# Science:Math Exam Resources/Courses/MATH105/April 2010/Question 05 (b)/Solution 2

From part (a) we have that

with the initial amount of money owed being . The goal is to solve this equation for the rate of so that we can actually obtain the quantity we want, which is the money owed after years. Consider the substitution,

which implies that

Therefore we can rewrite (and simplify) our differential equation as

which we recognize as the exponential model and so we know the solution is

with an arbitrary constant. This solution can also be determined from separation of variables and integration. Putting back into our original variable, , we get

which is our desired equation for the amount owed on the mortgage after time . Using the initial condition

we get that . Therefore we are able to write the amount of money owed solely in terms of the annual payments, ,

Now when , we want that the mortgage is paid off, i.e., we want the amount of money we owe to be zero. Therefore we seek that . Therefore,

Therefore, in order to pay off the mortgage in 25 years, we require that = 16818.61. Recall that when we started paying the mortgage at year zero we owed 80% of the value or 240 000 dollars. If we want to find out how much we really paid, we just need to multiply our annual rate, by 25 years to get $420 465.25. Add this to the $60 000 we already paid up front then our $300 000 dollar house has actually cost us $480 465.25!