The Proportion Formula states that two ratios are equal, expressed as a : b = c : d, or equivalently a/b = c/d, where the product of the means equals the product of the extremes. This fundamental concept is covered in NCERT Class 6 (Chapter 12) and revisited in Class 7 and Class 8 under Ratio and Proportion. It also forms the basis for direct and inverse proportion problems in Class 8. For JEE Main and NEET aspirants, proportion underpins mixture-and-alligation problems, percentage calculations, and data interpretation. This article covers the complete Proportion Formula, its derivation, formula sheet, three solved examples, CBSE exam tips, common mistakes, and JEE/NEET applications.
Key Proportion Formulas at a Glance
Quick reference for the most important proportion formulas used in CBSE and competitive exams.
- Basic proportion: \( \frac{a}{b} = \frac{c}{d} \) or \( a : b :: c : d \)
- Cross-multiplication (Fundamental Property): \( a \times d = b \times c \)
- Fourth proportional: \( d = \frac{b \times c}{a} \)
- Mean proportional: \( b = \sqrt{a \times c} \)
- Direct proportion: \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \)
- Inverse proportion: \( x_1 \times y_1 = x_2 \times y_2 \)
- Continued proportion: \( a : b :: b : c \Rightarrow b^2 = ac \)
What is the Proportion Formula?
The Proportion Formula is a mathematical statement that two ratios are equivalent. When four quantities a, b, c, and d are arranged so that the ratio of the first pair equals the ratio of the second pair, they are said to be in proportion. We write this as a : b :: c : d, which reads as “a is to b as c is to d.”
In NCERT Class 6 Mathematics, Chapter 12 (Ratio and Proportion), students first encounter this concept. It is further developed in Class 7 and Class 8. The four quantities a, b, c, and d are called the terms of the proportion. Here, a and d are the extremes, while b and c are the means.
The fundamental property of proportion states that the product of the extremes equals the product of the means. This is also known as the cross-multiplication rule. Proportion is not just a theoretical concept. It appears in everyday life — cooking recipes, map scales, currency exchange, speed-distance-time problems, and financial ratios all rely on proportion. Understanding the Proportion Formula deeply helps students solve a wide variety of problems efficiently in both board exams and competitive tests.
There are two main types of proportion: direct proportion (as one quantity increases, the other increases) and inverse proportion (as one quantity increases, the other decreases). A special case called continued proportion occurs when three quantities a, b, c satisfy a : b :: b : c.
Proportion Formula — Expression and Variables
The standard Proportion Formula is written as:
\[ \frac{a}{b} = \frac{c}{d} \quad \Rightarrow \quad a \times d = b \times c \]
This is also written in ratio notation as \( a : b :: c : d \). The cross-multiplication result \( ad = bc \) is the most useful form for solving unknowns.
| Symbol | Quantity | Role in Proportion |
|---|---|---|
| a | First term | First extreme |
| b | Second term | First mean |
| c | Third term | Second mean |
| d | Fourth term | Second extreme |
| ad | Product of extremes | Must equal bc |
| bc | Product of means | Must equal ad |
Additional Proportion Expressions
Several derived forms of the Proportion Formula are useful in different problem types:
- Fourth Proportional — Given a, b, c, find d: \( d = \dfrac{b \times c}{a} \)
- Third Proportional — Given a, b, find c such that a : b :: b : c: \( c = \dfrac{b^2}{a} \)
- Mean Proportional — Given a and c, find b such that a : b :: b : c: \( b = \sqrt{a \times c} \)
- Direct Proportion: \( \dfrac{x_1}{y_1} = \dfrac{x_2}{y_2} \)
- Inverse Proportion: \( x_1 \times y_1 = x_2 \times y_2 \)
- Continued Proportion: \( b^2 = a \times c \)
Derivation of the Cross-Multiplication Property
Start with the basic proportion statement: \( \dfrac{a}{b} = \dfrac{c}{d} \).
Step 1: Multiply both sides by b: \( a = \dfrac{b \times c}{d} \).
Step 2: Multiply both sides by d: \( a \times d = b \times c \).
This proves that the product of the extremes (a and d) always equals the product of the means (b and c) in any valid proportion. This property is the foundation for solving all proportion-based problems.
