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Operations on Rational numbers

Steps used for addition rational-numbers :

1) In a rational number, if the denominators are same then just add the numerators.

p / q + r / q = ( p + r ) / q

2) If the denominators are different then first find the LCM of denominators, make the denominators same and then add the numerators.
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Some solved examples on addition rational numbers
Same denominators :

1) 12 / 7 + 23 / 7 = ( 12 + 23 ) / 7 = 35 / 7

As the GCF(HCF) of 35 and 7 is 7 so we can write this rational number in lowest form.

35 / 7 = ( 35 ÷ 7) / ( 7 ÷ 7)

= 5 / 1 = 5
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2) 31 / 3 + 22 / 3 = ( 31 + 22 ) / 3 = 53 / 3

As the GCF(HCF) of 53 and 3 is 1 so 53 / 3 is the lowest form.
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Different denominators :

3) 5 / 12 + 3 / 8

Solution :

5/ 12 + 3 / 8 , here the denominators are different so we have to find the LCM of 12 and 8. LCM of 12 and 8 = 24

5 / 12 = ( 5 x 2 ) / ( 12 x 2) = 10 / 24

3 / 8 = ( 3 x 3 ) / ( 8 x 3) = 9 / 24

Now the denominators are same so add the numerators.

( 10 + 9 ) / 24 = 19 / 24
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4) 7 / 9 + 4

Solution :

7 / 9 + 4 / 1 ( since 4 = 4 /1 )

Here the denominators are different . LCM of 9 and 1 = 9

7 / 9 = ( 7 x 1) / ( 9 x 1) = 7 / 9

4 / 1 = ( 4 x 9) / ( 1 x 9) = 36 / 9

Now the denominators are same so add the numerators.

( 7 + 36 ) / 9 = 43 / 9
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5) Add 3 / 8 and – 5 / 12

Solution :

3/ 8 + (- 5 ) / 12 , here the denominators are different so we have to find the LCM of 12 and 8. LCM of 12 and 8 = 24

3 / 8 = ( 3 x 3 ) / ( 12 x 2) = 9 / 24

- 5/ 12 = ( -5 x 2 ) / ( 8 x 3) = - 10 / 24

Now the denominators are same so add the numerators.

[ ( 9 + ( - 10 )] / 24 = - 1 / 24
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Rational number

Representation of rational number on number line
Comparison of rational number