Chapter 2 of 7

29%

CA Foundation Quantitative Aptitude★ Roughly 14 marks of the 40-mark Business Mathematics section. High-frequency, mechanically solvable once the question type is recognised, and load-bearing for time value of money where logarithms solve for time.

Ratio, Proportion, Indices, Logarithms & Equations

The algebraic toolkit the rest of Business Mathematics runs on — comparing quantities, the laws that let exponents and logarithms be manipulated, and solving the equations that result.

⏱ 28 min readEasy difficulty✓ Free · exam-pattern aligned

Timed practiceMCQs from this chapter with instant scoring

FlashcardsFormulas & traps on spaced repetition

CA Foundation PYQsSolved previous-year questions

By the end of this chapter you'll be able to…

1Form inverse, duplicate, sub-duplicate, triplicate and compounded ratios, and build a continued ratio from two ratios sharing a term
2Solve ratio problems by introducing a constant of proportionality
3Compute mean, third and fourth proportionals and apply invertendo, alternendo, componendo, dividendo and componendo-dividendo by name
4Apply the laws of indices, including negative and fractional indices, and solve exponential equations by equating bases
5Apply the laws of logarithms and the change of base formula, and use logarithms to solve for an exponent
6Determine the characteristic and mantissa of a common logarithm
7Solve simultaneous linear equations and classify a pair as having a unique solution, no solution or infinitely many
8Use the discriminant to determine the nature of the roots of a quadratic without solving it
9Evaluate symmetric functions of the roots from their sum and product, and form an equation from given roots

💡

Why this chapter matters in CA Foundation

These four chapters supply the algebraic machinery the rest of Business Mathematics runs on. Logarithms are the only way to solve for an unknown sitting in an exponent, which is exactly the situation in every compound interest question asking how long a sum takes to grow to a target. The sum and product of roots let a whole family of quadratic questions be answered without solving the equation at all. And because the questions are short and mechanical, the marks turn on recognising the type quickly rather than on mathematical difficulty — which is why misreading what is asked costs more marks here than genuine ignorance does.

Before you start — revise these

🔗

How to Crack CA Foundation Quantitative Aptitude

Every question here is one mark with a 0.25 penalty, and the guessing arithmetic means none should be left blank.

🔗

School algebra

Manipulation of algebraic expressions and elementary factorisation are assumed.

Ratio, Proportion, Indices, Logarithms & Equations

Weightage: Roughly 14 marks of the Business Mathematics section. These chapters are high-frequency, mechanically solvable once the question type is recognised, and they feed directly into time value of money, where logarithms are used to solve for time.

Ratio

A ratio compares two quantities of the same kind by division. The ratio of $a$ to $b$ , written $a : b$ or $a / b$ , requires that both quantities be measured in the same units — a ratio of a length to a mass is meaningless.

The first quantity is the antecedent and the second the consequent. A ratio is unchanged when both terms are multiplied or divided by the same non-zero number, which is why ratios are normally reduced to their lowest terms.

Kinds of ratio

For a ratio $a : b$ :

Inverse ratio — $b : a$ .
Duplicate ratio — $a^{2} : b^{2}$ .
Sub-duplicate ratio — $a : b$ .
Triplicate ratio — $a^{3} : b^{3}$ .
Sub-triplicate ratio — $a^{1/3} : b^{1/3}$ .
Compounded ratio of $a : b$ and $c : d$ — $a c : b d$ .

A continued ratio compares three or more quantities, as $a : b : c$ . To combine $a : b$ and $b : c$ into a continued ratio, make the common term equal. If $a : b = 3 : 4$ and $b : c = 6 : 7$ , express $b$ consistently: multiply the first by 6 and the second by 4, giving $18 : 24$ and $24 : 28$ , so $a : b : c = 18 : 24 : 28 = 9 : 12 : 14$ .

