Comparing Fractions - Definition, Methods and Examples
Last Updated :
23 Jul, 2025
"Comparison of fractions" refers to the process of determining the larger or smaller fraction by comparing their numerators and denominators. This can be done for any fractions even if the fractions have different numerators and denominators.
Fractions
Before going through the concept of comparing two fractions, let us recall what a fraction is. A fraction is defined as the part of a whole that consists of two parts: the numerator and the denominator, where the numerator is the number above the fractional bar and the denominator is the number below the fractional bar.
Fraction RepresentationHow to Compare Two Fractions
To determine which of the two fractions is larger or smaller, one has to compare them. Based on the numerator and the denominator and the kind of fractions given there are different methods and rules to compare fractions. They are:
- Comparing Fractions with Same Denominators
- Comparing Fractions With Unlike Denominators
- Comparing Fractions using the Decimal Method
- Comparing Fractions using Visualization
- Comparing Fractions using the Cross Multiplication method
Learn all about Fractions : Fraction Tutorials
Comparing Fractions with Same Denominators
It is easy to find the greater or smaller fraction when the fractions have the same denominators. When comparing fractions, check whether the denominators are the same or not. If the denominators are equal, then the fraction with the bigger numerators is the bigger fraction. The fractions are equal if the numerators and denominators of both fractions are equal.
Example: Compare: 5/12 and 17/12.
Solution:
Step 1: First, observe the denominators of the given fractions, i.e., 5/12 and 17/12. Here, the denominators are the same for both fractions.
Step 2: Now, compare the numerators of the given fractions. We can observe that 17 > 5.
Step 3: We know that the fraction with the larger numerator is larger.
Hence, 5/12 < 17/12.
Comparing Fractions With Unlike Denominators
To compare fractions with unlike denominators, we have to convert them to like denominators for which we have to find the Least Common Multiple (LCM) of the denominators. As the denominators are made equal, we can compare the fractions with ease.
Example: Compare: 1/4 and 2/3.
Solution:
Step 1: First, observe the denominators of the given fractions, i.e., 1/4 and 2/3. Since the denominators are different make them equal by finding the LCM of 4 and 3. LCM(4, 3) = 12.
Step 2: Now, let us convert the given fraction in such a way that they have the same denominators. So, multiply the first fraction with 3/3, i.e., 1/4 × 3/3 = 4/12.
Step 3: Similarly, multiply the second fraction with 4/4, i.e., 2/3 × 4/4 = 8/12. Thus, the first fraction becomes 4/12 and the other becomes 8/12.
Step 4: Compare the obtained new fractions, i.e., 4/12 and 8/12. As the denominators are the same, we will compare the numerators. We can observe that 4 < 8.
Step 5: The fraction that has a large numerator is the larger fraction.
So, 8/12 > 4/12. So, 1/4 > 2/3.
Note: If the given fractions have the same numerators and different denominators, then we can compare them easily by looking at their denominators. The fraction that has a smaller denominator has a greater value, while the fraction that has a larger denominator has a smaller value. For example: 6/2 > 6/5.
Comparing Fractions using the Decimal Method
In this method, one can compare fractions by converting the fractions into decimal and comparing them. For this, divide the numerator by the denominator, and thus the fraction is converted into a decimal. Finally, compare their decimal values. Let us understand this by going through an example.
Example: Compare 3/5 and 2/4.
Solution:
Comparing fractions using Decimal methodStep 1: To write 3/5 and 2/4 in decimals, divide the numerator by the denominator. Divide 3 by 5, and 2 by 4.
Step 2: The obtained decimal values are 0.6 and 0.5.
Step 3: Finally, compare the decimal values. 0.6 > 0.5. The fraction that has a larger decimal value would be larger.
Hence, 3/5 > 2/4.
Comparing Fractions using Visualization
Compared to any other method, comparing fractions using visualization is easier.
To compare fractions visually we need to make two boxes of the same size are divided into parts based on the denominators of the fractions and shade the areas based on the numerators. Now, we can simply compare the shaded region to determine the larger or smaller fraction.
Comparing Fractions using VisualizationHere, we can easily see that 2/6 < 2/4, as the 2/4 covers a larger shaded area compared to the 2/6. The smaller fraction occupies a lesser area, while the larger fraction occupies a larger area of the same box.
Comparing Fractions using the Cross Multiplication method
To compare fractions using cross multiplication, we have to multiply the numerator of one fraction with the other fraction's denominator. Let us understand this by going through an example.
