Ratio and Proportion

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What: Ratio and Proportion covers simple ratios, compound ratios, direct and inverse proportion, partnership problems, mixture problems (alligation), and componendo-dividendo for advanced fraction manipulation.
Why it matters: 3–5 questions per CDS paper, with strong overlap into Percentage, Time-Speed-Distance, and Time and Work.
Key fact: If $a : b = c : d$, then by componendo $\tfrac{a+b}{b} = \tfrac{c+d}{d}$; by dividendo $\tfrac{a-b}{b} = \tfrac{c-d}{d}$; by componendo-dividendo $\tfrac{a+b}{a-b} = \tfrac{c+d}{c-d}$. One identity solves dozens of CDS algebra-flavoured ratio problems.

Ratio and Proportion is the connective tissue of CDS arithmetic. It underlies every speed problem, every work problem, every mixture problem, and every partnership question. Master the four classical proportion identities (componendo, dividendo, alternendo, invertendo) and you collapse most CDS ratio questions to a single line.

This page is built from CDS Previous Year Questions across 2000–2023 plus NCERT Class 8 Proportional Reasoning 1 and 2. Pair with Percentage, Time, Speed and Distance, and Time and Work.

What This Topic Covers

CDS scope: (1) definitions — ratio, antecedent, consequent, equivalent ratios; (2) compound and continued ratios — combining two or more ratios; (3) proportion — first/second/third/fourth proportional, mean proportional; (4) variation — direct, inverse, joint; (5) partnership — simple and compound (with time involvement); (6) mixtures (alligation) — replacement formula; and (7) the four classical identities — componendo, dividendo, alternendo, invertendo.

Why This Topic Matters

Ratio shows up directly in 3–5 CDS questions per paper.
Partnership and mixture problems are pure ratio applications — mastering ratio collapses both.
The componendo-dividendo identity is the highest-ROI algebra shortcut in CDS arithmetic.

Exam Pattern & Weightage

Year / Paper	No.	Subtopics Tested
2007-I/II	3	Simple ratio, partnership
2009-II	3	Compound ratio, mean proportional, mixture
2010-I/II	4	Partnership, alligation, third proportional
2011-I/II	4	Ratio with conditions, mixture replacement
2012-I/II	4	Inverse proportion, partnership with time
2013-I/II	4	Componendo-dividendo, ratio of variables
2014-I/II	4	Mean proportional, fourth proportional
2015-I/II	3	Compound ratio, partnership
2016-I/II	3	Mixture concentrations, alligation
2017-I/II	4	Componendo-dividendo, partnership
2018-I/II	3	Continued ratio, mixture
2019-II	3	Ratio modifications, mixture replacement
2020-I/II	4	Partnership, alligation
2021-I/II	3	Mean proportional, compound ratio
2023-I	2	Mixed application

⚡ CDS Alert

The mixture-replacement formula: if from a vessel of $N$ units, $k$ units are removed and replaced by pure solvent, after $n$ such operations, the original liquid remaining is $N \cdot (1 - k/N)^n$. CDS asks variants of this problem in nearly every other paper.

Core Concepts

Ratio Basics

A ratio $a : b$ compares two quantities of the same kind. The first term $a$ is the antecedent, the second $b$ the consequent. A ratio is unchanged if both terms are multiplied (or divided) by the same non-zero number: $2 : 3 = 4 : 6 = 10 : 15$.

Compound and Continued Ratios

Combining Ratios Compound: $(a : b) \cdot (c : d) = ac : bd$.
Continued: $a : b : c$ means $a : b$ and $b : c$ simultaneously.

To merge "$A : B = 2 : 3$" with "$B : C = 4 : 5$" into a single $A : B : C$: scale so $B$ matches. Multiply first by 4, second by 3 (LCM of $B$ terms): $A : B = 8 : 12$ and $B : C = 12 : 15$. So $A : B : C = 8 : 12 : 15$.

Proportion

Four numbers $a, b, c, d$ are in proportion if $a : b = c : d$, i.e. $\tfrac{a}{b} = \tfrac{c}{d}$, equivalently $ad = bc$.

Special Proportions Mean proportional of $a$ and $b$: $\sqrt{ab}$.
Third proportional of $a, b$: the $c$ such that $a : b = b : c$, i.e. $c = b^2/a$.
Fourth proportional of $a, b, c$: the $d$ such that $a : b = c : d$, i.e. $d = bc/a$.

