Page 29 - Start Up Mathematics_8 (Non CCE)

−5 1
(b) =− 2
2 2
1
⇒ –3 <− 2 <− 2
2
−5
i.e., lies between –3 and –2.
2
Since the denominator is 2, we divide the part between –3 and –2 into two equal parts.
−5
Starting from –2, one part to its left represents .
2
1 −5
−2 =
2 2

–3 –2 –1 0 1 2 3

Rational Numbers Between Two Rational Numbers
We already know that there is an infinite number of rational numbers between any two given rational numbers.
Let’s recall the method of finding such rational numbers.
2 7
For example, if and are two rational numbers with like denominators, then choose integers between 2
5 5
and 7 as numerators. Keep the denominator as 5.
3 4 5 6 2 3 4 5 6 7
So, the in-between rational numbers are , , , i.e., < < < < < .
5 5 5 5 5 5 5 5 5 5
However, for rational numbers having unlike denominators, follow these steps:

Step 1: Find the LCM of the denominators.
Step 2: Change the given rational numbers to their equivalent rational numbers with the LCM as the common
denominator.
Step 3: Keeping the LCM as denominator, choose numerators as integers lying in-between the numerators
of the equivalent rational numbers.
1 5
Example 45: Find rational numbers between and .
6 3
Solution: LCM of 6 and 3 = 6

1 1 5 52× 10
= = =
6 6 3 32× 6
The integers between 1 and 10 are 2, 3, 4, 5, 6, 7, 8, 9. 1 5
Taking these as numerators the rational numbers in-between and are:
6 3
2 3 4 5 6 7 8 9 i.e., 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 < 10
,
,
,
,
,
,
,
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
The above mentioned method yields limited number of in-between rational numbers. To find more
in-between rational numbers, take larger common denominators to form equivalent rational numbers.
This increases the range of choice of in-between rational numbers.
Let’s repeat the above example.
I. Choose a larger common denominator such as a multiple of 10 of the LCM. So, we get the common
denominator as 6 × 10 = 60.

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