Find five rational no between 1and2 …

Since we need to find 5 rational nubers between 1 and 2 so we will multiply and divide both numbers by 5+1 =6 

So 1=1×{tex}{6\over 6}{/tex}    and 2=2×{tex}{6\over6}{/tex}

=> 1={tex}{6\over6}{/tex}   and 2={tex}{12\over 6}{/tex} 

So five rational numbers between 1={tex}{6\over 6}{/tex}and 2 ={tex}{12\over 6}{/tex} are 

{tex}{7\over 6}{/tex},{tex}{8\over 6}{/tex},{tex}{9\over 6}{/tex},{tex}{10\over 6}{/tex} and {tex}{11\over 6}{/tex}

In general if need to find n rational numbers between  a and b then we multiply and divide both the numbers by n+1, then write numbers from a(n+1) to b(n+1) and after that divide it by (n+1) we get required rational numbers between a and b.

in above problem a=1 b=2  and n=5 so we multiply and  divide 1 and 2 by 5+1=6.

Now the numbers between a(n+1)=1×6=6 to b(n+1)=2×6=12 are 7,8,9,10 and 11.

So required rational numbers are 7/6,8/6,9/6,10/6 and 11/6.

We can write 1 as {tex}{1 \over 1}{/tex} and 2 as {tex}{2 \over 1}{/tex}

Multiplying denominator by 10, we have

{tex}{1 \over 10}{/tex} and {tex}{2 \over 10}{/tex}

Now, you can choose any five rational numbers between 

{tex}{1 \over 10}{/tex} and {tex}{2 \over 10}{/tex}

{tex}{2 \over {10}},{3 \over {10}},{4 \over {10}},{5 \over {10}},{6 \over {10}},{7 \over {10}},{8 \over {10}},{9 \over {10}}{/tex}

Let a = 1 , b = 2 , n = 5 ,

b - a = 1 , n + 1 = 6

Therefore 5 rational numbers between 1 and 2 will be

a + (b - a) / 6 , a + 2(b - a) / 6, a + 3(b - a) / 6 , a + 4(b - a) / 6 , a + 5(b - a)/ 6

1 + 1/6 , 1 + 2/6 , 1 + 3/6 , 1 + 4/6 , 1 + 5/6 , 

7/6 , 8/6 , 9 / 6 , 10 / 6 , 11 / 6

Hence required rational numbers are 

7/6, 4/3, 3/2, 5/3, 11/6