The Four Classical Identities

If $\tfrac{a}{b} = \tfrac{c}{d}$, then Invertendo: $\tfrac{b}{a} = \tfrac{d}{c}$
Alternendo: $\tfrac{a}{c} = \tfrac{b}{d}$
Componendo: $\tfrac{a+b}{b} = \tfrac{c+d}{d}$
Dividendo: $\tfrac{a-b}{b} = \tfrac{c-d}{d}$
Componendo-Dividendo: $\tfrac{a+b}{a-b} = \tfrac{c+d}{c-d}$

Variation

Direct: $y \propto x$ means $y = kx$. Doubling $x$ doubles $y$.
Inverse: $y \propto 1/x$ means $xy = k$. Doubling $x$ halves $y$.
Joint: $z \propto xy$ means $z = kxy$.

Partnership

Profit Ratio in Partnership Simple (same time): profit ratio = capital ratio.
Compound (different times): profit ratio = (capital × time) ratio.
Working / sleeping partners: agreed working-partner fee deducted from profit before sharing remainder.

Mixtures and Alligation

Alligation Rule $$\frac{\text{Quantity of cheaper}}{\text{Quantity of dearer}} = \frac{\text{Price of dearer} - \text{Mean price}}{\text{Mean price} - \text{Price of cheaper}}$$

Mixture Replacement $$\text{Original remaining} = N \left(1 - \frac{k}{N}\right)^n$$

Where $N$ is the vessel capacity, $k$ is the amount removed per operation, and $n$ is the number of operations.

Worked Examples

Example 1 — Compound Ratio (2009-II)

Q: $A : B = 2 : 3$ and $B : C = 4 : 5$. Find $A : B : C$.

Scale $B$ to common value. LCM of 3 and 4 is 12. Multiply first by 4, second by 3.
$A : B = 8 : 12$ and $B : C = 12 : 15$.
Combined: $A : B : C = 8 : 12 : 15$.

Example 2 — Mean Proportional (2014-I)

Q: Find the mean proportional of 16 and 25.

Mean proportional $= \sqrt{16 \cdot 25} = \sqrt{400} = 20$.

Example 3 — Partnership with Time (2012-II)

Q: A invested 20000 for 6 months; B invested 30000 for 8 months. Profit at year-end is 5500. Find each share.

Compute capital × time products. A: $20000 \cdot 6 = 120000$. B: $30000 \cdot 8 = 240000$.
Ratio A : B = $120000 : 240000 = 1 : 2$. Total parts = 3.
A's share $= \tfrac{1}{3} \cdot 5500 = 1833.\overline{3}$; B's share $= \tfrac{2}{3} \cdot 5500 = 3666.\overline{6}$.

Example 4 — Componendo-Dividendo (2017-II)

Q: If $\tfrac{x + 1}{x - 1} = \tfrac{5}{3}$, find $x$.

Apply componendo-dividendo in reverse: $\tfrac{x+1}{x-1} = \tfrac{5}{3}$ is already in componendo-dividendo form with $a = x, b = 1, c = 5, d = 3$ (or interpret as solving directly).
Cross-multiply: $3(x + 1) = 5(x - 1) \implies 3x + 3 = 5x - 5 \implies 2x = 8 \implies x = 4$.

Example 5 — Mixture Replacement (2011-II)

Q: A vessel contains 50 L of milk. 5 L is removed and replaced with water. This is repeated three times. How much milk remains?

Apply formula: $N(1 - k/N)^n = 50 \cdot (1 - 5/50)^3 = 50 \cdot (0.9)^3$.
$0.9^3 = 0.729$. So milk remaining $= 50 \cdot 0.729 = 36.45$ L.

Example 6 — Alligation (2016-II)

Q: In what ratio must a grocer mix two varieties of rice costing ₹15/kg and ₹20/kg to obtain a mixture worth ₹18/kg?

Apply alligation: $\tfrac{\text{cheaper}}{\text{dearer}} = \tfrac{20 - 18}{18 - 15} = \tfrac{2}{3}$.
Ratio = $2 : 3$.

Example 7 — Continued Ratio Application (2018-II)

Q: If $A : B = 3 : 4$ and $B : C = 5 : 6$, find $A : C$.

Make $B$ terms match. LCM of 4 and 5 = 20. Scale: $A : B = 15 : 20,\; B : C = 20 : 24$.
So $A : C = 15 : 24 = 5 : 8$.

How CDS Tests This Topic

Six recurring archetypes: (1) compound or continued ratios — scale to a common middle term, (2) mean/third/fourth proportional — apply the formula directly, (3) partnership with capital × time, (4) componendo-dividendo for algebraic fraction equations, (5) alligation for mixture pricing, and (6) mixture replacement $N(1 - k/N)^n$. Recognise the archetype and the formula is one line.

Exam Shortcuts (Pro-Tips)

Shortcut 1 — Componendo-Dividendo for Algebra

Whenever you see an expression like $\tfrac{x+a}{x-a} = k$, apply componendo-dividendo in reverse: this is equivalent to $x : a = (k+1) : (k-1)$. Saves two lines of algebra.