The standard technique

Almost every ratio problem is solved by introducing a constant of proportionality. If two quantities are in the ratio $3 : 5$ , write them as $3 k$ and $5 k$ . The condition given in the question then determines $k$ , and the individual quantities follow.

Proportion

Four quantities are in proportion when the ratio of the first to the second equals the ratio of the third to the fourth: $a : b = c : d$ , written $a : b :: c : d$ .

Here $a$ and $d$ are the extremes and $b$ and $c$ the means, and the defining property is:

$a d = b c$

the product of the extremes equals the product of the means.

Continued proportion. Three quantities $a, b, c$ are in continued proportion when $a : b = b : c$ , which gives $b^{2} = a c$ . Then $b$ is the mean proportional between $a$ and $c$ , so $b = a c$ , and $c$ is the third proportional to $a$ and $b$ .

The fourth proportional to $a$ , $b$ and $c$ is the quantity $d$ such that $a : b = c : d$ , so $d = b c / a$ .

Properties of proportion

Given $a / b = c / d$ , the following hold and are examined by name:

Invertendo — $b / a = d / c$ .
Alternendo — $a / c = b / d$ .
Componendo — $(a + b) / b = (c + d) / d$ .
Dividendo — $(a - b) / b = (c - d) / d$ .
Componendo and dividendo — $(a + b) / (a - b) = (c + d) / (c - d)$ .
Addendo — if $a / b = c / d = e / f$ , then each ratio equals $(a + c + e) / (b + d + f)$ .

Componendo and dividendo is the most useful in practice, because it converts an equation containing a sum and a difference into a single ratio, eliminating the need to expand.

Indices

For a positive real $a$ and integers or rationals $m, n$ :

$a^{m} \times a^{n} = a^{m + n}, \frac{a ^{m}}{a ^{n}} = a^{m - n}, (a^{m})^{n} = a^{mn}$

$(ab)^{n} = a^{n} b^{n}, (\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$

$a^{0} = 1 (a \neq = 0), a^{- n} = \frac{1}{a ^{n}}, a^{m / n} = n a^{m}$

These are not arbitrary rules but consequences of one idea. Since $a^{m}$ means $a$ multiplied by itself $m$ times, multiplying $a^{m}$ by $a^{n}$ places $m + n$ factors together, which is the first law. The definition $a^{0} = 1$ then follows from the second law with $m = n$ , since $a^{m} / a^{m} = 1$ and also equals $a^{0}$ . Negative and fractional indices are defined so that the same laws continue to hold, which is why $a^{- n} = 1/ a^{n}$ and $a^{1/2} = a$ .

Solving exponential equations. Where both sides can be expressed to the same base, equate the exponents. To solve $2^{x + 1} = 32$ , write $32 = 2^{5}$ , so $x + 1 = 5$ and $x = 4$ . Where a common base cannot be found, take logarithms of both sides.

Logarithms

The logarithm is the inverse of exponentiation. For $a > 0$ , $a \neq = 1$ , and $N > 0$ :

$lo g_{a} N = x ⟺ a^{x} = N$

So $lo g_{2} 32 = 5$ because $2^{5} = 32$ . Note that the logarithm of a negative number or of zero is not defined for a real base, and that $lo g_{a} 1 = 0$ for every valid base, since $a^{0} = 1$ .

Laws of logarithms

$lo g_{a} (mn) = lo g_{a} m + lo g_{a} n$ $lo g_{a} (\frac{m}{n}) = lo g_{a} m - lo g_{a} n$ $lo g_{a} (m^{p}) = p lo g_{a} m$ $lo g_{a} a = 1, lo g_{a} 1 = 0$

Each corresponds to a law of indices, which is exactly what one expects of an inverse operation. Writing $m = a^{x}$ and $n = a^{y}$ , we have $mn = a^{x + y}$ , so $lo g_{a} (mn) = x + y = lo g_{a} m + lo g_{a} n$ . The product law is the index addition law read backwards, and this is the historical reason logarithms were invented: they turn multiplication into addition.

