Ratio, Proportion, Indices and Logarithms for CA Foundation
Ratios compare quantities, indices handle powers, and logarithms turn multiplication into addition: three quick-scoring Paper 3 topics.
Quantitative AptitudePaper 3, Business Mathematics13 MCQs2 lessons2 flashcards3 memory tricks
Practise Ratio, Proportion, Indices and Logarithms free
What you'll learn
- Ratio and proportion
Combine ratios and find fourth, third and mean proportionals. - Indices and logarithms
Use the laws of indices and logarithms to simplify quickly.
Key formulas
| Fourth proportional to a, b, c | d = bc ÷ a |
|---|---|
| Third proportional to a, b | c = b² ÷ a |
| Mean proportional between a, b | √(ab) |
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power | (aᵐ)ⁿ = aᵐⁿ |
| Zero and negative | a⁰ = 1, a⁻ⁿ = 1/aⁿ |
| Fractional | a^(p/q) = (q-th root of a)ᵖ |
Concepts explained
Duplicate, triplicate and sub-duplicate ratios
Duplicate ratio means the ratio of the squares of two numbers: the duplicate ratio of a : b is a² : b².
What it means
- 'Duplicate' comes from old mathematical language meaning 'compounded with itself': a : b multiplied by a : b gives (a × a) : (b × b) = a² : b².
- Triplicate ratio is the ratio of the cubes: a³ : b³. Sub-duplicate is the ratio of the square roots: √a : √b. Sub-triplicate is the ratio of the cube roots: ∛a : ∛b.
- It is used in geometry: if the sides of two similar figures are in the ratio a : b, their areas are in the duplicate ratio a² : b², and their volumes (for solids) in the triplicate ratio a³ : b³.
- Original ratio: 2 : 3
- Square both terms: 2² : 3²
Answer: Duplicate ratio = 4 : 9
- Original ratio: 5 : 10
- Simplify: 1 : 2
- Square both terms: 1² : 2²
Answer: Duplicate ratio = 1 : 4
- Two square fields have sides in the ratio 4 : 5
- Area depends on side², so take the duplicate ratio: 4² : 5²
Answer: Ratio of areas = 16 : 25
Related terms
- Sub-duplicate ratio: ratio of the square roots, e.g. sub-duplicate of 9 : 16 is 3 : 4
- Triplicate ratio: ratio of the cubes, e.g. triplicate of 2 : 3 is 8 : 27
- Sub-triplicate ratio: ratio of the cube roots, e.g. sub-triplicate of 8 : 27 is 2 : 3
Exam tip: Doubling each term (3 : 4 → 6 : 8) is a trap option: it leaves the ratio unchanged. Duplicate means squaring.
Proportion: fourth, third and mean proportional
Four numbers a, b, c, d are in proportion when a : b = c : d, which means ad = bc (product of extremes = product of means).
What it means
- In a : b = c : d, a and d are the extremes, b and c are the means.
- Fourth proportional to a, b, c is d = bc ÷ a.
- Three numbers a, b, c are in continued proportion when a : b = b : c, so b² = ac. Then c is the third proportional (b² ÷ a) and b is the mean proportional (√ac).
- Find d in 4 : 6 = 10 : d
- 4d = 6 × 10 = 60
Answer: d = 15
- Third proportional to 4 and 6: 4 : 6 = 6 : c
- c = 6² ÷ 4 = 36 ÷ 4
Answer: c = 9
- Mean proportional between 9 and 16
- x = √(9 × 16) = √144
Answer: x = 12
Related terms
- Continued proportion: a : b = b : c, so b² = ac
- Arithmetic mean: (a + b) ÷ 2, not the same as the mean proportional
Exam tip: The mean proportional is the geometric mean √(ab). If an option shows the arithmetic mean (12.5 for 9 and 16), it's a trap.
Combining and compounding ratios
To combine a : b and b : c into a : b : c, make the common term equal. To compound ratios, multiply the first terms together and the second terms together.
What it means
- Combining: the common term (b) appears in both ratios; scale each ratio so b is the same (use the LCM).
- Compounding: (a : b) compounded with (c : d) = ac : bd.
- Inverse (reciprocal) ratio of a : b is b : a.
- a : b = 2 : 3 and b : c = 4 : 5
- LCM of 3 and 4 is 12: a : b = 8 : 12, b : c = 12 : 15
Answer: a : b : c = 8 : 12 : 15
- Compound 2 : 3 and 4 : 5
- (2 × 4) : (3 × 5)
Answer: 8 : 15
Related terms
- Inverse ratio: b : a
- Ratio of equality: a : b with a = b
Exam tip: Never place two ratios side by side (2 : 3 : 5) without matching the common term.
Laws of indices
With the same base, multiplying adds powers, dividing subtracts them, and a power of a power multiplies them.
What it means
- aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 and a⁻ⁿ = 1/aⁿ
- a^(p/q) = (q-th root of a)ᵖ: take the root first, then the power
- 2⁵ × 2⁻³
- Add powers: 5 + (−3) = 2
Answer: 2² = 4
- 8^(2/3)
- Cube root of 8 = 2
- Square it: 2²
Answer: 4
Related terms
- Surds: irrational roots such as √2
- Exponential equation: an equation with the unknown in the power, solved by equating powers of the same base
Exam tip: Write every number as a power of the same base (8 = 2³, 27 = 3³) and the question usually solves itself.
Logarithms
log_a N = x means aˣ = N: a logarithm is the power to which the base must be raised to give the number.
What it means
- log (mn) = log m + log n; log (m/n) = log m − log n; log mⁿ = n log m
- log_a 1 = 0 and log_a a = 1
- Change of base: log_b a = log a ÷ log b
- Common logs use base 10; log 2 = 0.3010 and log 3 = 0.4771 are worth remembering
- log 8 = log 2³ = 3 log 2
- = 3 × 0.3010
Answer: 0.9030
- log 2 + log 5 = log (2 × 5)
- = log 10
Answer: 1
- log₄ 64 = log 64 ÷ log 4
- = 6 log 2 ÷ 2 log 2
Answer: 3
Related terms
- Antilog: the reverse of log: antilog of 2 (base 10) is 100
- Characteristic and mantissa: the integer and decimal parts of a common log
Exam tip: log (m + n) is NOT log m + log n. Options built on that mistake are common.
Memory tricks
Duplicate a² : b², triplicate a³ : b³, sub-duplicate √a : √b, sub-triplicate ∛a : ∛b.
log mn = log m + log n; log m/n = log m − log n; log mⁿ = n log m.
Cross-multiply a : b = c : d (ad = bc) and solve for the missing term.
Try 3 CA Foundation-style questions
Q1. The duplicate ratio of 3 : 4 is
- 6 : 8
- √3 : 2
- 27 : 64
- 9 : 16
Show answer
Answer: (4) 9 : 16
Duplicate ratio means the ratio of squares: 3² : 4² = 9 : 16.
Q2. The fourth proportional to 4, 6 and 10 is
- 15
- 12
- 20
- 60
Show answer
Answer: (1) 15
4 : 6 = 10 : x, so 4x = 60 and x = 15.
Q3. Using the change-of-base rule, log₄ 64 equals
- 16
- 3
- 4
- 2
Show answer
Answer: (2) 3
log₄ 64 = log 64 ÷ log 4 = 6 log 2 ÷ 2 log 2 = 3. Check: 4³ = 64.
Practise all 13 questions Free · explanations for every option · see where you go wrong