If interest is simple then,
SI = prt/100
where, SI= simple interest, p=principal, r=rate and t=time
Given, SI = p and t=10
so, p = 10pr/100
=> 1 = r/10
or r=10
So rate must be 10%
If it is compound interest then
A = p(1+r/(n*100))^t
where A = new amount = 2p and n = compounding frequency
so, 2p = p(1+r/(n*100))^10n
2 = (1+r/(n*100))^10n
2^(1/10n) = 1 + r/(n*100)
(n*100)(2^(1/10n) - 1) = r
or r = (n*100)(2^(0.1n) - 1)