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Equivalent Rational Numbers

A rational number obtained by multiplying or dividing both the Remember
numerator and the denominator of a rational number by the same Equivalent rational
non-zero integer, is said to be an equivalent form of the given rational numbers have the same
1 2 4
number. For example, ,, and are equivalent rational numbers, value, even though they
3 6 12 may look different.
×
×
1 12 2 1 14 4
because = = , = = .
×
3 32 6 3 34 12
×
p p pk pk
×
÷
In general, if is a rational number and k is a non-zero integer, then = = .
q q qk qk
÷
×
−12
Example 1: Express as a rational number with the numerator 28.
36
Solution: Let the equivalent rational number of −12 with numerator 28 be 28 .
36 x
−12 28  a c 
Then, = ⇒ –12 × x = 36 × 28  Q = ⇒ ad = bc 
36 x  b d 
36 28
×
⇒ x = = –84
− 12
28
Therefore, the required rational number is .
− 84
24
Example 2: Express as a rational number with the denominator 42.
56
Solution: Let the equivalent rational number of 24 with denominator 42 be x .
56 42
24 x  a c 
Then, = ⇒ 24 × 42 = 56 × x  Q = ⇒ ad = bc 
56 42  b d 
×
24 42
⇒ x = = 18
56
18
Therefore, the required rational number is .
42
Standard Form of a Rational Number

A rational number is said to be in standard form if the numerator and the denominator are
co-prime and the denominator is positive. To reduce a rational number into its standard form, we
• make the denominator of the given rational number positive.
• divide the numerator and the denominator by their HCF.
Remember
6 6 ×− ( 1) − 6
For example, = = If the denominator of a rational number
− 36 − 36 × − 1) 36 is negative, change the signs of both the
(
Next, the HCF of 6 and 36 is 6. So, divide the numerator and numerator and denominator to make
the denominator by 6. the denominator positive. As
a − a
−6 = −÷66 = −1 . Hence, the standard form of 6 is −1 . =
36 36 ÷ 6 6 − 36 6 − b b

9 Rational Numbers