2 wires A &B are of the same metal and the same area of cross section & have their lengths in the ratio of 1:2. what will be the ratio of currents flowing through them respectively when the same potential difference is applied across length of each of them?

we have

R = ρL/A

now, 

for wire 1

R1 = ρL₁/A

and

for wire 2

R2 = ρL₂/A

now,

from Ohms Law we have

V = IR

or

I = V/R

so,

I₁ = V₁/R₁ = V/R₁

and

I₂ = V₂/R₂ = V/R₂

so,

the ratio of currents will be

I₁ / I₂ = ( V/R₁) / (V/R₂)

or

I₁ / I₂ = (R₂ / R₁) = [(ρL₂/A) / (ρL₁/A) ]

or

I₁ / I₂ = L₂ / L₁

and we are given -  L₁/ L₂ = 1/2

so, current ratio

I₁ / I₂ = 2 / 1