Paying Off a Loan

Interest grows the balance while payments shrink it. If the payments beat the interest, the loan dies faster and faster; if not, it grows forever no matter how long you pay. One balance sits exactly on the knife edge.

Click or drag anywhere to start a new solution there.

How it works

Modeling with first-order equations

In one sentence

Modeling with a first-order equation means writing (rate in) − (rate out); here $A' = rA - p$ , a linear equation with an unstable equilibrium at $p/r$ .

General formula

\frac{d{\htmlData{sym=0}{A}}}{dt} = {\htmlData{sym=1}{r}}{\htmlData{sym=0}{A}} - {\htmlData{sym=2}{p}} \quad\Rightarrow\quad {\htmlData{sym=0}{A}}(t) = \frac{\htmlData{sym=2}{{p}}}{\htmlData{sym=1}{{r}}} + \left({\htmlData{sym=3}{A_0}} - \frac{\htmlData{sym=2}{{p}}}{\htmlData{sym=1}{{r}}}\right)e^{{\htmlData{sym=1}{r}}t}

Each year the balance changes by interest added minus payment made. Whichever is bigger wins, and the gap between them grows like $e^{rt}$ .

What the symbols mean

$A$
Balance
How much is still owed, in thousands.
$r$
Interest rate
Yearly rate, compounded continuously.
$p$
Payment rate
Thousands paid per year, spread evenly.
$A_0$
Amount borrowed
The balance at the start.

Interest adds $rA$ per year to the balance; payments remove $p$ per year.
The equilibrium $A^* = p/r$ is where interest exactly eats the payments: the balance never shrinks.
Below $A^*$ the balance falls ever faster to zero; the payoff time is $\frac1r\ln\frac{p}{p - rA_0}$ .
Above $A^*$ the debt grows forever: the payments never catch up.

Learn the math

Modeling with First Order EquationsDiff Eq · Lesson 10, First Order Differential Equations