The total amount paid on a 20-year loan was 5,800 dollars. If the interest rate was 6.6 percent compounded monthly, what was the principal?

Step-by-step solution

1. 1

   Pick the formula that links "amount paid" to "principal". A single lump sum growing at a fixed periodic rate obeys the compound-interest law

   $A=P\left(1+\frac{r}{n}\right)^{nt}$

   Everything except $P$ is given, so this is a one-unknown equation — no annuity formula is needed because the problem describes one final payment, not a stream of them.

2. 2

   Convert the quoted annual rate into a periodic rate and a period count. With $r=0.066$ and $n=12$ compounding periods per year,

   $i=\frac{0.066}{12}=0.0055,\qquad N=nt=12\times 20=240$

   The $6.6\%$ is a nominal annual rate, so it must be split by $12$ before it can be used as an exponent base.

3. 3

   Build the growth factor $(1.0055)^{240}$ carefully. Working through logarithms keeps the arithmetic checkable:

   $240\ln(1.0055)=240\times 0.00548492=1.3163808$
   $e^{1.3163808}=3.7299068$

   So the money multiplies by $3.7299068$ over the 240 months. A frequently seen slip is to write this factor as $3.6521$; that value is simply wrong and it pulls the final principal up by roughly \34$.

4. 4

   Divide the final amount by the growth factor. Rearranging $A=P(1+i)^N$ gives

   $P=\frac{A}{(1+i)^N}=\frac{5800}{3.7299068}=1554.9986$

   Rounded to the cent, the principal is \1{,}555.00$.

5. 5

   Push $1{,}555.00$ forward again as a check. Multiplying back,

   $1555.00\times 3.7299068=5800.00$

   which reproduces the given total, so the discounting was done in the right direction. For contrast, the bad factor $3.6521$ would have returned 5800/3.6521=\1{,}588.13$, and growing that at the true factor gives$1588.13\times 3.7299068=$5{,}924.50$— not$$5{,}800$, which is how you catch the error without a second opinion.