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\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________

Subtraction of Rational Numbers

We know that subtraction is the inverse of addition. Thus, while subtracting two rational numbers, the
additive inverse of the rational number that is being subtracted is added to the other rational number.
3 5
Example 11: Subtract from .
4 6
5 3 5  − 3 3 −3
Solution: We have, − = +   (Since, the additive inverse of is .)
6 4 6  4  4 4
Here, LCM of 6 and 4 is 12. So, make both the denominators equal to 12.

×
5 = 52 = 10 and −3 = −×33 = −9
×
×
6 62 12 4 43 12
5 3 10  − 9 10 +− ( 9) 1
Now, – = +   = =
6 4 12  12  12 12
Example 12: Subtract −1 from −1 .
4 8
−  1

Solution: We have,  −  1   −  −  1   =    8   + Additive inverse of −  1   =  −  1   +   1
 




4
 8
 4
 8

  4

Here, 8 is a multiple of 4. So, their LCM = 8.
× 
 −  1   1  −  1  12 −1 2 −+12 1
\   8   +   =   8   +  ×  = 8 + 8 = 8 = 8

 42
  4
2 −1
Example 13: Subtract from .
5 7
−  1

Solution: We have,  −  1   −   2 =    7   + Additive inverse of  2  =  − −  1   +  −  2  


 



 5
 7
 5
  5
 7
Here, 5 and 7 have no common factor. So, their LCM = 5 × 7 = 35
5
9
−  1
−14
−× 
−× 
\    7   +  −  2  =  15   +  27   = −5 +    35    = −+ − ( 14 ) = −19




35
35
35
 5
 57
 75
×
×
Properties of Subtraction of Rational Numbers
Closure Property
Let us subtract a few pairs of rational numbers.
5 − 3 = 2 (a rational number), −3 − 9 = −96 (a rational number), −7 −13 (a rational number).
1
9 9 9 7 11 77 6 −= 6
From these examples, we observe that subtracting two rational numbers always results in another
rational number. Hence, rational numbers holds the closure property for the subtraction. In other
words, rational numbers are closed under subtraction.
In general,
p r  p r 
if and are two rational numbers, then − is also a rational number.

q s  q s

19 Rational Numbers