Ratio and Proportion: Formulas, Concepts, Tricks & Examples

Ratio and Proportion is one of the most scoring, logic-based, and formula-driven topics in Arithmetic. Yet many students find it confusing because ratios behave differently from normal numbers, proportions involve multiple terms, and comparisons depend on simple but powerful rules.

If you understand these three ideas:
✔ Ratio meaning and simplification
✔ Equality of ratios (Proportion)
✔ Relationship between means and extremes

Then you can solve ANY Ratio–Proportion question effortlessly.

This complete Ratio and Proportion guide covers all formulas, concepts, shortcuts, cases, diagrams, FAQs, and exam tips, making it the perfect one-stop resource for competitive exams.

Quick Overview: Ratio & Proportion Formulas

Concept / Situation	What is Considered	What is Used	Formula (With Meaning of Symbols Inside Row)
Basic Ratio	Comparison of two same-unit quantities	Fraction form	Ratio = a : b = a/b (a = antecedent, b = consequent)
Simplifying Ratio	Common factor of a & b	Divide both terms	a : b = (a/k) : (b/k) (k = non-zero common factor)
Proportion	Equality of two ratios	Cross-multiplication	a : b :: c : d ⇒ ad = bc (a,d = extremes; b,c = means)
Fourth Proportional	a : b = c : d	Missing last term	d = (b × c) / a
Third Proportional	a : b = b : c	Missing last term	c = b² / a
Mean Proportional	Between a & b	Geometric mean	x = √(ab)
Comparison of Ratios	Two ratios a/b, c/d	Cross-multiply	a/b > c/d ⇔ ad > bc
Compounded Ratio	Multiple ratios	Multiply terms	(a : b), (c : d), (e : f) → ace : bdf
Duplicate Ratio	Square of ratio	Square numerator & denominator	a² : b²
Sub-duplicate Ratio	Square root	Take √ on both terms	√a : √b
Triplicate Ratio	Cube	a³ : b³
Sub-triplicate Ratio	Cube root	a^(1/3) : b^(1/3)
Componendo & Dividendo	a/b = c/d	Transform ratio	(a+b)/(a-b) = (c+d)/(c-d)
Direct Variation	x increases with y	x ∝ y	x = ky
Inverse Variation	x decreases when y increases	x ∝ 1/y	xy = k

Formulas for Ratio & Proportion

1. Basic Ratio Formula (Foundation of Ratio Problems)

Ratio compares two quantities of the same unit.
If two numbers a and b are compared, the ratio is written as:

Ratio = a : b = a/b

Meaning of symbols

a = antecedent (first term)
b = consequent (second term)

Why this formula works

A ratio expresses how many times one quantity contains the other.
It does not depend on absolute values but on relative values.

Common mistakes students make

Comparing values with different units (rupees vs meters)
Not simplifying ratios

Key tip

Always simplify ratios before using them in equations.

2. Simplifying a Ratio (Zero-Change Rule of Multiplying/Dividing)

Multiplying or dividing each term of the ratio by the same non-zero number does not change the ratio.

Formula:
a : b = ka : kb (k ≠ 0)
a : b = (a/k) : (b/k)

Why this formula works

Both quantities expand or shrink in the same proportion, so their relationship stays the same.

Example

4 : 5 = 8 : 10 = 12 : 15
4 : 6 = 2 : 3

Common errors

Dividing only one term
Forgetting to eliminate common factors

3. Proportion Formula (Equality of Two Ratios)

Two ratios are in proportion if:

a : b = c : d
or
a : b :: c : d

Core Formula

Product of Means = Product of Extremes
b × c = a × d

Meaning of symbols

a and d = extremes
b and c = means

Why this formula works

Cross-multiplication gives equality between the two ratios, forming the basis of all proportion problems.

Common errors

Mixing numerator/denominator
Using wrong term as “mean”

4. Fourth Proportional (Missing Last Term in Proportion)

If:

a : b = c : d

Then d is the fourth proportional.