Shortcut 2 — Alligation as a "+/−" Diagram

Draw the dearer and cheaper prices at the top; mean price in the middle. Subtract diagonally. The differences (always positive) give the mixing ratio.

Shortcut 3 — Partnership in 10 Seconds

Profit ratio = (capital × time) ratio. Compute the products mentally, simplify the ratio, distribute the total profit.

Shortcut 4 — Equivalent Ratio Detection

Two ratios are equivalent if their cross-products are equal. $a : b = c : d \iff ad = bc$.

Shortcut 5 — Mixture Replacement: Memorise Powers

For replacement problems, $(1 - k/N)$ is the "retention factor". After $n$ operations, retained amount = $N \cdot (\text{factor})^n$. Common retention factors: $0.9^2 = 0.81,\; 0.9^3 = 0.729,\; 0.8^2 = 0.64,\; 0.8^3 = 0.512$.

Common Question Patterns

Pattern 1 — Find the Combined Ratio

Given $A : B$ and $B : C$ separately, find $A : B : C$ or $A : C$. Scale $B$ terms to a common value.

Pattern 2 — Mean / Third / Fourth Proportional

Apply the relevant formula. Mean proportional of $a$ and $b$ is $\sqrt{ab}$; third proportional is $b^2/a$; fourth proportional is $bc/a$.

Pattern 3 — Partnership Share Distribution

Capital × time gives the profit ratio. Distribute total profit in that ratio. Watch out for partners who join or leave mid-year — adjust their effective time.

Pattern 4 — Mixture Alligation

Two prices or concentrations, one target mean. Apply the alligation rule. The ratio gives quantities of cheaper to dearer.

Pattern 5 — Mixture Replacement

Repeated removal and replacement with pure solvent. Apply $N (1 - k/N)^n$ for original liquid remaining.

Preparation Strategy

Week 1. Definitions, compound ratios, and the four classical identities (componendo, dividendo, alternendo, invertendo). Drill 20 problems on combining ratios and on the special proportions (mean, third, fourth).

Week 2. Partnership and mixtures. Drill the capital × time formula on 10 partnership variants. Master alligation as the +/− diagram. Practice mixture replacement with the $N(1 - k/N)^n$ formula until it is reflex.

Mock testing. Take timed CDS papers, tagging every ratio question. The chapter is mechanical — speed is the only limit. Use CDS mock tests to build pace.

Cross-train with Percentage (overlap on mixtures), Time-Speed-Distance (ratio of speeds), and Time and Work (ratio of efficiencies).

Drill Ratio and Proportion at Speed

CDS mock papers loaded with partnership, alligation, and mixture-replacement problems. Six archetypes — six shortcuts — practice until reflex.

Start Free Mock Test

Frequently Asked Questions

What is the difference between ratio and proportion?

A ratio compares two quantities of the same kind ($a : b$). A proportion states that two ratios are equal ($a : b = c : d$). Every proportion involves two ratios; every ratio can participate in many proportions.

What is the mean proportional of $a$ and $b$?

It is the value $x$ such that $a : x = x : b$, i.e. $x^2 = ab$, so $x = \sqrt{ab}$. Example: mean proportional of 16 and 25 is $\sqrt{400} = 20$. This formula appears in 2 of every 3 CDS sittings.

What is componendo-dividendo and when do I use it?

If $\tfrac{a}{b} = \tfrac{c}{d}$, then $\tfrac{a+b}{a-b} = \tfrac{c+d}{c-d}$. Use it whenever an equation has the structure $\tfrac{x + k}{x - k}$ on one side and a numerical ratio on the other — convert in one line to $x : k = $ (sum) : (difference). Saves two or three lines of algebra.

How do I solve partnership problems?

Compute capital × time for each partner. The ratio of these products is the profit-share ratio. If a partner joins mid-year, their effective time is the months they were invested. Apply the ratio to distribute total profit.

What is the alligation rule and when does it apply?

Alligation gives the mixing ratio of two ingredients (at known prices or concentrations) to achieve a target mean: $\tfrac{\text{cheaper}}{\text{dearer}} = \tfrac{\text{dearer price} - \text{mean}}{\text{mean} - \text{cheaper price}}$. It applies to price mixtures, concentration mixtures, and even some speed/time problems with average speed as the "mean".

How does the mixture-replacement formula work?

From a vessel of $N$ units of pure liquid, you repeatedly remove $k$ units and replace with water. After $n$ operations, the original liquid remaining is $N (1 - k/N)^n$. The "retention factor" $1 - k/N$ is the fraction kept per operation.

Which CDS Maths topics rely on Ratio and Proportion?

Almost all of arithmetic. Percentage is a ratio in disguise. Time-Speed-Distance uses ratio of speeds. Time and Work uses ratio of efficiencies. Profit and Loss uses cost-to-selling ratios. Master this chapter and the rest of arithmetic gets easier.