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Example 1: Write three equivalent rational numbers for the following rational numbers.

2 −3 1
(a) (b) (c)
7 5 − 10
×
×
×
Solution: (a) 2 = 22 = 4 , 2 = 23 = 6 , 2 = 24 = 8
7 72 14 7 73 21 7 74 28
×
×
×
−3 −×32 −6 −3 −×33 −9 −3 −×34 −12
(b) = = , = = , = =
5 52 10 5 53 15 5 54 20
×
×
×
×
1 12 2 1 13 3 1 14 4
×
×
(c) = = , = = , = =
− 10 − 10 2 − 20 − 10 − 10 3 − 30 − 10 − 10 4 − 40
×
×
×
−3
Example 2: Express as a rational number with numerator 21.
8
Solution: We know that (–3) × (–7) = 21
3
−× − ( ) 7 21
\ =
8 ×− ( ) 7 −56
−33 Think and Answer
Example 3: Express as a rational number
55
with denominator 5. Write the next four rational numbers
to complete the following pattern:
Solution: We know that 55 ÷ 11 = 5 −3 −9 −27 −81 ___, ___, ___, ___
,
,
,
−33 11 −3 5 15 45 135 ,
÷
\ =
55 ÷ 11 5
Standard Form of a Rational Number
p
A rational number is said to be in standard form if q is always positive and the numerator and
q
denominator have no common factor other than 1.
18
For example: Let us reduce the rational number to its standard form
− 56
The HCF of 18 and 56 is 2. So, divide the numerator and denominator by 2.

÷
18 18 2 9 Remember
\ = =
÷
− 56 − 56 2 − 28 If the denominator of a rational
number is negative, then to write
Here, denominator is negative. Now, multiply the numerator its standard form we multiply
and denominator by (–1). numerator and denominator by
9 ×− ( 1) − 9 (–1) to make the denominator
positive. For example,
− 28 × − ( 1) = 28 a a ×− ( 1) − a
b 1 ( )
Thus, the standard form of the rational number 18 is −9 . − b = −× − = b
− 56 28
Example 4: Express the following rational into their standard form.
−27 −78 4
(a) (b) (c)
144 −169 − 11

57 Rational Numbers