Find Rational Numbers between Two Rational Numbers?
Between two rational numbers there exists infinite rational numbers. However, we can find a rational number between two rational numbers using the formula 1/2(a + b) where a and b are rational numbers. Let’s say we have to find rational numbers between 2 and 3 then a rational number between 2 and 3 is given as 1/2(2 + 3) = 5⨯1/2 = 5/2
However, other methods also exist to find rational numbers between two rational numbers.
Method 1: To find rational numbers between two rational numbers with like denominators.
In this, we need to multiply the numerator and denominator of rational numbers with a larger number to create a gap between numerators.
Once the gap is created write the in-between rational numbers just increasing the numerator by 1 and keeping the denominators same.
Example: Find 10 rational numbers between 4/5 and 6/5.
Solution:
In this case, we see that we can only find only one rational number between 4/5 and 6/5 which is 5/5. But here we need to find 10 rational numbers.
Hence, we would multiply the numerator and denominator in both the rational number by 10. Hence we have to find 10 rational numbers between (4⨯10)/(5⨯10) and (6⨯10)/(5⨯10) i.e. 40/50 and 60/50.
Hence, ten rational numbers between 40/60 and 50/60 are 41/50, 42/50, 43/50, 44/50, 45/50, 46/50, 47/50, 48/50, 49/50, 50/50.
If we need more we would multiply by a larger number. For simplicity, you can multiply by 10, 100, etc.
Method 2: To find a rational number between two rational numbers with unlike denominators
In this, we first convert the unlike denominators to like decimals then follow the same method as followed in the case of like denominators
Example: Find 5 rational numbers between 4/3 and 6/5
Solution:
Here we will first make the denominators like, by taking the LCM of denominators 3 and 5. Hence, the LCM of 3 and 5 is 15. Therefore our new equivalent rational numbers will be (4⨯5)/(3⨯5) and (6⨯3)/(5⨯3) i.e. 20/15 and 18/15.
Still, we see that gap is of two only between 18 and 20. Hence, we will multiply with a larger number say 5.
Hence, we have to find a rational number between 20⨯5/15/⨯5 and 18⨯5/15⨯5 i.e. 100/75 and 90/75. Hence, 5 rational numbers between 90/75 and 100/75 are 91/75, 92/75, 93/75, 94/75 and 95/75.
Method 3: To find ‘n’ rational numbers between two rational numbers x and y with unlike denominators such that x < y
In this case, first calculate, d = (y – x)/(n + 1) then find the rational numbers between two rational numbers as (x + d), (x + 2d), (x + 3d),…..,(x + nd)
Example: Find five rational numbers between 1/3 and 2/5.
Solution:
x = 1/3, y = 2/5, n = 5
d = (y – x)/(n + 1) = (2/5 – 1/3)/(5 + 1) = 1/15/6 = 1/90
Five rational numbers between 1/3 and 2/5 are given as
(x + d), (x + 2d), (x + 3d), (x + 4d) and (x + 5d)
(1/3 + 1/90), (1/3 + 2/90), (1/3 + 3/90), (1/3 + 4/90) and (1/3 + 5/90)
(31/90), (32/90), (33/90), (34/90) and (35/90)
Rational Numbers: Definition, Examples, Worksheet
Rational Numbers are numbers written in terms of the ratio of two integers, where the denominator is not zero. In maths, Rational numbers are a type of real numbers that can be written in the form of p/q, where q ≠0. Any fraction is a rational number provided its denominator should not be zero.
Examples of Rational Numbers include 12/21​, 34/2, -22 etc. In other words, a rational number is any number that can be written in the form of a/b​, where a and b are integers and b is not equal to zero.
In this article, we have provided everything related to Rational numbers including, definitions, examples, types, a list of rational numbers, and how to identify rational numbers.
Table of Content
- What is a Rational Numbers?
- Examples of Rational Numbers
- Representation of Rational Numbers
- Types of Rational Numbers
- How to Identify Rational Numbers?
- List of Rational Numbers in Number System
- Arithmetic Operations on Rational Numbers
- Addition of Rational Numbers
- Subtraction of Rational Numbers
- Multiplication of Rational Numbers
- Division of Rational Numbers
- Equivalent Rational Numbers
- Decimal Expansion of Rational Numbers
- Multiplicative Inverse of a Rational Number
- Rational Numbers Properties
- Find Rational Numbers between Two Rational Numbers?
- Representing Rational Numbers on Real Line
- Rational and Irrational Numbers