third proportional: if a : b :: b : x → x = b²/a.
- Fourth proportional: if a : b :: c : x → x = bc/a.
- Componendo: if a/b = c/d then (a+b)/b = (c+d)/d.
- Dividendo: if a/b = c/d then (a−b)/b = (c−d)/d.
- Componendo-dividendo: (a+b)/(a−b) = (c+d)/(c−d).
Componendo-Dividendo application
If x : y = 3 : 4, find (5x + 3y) : (5x − 3y).
- →Let x = 3k, y = 4k.
- →5x + 3y = 15k + 12k = 27k.
- →5x − 3y = 15k − 12k = 3k.
- →Required ratio = 27k : 3k = 9 : 1.
Study tip: Assume x = 3k and y = 4k ("k-trick") whenever you see a ratio. It converts ratio problems into simple algebra and saves time in MCQs.
03Indices (Exponents)
If a is any real number and n is a positive integer, aⁿ = a × a × ... × a (n times). The six laws of indices are the foundation of all compound-growth calculations.
| Law | Statement | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 3⁵ ÷ 3² = 3³ = 27 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (5²)³ = 5⁶ |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 100⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional exponent | aᵐ/ⁿ = ⁿ√(aᵐ) | 8²/³ = (∛8)² = 4 |
Compound interest as an index
HDFC Bank FD: ₹50,000 at 7% p.a. compounded annually for 3 years. Find maturity amount.
- →A = P × (1 + r)ⁿ = 50,000 × (1.07)³
- →(1.07)³ = 1.07 × 1.07 × 1.07 = 1.225043
- →A = 50,000 × 1.225043 ≈ ₹61,252
Exam tip: Surds are irrational numbers of the form ⁿ√a where the result is not a whole number. Rationalise by multiplying numerator and denominator by the conjugate. E.g., 1/(√5 + √3) × (√5 − √3)/(√5 − √3) = (√5 − √3)/2.
04Logarithms
If aˣ = N then x = logₐN. In words: "x is the logarithm of N to base a." The two most-used bases in CA Foundation: base 10 (common logarithm, written log₁₀ or just log) and base e ≈ 2.718 (natural logarithm, written ln).
- log(MN) = log M + log N [product rule]
- log(M/N) = log M − log N [quotient rule]
- log(Mⁿ) = n × log M [power rule]
- logₐa = 1 [log of base to itself = 1]
- logₐ1 = 0 [log of 1 to any base = 0]
- Change of base: logₐb = log b / log a
Characteristic and Mantissa
A common logarithm has two parts: (1) Characteristic — the integer part, determined by the number of digits before the decimal point. For numbers ≥ 1: characteristic = (digits before decimal − 1). For numbers < 1: characteristic is negative, = −(number of zeros after decimal point + 1). (2) Mantissa — the decimal part, always positive, looked up in log tables.
| Number | Characteristic | Mantissa (from table) | log value |
|---|---|---|---|
| 347.2 | 2 | 5403 | 2.5403 |
| 34.72 | 1 | 5403 | 1.5403 |
| 3.472 | 0 | 5403 | 0.5403 |
| 0.3472 | −1 (written 1̄) | 5403 | 1̄.5403 |
| 0.03472 | −2 (written 2̄) | 5403 | 2̄.5403 |
Doubling time with logarithms
In how many years does ₹1,00,000 double at 8% p.a. compounded annually?
- →1,00,000 × (1.08)ⁿ = 2,00,000
- →(1.08)ⁿ = 2
- →n × log(1.08) = log 2
- →n = log 2 / log 1.08 = 0.3010 / 0.0334 ≈ 9.01 years
- →Rule of 72 approximation: 72 / 8 = 9 years (matches!)
Exam tip: ICAI gives log tables in the exam. Learn to read: log 2 = 0.3010, log 3 = 0.4771, log 7 = 0.8451. Common trap: the characteristic of log(0.02) = 2̄ (bar 2), not −2 written plainly, because the mantissa stays positive.
Chapter Summary
- 1Ratio a:b compares quantities of the same unit; simplify by dividing by HCF. Key derived ratios: duplicate (a²:b²), compound (ac:bd).
- 2Proportion a:b :: c:d implies ad = bc. Componendo-dividendo shortcut: (a+b)/(a−b) = (c+d)/(c−d).
- 3Six laws of indices: product (aᵐaⁿ = aᵐ⁺ⁿ), quotient, power of power, zero, negative, and fractional exponents.
- 4Surds are irrational roots; rationalise by multiplying by the conjugate of the denominator.
- 5Log rules: product → add, quotient → subtract, power → multiply. logₐa = 1, logₐ1 = 0.
- 6Characteristic = digits before decimal − 1 (for numbers ≥ 1); negative for numbers < 1.
- 7Doubling-time problems: n = log2 / log(1+r). Rule of 72 gives a quick approximation.
Key Terms
Ratio
A comparison of two quantities of the same kind by division; expressed as a:b or a/b (dimensionless).
Proportion
Equality of two ratios: a:b :: c:d means a/b = c/d.
Componendo-Dividendo
If a/b = c/d then (a+b)/(a−b) = (c+d)/(c−d). Used to simplify proportion equations quickly.
Index (Exponent)
The power to which a base is raised; aⁿ means a multiplied by itself n times.
Surd
An irrational root of a rational number (e.g., √2, ∛5). Cannot be expressed as a finite decimal.
Logarithm
The inverse of exponentiation: logₐN = x means aˣ = N.
Characteristic
The integer part of a common logarithm; determined by the position of the decimal point.
Mantissa
The non-negative decimal part of a common logarithm; looked up in log tables.
Antilogarithm
The reverse of taking a logarithm: antilog(x) = 10ˣ.
Natural Logarithm
Logarithm to base e (≈ 2.718); written ln. Used in continuous compounding.