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1. Class IX
Mathematics

2. FINDING RATIONAL NUMBERS BETWEEN TWO GIVEN RATIONAL
NUMBERS
METHOD 1 Suppose we are required to find one rational number between two
rational numbers and such that Then, is a rational number
lying between and
Find a rational number lying between (i) and ; (ii) and
(i) Let and required rational number lying between and
(ii) Let and required rational number lying between and

3. Find three rational numbers between
A rational number lying between and is
Now, a rational number lying between and is
= =
And, a rational number lying between and is
= =

4. METHOD 2 Suppose we are required to find rational numbers between two
rational numbers, and with like denominators.
Then, we convert the given rational numbers into equivalent rational numbers by
multiplying the numerator and denominator by a suitable number, usually
.
Now, the required rational numbers may be manually chosen.
Find five rational numbers between and
Let We convert and into equivalent rational numbers by multiplying
the numerator and denominator by .
Thus, and
Clearly, we have

5. Find six rational numbers between 3 and 4.
Let We convert 3 and 4 into equivalent rational numbers using
as multiplying factor.
Thus, and
or
Insert 10 rational numbers between and
We have
Insert 100 rational numbers between and
We have and
Now,

6. METHOD 3 Suppose we are required to find rational numbers between two
given rational numbers and (especially those with unlike denominators)
such that
Let
Then, rational numbers lying between and are
, , .
There are infinitely many rational numbers between any two given rational
numbers.

7. Insert six rational numbers between and
Let and Clearly,
Let
So, the six rational numbers between and are
and
i.e. , and
i.e. ,
, ,

8. Find nine rational numbers between and .
Here and
Hence, the required numbers between and are
,
,
i.e. and ,
i.e. and
i.e. and