Question

if in an ap ratio of 4th term and 9 term is 1 :3 find the ratio of 12th and 5th term.

Answer 1

this solution and ratio is 3:1

Answer 2

4th term = a+3d 
9th term = a +8d    
it is given that ratio is 1:3
thus,

a  + 3d           1
----------  =     ----
a + 8 d           3
cross multiple we get,

3(a+3d) = a+8d
3a+9d=a+8d
2a+d=0

d= - 2a  ------------------1

 a12 :   a5
         
12 th term= a+11d                            
5 th term     a + 4d
 
               a + 11d
 ratio =   -----------  ------------2
                a + 4d

substitute 1 in 2 we get,

 =  a+( 11* -2a)
    -------------------
     a +(4* -2a)


=   a-22a
     ----------
    a - 8a

      -21a
 = ----------
       -7a

=    3/1
thus,
the ration between 12th and 5th term is also 3:1

hope it helps!!!!