Complete Algebra Formula Sheet — Ratio and Proportion
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Basic Proportion | \( a : b :: c : d \) | a, b, c, d are quantities | Dimensionless | Class 6, Ch 12 |
| Cross-Multiplication Rule | \( ad = bc \) | a, d = extremes; b, c = means | Dimensionless | Class 6, Ch 12 |
| Fourth Proportional | \( d = \frac{bc}{a} \) | a, b, c known; d unknown | Same as c/a | Class 6, Ch 12 |
| Third Proportional | \( c = \frac{b^2}{a} \) | a : b :: b : c | Same as b²/a | Class 7, Ch 8 |
| Mean Proportional | \( b = \sqrt{ac} \) | a : b :: b : c | Same as √(ac) | Class 7, Ch 8 |
| Direct Proportion | \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \) | x, y increase together | Dimensionless | Class 8, Ch 13 |
| Inverse Proportion | \( x_1 y_1 = x_2 y_2 \) | x increases, y decreases | Dimensionless | Class 8, Ch 13 |
| Continued Proportion | \( b^2 = ac \) | a : b :: b : c | Dimensionless | Class 8, Ch 13 |
| Componendo Rule | \( \frac{a+b}{b} = \frac{c+d}{d} \) | Derived from a/b = c/d | Dimensionless | Class 11, Algebra |
| Dividendo Rule | \( \frac{a-b}{b} = \frac{c-d}{d} \) | Derived from a/b = c/d | Dimensionless | Class 11, Algebra |
Proportion Formula — Solved Examples
Example 1 (Class 6-8 Level)
Problem: Determine whether 4, 8, 6, and 12 are in proportion.
Given: a = 4, b = 8, c = 6, d = 12
Step 1: Write the proportion check using cross-multiplication: \( a \times d = b \times c \)
Step 2: Calculate the product of the extremes: \( 4 \times 12 = 48 \)
Step 3: Calculate the product of the means: \( 8 \times 6 = 48 \)
Step 4: Since \( 48 = 48 \), the four numbers are in proportion.
Answer
Yes, 4 : 8 :: 6 : 12. The numbers are in proportion because the product of the extremes equals the product of the means (48 = 48).
Example 2 (Class 9-10 Level)
Problem: A car travels 150 km in 3 hours. How far will it travel in 7 hours at the same speed? Use the direct proportion formula.
Given: \( x_1 = 150 \) km, \( y_1 = 3 \) hours, \( y_2 = 7 \) hours, \( x_2 = ? \)
Step 1: Since distance and time are directly proportional, apply: \( \dfrac{x_1}{y_1} = \dfrac{x_2}{y_2} \)
Step 2: Substitute the known values: \( \dfrac{150}{3} = \dfrac{x_2}{7} \)
Step 3: Cross-multiply: \( 150 \times 7 = 3 \times x_2 \)
Step 4: Simplify: \( 1050 = 3 x_2 \)
Step 5: Solve for \( x_2 \): \( x_2 = \dfrac{1050}{3} = 350 \) km
Answer
The car will travel 350 km in 7 hours.
Example 3 (JEE/NEET Level)
Problem: Two alloys A and B contain gold and silver in the ratios 3 : 5 and 5 : 3, respectively. In what proportion should alloys A and B be mixed so that the resulting mixture contains gold and silver in equal amounts (ratio 1 : 1)?
Given: Alloy A — gold : silver = 3 : 5 (gold fraction = 3/8). Alloy B — gold : silver = 5 : 3 (gold fraction = 5/8). Required gold fraction in mixture = 1/2.
Step 1: Let the proportion of alloy A in the mixture be \( p \) and alloy B be \( q \). The gold content equation is:
\[ p \times \frac{3}{8} + q \times \frac{5}{8} = (p + q) \times \frac{1}{2} \]
Step 2: Multiply throughout by 8: \( 3p + 5q = 4(p + q) \)
Step 3: Expand the right side: \( 3p + 5q = 4p + 4q \)
Step 4: Rearrange: \( 5q – 4q = 4p – 3p \Rightarrow q = p \)
Step 5: Therefore \( p : q = 1 : 1 \). The two alloys must be mixed in equal proportions.
Answer
Alloys A and B must be mixed in the ratio 1 : 1 to obtain a mixture with equal amounts of gold and silver.
CBSE Exam Tips 2025-26
- Always verify using cross-multiplication: In CBSE exams 2025-26, proportion problems often ask you to check whether four numbers form a proportion. Always use \( ad = bc \) as your first step. This earns full method marks even if arithmetic slips occur.