Change of base

$lo g_{a} N = \frac{l o g _{b} N}{l o g _{b} a}$

Two consequences follow and are worth memorising separately:

$lo g_{a} b \times lo g_{b} a = 1, equivalently lo g_{a} b = \frac{1}{l o g _{b} a}$

$lo g_{a} b \times lo g_{b} c \times lo g_{c} d = lo g_{a} d$

Common logarithms use base 10 and are written $lo g N$ . Natural logarithms use base $e \approx 2.71828$ and are written $ln N$ .

Characteristic and mantissa

For a common logarithm, the integral part is the characteristic and the decimal part the mantissa. The mantissa is always positive and depends only on the digits of the number; the characteristic depends only on the position of the decimal point.

For a number greater than 1, the characteristic is one less than the number of digits before the decimal point. For a number less than 1, the characteristic is negative, and is one more than the number of zeros immediately after the decimal point, written with a bar over it. So $lo g 234.5$ has characteristic 2, and $lo g 0.02345$ has characteristic $\overset{ˉ}{2}$ .

Equations

Linear simultaneous equations

Three methods are available, and choosing the right one saves time.

Elimination — multiply the equations so that one variable has the same coefficient, then add or subtract. Fastest when coefficients are small.

Substitution — express one variable in terms of the other and substitute. Fastest when one coefficient is 1.

Cross-multiplication — for $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ :

$\frac{x}{b _{1} c _{2} - b _{2} c _{1}} = \frac{y}{c _{1} a _{2} - c _{2} a _{1}} = \frac{1}{a _{1} b _{2} - a _{2} b _{1}}$

The consistency of a pair of linear equations is determined by comparing coefficients. Writing the equations as $a_{1} x + b_{1} y = c_{1}$ and $a_{2} x + b_{2} y = c_{2}$ :

If $a_{1} / a_{2} \neq = b_{1} / b_{2}$ , there is a unique solution — the lines intersect.
If $a_{1} / a_{2} = b_{1} / b_{2} \neq = c_{1} / c_{2}$ , there is no solution — the lines are parallel.
If $a_{1} / a_{2} = b_{1} / b_{2} = c_{1} / c_{2}$ , there are infinitely many solutions — the lines coincide.

Quadratic equations

For $a x^{2} + b x + c = 0$ with $a \neq = 0$ :

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

The quantity $D = b^{2} - 4 a c$ is the discriminant, and it determines the nature of the roots without solving:

$D > 0$ and a perfect square — roots are real, distinct and rational.
$D > 0$ but not a perfect square — roots are real, distinct and irrational, occurring in conjugate pairs.
$D = 0$ — roots are real and equal, each $- b /2 a$ .
$D < 0$ — roots are imaginary, occurring in conjugate pairs.

Sum and product of roots. If $α$ and $β$ are the roots:

$α + β = - \frac{b}{a}, α β = \frac{c}{a}$

These follow from writing $a x^{2} + b x + c = a (x - α) (x - β)$ and comparing coefficients, and they are the single most useful pair of results in the topic. Many questions ask for a symmetric function of the roots — $α^{2} + β^{2}$ , or $1/ α + 1/ β$ — which can be evaluated from the sum and product without ever finding the roots:

$α^{2} + β^{2} = (α + β)^{2} - 2 α β, \frac{1}{α} + \frac{1}{β} = \frac{α + β}{α β}$

Forming an equation from its roots. If the roots are $α$ and $β$ :

$x^{2} - (α + β) x + α β = 0$

that is, $x^{2} - (sum) x + (product) = 0$ .

Cubic equations

For $a x^{3} + b x^{2} + c x + d = 0$ with roots $α, β, γ$ :

$α + β + γ = - \frac{b}{a}, α β + β γ + γ α = \frac{c}{a}, α β γ = - \frac{d}{a}$

At Foundation level, cubic equations are usually solved by finding one root by inspection — testing the factors of the constant term — and then factorising to a quadratic.

Applications

These topics appear in disguise more often than directly.

