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Ordering Rational Numbers

Using the concept of comparing rational numbers, we can arrange the given rational numbers in
ascending or descending order.

−2 3 −3 −4
Example 5: Arrange –1, ,, 1, 0 , , and in descending order.
3 5 4 5
Solution: LCM of the denominators 3, 5 and 4 is 60. So, we first write the equivalent rational

numbers having the same denominator.
×
315
2
−= −60 , −2 = −× 20 = −40 , 3 = 312 = 36 , 1 = 60 − 3 = −× = − 45 ,
,
1
×
×
60 3 3 × 20 60 5 512 60 60 4 415 60
412
0 = 0 and − 4 = −× = − 48
×
60 5 512 60
Now, compare the numerators: 60 > 36 > 0 > –40 > –45 > –48 > –60
Therefore, 60 > 36 > 0 > − 40 > − 45 > − 48 > − 60
60 60 60 60 60 60 60
Thus, the correct descending order of the given rational numbers is
3 − 2 − 3 − 4
1 > > 0 > > > >− 1
5 3 4 5

Rational Numbers Between Two Rational Numbers

There are infinitely many rational numbers between any two rational numbers, as we can always
find more rational numbers between any two given rational numbers.
• By calculating the mid point

+
If a and b are any two rational numbers, then ab is a rational number that always lies between
2
them.
1 3
For example, let us find three rational numbers between and .
3 5

+
1 + 3 = 59 = 14 (Since LCM of 3 and 5 is 15)
3 5 15 15
1 3 1  14 14 7
A rational number between and = ×   = = .
3 5 2  15 30 15

1 7 3
Clearly, < < .
3 15 5
1 7
Next, find the rational number between and .
3 15
+
1   1 + 7   = 1   57  = 1 × 12 = 2
×
×
2  3 15 2  15  2 15 5
1 7 2
\ The rational number between and is .
3 15 5

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