Question

A uniform wire of resistance 100ohms is melted & recast into a wire of half length & double cross sectional area that of the original.What would be the resistance of new wire formed?

Answer 1

The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area.
When the length of the wire is halved, the resistance is also halved and when its area is doubled also, its resistance is further halved and becomes 1/4 th of the original value.
Thus, the resistance of the new wire is 1/4 of 100 ohms = 25 ohms.

Answer 2

The volume of the wire remains same when it is recast.
       A₂ = 2 *A₁       and           L₂ = L₁ / 2          
 
 Resistance R = ρ * L / A
      R₂ = ρ L₂ / A₂ = ρ L₁ / (2 2 A₁)  = R₁ / 4   = 100 Ω / 4 = 25 Ω

  Hence resistance becomes 1/4 th.
 
You can also say that, resistance is proportional to the length and inversely proportional to the area of cross section.  Hence, the resistance becomes 1/2 * 1/2 times initial.