Proportions and Variations: Definition, Formula, Questions, Examples

What are Proportions and Variations in Mathematics?

Proportions and variations are important concepts in mathematics that explain the relationship between quantities. These concepts help us understand how values compare with each other and how one quantity changes when another quantity changes. Topics like ratio, direct proportion, inverse variation, and proportional relationships are widely used in arithmetic, algebra, and quantitative aptitude.

In daily life, proportions and variations appear in situations such as calculating speed and distance, dividing expenses, scaling recipes, managing time and work, and comparing prices. In competitive exams, these concepts are important because they test logical thinking, calculation skills, and problem-solving ability.

Proportion and Variation Definition with Simple Explanation

What is proportion in maths?

A proportion is a statement that shows two ratios are equal. It represents equality between two comparisons.

Example:

$\frac{2}{5} = \frac{4}{10}$

Both ratios are equal because they represent the same relationship.

Important points about proportion:

• It compares two equal ratios
• Usually written in fractional form
• Solved using cross multiplication
• Commonly used in ratio and proportion questions

Basic proportion formula:

What is variation in mathematics?

A variation explains how one quantity changes with another quantity. In simple words, it studies the dependency between variables.

For example:

• If speed increases, distance covered in fixed time increases
• If the number of workers increases, the time required to complete work decreases

Types of variation:

• Direct variation
• Inverse variation

In direct variation:

• Both quantities increase or decrease together

Example:

• More items purchased → higher total cost

In inverse variation:

• One quantity increases while the other decreases

Example:

• More workers → less time required

Why proportion and variation are important:

• Used in quantitative aptitude and arithmetic problems
• Helps solve time, speed, work, and percentage questions
• Frequently asked in exams like CUET, SSC, CAT, IPMAT, and banking exams
• Builds the foundation for algebra and advanced mathematics

Difference Between Ratio, Proportion, and Variation

Ratio, proportion, and variation are related concepts, but they are not the same. Understanding the difference between them is important for solving aptitude and maths questions correctly.

Understanding ratio:

• Ratio compares two quantities using division
• Written as $a:b$ or $\frac{a}{b}$
• Used to compare sizes or values

Example:

• Ratio of boys to girls = $3:2$

Understanding proportion:

• Proportion states that two ratios are equal
• Solved using cross multiplication

Example:

$\frac{3}{4} = \frac{6}{8}$

Cross multiplication:

$3 \times 8 = 4 \times 6$

$24 = 24$

Understanding variation:

• Variation studies how quantities change together
• Used in direct variation and inverse variation problems

Examples:

• Higher speed → more distance covered
• More workers → less time required

Quick comparison table:

Types of Proportion and Variation

In mathematics, proportions and variations are classified into different types based on how quantities are related to each other. Understanding these types is important for solving quantitative aptitude questions, arithmetic problems, and real-life mathematical situations. Topics like direct proportion, inverse proportion, and compound proportion are frequently asked in exams such as CUET, SSC, IPMAT, CAT, and banking exams.

Learning the different types of variation helps students quickly identify the relationship between quantities and apply the correct formula without confusion.

Direct Proportion - Definition, Formula, and Examples

In direct proportion, two quantities increase or decrease together in the same ratio. If one quantity increases, the other also increases. Similarly, if one decreases, the other also decreases.

Definition:
Two quantities are said to be in direct proportion if their ratio remains constant.

Direct proportion formula:

$\frac{x_1}{y_1} = \frac{x_2}{y_2}$

or

$y \propto x$

which means

$y = kx$

where:

• $k$ = constant of proportionality

Example:
If 2 notebooks cost ₹40, then 4 notebooks will cost ₹80.

As the number of notebooks increases, the total cost also increases.

Real-life examples of direct proportion:

• More items purchased → higher total cost

• Higher speed → more distance covered in fixed time

• More hours worked → higher wages

Important characteristics of direct variation:

• Both quantities move in the same direction

• Ratio remains constant

• Graph forms a straight line through origin

Inverse Proportion - Meaning, Formula, and Questions

In inverse proportion, one quantity increases while the other decreases. The product of the two quantities remains constant.

Definition:
Two quantities are said to be in inverse proportion if increasing one quantity causes the other quantity to decrease proportionally.

Inverse proportion formula:

$x_1y_1 = x_2y_2$

or

$y \propto \frac{1}{x}$

which means

$y = \frac{k}{x}$

where:

• $k$ = constant

Example:
If 4 workers complete a task in 12 days, then 8 workers will complete it in fewer days.

Using inverse proportion:

$4 \times 12 = 8 \times x$

$48 = 8x$

$x = 6$

So, 8 workers complete the work in 6 days.

