**How to Find Rational Numbers Between
Two Rational Numbers?**

Can
you find the integers between –4 and 1? They are –3, –2, –1 and 0. It is
observed

that
there are finite number of integers lying between two given integers.

Now,
let us find the rational numbers lying between 3/10 and 7/10.

You can find that they are 4/10, 5/10 and 6/10.

You
can also write 3/10 as 30/100 and 7/10 as 70/100.

Now the numbers 31/100, 32/100, 33/100, … , 69/100 are all lying between 3/10 and 7/10, i.e., there are 39 rational numbers.

In the same way, we can go on inserting more and more rational numbers between two given rational numbers. Thus, the number of rational numbers lying between two rational numbers is not finite as there are countless rational numbers lying between two given rational numbers.

Now,
to find a rational number between any two given rational numbers, follow these
steps:

**Method
1: **If *a *and *b *are
two rational numbers and *a *< *b*, then ½(*a *+ *b*)
is a rational number lying between *a *and *b*.

Further,
a rational number between *a *and ½(*a *+ *b*)
is ½[*a *+ ½(*a *+ *b*)]. Also a rational number
between ½(*a *+ *b*) and *b *is ½[½(*a *+ *b*)
+ *b*].

Thus,
we can find infinite number of rational numbers between two given rational
numbers.

**Method
2: **Write the given rational numbers as equivalent
rational numbers by making the same denominator which is equal to the LCM of
both the denominators.

Write
the required number of rational numbers by choosing numbers lying between the
numbers.

**Find Five
Rational Numbers Between 3/5 and 4/5**

Here, we will use Method 2 to find five
rational numbers between 3/5 and 4/5. We should use method 1 in case we have to
find 2 to 3 rational numbers between the given rational numbers. But here we
have to find 5 rational numbers between 3/5 and 4/5.

Using method 2, write the equivalent
rational numbers of 3/5 and 4/5 with denominator 50.

So, multiply the numerator and the
denominator of 3/5 and 4/5 by 10 to get 30/50 and 40/50.

Now, five rational numbers between
30/50 and 40/50 are:

31/50, 32/50, 33/50, 34/50, 35/50

These rational numbers lie between 3/5
and 4/5.

So, five rational numbers between 3/5
and 4/5 are 31/50, 32/50, 33/50, 34/50 and 35/50.

**How many rational
numbers lies between two rational numbers?**

How many rational numbers can you write
between 2/5 and 4/5?

Of course, we can write one rational
number 3/5 between 2/5 and 4/5.

If we write the equivalent rational
numbers of 2/5 and 4/5 by multiplying the numerator and the denominator of both
the rational numbers by 10, we get 20/50 and 40/50.

Between 20/50 and 40/50, we write 19
rational numbers, i.e., 21/50, 22/50, 23/50, 24/50, ……., 39/50.

If we write the equivalent rational
numbers of 2/5 and 4/5 by multiplying the numerator and the denominator of both
the rational numbers by 100, we get 200/500 and 400/500.

Between 200/500 and 400/500, we can
write 199 rational numbers, i.e., 201/500, 202/500, 203/500, 204/500, ……., 399/500.

Similarly, between 2000/5000 and 4000/5000,
we can write 1999 rational numbers, i.e., 2001/5000, 2002/5000, 2003/5000, 2004/5000,
……., 3999/5000.

In the same way, we can write infinite
number of rational numbers between these two rational numbers.

Thus, **there are infinite rational
numbers lie between two given rational numbers**.