Formula

d = (b × c) / a

Why this formula works

Cross-multiplying:
a/b = c/d → ad = bc → d = bc/a

Example

2 : 4 = 6 : x
x = (4 × 6)/2 = 12

5. Third Proportional (Special Case Ratio)

If:

a : b = b : c

Then c is the third proportional.

Formula

c = b²/a

Why this formula works

a/b = b/c → ac = b² → c = b²/a

Example

3 : 6 = 6 : x
x = 6²/3 = 12

6. Mean Proportional (Geometric Mean Rule)

Mean proportional between a and b is:

Formula

x² = ab → x = √(ab)

Why this works

a/x = x/b → x² = ab

Example

Between 4 and 9 → √(36) = 6

7. Comparing Two Ratios (Cross-Multiplication Rule)

To compare a : b and c : d:

Formula

a/b > c/d ⇔ ad > bc

Why this formula works

Brings both ratios to a common base without converting to decimals.

Common errors

Attempting decimal conversion
Forgetting to compare cross-products

8. Compounded Ratio (Like Adding Train Length + Platform)

Multiple ratios can be combined by multiplying corresponding terms.

Formula

(a : b), (c : d), (e : f) → ace : bdf

Why this formula works

Each ratio scales the next, just like adding lengths in trains.

Example

2 : 3 and 4 : 5 → 2×4 : 3×5 = 8 : 15

9. Duplicate / Triplicate / Sub-duplicate / Sub-triplicate

These represent squared or cubed ratio relationships.

Formulas

Duplicate: a² : b²
Sub-duplicate: √a : √b
Triplicate: a³ : b³
Sub-triplicate: a^(1/3) : b^(1/3)

Why these formulas work

They reflect changes in power values, used in geometry and advanced ratio questions.

10. Componendo & Dividendo (Most Powerful Ratio Tool)

If a/b = c/d then:

Formula

(a + b)/(a – b) = (c + d)/(c – d)

Why this works

It transforms the proportion to isolate variables easily.

Common errors

Adding/subtracting wrong terms
Using when b = 0 (not valid)

11. Direct Variation (x ∝ y)

Formula

x = ky

Why this works

If one value rises, the other rises proportionally.

Example

Cost ∝ Quantity

12. Inverse Variation (x ∝ 1/y)

Formula

xy = k

Why this works

When one quantity increases, the other decreases proportionally.

Example

Workers ∝ 1/Days

Smart Tips and Practical Tricks for Solving Ratio & Proportion

Mastering Ratio and Proportion becomes simple when you understand how comparisons, equal ratios, and variations work together. Most students make mistakes not because formulas are difficult, but because they apply them without understanding which rule fits the situation. This section breaks down the most important concepts into clear, actionable tips so you can solve questions faster and more accurately.

1. Always Simplify Ratios Before Using Them

A large number of questions become easy once the ratio is reduced to its simplest form.
If a ratio is not simplified, calculations become longer and more error-prone.

Example:
18 : 24 = 3 : 4

This single step reduces complexity in every proportion problem.

2. Check That Both Quantities Have the Same Units

Ratios compare similar quantities.
If units differ, the comparison becomes invalid until converted.

Examples:
✔ Convert rupees into paise
✔ Convert meters into centimeters
✔ Convert hours into minutes

Ignoring unit conversion is one of the most common causes of errors.

3. Use Cross-Multiplication for Quick Comparison

When comparing two ratios, students often convert them into decimals, which wastes time.
Instead, use cross-multiplication:

a/b > c/d
⇔ ad > bc

This shortcut removes the need for division entirely and works instantly.

4. Apply Means–Extremes Rule for Missing Terms

Most proportion problems revolve around the fundamental identity:

Means = Extremes
b × c = a × d

Whether the problem is about partnership, mixture, population growth, or speed-based reasoning, this property solves missing-term questions in one step.

5. Use Componendo and Dividendo to Reduce Long Expressions

This underrated trick simplifies complicated ratios quickly.
If:

a/b = c/d

Then:

(a + b) / (a – b) = (c + d) / (c – d)

This eliminates large fractions and is widely used in age problems, profit & loss, mixture, and algebra-based ratio questions.

