Finding interest rate without principal

Question

So, I recently sat a year ten exam and one of the questions was:

Bob invested one half of his savings in an account that paid simple interest for two years and received $550$ dollars as interest. He invested the remaining in an account that paid compound interest (compounded annually) for the same two years at the same rate of interest and received $605$ dollars as interest. What is the annual rate of interest?

I have tried to simplify $p-p(1+r)^2-2pr=55$ ($p=$ principal investment, $r=$ interest rate) but I have not been able to eliminate $p$ from the equation. I am interested to see how you would go about proving the answer to this question.

Answer 1

$2rP = 550$, $((1+r)^2-1)P= 605$, hence ${(1+r)^2-1 \over 2r } = {605 \over 550}$.

This gives ${2+r \over 2} = {605 \over 550}$. from which we get $r = 2 {605 \over 550} -2 = {1 \over 5}$.

Answer 2

Let $2P$ be the total savings, then Bob invested $P$ in an account with simple interest and $P$ in an account with compound interest. Let $R$ be the interest rate and $T$ the number of years.

For simple interest, $$550 = P\cdot R\cdot T = 2\cdot P\cdot R \implies P\cdot R = 275\tag{1}$$

For compound interest, $$605 + P = P(1 + R)^2\tag{2}$$

Solve $(1)$ and $(2)$ simultaneously to find $P$ and $R$.