The mass of three wires is in the ratio 1:3:5 & their length is in the ratio 5:3:1. What is the ratio of their resistance?

The resistance of a wire is given by rho*l/A , where rho
is the resistivity, l is the length of the wire and A is the area of
cross-section.

Now, it is assumed that the three wires are
of the same material and hence the resistivity and density is the
same.

The mass of a wire is the product of the density, the
area of cross-section and the length.

Let us denote the
three wires as W1, W2 and W3. Their masses are in the ratio 1:3:5 and their lengths are
in the ratio 5:3:1.

If the mass of W1 is M , W2 weights 3M
and W3 weighs 5M. If W3 has a length L, W2 has a length 3L and W1 has a length
5L.

Using this notation, we write the area of cross-section
of W1 as M/5L, the area of cross-section of W2 by 3M/3L and the area of cross-section of
W3 by 5M/L.

So the resistance of W1 is rho*5L/ (M/5L), W2
has a resistance rho*3L/(3M/3L) and W3 has a resistance
rho*L/(5M/L)

The ratio of resistance is rho*5L/ (M/5L) :
rho*3L/(3M/3L):rho*L/(5M/L)

=> 5L/ (M/5L):
3L/(3M/3L) : L/(5M/L)

=> 25L^2/M : 9L^2/3M :
L^2/5M

=> 25 : 9/3 :
1/5

=> 25 : 3 :
(1/5)

Therefore the ratio of their resistance
is 125 : 15 : 1.