6. Identify Whether Variation is Direct or Inverse

Many ratio problems are disguised variation questions.
Identifying the correct type is the key:

✔ Direct variation (x ∝ y) → both rise or fall together
✔ Inverse variation (x ∝ 1/y) → one rises when the other falls

Correctly recognizing variation type leads you to the right equation without confusion.

7. Understand Compound, Duplicate, and Triplicate Ratios Clearly

Higher-level exams frequently use:

✔ Compounded ratio
✔ Duplicate ratio
✔ Triplicate ratio

These help combine multiple comparisons or handle squared and cubed quantities.
Understanding these patterns prevents mistakes in chain-rule and partnership problems.

FAQs About Ratio & Proportion

Q1. Why must both numbers in a ratio have the same unit?

A ratio compares two similar quantities. If the units differ, the comparison becomes meaningless. Converting them to identical units ensures a correct relationship.

Q2. Why does simplifying a ratio help in solving problems faster?

Simplified ratios reduce the size of numbers, making multiplication, division, and proportion calculations easier and less error-prone.

Q3. Why is cross-multiplication used to compare ratios?

It avoids converting ratios into decimals. Cross-multiplying instantly shows which ratio is larger because ad and bc are whole numbers.

Q4. What makes proportion essential in competitive exams?

Proportion converts comparisons into equations using the means–extremes relationship. This simplifies partnership, percentage, and mixture questions dramatically.

Q5. Why is the means = extremes rule important?

In a proportion a : b :: c : d, the identity bc = ad allows you to find any missing term quickly. It is the backbone of proportion questions.

Q6. When is componendo and dividendo used?

It is used when both numerator and denominator have addition or subtraction patterns. The rule simplifies expressions and saves long algebraic steps.

Q7. Why do students confuse direct and inverse proportions?

They look only at numbers without analyzing how one quantity affects the other. Identifying whether both quantities increase together or in opposite directions is essential.

Q8. How does compounded ratio help in exams?

It combines multiple ratios into one. This is useful in chain-rule, mixture, and partnership problems where several comparisons need to be merged.

Q9. Why is √(ab) called the mean proportional?

Because it satisfies the relationship a/x = x/b, giving x² = ab → x = √(ab). It balances both quantities equally.

Q10. What is the simplest way to master Ratio & Proportion?

Recognize patterns: simplify → compare → apply proportion → use variation rules. Once you see the structure, solving becomes automatic and effortless.

Aptitude

Ratio and Proportion: Formulas, Concepts, Tricks & Examples

If you understand these three ideas:
✔ Ratio meaning and simplification
✔ Equality of ratios (Proportion)
✔ Relationship between means and extremes

Then you can solve ANY Ratio–Proportion question effortlessly.

This complete Ratio and Proportion guide covers all formulas, concepts, shortcuts, cases, diagrams, FAQs, and exam tips, making it the perfect one-stop resource for competitive exams.

Quick Overview: Ratio & Proportion Formulas

Concept / Situation	What is Considered	What is Used	Formula (With Meaning of Symbols Inside Row)
Basic Ratio	Comparison of two same-unit quantities	Fraction form	Ratio = a : b = a/b (a = antecedent, b = consequent)
Simplifying Ratio	Common factor of a & b	Divide both terms	a : b = (a/k) : (b/k) (k = non-zero common factor)
Proportion	Equality of two ratios	Cross-multiplication	a : b :: c : d ⇒ ad = bc (a,d = extremes; b,c = means)
Fourth Proportional	a : b = c : d	Missing last term	d = (b × c) / a
Third Proportional	a : b = b : c	Missing last term	c = b² / a
Mean Proportional	Between a & b	Geometric mean	x = √(ab)
Comparison of Ratios	Two ratios a/b, c/d	Cross-multiply	a/b > c/d ⇔ ad > bc
Compounded Ratio	Multiple ratios	Multiply terms	(a : b), (c : d), (e : f) → ace : bdf
Duplicate Ratio	Square of ratio	Square numerator & denominator	a² : b²
Sub-duplicate Ratio	Square root	Take √ on both terms	√a : √b
Triplicate Ratio	Cube	a³ : b³
Sub-triplicate Ratio	Cube root	a^(1/3) : b^(1/3)
Componendo & Dividendo	a/b = c/d	Transform ratio	(a+b)/(a-b) = (c+d)/(c-d)
Direct Variation	x increases with y	x ∝ y	x = ky
Inverse Variation	x decreases when y increases	x ∝ 1/y	xy = k

