

Rational Numbers – An introduction
A rational number is defined as a fraction of two numbers in the form of \[\dfrac{p}{q}\] where p and q can be any integer but q is not equal to 0. Rational numbers consist of all types of numbers including Natural numbers, Whole numbers, 0 etc. Integers consist of both positive and negative numbers. Rational can also be compared just to fractions and integers.
Comparison of Rational Numbers
Comparison of rational numbers means to determine that the given two or more rational numbers are equal, smaller and greater than each other. It is done in the same way as the fractions and integers are compared. Every positive rational number is always greater than zero and negative rational numbers and in the same way, negative rational numbers are always smaller than zero and positive rational numbers.
Rational numbers can be compared by numerators and denominators. Here, we only compare the values of numerators and denominators. There are two methods to compare rational numbers making either the numerator equal or making the denominator equal. Comparing rational numbers is easy; it is similar to fractions and integers as you have done these in previous classes.
How to Compare Rational Numbers
There are two methods to compare rational numbers, i.e., the Decimal Method and the Same Denominator Method.
Decimal Method
In this method of comparison of rational numbers, we convert the rational numbers into their decimal expansions. Then compare the decimal numbers to find out which is larger and which is smaller.
For example, Suppose we need to check the numbers \[\dfrac{3}{5}\] and \[\dfrac{7}{10}\].
We will find the decimal expansion of both the numbers:
\[\dfrac{3}{5}=0.6\]
and
\[\dfrac{7}{10}=0.7\]
Now, we will compare the decimals.
Clearly, the whole number part is the same in both numbers, so we will focus on the decimal part.
\[7>6\]
Hence,
\[\dfrac{7}{10}>\dfrac{3}{5}\]
Same Denominator Method
In this method of comparison, we check if the denominator of the given numbers is equal. If they have the same denominator, comparing just the numerator will give the result for comparing the numbers. If the denominators are different, then we use the HCF of both denominators and make the denominators the same using multiplicative factor \[\dfrac{x}{x}\] where \[x\] is the number which when multiplied to one of the denominator results in the HCF.
Example: Let us again take the same example of \[\dfrac{3}{5}\] and \[\dfrac{7}{10}\].
Here, the denominators in both fractions are not equal, thus we will find the HCF of the denominators.
\[HCF\left(5,10\right)=10\]
Now, we will convert the fractions so that they both have the same denominator, i.e., their HCF.
1st fraction:
\[\dfrac{3}{5}\times\dfrac{2}{2}=\dfrac{6}{10}\]
2nd fraction:
This fraction already has the same denominator, thus we move on to the next step, which is comparing the numerators.
\[7>6\]
Hence,
\[\dfrac{7}{10}>\dfrac{6}{10}\]
\[\Rightarrow\dfrac{7}{10}>\dfrac{3}{5}\]
Finding Rational Numbers Greater or Smaller than a Given Number
Finding rational numbers greater or smaller than a given number can be done using the same denominator method of comparison. To find the numbers greater than the given number, we can just simply increase or decrease, respectively, the number in the numerator, keeping the denominator the same.
Example: Find five rational numbers greater than and 5 rational numbers smaller than 2.
The number 2 can also be written as \[\dfrac{2}{1}\]. So, to write five rational numbers greater than 2 or \[\dfrac{2}{1}\], we will increase the numerator without changing the denominator. Thus, 5 rational numbers greater than \[\dfrac{2}{1}\] are:
\[\dfrac{3}{1},\dfrac{4}{1},\dfrac{5}{1},\dfrac{6}{1},\dfrac{7}{1}\]
Similarly, to find the rational numbers smaller than 2 or \[\dfrac{2}{1}\], we will decrease the numerator without changing the denominator.
Hence, five rational numbers less than 2 or \[\dfrac{2}{1}\] are:
\[\dfrac{1}{1},\dfrac{0}{1},\dfrac{-1}{1},\dfrac{-2}{1},\dfrac{-3}{1}\]
Facts About Rational Numbers
Rational numbers are expressed as \[\dfrac{p}{q}\] such that \[q \ne 0\] where p and q are coprime integers.
All fractions can be rational numbers but all rational numbers cannot be a fraction.
Rational numbers can also be shown on a number line.
Rational numbers are mainly of 3 types: Positive rational numbers, negative rational numbers and zero.
Whole numbers, integers, fractions and natural numbers are all included in rational numbers.
Uses of Rational Numbers
Rational numbers are used in many instances of daily life and Mathematics; some of them are given below:
Rational numbers are used to calculate simple interest and compound interest.
Rational numbers are used in finding the values of trigonometric values.
Used in sharing of anything like land, food, money etc.
Rational numbers are used to show the proportion of anything like one-third of milk.
Rational numbers are also used for measurements.
Interesting Facts
All fractions can be rational numbers but all rational numbers cannot be a fraction.
Rational numbers are only created for inadequate measurements and are created by the Greek mathematician Pythagoras.
Rational numbers can be extended up to infinity.
Every rational number falls in real numbers.
Solved Problems
Find the positive difference between the rational numbers \[\dfrac{5}{8}\] and \[\dfrac{2}{3}\].