Ratio appears as partnership profit sharing, mixture problems, and the division of a sum among persons in a stated proportion. Where a mixture question asks how much of one component must be added to change a ratio, set up the new quantities in terms of the added amount and equate to the required ratio.

Proportion appears in problems of direct and inverse variation. If $y$ varies directly as $x$ , then $y = k x$ ; if inversely, $y = k / x$ . Men-and-work and time-and-distance questions are inverse variation problems.

Logarithms appear whenever an unknown sits in an exponent, which is why they matter for compound interest. To find how long a sum takes to double at compound interest, the equation $2 = (1 + i)^{n}$ is solved as $n = lo g 2/ lo g (1 + i)$ .

Quadratic equations appear in break-even and profit-maximisation problems, and in any situation where a rate and a time multiply to a fixed quantity.

How this chapter is examined

Expect questions asking for a fourth proportional or a mean proportional; for a compounded, duplicate or sub-duplicate ratio; for the value of a logarithmic expression using the laws; for the nature of the roots of a quadratic from its discriminant; for a symmetric function of the roots without solving; and for the value of a constant that makes a pair of linear equations inconsistent or dependent.

The questions are short and mechanical once the type is recognised, which makes recognition the skill to drill. Read the question to identify what is being asked before computing anything — a substantial share of lost marks in this section comes from computing the sum of the roots when the product was asked, or the third proportional when the fourth was.

Key formulas & results

Everything to memorise for the exam hall, in one card. Screenshot this for revision.

Kinds of ratio

Inverse b:a. Duplicate a²:b². Sub-duplicate √a:√b. Triplicate a³:b³. Sub-triplicate a^(1/3):b^(1/3). Compounded of a:b and c:d is ac:bd.

Duplicate squares the terms; sub-duplicate takes square roots. Candidates routinely reverse the two.

Proportion

If a:b :: c:d then ad = bc — the product of the extremes equals the product of the means

This single identity solves most proportion questions without further manipulation.

Mean, third and fourth proportional

Mean proportional between a and c: b = √(ac). Third proportional to a and b: c = b²/a. Fourth proportional to a, b, c: d = bc/a.

The mean proportional involves two quantities, the third involves two, the fourth involves three — checking how many quantities are given identifies which is wanted.

Componendo and dividendo

If a/b = c/d then (a+b)/(a−b) = (c+d)/(c−d)

The most useful property in practice, because it converts an expression containing a sum and a difference into a single ratio without expansion.

Laws of indices

aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(m/n) = ⁿ√(aᵐ)

All follow from the first law. Negative and fractional indices are defined precisely so that the laws continue to hold.

Laws of logarithms

log(mn) = log m + log n; log(m/n) = log m − log n; log(mᵖ) = p log m; log_a a = 1; log_a 1 = 0

Each is an index law read backwards, which is why logarithms convert multiplication into addition.

Change of base

log_a N = log_b N ÷ log_b a; hence log_a b × log_b a = 1 and log_a b × log_b c × log_c d = log_a d

The reciprocal relation and the chaining relation are asked directly and are worth memorising separately.

Quadratic roots and discriminant

x = [−b ± √(b² − 4ac)] / 2a, with D = b² − 4ac. D > 0 perfect square: real, distinct, rational. D > 0 not a perfect square: real, distinct, irrational. D = 0: real and equal. D < 0: imaginary.

The nature of the roots can always be settled from D alone, without solving.

Sum and product of roots

α + β = −b/a and αβ = c/a. Equation from roots: x² − (sum)x + (product) = 0.

Symmetric functions follow: α² + β² = (α+β)² − 2αβ, and 1/α + 1/β = (α+β)/αβ. These avoid solving the equation entirely.

Consistency of linear equations

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: unique solution if a₁/a₂ ≠ b₁/b₂; no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂; infinitely many if a₁/a₂ = b₁/b₂ = c₁/c₂

Geometrically: intersecting lines, parallel lines, coincident lines.