Formulas for Ratio & Proportion

1. Basic Ratio Formula (Foundation of Ratio Problems)

Ratio compares two quantities of the same unit.
If two numbers a and b are compared, the ratio is written as:

Ratio = a : b = a/b

Meaning of symbols

a = antecedent (first term)
b = consequent (second term)

Why this formula works

A ratio expresses how many times one quantity contains the other.
It does not depend on absolute values but on relative values.

Common mistakes students make

Comparing values with different units (rupees vs meters)
Not simplifying ratios

Key tip

Always simplify ratios before using them in equations.

2. Simplifying a Ratio (Zero-Change Rule of Multiplying/Dividing)

Multiplying or dividing each term of the ratio by the same non-zero number does not change the ratio.

Formula:
a : b = ka : kb (k ≠ 0)
a : b = (a/k) : (b/k)

Why this formula works

Both quantities expand or shrink in the same proportion, so their relationship stays the same.

Example

4 : 5 = 8 : 10 = 12 : 15
4 : 6 = 2 : 3

Common errors

Dividing only one term
Forgetting to eliminate common factors

3. Proportion Formula (Equality of Two Ratios)

Two ratios are in proportion if:

a : b = c : d
or
a : b :: c : d

Core Formula

Product of Means = Product of Extremes
b × c = a × d

Meaning of symbols

a and d = extremes
b and c = means

Why this formula works

Cross-multiplication gives equality between the two ratios, forming the basis of all proportion problems.

Common errors

Mixing numerator/denominator
Using wrong term as “mean”

4. Fourth Proportional (Missing Last Term in Proportion)

If:

a : b = c : d

Then d is the fourth proportional.

Formula

d = (b × c) / a

Why this formula works

Cross-multiplying:
a/b = c/d → ad = bc → d = bc/a

Example

2 : 4 = 6 : x
x = (4 × 6)/2 = 12

5. Third Proportional (Special Case Ratio)

If:

a : b = b : c

Then c is the third proportional.

Formula

c = b²/a

Why this formula works

a/b = b/c → ac = b² → c = b²/a

Example

3 : 6 = 6 : x
x = 6²/3 = 12

6. Mean Proportional (Geometric Mean Rule)

Mean proportional between a and b is:

Formula

x² = ab → x = √(ab)

Why this works

a/x = x/b → x² = ab

Example

Between 4 and 9 → √(36) = 6

7. Comparing Two Ratios (Cross-Multiplication Rule)

To compare a : b and c : d:

Formula

a/b > c/d ⇔ ad > bc

Why this formula works

Brings both ratios to a common base without converting to decimals.

Common errors

Attempting decimal conversion
Forgetting to compare cross-products

8. Compounded Ratio (Like Adding Train Length + Platform)

Multiple ratios can be combined by multiplying corresponding terms.

Formula

(a : b), (c : d), (e : f) → ace : bdf

Why this formula works

Each ratio scales the next, just like adding lengths in trains.

Example

2 : 3 and 4 : 5 → 2×4 : 3×5 = 8 : 15

9. Duplicate / Triplicate / Sub-duplicate / Sub-triplicate

These represent squared or cubed ratio relationships.

Formulas

Duplicate: a² : b²
Sub-duplicate: √a : √b
Triplicate: a³ : b³
Sub-triplicate: a^(1/3) : b^(1/3)

Why these formulas work

They reflect changes in power values, used in geometry and advanced ratio questions.

10. Componendo & Dividendo (Most Powerful Ratio Tool)

If a/b = c/d then:

Formula

(a + b)/(a – b) = (c + d)/(c – d)

Why this works

It transforms the proportion to isolate variables easily.

Common errors

Adding/subtracting wrong terms
Using when b = 0 (not valid)

11. Direct Variation (x ∝ y)

Formula

x = ky

Why this works

If one value rises, the other rises proportionally.

