How to find number of years and interest to break even using annuity

Question

Say you have a cash flow diagram with all the outflow and inflow laid out and need to calculate the break even point/rate.

1. How would you use the annuity formula to find the number of years?
2. How would you use the annuity formula to find the interest rate given a certain number of years?

For example:

Break Even is inflow = outflow (Economic Equivalence)

And using this formula.

What would N be or what would i be?

Answer 1

$$P=A{(1+i)^N-1\over i(1+i)^N}$$ To solve for $N$, divide by $A$, multiply by $i$: $${Pi\over A}={(1+i)^N-1\over(1+i)^N}=1-(1+i)^{-N}$$ Rewrtie as $$(1+i)^{-N}=1-{Pi\over A}$$ Take logs: $$-N\log(1+i)=\log\left(1-{Pi\over A}\right)$$ Now divide both sides by $-\log(1+i)$.