Real-life examples of inverse variation:

• More workers → less time required

• Higher speed → less travel time

• More machines → less production time

Important characteristics of inverse proportion:

• Quantities move in opposite directions

• Product remains constant

• Commonly used in time and work problems

Direct Variation vs Inverse Variation - Key Differences

Direct variation and inverse variation are often confused by students. Understanding the difference between them is essential for solving quantitative aptitude questions correctly.

Example of direct variation:
If petrol consumption increases, total cost also increases.

Example of inverse variation:
If the number of workers increases, the time taken decreases.

Common mistake students make:
Applying direct proportion formula in inverse variation problems.

Quick identification trick:

• Same direction movement → direct variation

• Opposite direction movement → inverse variation

Compound Proportion with Solved Examples

Compound proportion involves more than two quantities and combines multiple proportional relationships in a single problem.

Definition:
Compound proportion is used when one quantity depends on two or more other quantities.

These problems are commonly asked in quantitative aptitude and competitive exams.

Example:
If 5 workers can build 20 walls in 8 days, how many walls can 10 workers build in 4 days?

Step 1:
More workers → more walls
Direct proportion

Step 2:
Fewer days → fewer walls
Direct proportion

Step 3:
Set up equation:

$\frac{10 \times 4 \times x}{5 \times 8 \times 20}$

Simplify:

$\frac{10 \times 4}{5 \times 8} \times 20$

$= \frac{40}{40} \times 20$

$= 20$

So, 10 workers can build 20 walls in 4 days.

Applications of compound proportion:

• Time and work problems

• Speed and distance questions

• Production and efficiency calculations

Important Proportion and Variation Formulas

Formulas play a crucial role in solving proportion and variation questions quickly. A clear understanding of these formulas improves speed, accuracy, and problem-solving ability in competitive exams.

Direct Proportion Formula

In direct proportion, the ratio between two quantities remains constant.

Formula:

$\frac{x_1}{y_1} = \frac{x_2}{y_2}$

or

$y = kx$

where:

• $k$ = constant of proportionality

Example:

If 3 pens cost ₹30, then 6 pens cost:

$\frac{3}{30} = \frac{6}{x}$

$3x = 180$

$x = 60$

Inverse Proportion Formula

In inverse proportion, the product of two quantities remains constant.

Formula:

$x_1y_1 = x_2y_2$

or

$y = \frac{k}{x}$

Example:

If 6 workers complete a task in 10 days, then 12 workers complete it in:

$6 \times 10 = 12 \times x$

$60 = 12x$

$x = 5$

Continued Proportion Formula

Three quantities are said to be in continued proportion if the ratio of the first and second equals the ratio of the second and third.

Formula:

$a:b = b:c$

or

$\frac{a}{b} = \frac{b}{c}$

which gives

$b^2 = ac$

Example:

If $a = 4$ and $c = 16$

$b^2 = 4 \times 16 = 64$

$b = 8$

Mean Proportion Formula

The mean proportional between two numbers is the square root of their product.

Formula:

$\text{Mean Proportion} = \sqrt{ab}$

Example:

Find mean proportion between 9 and 25:

$\sqrt{9 \times 25}$

$= \sqrt{225}$

$= 15$

Variation Formula Cheat Sheet for Quick Revision

Quick revision tips:

• Use ratio method for direct proportion

• Use product method for inverse proportion

• Identify relationship type before applying formula

• Practice real-life word problems regularly

Mastering these proportion and variation formulas helps solve quantitative aptitude questions faster and improves exam performance significantly.

Properties of Proportion in Quantitative Aptitude

Properties of proportion help simplify calculations and solve aptitude questions faster.

Addendo Property

If $a:b = c:d$, then:

$(a+c):(b+d)$

Subtrahendo Property

If $a:b = c:d$, then:

$(a-c):(b-d)$

Dividendo Property

If $a:b = c:d$, then:

$(a-b):b = (c-d):d$

Componendo Property

If $a:b = c:d$, then:

$(a+b):b = (c+d):d$

Alternendo Property

If $a:b = c:d$, then:

$a:c = b:d$

Invertendo Property

If $a:b = c:d$, then:

$b:a = d:c$

Componendo and Dividendo Property

If $a:b = c:d$, then:

$(a+b):(a-b) = (c+d):(c-d)$

Importance of Properties of Proportion

• Reduce lengthy calculations

• Useful in ratio and proportion shortcut methods

• Important for quantitative aptitude exams

• Improve problem-solving speed and accuracy

Properties of Direct and Inverse Variation

Understanding the properties of direct and inverse variation helps identify the correct relationship between quantities.