Example

Cost ∝ Quantity

12. Inverse Variation (x ∝ 1/y)

Formula

xy = k

Why this works

When one quantity increases, the other decreases proportionally.

Example

Workers ∝ 1/Days

Smart Tips and Practical Tricks for Solving Ratio & Proportion

1. Always Simplify Ratios Before Using Them

A large number of questions become easy once the ratio is reduced to its simplest form.
If a ratio is not simplified, calculations become longer and more error-prone.

Example:
18 : 24 = 3 : 4

This single step reduces complexity in every proportion problem.

2. Check That Both Quantities Have the Same Units

Ratios compare similar quantities.
If units differ, the comparison becomes invalid until converted.

Examples:
✔ Convert rupees into paise
✔ Convert meters into centimeters
✔ Convert hours into minutes

Ignoring unit conversion is one of the most common causes of errors.

3. Use Cross-Multiplication for Quick Comparison

When comparing two ratios, students often convert them into decimals, which wastes time.
Instead, use cross-multiplication:

a/b > c/d
⇔ ad > bc

This shortcut removes the need for division entirely and works instantly.

4. Apply Means–Extremes Rule for Missing Terms

Most proportion problems revolve around the fundamental identity:

Means = Extremes
b × c = a × d

Whether the problem is about partnership, mixture, population growth, or speed-based reasoning, this property solves missing-term questions in one step.

5. Use Componendo and Dividendo to Reduce Long Expressions

This underrated trick simplifies complicated ratios quickly.
If:

a/b = c/d

Then:

(a + b) / (a – b) = (c + d) / (c – d)

This eliminates large fractions and is widely used in age problems, profit & loss, mixture, and algebra-based ratio questions.

6. Identify Whether Variation is Direct or Inverse

Many ratio problems are disguised variation questions.
Identifying the correct type is the key:

✔ Direct variation (x ∝ y) → both rise or fall together
✔ Inverse variation (x ∝ 1/y) → one rises when the other falls

Correctly recognizing variation type leads you to the right equation without confusion.

7. Understand Compound, Duplicate, and Triplicate Ratios Clearly

Higher-level exams frequently use:

✔ Compounded ratio
✔ Duplicate ratio
✔ Triplicate ratio

These help combine multiple comparisons or handle squared and cubed quantities.
Understanding these patterns prevents mistakes in chain-rule and partnership problems.

FAQs About Ratio & Proportion

Q1. Why must both numbers in a ratio have the same unit?

A ratio compares two similar quantities. If the units differ, the comparison becomes meaningless. Converting them to identical units ensures a correct relationship.

Q2. Why does simplifying a ratio help in solving problems faster?

Simplified ratios reduce the size of numbers, making multiplication, division, and proportion calculations easier and less error-prone.

Q3. Why is cross-multiplication used to compare ratios?

It avoids converting ratios into decimals. Cross-multiplying instantly shows which ratio is larger because ad and bc are whole numbers.

Q4. What makes proportion essential in competitive exams?

Proportion converts comparisons into equations using the means–extremes relationship. This simplifies partnership, percentage, and mixture questions dramatically.

Q5. Why is the means = extremes rule important?

In a proportion a : b :: c : d, the identity bc = ad allows you to find any missing term quickly. It is the backbone of proportion questions.

Q6. When is componendo and dividendo used?

It is used when both numerator and denominator have addition or subtraction patterns. The rule simplifies expressions and saves long algebraic steps.

Q7. Why do students confuse direct and inverse proportions?

They look only at numbers without analyzing how one quantity affects the other. Identifying whether both quantities increase together or in opposite directions is essential.

Q8. How does compounded ratio help in exams?

It combines multiple ratios into one. This is useful in chain-rule, mixture, and partnership problems where several comparisons need to be merged.

Q9. Why is √(ab) called the mean proportional?

Because it satisfies the relationship a/x = x/b, giving x² = ab → x = √(ab). It balances both quantities equally.

Q10. What is the simplest way to master Ratio & Proportion?