- Identify the type of proportion first: Before solving, determine whether the problem involves direct proportion, inverse proportion, or continued proportion. Misidentifying the type is the most common reason for losing marks. We recommend writing the type explicitly in your answer sheet.
- Show the proportion statement clearly: Write the full proportion \( a : b :: c : d \) before applying cross-multiplication. CBSE examiners award step marks for this.
- Use the unitary method as a check: After finding an unknown using the Proportion Formula, verify your answer using the unitary method. This takes 10 seconds and prevents errors.
- Memorise the mean proportional formula: The mean proportional \( b = \sqrt{ac} \) appears frequently in Class 10 and Class 11 questions. Our experts suggest practising at least five problems on this variant before the board exam.
- Componendo-Dividendo for Class 11: In Class 11 algebra, the componendo and dividendo rules save significant time. Practice recognising when to apply \( \frac{a+b}{a-b} = \frac{c+d}{c-d} \) in algebraic proportion problems.
Common Mistakes to Avoid
- Confusing means and extremes: Many students mix up which terms are the means (inner terms) and which are the extremes (outer terms). In \( a : b :: c : d \), the extremes are a and d, and the means are b and c. Getting this wrong leads to incorrect cross-multiplication.
- Applying direct proportion to inverse relationships: If more workers complete a job faster, the relationship is inverse, not direct. Always ask: “Does increasing one quantity increase or decrease the other?” before choosing the formula.
- Forgetting to simplify ratios before comparing: When checking if four numbers are in proportion, simplify each ratio to its lowest terms first. For example, 6 : 10 and 9 : 15 are both equal to 3 : 5, confirming they are in proportion.
- Incorrect application of continued proportion: In continued proportion \( a : b :: b : c \), students often forget that b is the geometric mean of a and c, and that \( b^2 = ac \) — not \( b = (a + c)/2 \) (which would be the arithmetic mean).
- Unit mismatch: When setting up a proportion for real-world problems (speed, cost, weight), ensure both ratios use the same units. Mixing kilometres with metres or rupees with paise leads to wrong answers.
JEE/NEET Application of Proportion Formula
In our experience, JEE aspirants encounter the Proportion Formula most frequently in the context of mixture and alligation problems, percentage composition, and coordinate geometry. NEET aspirants use proportion heavily in concentration calculations, dilution problems, and pharmacology dosage questions. Here are the key application patterns:
Application 1 — Mixture and Alligation (JEE Main)
Alligation problems are essentially inverse proportion problems. When two solutions of different concentrations are mixed, the ratio of their volumes is inversely proportional to the difference of their concentrations from the desired concentration. The formula is:
\[ \frac{\text{Volume of solution 1}}{\text{Volume of solution 2}} = \frac{C_2 – C_m}{C_m – C_1} \]
where \( C_1 \), \( C_2 \) are the two concentrations and \( C_m \) is the desired mixture concentration. This is a direct application of the inverse proportion principle.
Application 2 — Coordinate Geometry Section Formula (JEE)
The section formula in coordinate geometry divides a line segment in a given ratio. If a point P divides the line joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio m : n, then:
\[ P = \left( \frac{mx_2 + nx_1}{m+n},\ \frac{my_2 + ny_1}{m+n} \right) \]
This is a direct extension of the Proportion Formula. JEE questions often ask for the ratio in which a point divides a segment, requiring reverse application of this formula.
Application 3 — Concentration and Dilution (NEET)
In NEET chemistry and biology, proportion governs dilution calculations. When a solution is diluted, the amount of solute remains constant. This gives the inverse proportion relationship \( C_1 V_1 = C_2 V_2 \), which is the same as \( x_1 y_1 = x_2 y_2 \). Recognising this as an inverse proportion allows students to solve dilution problems quickly without memorising a separate formula.
In our experience, JEE aspirants who master the Proportion Formula and its variants — especially componendo-dividendo and the mean proportional — gain a significant speed advantage in the quantitative aptitude and mathematics sections.
FAQs on Proportion Formula
We hope this comprehensive guide to the Proportion Formula has helped you understand the concept deeply. For more related formulas, explore our Complete Algebra Formulas hub. You may also find our articles on the Cross Product Formula and the Chi-Square Formula useful for building your quantitative skills. For official NCERT textbook references, visit ncert.nic.in.