Page 20 - ICSE Math 8

 2  − 4  1 2   − 4  1
Example 14: Check if  −   − = −    −  .
 3  5  2 3   5  2 

 2  − 4  1 { 10 −− ( 12) } 1 { 10 12+ } 1 22 1 44 15− 29
Solution: LHS =  −   − = − = − = − = =
 3  5  2 15 2 15 2 15 2 30 30

2   − 4  1 2 { } 2  − 13  20 −−( 39) 20 39+ 59
−−
85
RHS = −    −  = − = −   = = =
3   5  2  3 10 3  10  30 30 30
 2  − 4  1 2   − 4  1
So,  −   − ≠ −    −  (Not associative)
 3  5  2 3   5  2 
Thus, Associative property does not hold true for subtraction of rational numbers.

IV. Existence of Right Identity: The Role of ‘Zero’
p p p p –p p
For any rational number , we have – 0 = but 0 – = (not equal to ). Therefore, only right
q q q q q q
identity exists for subtraction of rational numbers.
7 7 −2  2
− 
0
Example 15: Find if: (a) −= 0 − (b) − 0 = −  
0
3 3 5  5 
7 7 7 7 −2 2  2 02 2
+
− 
Solution: (a) −= and 0 − = − (b) − 0 = − and 0 −   = =
0
3 3 3 3 5 5  5  5 5
7 7 −2  − 2 
So, − 0 ≠ 0 − So, − 0 ≠ 0 −  
3 3 5  5 
Addition and Subtraction of Two or More Rational Numbers

Follow the given steps to add or subtract two or more rational numbers.
Step 1: Find the LCM of the denominators of all the rational numbers. Make the LCM the common denominator
of the resulting answer.
Step 2: Taking one rational number at a time, divide the LCM by the denominator of the first rational number.
Multiply the quotient so obtained with the numerator of the first rational number. Retain the signs.
Step 3: Repeat step 2 for every rational number.

Step 4: Simplify by taking the appropriate signs among all the products. This is the numerator of the resulting
answer.
Step 5: Arrange the numerator and denominator of the result in the rational number form.
Step 6: Reduce the result to its lowest form, if required.

In case the denominator of any rational number is negative, first make it positive.

3  − 6  8  − 5 
Example 16: Find +   + +   .
7  11  21  22 
7 7, 11, 21, 22
Solution: LCM of 7, 11, 21, 22 = 7 × 11 × 3 × 2 = 462 11 1, 11, 3, 22
3  − 6  8  − 5 
∴ +   + +   1, 1, 3, 2
7  11  21  22 

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