2. Rational numbers between two or more rational numbers

Rational numbers between two or more rational numbers

Rational numbers between two rational numbers

When Denominators are the same
Step 1: Check the values on the numerators of the rational numbers
Step 2: Find by how many values, the numerators differ from each other
Step 3: Since, the denominators are the same for the two rational numbers, therefore, we can write the rational numbers between the two given rationals, in the increasing order of numerator, if the difference between the two numerators is more.
Step 4: If the difference between two numerators is less, and we need to find more rational numbers, then multiply the numerator and denominator of the given rational numbers by multiples of 10.

When Denominators are Different
Suppose we have two rational numbers with different denominators, then follow the below steps to find the rational numbers between them.
Step 1: Find the LCM of two rational numbers first.
Step 2: Multiply and divide the two rational numbers, by the value that results in the denominators equal to the obtained LCM.
Step 3: Once the denominators become the same, follow the same rules as we have discussed for the rational numbers with the same denominators.
Step 4: If the difference between two numerators is less, and we need to find more rational numbers, then multiply the numerator and denominator of the given rational numbers by multiples of 10.

Visual representation of rational numbers between two rational numbers

if we want to find more number of rational numbers between  3/ 5and 4/5. We should write the multiple of the rational numbers, that is,
3/5can be written as 30/ 50 and 4/5can be written as 40/50
From, this we can understand that an infinite (or) an uncountable number of rational numbers are present between any two rational numbers.

Using average method to find rational numbers between two rational numbers

Let us take two numbers as x and y, then
average = x + y²
Let, x and y be any two rational numbers, then we can find as many rational numbers between x and y by using the average concept as e1,e2,e3,e4,e5....etc
e₁=x+y₂;e₂=x+e12;e₃=x+e22. (e2,e3) represents the rational numbers present to the left of the average of x and y, which is to the left of e1 in the number line.

Rational numbers following a pattern

Aasha was asked to write 3 more rational numbers similar to 12,24,36,48,..... She noticed that
all these numbers surprisingly follow a pattern like, 1×22×2=24,1×32×3=36,1×42×4=48.
Since she identified the pattern in which the numbers are written, it became easy for her to write 3 more rational numbers.
It is 1×52×5=24,1×62×6=612,1×72×7=712.