Quick Identification Trick

• Same direction movement → direct variation

• Opposite direction movement → inverse variation

Common Mistakes Students Make

• Applying direct proportion formula in inverse variation questions

• Ignoring units while comparing quantities

• Using incorrect proportional relationships

A strong understanding of proportion and variation concepts improves speed, logical thinking, and accuracy in quantitative aptitude and competitive exam mathematics.

Practice Questions/Solved Examples on Proportion and Variation

Practising solved examples on proportion and variation helps students understand how formulas and concepts are applied in real mathematical problems. These quantitative aptitude questions improve calculation speed, logical thinking, and accuracy for exams like CUET, SSC, IPMAT, CAT, Banking, and other competitive exams.

Q.1. If $(2x-y)^{2}+(3y-2z)^{2}=0$, then the ratio $x:y:z$ is:

A. $1:3:2$

B. $1:2:3$

C. $3:1:2$

D. $3:2:1$

Hint: If the sum of squares of two terms is equal to zero, then each individual term is also equal to zero.

Solution:

Given:

$(2x-y)^2+(3y-2z)^2=0$

Since the sum of two squares is zero, each squared term must be zero.

Therefore:

$2x-y=0$

and

$3y-2z=0$

From $2x-y=0$:

$2x=y$

Divide both sides by 2:

$x=\frac{y}{2}$

Therefore:

$x:y=1:2$

Now, from $3y-2z=0$:

$3y=2z$

Divide both sides by 2:

$y:z=2:3$

Combining both ratios:

$x:y:z=1:2:3$

Correct Answer: Option B

Q.2. The speeds of three cars are in the ratio $1:3:5$. The ratio of the time taken by these cars to travel the same distance is:

A. $3:5:15$

B. $15:3:5$

C. $15:5:3$

D. $5:3:1$

Hint: Time is inversely proportional to speed.

$\text{Time} \propto \frac{1}{\text{Speed}}$

Solution:

Given speed ratio:

$1:3:5$

Since time varies inversely with speed:

Required time ratio:

$\frac{1}{1}:\frac{1}{3}:\frac{1}{5}$

Take LCM of 1, 3, and 5.

LCM $=15$

Multiply each term by 15:

$1\times15:\frac{1}{3}\times15:\frac{1}{5}\times15$

$=15:5:3$

Correct Answer: Option C

Q.3. If a 7-storey building has a 28 m long shadow, then the number of storeys of a building whose shadow is 48 m long is:

A. $14$

B. $24$

C. $16$

D. $12$

Hint: The length of the shadow is directly proportional to the number of storeys.

Solution:

Given:

7-storey building $\rightarrow$ 28 m shadow

Let the number of storeys of the second building be $x$.

Then:

$\frac{7}{28}=\frac{x}{48}$

Cross multiply:

$28x=7\times48$

$28x=336$

Divide both sides by 28:

$x=\frac{336}{28}$

$x=12$

Therefore, the building has 12 storeys.

Correct Answer: Option D

Q.4. The force (in pound-force) needed to keep a car from skidding on a curve varies directly with the weight of the car and the square of its speed, and inversely with the radius of the curve. Suppose 6125 pound-force is required to keep a 2750-pound car travelling at 35 mph from skidding on a curve of radius 550 feet. How much pound-force is required to keep a 3600-pound car travelling at 50 mph from skidding on a curve of radius 750 feet?

A. $11960$

B. $12150$

C. $12240$

D. $12000$

Hint:

$F=k\left(\frac{ws^2}{r}\right)$

where:

• $F$ = force

• $w$ = weight

• $s$ = speed

• $r$ = radius

Solution:

Given formula:

$F=k\left(\frac{ws^2}{r}\right)$

Substitute the first set of values:

$6125=k\left(\frac{2750\times35^2}{550}\right)$

First calculate $35^2$:

$35^2=1225$

Substitute:

$6125=k\left(\frac{2750\times1225}{550}\right)$

Simplify:

$\frac{2750}{550}=5$

So:

$6125=k(5\times1225)$

$6125=k(6125)$

Divide both sides by 6125:

$k=1$

Now use the formula again:

$F=\frac{3600\times50^2}{750}$

Calculate $50^2$:

$50^2=2500$

Substitute:

$F=\frac{3600\times2500}{750}$

Simplify:

$\frac{3600}{750}=4.8$

So:

$F=4.8\times2500$

$F=12000$

Correct Answer: Option D

Q.5. If the cost of 120 m of cloth is INR 9600, then what will be the cost of 147 m of cloth?

A. INR 16170

B. INR 11670

C. INR 11760

D. INR 17160

Hint: Cost varies directly as the length of the cloth.

Solution:

Given:

120 m cloth costs INR 9600

First find cost per metre:

$\frac{9600}{120}=80$

So, cost of 1 m cloth = INR 80

Now find cost of 147 m cloth:

$147\times80$

$=11760$

Therefore, the required cost is INR 11760.

