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(d) 5 + 3 ; here, LCM of 21 and 14 is 42. We find the equivalent rational numbers with
21 14
the common denominator and add them.
×
5 + 3 = 52 + 33 = 10 9 = 19
×
+
×
×
21 14 21 2 14 3 42 42
Example 7: Simplify:
5 3 5 3 2
(a) −5 + 2 (b) 4 + 2 + 3
7 7 7 7 9
5 3 5 3  −5  3  2  2
52)
(
2
5
Solution: (a) −5 + 2 =− − + + = −+ +  +  =− +−  =−3
(
3)

7 7 7 7  7 7   7  7 7
[By grouping integral parts and fractions separately]
Alternative method:
+
−5 5 + 2 3 = −40 + 17 = −40 17 = −23 =−3 2 Maths Talk
7 7 7 7 7 7 7 For any two rational numbers a and
5 3 2  5 3 2 b, check whether |a + b| = |a| + |b|.
+ )
+
(b) 4 + 2 + 3 = ( 42 3 +  + +  Discuss your observations in the class.
7 7 9  7 7 9
[By grouping integral parts and fractions separately]
+
+
9
= 9 +  45 27 14 =+    86  (Q LCM of 7 and 9 is 63)


63


63
23 23
= 91++ = 10
63 63
Properties of Addition of Rational Numbers
The properties of addition of rational numbers are similar to those of integers. Let us examine
them one by one.
Closure Property

Let us add a few pairs of rational numbers and think about the result.
5 + 3 = 8 (a rational number), −3 + 9 = 30 (a rational number), −7 += −1 (a rational number)
1
9 9 9 7 11 77 6 6

From these examples, we observe that adding two rational numbers always results in another
rational number. Thus, rational numbers, holds the closure property for addition. In other words,
rational numbers are closed under addition.
In general,
p r  p r ps qr
+
if and are two rational numbers, then + = is also a rational number.
q s  q s qs 


Commutative Property

To illustrate this, we can add a few pairs of rational numbers in different orders and check that
the results are the same.

15 Rational Numbers