Resistance in two identical wires of different densities

Here is a question >

Two wires A and B of same material and mass have their lengths in the ratio $1:2$ .On connecting them to the same source , the rate of heat dissipation in B is $5W$ . What will the rate of heat dissipation in $A$ ?

My theory of doing it :

1. We know $R = frac{rho l}{a}$
2. Both wires will have different densities. Resistivity ($rho$) is independent of density of the material.
3. Therefore $Rpropto l$
4. Find the ratio and get the answer to be $10W$

The book’s way of doing it:

1. $Rpropto frac{l}{a}$
2. Let the density be denoted by $beta$. Multiplying Numerator and denominator by $lbeta$
3. $R propto frac{l^2beta}{albeta}$ . Now $albeta$ is mass. If mass is constant , then
4. $Rpropto l^2$
5. By taking ratio , we get answer to be $25W$

I dont know if my calculation/theory is correct but in step 3 (book’s way) , Why was the density factor not taken into consideration? The step 3 should have been $R propto l^2 beta$ . Clearly the length would have been squared but the density would also be halfed so both the book and me should end up getting the same answer $10W$.

Was my method correct and is $10W$ the correct answer?