Ans: To find the positive difference between the two numbers, we first need to check which number is larger. We can use the same denominator method to check the larger number.
\[HCF\left(8,3\right)=24\]
We will convert the denominator of both fractions to 24 using respective multiplicative factors.
\[\dfrac{5}{8}\times\dfrac{3}{3}=\dfrac{15}{24}\]
\[\dfrac{2}{3}\times\dfrac{8}{8}=\dfrac{16}{24}\]
Clearly, the fraction \[\dfrac{16}{24}\] is greater than the fraction \[\dfrac{15}{24}\]. Thus, \[\dfrac{2}{3}\] is larger than \[\dfrac{5}{8}\].
Now subtracting \[\dfrac{5}{8}\] from \[\dfrac{2}{3}\], we get
\[\dfrac{2}{3} - \dfrac{2}{3} = \dfrac{16}{24} - \dfrac{15}{24} = \dfrac{16-15}{24} = \dfrac{1}{24}\]
Hence, the positive difference between \[\dfrac{5}{8}\] and \[\dfrac{2}{3}\] is \[\dfrac{1}{24}\].
Find five rational numbers greater than \[-7\] but numbers should be consecutive.
Ans: Rational numbers greater than \[-7\] are as follows: \[-5,-2,0,4,2\], etc. as you can see these rational numbers are not consecutive.
Find out which of the two rational numbers is greater \[\dfrac{2}{3}\] or \[\dfrac{3}{4}\] using the decimal method?
Ans: Given rational numbers are \[\dfrac{2}{3}\], \[\dfrac{3}{4}\].
We will find the decimal expansions of both rational numbers,
\[\dfrac{2}{3} = 0.666…=0.{\overline{6}}\]
and
\[\dfrac{3}{4} = 0.75\]
Clearly, both decimal expansions have the same whole number, i.e., 0. So, it comes down to the decimals, i.e., 7 is greater than 6.
Thus, \[\dfrac{3}{4}\] is greater than \[\dfrac{2}{3}\].
Key Features
A rational number is the ratio of two integers, where the denominator of that ratio should not be zero.
Rational numbers can be compared just like fractions with the help of their numerators and denominators.
Natural numbers, whole numbers, integers and fractions with integers are rational numbers. And all these numbers fall in the category of real numbers.
Practice Questions
1. Find 5 rational numbers greater than \[\dfrac{3}{4}\].
Ans: \[1,2,\dfrac{5}{4},\dfrac{7}{4},4\]
2. Which rational number is smaller \[\dfrac{7}{18}\] and \[\dfrac{5}{9}\]?
Ans: \[\dfrac{5}{9}\]
FAQs on Comparison of Rational Numbers
1. What is the most effective method to compare two rational numbers quickly in exams?
The most effective method depends on the denominators. If denominators are small and easy to convert to decimals, use the decimal method. If denominators are large or not easy to convert, the same denominator method is often quicker as per the CBSE 2025–26 syllabus. Choose the approach that minimizes calculation steps during exams.
2. Can you explain how to use the same denominator method for comparing rational numbers with an example?
To use the same denominator method, first find the least common multiple (LCM) of the denominators. Adjust each fraction so they have this common denominator, then compare the numerators. For example, to compare 3/5 and 7/10:
- LCM of 5 and 10 is 10.
- Convert 3/5 to 6/10 by multiplying numerator and denominator by 2.
- 7/10 remains the same.
- Compare: 6/10 < 7/10, so 7/10 is greater.
3. What are some common mistakes students make when comparing rational numbers?
Common mistakes include:
- Not converting both numbers to the same denominator.
- Incorrectly calculating decimal values.
- Assuming the number with the bigger numerator is always bigger, which only works if denominators are equal.
- Overlooking negative signs, which affect order.
Careful step-by-step checking prevents these errors in board exams.
4. Why is it important to understand the comparison of rational numbers for higher-level mathematics?
Understanding comparison of rational numbers builds foundation for topics like inequalities, algebraic manipulation, and number systems in higher classes. Mastery of this concept improves analytical skills and ensures accuracy in multi-step problems in competitive exams.
5. How do you determine which rational number is greater when both have the same numerator?
When rational numbers have the same numerator, the number with the smaller denominator is greater. For example, between 2/3 and 2/5, 2/3 is greater because 3 is less than 5.
6. What strategy should be followed to find several rational numbers greater than a given integer?
Write the integer in fractional form, such as 2 = 2/1. To find numbers greater than this, increase the numerator and keep the denominator constant. For example, 3/1, 4/1, 5/1, etc., are greater than 2/1. This method is helpful for quick listing during CBSE board exams.
7. In what real-life situations can comparing rational numbers be useful?
Comparing rational numbers helps in situations like evaluating discounts, dividing resources, comparing rates, or analyzing fractions in science experiments. This practical skill links mathematics to everyday decision-making.
8. Is there a smallest or largest rational number?
There is no smallest or largest rational number because rational numbers extend infinitely in both positive and negative directions. Between any two rational numbers, you can always find another rational number.
9. How can you verify your answer when comparing two rational numbers using both decimal and same denominator methods?
After comparing by one method, use the other method to confirm your result. For example, compare 5/8 and 2/3:
- Decimal method: 5/8 = 0.625 and 2/3 ≈ 0.666...
- Same denominator: 5/8 = 15/24, 2/3 = 16/24
Both methods show 2/3 is greater, ensuring your answer is correct for the CBSE 2025–26 Maths exam.
10. What is the application of rational number comparison in solving algebraic inequalities?
Comparing rational numbers is essential in solving algebraic inequalities, as it allows you to determine which values satisfy a given condition. For instance, solving for x in inequalities like x/5 > 3/10 requires understanding how to compare and manipulate rational expressions.





