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How long will it take to pay off a debt of $5000$, with an annual interest rate of $19\%$, compounded monthly, if you make a monthly payment of $100?

I've been using the Loan Repayment formula, where I look for the value of n.

So far I have: $$100= \frac{\left(1+\frac{19}{1200}\right)^{12n} }{ \left(1+\frac{19}{1200}\right)^{12n}-1} \times 5000 \times \left(\frac{19}{1200}\right)$$

I've tried rearranging the terms to find n, but keep getting stuck.

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Solve the equation for $(1+19/1200)^{12n}$, i.e. $$(1+19/1200)^{12n} = \text{something that does not contain $n$}.$$ You can solve for $n$ by taking the logarithm on each side.

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In your expression, if you multiply the numerator and denominator of the term $$ \frac{\left(1+\frac{19}{1200}\right)^{12n} }{ \left(1+\frac{19}{1200} \right)^{12n}-1}\;\; by\;\; \left(1+\frac{19}{1200}\right)^{-12n}$$

it will simplify to $\;\; \frac{1}{\left(1-\frac{19}{1200}\right)^{-12n}}$

It should be easy now to isolate $n$

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