Recognize patterns: simplify → compare → apply proportion → use variation rules. Once you see the structure, solving becomes automatic and effortless.

Aptitude

Table of contents

Ratio and Proportion: Formulas, Concepts, Tricks & Examples

Quick Overview: Ratio & Proportion Formulas

Formulas for Ratio & Proportion

1. Basic Ratio Formula (Foundation of Ratio Problems)

2. Simplifying a Ratio (Zero-Change Rule of Multiplying/Dividing)

3. Proportion Formula (Equality of Two Ratios)

4. Fourth Proportional (Missing Last Term in Proportion)

5. Third Proportional (Special Case Ratio)

6. Mean Proportional (Geometric Mean Rule)

7. Comparing Two Ratios (Cross-Multiplication Rule)

8. Compounded Ratio (Like Adding Train Length + Platform)

9. Duplicate / Triplicate / Sub-duplicate / Sub-triplicate

10. Componendo & Dividendo (Most Powerful Ratio Tool)

11. Direct Variation (x ∝ y)

12. Inverse Variation (x ∝ 1/y)

Smart Tips and Practical Tricks for Solving Ratio & Proportion

1. Always Simplify Ratios Before Using Them

2. Check That Both Quantities Have the Same Units

3. Use Cross-Multiplication for Quick Comparison

4. Apply Means–Extremes Rule for Missing Terms

5. Use Componendo and Dividendo to Reduce Long Expressions

6. Identify Whether Variation is Direct or Inverse

7. Understand Compound, Duplicate, and Triplicate Ratios Clearly

FAQs About Ratio & Proportion

Q1. Why must both numbers in a ratio have the same unit?

Q2. Why does simplifying a ratio help in solving problems faster?

Q3. Why is cross-multiplication used to compare ratios?

Q4. What makes proportion essential in competitive exams?

Q5. Why is the means = extremes rule important?

Q6. When is componendo and dividendo used?

Q7. Why do students confuse direct and inverse proportions?

Q8. How does compounded ratio help in exams?

Q9. Why is √(ab) called the mean proportional?

Q10. What is the simplest way to master Ratio & Proportion?

Related Articles

Table of contents

Ratio and Proportion: Formulas, Concepts, Tricks & Examples

Quick Overview: Ratio & Proportion Formulas

Formulas for Ratio & Proportion

1. Basic Ratio Formula (Foundation of Ratio Problems)

2. Simplifying a Ratio (Zero-Change Rule of Multiplying/Dividing)

3. Proportion Formula (Equality of Two Ratios)

4. Fourth Proportional (Missing Last Term in Proportion)

5. Third Proportional (Special Case Ratio)

6. Mean Proportional (Geometric Mean Rule)

7. Comparing Two Ratios (Cross-Multiplication Rule)

8. Compounded Ratio (Like Adding Train Length + Platform)

9. Duplicate / Triplicate / Sub-duplicate / Sub-triplicate

10. Componendo & Dividendo (Most Powerful Ratio Tool)

11. Direct Variation (x ∝ y)

12. Inverse Variation (x ∝ 1/y)

Smart Tips and Practical Tricks for Solving Ratio & Proportion

1. Always Simplify Ratios Before Using Them

2. Check That Both Quantities Have the Same Units

3. Use Cross-Multiplication for Quick Comparison

4. Apply Means–Extremes Rule for Missing Terms

5. Use Componendo and Dividendo to Reduce Long Expressions

6. Identify Whether Variation is Direct or Inverse

7. Understand Compound, Duplicate, and Triplicate Ratios Clearly

FAQs About Ratio & Proportion

Q1. Why must both numbers in a ratio have the same unit?

Q2. Why does simplifying a ratio help in solving problems faster?

Q3. Why is cross-multiplication used to compare ratios?

Q4. What makes proportion essential in competitive exams?

Q5. Why is the means = extremes rule important?

Q6. When is componendo and dividendo used?

Q7. Why do students confuse direct and inverse proportions?

Q8. How does compounded ratio help in exams?

Q9. Why is √(ab) called the mean proportional?

Q10. What is the simplest way to master Ratio & Proportion?

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