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Rational numbers between two numbers are basically numbers that are located between two rational numbers. These numbers can be expressed in terms of the fraction. Rational numbers are a subtype of the broad class of numbers.
- Rational numbers between two rational numbers contain countless rational numbers between them.
- The fraction possesses a numerator and a denominator.
- The denominator should be non-zero type.
- There can be several rational numbers between two rational numbers, too.
- Natural numbers, integers, whole numbers and decimals are included in rational numbers.
- Checking the temperature of the day is a real-life example of these numbers.
- As a result, rational numbers can be represented as:
a/b
- where a and b can be any integer and b ≠ 0.
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Key Terms: Rational Numbers between Two Numbers, Rational Numbers, Numerator, Denominator, Fraction, Natural Numbers, Decimals, Whole Numbers, Integers, Numbers
How to find Rational Number Between Two Rational Numbers?
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There are several rational numbers between two rational numbers, just like there are many numbers between two numbers. The process to find rational numbers between two rational numbers needs to consider the nature of denominators.
- There are two sets of processes, they are for the following cases:
Case 1: When the Numbers have same denominators
The process to find out the rational number between two rational numbers having the same denominators are as follows.
- Since both the denominators are the same, we have to check the value of the numerators.
- Determine the difference between the numerators is calculated.
- This difference gives the numerators which give us the number of rational numbers.
- The numbers are then written.
Example of When the Numbers have same DenominatorsExample: Let us find out the total number of rational numbers between 3/8 and 7/8. Here the denominators are equal, i.e., 8. Now we can consider 3 and 7 as integers and 7 is greater than 3. Thus, there are 3 rational numbers between 3/8 and 7/8 which are 4/8, 5/8 and 6/8. |
Case 2: When Numbers have different Denominators
The process to find out the rational number between two rational numbers having different denominators are as follows.
- Since both denominators are different, we have to make them equal.
- We need to find out their least common multiple (L.C.M) for this.
- To find the L.C.M., we need to multiply it with both the numerator and the denominator.
- The denominator is thus equated.
- Now, we have to check the value of the numerators.
- Determine the difference between the numerators is calculated.
- This difference gives the numerators which give us the number of rational numbers.
- The numbers are then written.
Example of When Numbers have different DenominatorsExample: Let us find out the rational numbers between 1/3 and 5/8 Since the denominators are different, the L.C.M is supposed to be taken, the L.C.M of 3 and 8 is 24. 1/3 X 8/8 = 8/24, and 5/8 X 3/3 = 15/24. So, the two fractions become 8/24 and 15/24. The difference between the numerators 15 and 8 is 6. Thus, there are 6 rational numbers between them, 9/24, 10/24, 11/24, 12/24, 13/24, and 14/24. |
Rational numbers between two numbers
What are Rational Numbers?
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Rational numbers are class of numbers which can be expressed in the form of x/y where y ≠ 0. They are the numbers that are presented in the form of fractions, also known as proportional numbers.
- The fractions have two major components, the numerator, and a non-zero denominator.
- It can also be represented in terms of decimals.
- The word rational is derived from the word ratio.
- When these numbers are divided it can be represented in decimals form.
- A numerator is the part of the fraction present on the upper side of the bar.
- The denominator is present on the lower side of the fraction.
Example of What are Rational Numbers?Example: In ¾, 3 is the numerator, and 4 is the denominator.
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Types of Rational Numbers
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Rational numbers are broadly divided into categories namely, positive rational numbers, and negative rational numbers.
Positive Rational Numbers
Positive rational numbers are numbers where the numerator and the denominator are positive numbers. All the numbers are greater than zero.
Example of Positive Rational NumbersExample: ¾, 5/7, 8/9, etc. are positive rational numbers. |
Negative Rational Numbers
Negative rational numbers are numbers where the numerator and the denominator are of opposite signs. All the numbers are less than zero. At least one of the two, the numerator, or the denominator is negative, e.g., - ½, -3/4, etc.
Example of Negative Rational NumbersExample: - ½, -3/4, etc are negative rational numbers. |
Properties of Rational Numbers
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Some important properties of rational numbers are as follows:
- They are expressed in fractions and can be converted into decimal forms.
- These numbers can be greater than or less than 0.
- They undergo all kinds of arithmetic operations.
- When any rational numbers are added, subtracted, or multiplied, the result will also be a rational number.
- The results remain the same when three numbers are added.
Read More:
| Class 10 Mathematics Related Concept | ||
|---|---|---|
| Real Numbers | Operations on Rational Numbers | Definite Integrals |
| Rational Numbers NCERT Solutions | Degree of polynomial | Number Systems |
Things to Remember
- Rational numbers between two rational numbers contain countless rational numbers.
- These are found in two cases: numbers having the same denominators and numbers having different denominators.
- In the case of different denominators, the least common multiple is supposed to be calculated.
- Rational numbers are the numbers that can be expressed in terms of fractions.
- Fractions have two major components: the numerator and the denominator.
- These numbers are divided into two types, namely positive rational numbers and negative rational numbers.
Sample Questions
Ques. What are rational numbers? (2 marks)
Ans. Rational numbers are a subtype of the broad class of numbers. Rational numbers are the numbers that can be expressed in terms of the fraction. The fraction possesses a numerator, and a denominator, the denominator should be non-zero type. For example, in 1/4, 1 is the numerator, and 4 is the denominator.
