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\ 15-Nov-2024 Surender Prajapati Proof-6 Reader’s Sign _______________________ Date __________
Enrichment
ab ab
+
+
If a and b are two rational numbers, then will be a rational number between a and b such that a < < b .
2 2
Continuing in this manner, we can find as many rational numbers as we want between any two given rational numbers.
1 4
Example: Let us find three rational numbers between and .
3
5
1 + 4 = 512 = 17 (Since, LCM of 3 and 5 is 15)
+
3 5 15 15
1 4 1 17 17 1 17 4
\ A rational number between and = × = As, < <
3 5 2 15 30 3 30 5
1 17
Now, find the rational number between and .
3 30
1 27
+
1
1
1 1 17 =× 10 + 17 =× 10 17 =× = 27
+
×
2 3 30 2 30 30 2 30 2 330 60
1 17 27 1 27 17
\ Another rational number between and is . As, < <
3 30 60 3 60 30
17 4
Now, find the rational number between and .
30 5
+
1 17 4 1 17 24 1 17 24 1 41 41
×
2
2
2 30 + 5 =× 30 + 30 =× 30 =× = 60
2 330
17 4 41 17 41 4
\ Another rational number between and is . As, < <
30 5 60 30 60 5
1 4 27 17 41
Thus, three rational numbers between and are: , and .
3 5 60 30 60
4
1
Continuing in this manner, we can find as many rational numbers as we want between the rational numbers and .
3 5
Practice Time 3B
1. Which is greater?
−3 −7 −5 −3 7 −7 5
(a) , (b) −1, (c) , (d) ,
11 11 7 −8 12 15 −9
2. Fill in the blanks by the correct symbol out of >, < or =
−8 1 −4 5 − 12 4 20
(a) ....... (b) ....... (c) 0 ........ (d) .......
13 −3 −7 9 13 − 9 − 45
3. Arrange the following rational numbers in ascending order.
(a) −1 , −5 4 , 1 , −7 (b) 1 − 5 − 3 2 − 3
,
,
,
,
,
4 6 −9 −8 12 − 2 8 5 − 3 10
4. Arrange the following rational numbers in descending order.
,
(a) −2 5 1 , −9 1 (b) 7 − 5 2 , 1 − 3
,
,
,,
,
5 7 −35 14 −2 12 16 − 9 − 3 8
5. Write four rational numbers between two given rational numbers.
(a) –2 and −4 (b) –2 and –1 (c) −4 and −2
7 5 3
63 Rational Numbers