Correct Answer: Option C

Q.6. $A$ varies directly as $(B+18)$ and $A=108$ when $B=36$. Find the value of $A$ when $B=68$.

A. $75$

B. $86$

C. $127$

D. $172$

Hint:

If $A$ varies directly as $(B+18)$, then:

$\frac{A_1}{A_2}=\frac{B_1+18}{B_2+18}$

Solution:

Given:

$A_1=108$

$B_1=36$

$B_2=68$

Using the formula:

$\frac{108}{A_2}=\frac{36+18}{68+18}$

Simplify numerator and denominator:

$\frac{108}{A_2}=\frac{54}{86}$

Cross multiply:

$108\times86=54\times A_2$

$9288=54A_2$

Divide both sides by 54:

$A_2=\frac{9288}{54}$

$A_2=172$

Correct Answer: Option D

Q.7. The third proportional of the numbers $(x-y)^2$ and $(x^2-y^2)^2$ is:

A. $(x+y)^3(x-y)^2$

B. $(x+y)^4(x-y)^2$

C. $(x+y)^2(x-y)^2$

D. $(x+y)^2(x-y)^3$

Hint:

Third proportional $=\frac{b^2}{a}$

Solution:

Let:

$a=(x-y)^2$

$b=(x^2-y^2)^2$

We know:

$x^2-y^2=(x-y)(x+y)$

Therefore:

$b=[(x-y)(x+y)]^2$

$b=(x-y)^2(x+y)^2$

Now find third proportional:

$c=\frac{b^2}{a}$

Substitute values:

$c=\frac{[(x-y)^2(x+y)^2]^2}{(x-y)^2}$

Expand numerator:

$c=\frac{(x-y)^4(x+y)^4}{(x-y)^2}$

Subtract powers:

$c=(x-y)^2(x+y)^4$

Therefore:

$c=(x+y)^4(x-y)^2$

Correct Answer: Option B

Q.8. What is the third proportional to 10 and 20?

A. $30$

B. $25$

C. $50$

D. $40$

Hint:

If $x$ is the third proportional of $a$ and $b$, then:

$a:b::b:x$

Solution:

Let the third proportional be $x$.

Then:

$10:20::20:x$

Convert into fraction form:

$\frac{10}{20}=\frac{20}{x}$

Cross multiply:

$10x=20\times20$

$10x=400$

Divide by 10:

$x=40$

Correct Answer: Option D

Q.9. What is the fourth proportional to 189, 273, and 153?

A. $117$

B. $299$

C. $221$

D. $187$

Hint:

In $a:b::c:d$, $d$ is called the fourth proportional.

Solution:

Let the fourth proportional be $x$.

Then:

$189:273::153:x$

Convert into fraction form:

$\frac{189}{273}=\frac{153}{x}$

Cross multiply:

$189x=273\times153$

$189x=41769$

Divide both sides by 189:

$x=\frac{41769}{189}$

$x=221$

Correct Answer: Option C

Q.10. Find two numbers such that their mean proportion is 16 and the third proportion is 1024.

A. $4$ and $32$

B. $4$ and $64$

C. $8$ and $64$

D. $8$ and $32$

Hint:

Mean proportion of $a$ and $b$:

$\sqrt{ab}$

Third proportional:

$\frac{b^2}{a}$

Solution:

Let the numbers be $a$ and $b$.

Mean proportion is 16:

$\sqrt{ab}=16$

Square both sides:

$ab=256 \quad ...(1)$

Third proportional is 1024:

$\frac{b^2}{a}=1024$

Cross multiply:

$b^2=1024a \quad ...(2)$

From equation (1):

$a=\frac{256}{b}$

Substitute into equation (2):

$b^2=1024\left(\frac{256}{b}\right)$

Multiply both sides by $b$:

$b^3=1024\times256$

$b^3=262144$

Take cube root:

$b=64$

Substitute into equation (1):

$64a=256$

Divide by 64:

$a=4$

Therefore, the two numbers are 4 and 64.

Correct Answer: Option B

Best Books for Ratio, Proportion, and Variation

Choosing the right book for ratio, proportion, and variation helps build strong quantitative aptitude and arithmetic problem-solving skills. A good book should include concept explanations, formulas, solved examples, shortcut tricks, and practice questions for competitive exams like CUET, SSC, CAT, Banking, and other aptitude tests.

Related Quantitative Aptitude Topics

To strengthen your understanding of proportion and variation, it is important to study related quantitative aptitude topics that use similar formulas, logical relationships, and arithmetic concepts. These topics help improve problem-solving speed and accuracy in exams like CUET, SSC, CAT, IPMAT, Banking, and other competitive aptitude exams.