# Rational  Number

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A Rational Number of the form or a number which can be expressed in the form of , where p and q are integers and q ≠ zero, is called a
Rational - Number.

Example : 2 / 3 , -5 / 7, -10 / -3 are Rational Number.

 Some useful results 1) Every natural number is a Rational No.but a rational no. need not be a natural number. 2) Zero is a rational no. 3) Every integer is a rational no. but a rational no. need not be an integer. 4) Every fraction is a rational no. but a rational no. need not be a fraction.

If p /q is a rational no., then p - > Numerator and q - > denominator.

If both numerator and denominator are positive or negative then that rational no.called Positive rational no.

If either numerator or denominator is negative then the rational no. is called a Negative rational no.

The rational no. zero is neither positive nor negative.

 Properties of rational no. 1) If p / q is a rational no.and m is non- zero integer then p / q = ( p x m) / ( q x m) This ( p x m) / ( q x m) is rational no. equivalent to p / q. 2) If p / q is a rational number and m is non- zero integer then p / q = ( p ÷ m) / ( q ÷ m)
Some solved examples

1) Express 3 / 4 as a rational no. with denominator 36.

Solution :

In order to express 3 / 4 as rational no. with denominator 36, we first find a number which when multiplied with 4 gives 36.

So, that number is 9. So multiply numerator and denominator both by 9.

3 / 4 = ( 3 x 9) / (4 x 9)

3 /4 = 27 / 36
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2) Express -4 / 5 as rational no. with denominator -30.

Solution :

In order to express -4 / 5 as rational no. with denominator - 30, we first find a number which when multiplied with 5 gives - 30.

So, that number is - 6. So multiply numerator and denominator both by - 6.

-4 / 5 = ( - 4 x – 6 ) / (5 x - 6 )

-4 / 5 = 24 / - 30
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3) Express 42 / -63 as a rational no. of denominator 3.

Solution :

In order to express 42 / -63 as rational no. with denominator 3, we first find a number which when divide with -63 gives 3.

So, that number is -21. So divide numerator and denominator both by - 21.

42 / - 63 = ( 42 ÷ - 21) / (- 63 ÷ -21)

42 / - 63 = -2 / 3
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Lowest form of rational no. :

Every rational no. can be put in the lowest form by using the following steps :

1) Obtain the rational no. p / q .
2) Find the GCF or HCF of p and q.
3) If m = 1 then p / q is the lowest form.
4) If m ≠ 1, then ( p ÷ m) / ( q ÷ m) is the lowest form of p / q.
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Some solved example

1) Write in lowest form : i) 17 / 79 ii) -60 / 72
Solution : i) 17 / 79

As GCF of 17 and 79 is 1 so 17 / 79 is in the lowest form.

ii) -60 / 72

GCF of 60 and 72 is 12.

-60 ÷ 12 = - 5 and 72 ÷ 12 = 6.

∴ -60 / 72 = - 5 / 6.
Rational number

Representation of rational-number on number line
Comparison of rational-number