Steps to compare fractions Using Cross Multiplication method
Step 1: Multiply the numerator of the first fraction with the second fraction's denominator. And write the product on the side of the selected numerator.
Step 2: Similarly, multiply the second fraction's numerator with the first fraction's denominator, and write the product on the side of the selected numerator.
Step 3: Now, compare both products and the fraction on the side with greater product will be bigger than the other fraction.
Example: Compare 3/8 and 4/5.
Observe the figure given below which explains the concept of cross multiplication better.
Comparing fractions using cross multiplication methodStep 1: Multiply the numerator of the first fraction with the second fraction's denominator. Here, the product is 3 × 5 = 15, which we write near the first fraction.
Step 2: Multiply the second fraction's numerator with the first fraction's denominator. Here, the product is 4 × 8 = 32, which we will write near the second fraction.
Step 3: Now, compare both products, i.e., 15 and 32. Since 15 < 32, the respective fractions can be easily compared, i.e., 3/8 < 4/5.
Hence, 3/8 < 4/5.
Solved Examples on Comparison of Fractions
Example 1: Which of the following fractions is larger: 6/11 or 8/15?
Solution:
Given fractions: 6/11 and 8/15
The denominators of the given fractions are different. So, find out the LCM of the denominators, i.e., LCM(11, 15) = 165.
Now, multiply 6/11 with 15/15 and 8/15 with 11/11.
6/11 × 15/15 = 90/165
8/15 × 11/11 = 88/165
Compare the numerators now, as the denominators are the same.
So, 90 > 88, i.e., 90/165 > 88/165.
Hence, 6/11 > 8/15, i.e., 6/11 is the larger fraction.
Example 2: Which of the given fractions is smaller: 13/85 or 21/85?
Solution:
Given fractions: 13/85 and 21/85.
The denominators of the given fractions are the same. So, compare the numerators of the given fractions.
13 < 21.
So, 13/85 < 21/85.
Hence, 13/85 is the smaller fraction.
Example 3: Compare the fractions 4/25 and 33/100.
Solution:
To compare the given fractions, find their decimal values. So, divide 4 by 25 and 33 by 100.
4/25 = 0.16
33/100 = 0.33
From the decimal values, we can conclude that 0.33 > 0.16. So, 33/100 is greater than 4/25.
Therefore, 33/100 is greater than 4/25.
Example 4: Mrunal Murthi was asked to prove that the given fractions are equal: 30/90 and 25/75. Can you prove the given statement using the LCM method?
Solution:
Given fractions: 30/90 and 25/75.
The denominators of the given fractions are different. So, find out the LCM of the denominators, i.e., LCM(90, 75) = 450.
Now, multiply 30/90 with 5/5 and 25/75 with 6/6.
30/90 × 5/5 = 150/450
25/75 × 6/6 = 150/450
Compare the numerators now, as the denominators are the same.
So, 150 = 150, i.e., 150/450 = 150/450.
Thus, 30/90 = 25/75, i.e., both the given fractions are equal.
Hence, proved.
Example 5: Which of the following fractions is larger: 27/41 or 27/67?
Solution:
Given fractions: 27/41 and 27/67.
Here, the numerators of both fractions are the same but the denominators are different.
We know that the fraction that has a smaller denominator has a greater value, while the fraction that has a larger denominator has a smaller value.
Here, 41 < 67.
So, 27/41 > 27/67
Therefore, 27/41 is the larger fraction.
Read More,
Similar Reads
Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields
5 min read
Basic Arithmetic
What are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or
15+ min read
Arithmetic OperationsArithmetic Operations are the basic mathematical operations—Addition, Subtraction, Multiplication, and Division—used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in
9 min read
Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E
7 min read
What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t
10 min read
ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t
9 min read
PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a
5 min read
Algebra
Variable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch
5 min read
Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one
9 min read
CoefficientA coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 — it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is
8 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (−)
3 min read
Geometry
Lines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang
9 min read
Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two
2 min read
Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of
10 min read
Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the
10 min read
Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction
14 min read
Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su
6 min read
Trigonometry & Vector Algebra
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might
9 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in
6 min read
Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
11 min read
Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv
9 min read
Calculus
Introduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia
6 min read
Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp
12 min read
Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li
13 min read
DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee
2 min read
Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f
11 min read
IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y
3 min read
Probability and Statistics
Basic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors
7 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
5 min read
What is Inferential Statistics?Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi
7 min read
Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o
11 min read
Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
Practice