How much more than will you pay on a 30 year mortgage than if you paid it all up front? There are hundreds of loan calculators online but none of them tell me this,
Say I take out a 30 year mortgage on a 80,000 dollar house for 4% interest. How much more than $80k will I spend at the end of the 30 years? Assume the normal compounding, etc.  
 A: The loan calculator will tell you the monthly payment (or the biweekly payment, if you opt for that).
If the monthly payment is $M$, then to find your total payments, multiply $M$ by $(30)(12)$. Subtract $80000$ to find the answer to your question.  
With biweekly payments $B$, to find the total payments we again multiply $B$ by the number of two week periods. This could be a little larger than $(30)(24)$, since the extra $1$ or two days in the year add up to more than a month. Your mortgage document will specify the exact number of payments.
A: First determine the payment $A$ given the initial principal $G = 80,000$ with yearly interest rate $E = 0.04$ -- so the effective payment (monthly, bi-weekly, etc.) interest rate is $R = 1 + E/W$, where $W$ is the number of payments per year -- over $N = Y W$ payments, where $Y$ is the number of years (typically 15 or 30). Let $P\left(t\right)$ be the principle remaining at the $t$-th payment, so that $P\left(0\right) = G$ and $P\left(N\right) = 0$. Then
$$
\begin{eqnarray}
P\left(1\right) &=& P\left(0\right) R - A = GR - A \\
P\left(2\right) &=& P\left(1\right) R - A = GR^2 - AR - A \\
&...& \\
P\left(t\right) &=& G R^t - A \sum_{k=0}^{t-1} R^k = G R^t - A\left(\frac{R^t-1}{R-1}\right)\\
\end{eqnarray}
$$
Using $P\left(N\right) = 0$:
$$
\begin{eqnarray}
A = G R^N \left(\frac{R-1}{R^N-1}\right)
\end{eqnarray}
$$
The total amount paid over the life of the loan is then $AN$, and the total interest paid is $AN-G$. With W = 12 and Y = 30, it looks like the total interest paid is $57,495.61.
You can actually calculate the total interest paid as a fraction of the loan amount $G$:
$$
\begin{eqnarray}
\frac{AN-G}{G} = \frac{AN}{G} - 1 = N R^N \left(\frac{R-1}{R^N-1}\right) - 1
\end{eqnarray}
$$
