Helen borrows a sum of money from a bank at 12% convertible monthly and wishes to repay it by 24 monthly payments. In total, she will pay 584 of interest. Determine the size of the loan.
I have started by doing this:
The total amount paid back is given by $Pi(1+i)^n/(1+i)^n-1$ so the total interest would be this minus principal $P$. Given $i=0.12$ and given $n=2$. I substitute into our equation and get: $$584=((P(0.12)(1+0.12)^2)/(1+0.12)^2-1)-P$$ but I am not sure how to go farther with this, or if I am doing it right. I got $-1430.31$.
The monthly interest rate is $i=\frac{i^{(12)}}{12}=\frac{12\%}{12}=1\%$.
The total interest is $I=nP-L$, where $P$ is the monthly payment, $n$ is the number of months and $L$ is the loan.
Then $$\left\{ \begin{align} I&=P(n-a_{\overline{n}|i})\\ L&=Pa_{\overline{n}|i} \end{align}\right.\qquad \Longrightarrow\quad \boxed{L=I\cdot\frac{a_{\overline{n}|i}}{n-a_{\overline{n}|i}}=584\cdot\frac{a_{\overline{24}|1\%}}{24-a_{\overline{24}|1\%}}\approx 4,500.5} $$ where $a_{\overline{n}|i}=\frac{1-(1+i)^{-n}}{i}$.
We can also find $P=\frac{L}{a_{\overline{n}|i}}=\frac{4,500.5}{21.24}\approx 211.85$.