Ques. What are the different types of rational numbers? (2 marks)
Ans. Rational numbers are broad of two types, positive rational numbers, and negative rational numbers.
Positive rational numbers: The numerator and the denominator, both are positive numbers or negative numbers, e.g., ¾, 5/7, 8/9, etc.
Negative rational numbers: At least one of the two, the numerator, or the denominator is negative, e.g., - ½, -3/4, etc.
Ques. Find out the rational numbers between 4/11 and 9/11? (2 marks)
Ans. The denominators are the same, 11. Thus, the numerators are considered. There are 4 numbers between 4 and 9, thus there are 4 rational numbers between these two fractions, and they are:
5/11, 6/11, 7/11, and 8/11.
Ques. Find out the rational numbers between 1/7 and 1? (2 marks)
Ans. 1 can be written as 7/7, thus the fractions become 1/7 and 7/7.
The denominators are the same, and the numerators are to be considered. There are 5 numbers between 1 and 7, thus the rational numbers become 2/7, 3/7, 4/7, 5/7, and 6/7.
Ques. Find out 5 rational numbers between ½ and 3/2? (2 marks)
Ans. The denominators are the same, and the numerators are 1 and 3. But there are only 5 numbers between 1 and 3. Thus, we need to solve it as:
½ x 5/5 = 5/10, and 3/2 x 5/5 = 15/10.
Now there are 10 numbers between 5 and 15. Thus the first 5 rational numbers would be, 6/10, 7/10, 8/10, 9/10, and 10/10.
Ques. Find six rational digits between ¼ and 2/5? (2 marks)
Ans. Since the denominators are different, the L.C.M is supposed to be taken, the L.C.M of 4 and 5, which is 20.
1/4 X 5/5 = 5/20, and
2/5 X 4/4 = 8/20. So, the two fractions become 5/20 and 8/20.
Since there are 3 integers between 5 and 8, and we need 6 so,
5/20 x 6/6 = 30/120, and 8/20 x 6/6 = 48/120.
Thus the 6 rational numbers can be, 31/120, 32/120, 33/120, 34/120, 35/120, and 36/120.
Ques. Find out two rational numbers between 1/7 and 2/9? (2 marks)
Ans. Since the denominators are different, the L.C.M is supposed to be taken, the L.C.M of 7 and 9, which is 63.
1/7 X 9/9 = 9/63, and
2/9 X 7/7 = 14/63. So, the two fractions become 9/63 and 14/63.
The first two rational numbers are 10/63 and 11/63.
Ques. Write the next 4 rational numbers in the sequence, ½, 2/4, 3/6, _,_,_,_? (2 marks)
Ans. The sequence is as follows:
½ x 1/1 = ½
½ x 2/2 = 2/4
½ x 3/3 = 3/6
½ x 4/4 = 4/8
½ x 5/5 = 5/10
½ x 6/6 = 6/12
½ x 7/7 = 7/14
Thus, the sequence is, 4/8, 5/10, 6/12, and 7/14
Ques. Find out the total number of rational numbers between 2/7 and 5/7? (2 marks)
Ans. Here the denominators are equal, i.e., 7.
Now we can consider 2 and 5 as integers and 5 are greater than 2.
The rational numbers are, 3/7, 4/7
Ques. Find out the total number of rational numbers between 5/9 and 7/9? (2 marks)
Ans. Here the denominators are equal, i.e., 9.
Now we can consider 5 and 7 as integers and 7 is greater than 5.
Thus, there is 1 rational number between 5/9 and 7/9, that is 6/9, or 2/3.
Ques. Find out the rational numbers between 1/2 and 2/3? (2 marks)
Ans. Since the denominators are different, the L.C.M is supposed to be taken, the L.C.M of 3 and 2, which is 6.
1/2 X 3/3 = 3/6, and
2/3 X 2/2 = 4/6. So, the two fractions become 3/6 and 4/6.
Since there are no integers between 2 and 3, there are no irrational numbers between 3/6 and 4/6.
Ques. Find the rational number between 6/11 and 9/11? (2 marks)
Ans. Since, the rational numbers are having same denominators, therefore, we will compare the numerators here.
Numerator 6 < Numerator 9
The numbers between 6 and 9 are 7 and 8.
Therefore, the rational numbers between 6/11 and 9/11 are 7/11and 8/11.
Ques. Find out the total number of rational numbers between 2/13 and 5/13? (2 marks)
Ans. Here the denominators are equal, i.e., 7.
Now we can consider 2 and 5 as integers and 5 are greater than 2.
The rational numbers are, 3/13, 4/13
Ques. Find out the rational numbers between 4/17 and 9/17? (2 marks)
Ans. The denominators are the same, 17. Thus, the numerators are considered. There are 4 numbers between 4 and 9, thus there are 4 rational numbers between these two fractions, and they are:
5/17, 6/17, 7/17, and 8/17.
Ques. Find out the rational numbers between 1/13 and 1? (2 marks)
Ans. 1 can be written as 13/13, thus the fractions become 1/13 and 13/13.
The denominators are the same, and the numerators are to be considered. There are 5 numbers between 1 and 13, thus the rational numbers become 2/13, 3/13, 4/13, 5/13, and 6/13.
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