FOUNDATION EXAMINATIONCA Foundation

1. The duplicate ratio of 4 : 9 is: (a) 8 : 18 (b) 16 : 81 (c) 2 : 3…[1]

The duplicate ratio of a : b is a² : b².…

1/1

Tap the paper to answer it.

Practice Studio

CA Foundation answers are written, not ticked

Write one on Ratio, Proportion, Indices, Logarithms &… and it comes back marked to the CA Foundation scheme in about twenty seconds.

Write one answer

⚠️

Traps CA Foundation sets — and how to dodge them

These are the exact option-traps and misreads that cost marks under negative marking.

WATCH OUT

✗ Confusing duplicate ratio with sub-duplicate ratio

✓ Duplicate squares the terms, giving a²:b². Sub-duplicate takes square roots, giving √a:√b. The prefix 'sub' always means the inverse operation.

WATCH OUT

✗ Confusing the third proportional with the fourth proportional

✓ A third proportional is computed from two quantities (c = b²/a); a fourth proportional from three (d = bc/a). Count the quantities the question gives.

WATCH OUT

✗ Comparing quantities in different units in a ratio

✓ A ratio requires both quantities in the same unit. Convert first — a ratio of 2 metres to 50 centimetres is 200:50, not 2:50.

WATCH OUT

✗ Taking the logarithm of a negative number or of zero

✓ These are undefined for a real base. If a computation produces such a step, an earlier error has occurred.

WATCH OUT

✗ Writing log(m + n) as log m + log n

✓ The product law applies to a product, not a sum. There is no law simplifying the logarithm of a sum, which is why questions containing sums usually need componendo-dividendo or factorisation first.

WATCH OUT

✗ Treating the mantissa as negative for a number less than 1

✓ The mantissa is always positive; only the characteristic is negative, and it is written with a bar. The characteristic of log 0.02345 is 2-bar, not −2 applied to the whole logarithm.

WATCH OUT

✗ Solving a quadratic to determine the nature of its roots

✓ The discriminant settles it without solving. Computing D is one step; solving is three or four and risks arithmetic error.

WATCH OUT

✗ Finding the roots in order to evaluate α² + β² or 1/α + 1/β

✓ Every symmetric function of the roots can be expressed in terms of the sum and product, which are read directly off the coefficients.

WATCH OUT

✗ Computing the sum of the roots when the product was asked, or the third proportional when the fourth was

✓ Read the question twice and underline what is asked before computing. In this section, misreading loses more marks than not knowing the method.

Exam-pattern practice

PYQ-style questions with full solutions. Work through them as a readiness check — mark yourself honestly and get your gap report at the end.

Readiness check

Are you exam-ready for Ratio, Proportion, Indices, Logarithms & Equations?

15 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

15 questions~11 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•A ratio requires both quantities in the same units, and is unchanged when both terms are multiplied by the same number.
•Duplicate squares the terms; sub-duplicate takes square roots; 'sub' always means the inverse operation.
•In a proportion, the product of the extremes equals the product of the means.
•Mean proportional between a and c is √(ac); third proportional to a and b is b²/a; fourth proportional to a, b, c is bc/a.
•Componendo and dividendo converts (a+b)/(a−b) into a single ratio without expansion.
•Negative indices mean reciprocals, never negative values; for a positive base no index gives a negative result.
•To solve an exponential equation, express both sides to a common base and equate exponents; if none exists, take logarithms.
•There is no law for the logarithm of a sum — the product law applies only to products.
•log_a b × log_b a = 1, and log_a b × log_b c × log_c d = log_a d.
•Number of digits in N equals the characteristic of log N plus one.
•The mantissa is always positive; only the characteristic can be negative.
•D = b² − 4ac settles the nature of the roots without solving.
•α + β = −b/a and αβ = c/a; every symmetric function of the roots follows from these.
•An equation from its roots is x² − (sum)x + (product) = 0.
•Reversing the coefficients of ax² + bx + c = 0 gives the equation whose roots are the reciprocals.
•Linear pairs: unique if a₁/a₂ ≠ b₁/b₂; none if the coefficient ratios match but the constant ratio differs; infinite if all three match.

