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Q.



The six rational numbers between 3 and 4 are:

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a

5 2 , 13 4 , 15 4 , 23 8 , 31 8 , 63 16  

b

7 2 , 13 4 , 15 4 , 25 8 , 31 8 , 63 16  

c

7 2 , 13 4 , 15 4 , 25 8 , 31 8 , 61 16  

d

7 2 , 13 4 , 17 4 , 25 8 , 31 8 , 63 16   

answer is B.

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Detailed Solution

Given numbers are 3 and 4.
Mid-point Method:
Let x and y be two numbers, such that x < y. Then is a rational number lying between x and y. i.e. x <   < y
Applying mid-point method:
A first rational number between 3 and 4.
Q 1 = 3+4 2    Q 1 = 7 2  
Applying the midpoint method:
A second rational number between 3 and 7 2  .
Q 2 = 3+ 7 2 2  
Q 2 = 6+7 2 2  
Q 2 = 13 2 × 1 2  
Q 2 = 13 4   Applying the midpoint method:
The third rational number between 7 2  and 4.
Q 3 = 7 2 +4 2  
Q 3 = 1 2 (7+8) 2    Q 3 = 1 2 × 15 2  
Q 3 = 15 4   Applying the midpoint method:
A fourth rational number between 3 and 13 4  .
Q 4 = 1 2 3+ 13 4  
Q 4 = 1 2 × (12+13) 4    Q 4 = 1 2 × 25 4  
Q 4 = 25 8   Applying the midpoint method:
Fifth rational number between 15 4  and 4.
Q 5 = 1 2 15 4 +4  
Q 5 = 1 2 (15+16) 4  
Q 5 = 1 2 × 31 4    Q 5 = 31 8  
Applying the midpoint method:
The sixth rational number between 31 8  and 4.
Q 6 = 1 2 31 8 +4  
Q 6 = 1 2 (31+32) 8    Q 6 = 1 2 × 63 8  
Q 6 = 63 16  
Hence, six rational numbers between 3 and 4 are:
7 2 , 13 4 , 15 4 , 25 8 , 31 8 , 63 16   Therefore, the correct option is 2.
 
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