CA Foundation question blueprint

How this topic is asked, tier by tier — so you can prep to the pattern.

Typical weightage: 14

Exam-hall strategy

Battle-tested tips from mentors and toppers for this topic under the sectional clock.

Identify the question type before computing — most questions here are mechanical once recognised.
Underline what is asked: sum or product, third or fourth proportional, duplicate or sub-duplicate.
Answer nature-of-roots questions from the discriminant alone; never solve the equation.
Evaluate symmetric functions of roots from the sum and product rather than by finding the roots.
For reciprocal-root questions, reverse the coefficients rather than deriving from scratch.
Check a symbolic answer by substituting a simple numerical case; it takes seconds and catches sign errors.
Sanity-check logarithm answers against the characteristic — a number between 10 and 100 must have characteristic 1.
In mixture problems, identify the component that stays constant and use it as the anchor.

Beyond the exam

Where this skill shows up in the job you're competing for — and in life.

Ratio analysis of financial statements

Ratio analysis of financial statements — current ratio, debt-equity, gross profit ratio — is the same comparison of two quantities in the same units, and is central to CA Intermediate and Final.

Logarithms convert the compound interest equation into a …

Logarithms convert the compound interest equation into a solvable one whenever the unknown is time, which is how doubling periods and loan tenures are computed.

Proportional division underlies partnership profit sharing

Proportional division underlies partnership profit sharing, the apportionment of overheads in cost accounting, and the allocation of joint costs.

Quadratic equations arise in break-even analysis and in a…

Quadratic equations arise in break-even analysis and in any relationship where a rate and a quantity multiply to a fixed total.

Where else this topic is tested

Prepare once, score in every exam that asks it.

CA Intermediate Paper 6 — Financial Management, which applies logarithms and equations to valuation

CS Executive and CMA Foundation quantitative papers

CAT, XAT and banking aptitude tests, which examine ratio, proportion and equations at comparable depth

Class 11 and 12 Mathematics under CBSE and ISC

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions aspirants ask

Pulled from the Q&A community and mentor sessions.

Count the quantities the question gives you. A third proportional is computed from two quantities and is c = b²/a. A fourth proportional is computed from three and is d = bc/a. The mean proportional is also computed from two but is √(ac) — the distinction from the third proportional is that the mean sits between the two given quantities while the third comes after them.

Whenever an equation contains a sum and a difference of the same two quantities, most obviously in the form (a+b)/(a−b). Applying the property converts it directly into a/b without expanding or clearing denominators. It is worth learning by name rather than re-deriving, because recognising it saves half a minute on a paper where the average allowance is seventy-two seconds.

Because a negative index denotes a reciprocal, by definition: a⁻ⁿ = 1/aⁿ. The definition is chosen precisely so that the law aᵐ × aⁿ = aᵐ⁺ⁿ continues to hold when the exponents are negative. For a positive base, every power is positive, so no index of any kind can produce a negative value.

No. The product law converts the logarithm of a product into a sum of logarithms; there is no corresponding law for the logarithm of a sum. Where a question presents a sum inside a logarithm, the usual route is to factorise the expression first, or to apply componendo and dividendo before taking logarithms.

No, and you should not. The discriminant D = b² − 4ac settles it in one step: positive and a perfect square means real, distinct and rational; positive but not a perfect square means real, distinct and irrational; zero means real and equal; negative means imaginary. Solving takes three or four steps and risks arithmetic error for no additional information.

Wherever an unknown sits in an exponent, which is the situation in every compound interest problem asking for time. The equation A = P(1 + i)ⁿ cannot be solved for n by algebra alone; taking logarithms gives n = [log A − log P] / log(1 + i). This is why the logarithm chapter precedes time value of money in the syllabus.