I need help figuring out what I think is an algebra problem, but I don't know how to set it up.

I have a population starting at 1.6 billion. The annual growth rate is 2%. The annual mortality rate is 0.6%.

The problem is this: In addition to the above parameters, there is a constant loss of 52,560,000 people per year.

How many years would it take for the population to drop to 620,000,000?

(also could you please show how you did it?)

Thanks!

Would it help to cast it in a different light: You borrow $1.6 billion dollars at an interest rate of [(1-0.006)*(1+0.02)=] 1.388% and make payments of $52,560,000 per year. When will your debt be 620,000,000?

If you cast it in this light there may be some on-line financial calculators which would help if you decide to change the numbers.

If you do want to do it yourself, you would have the following: The interest rate is i [in this case 1.388%], the loan amount is A [in this case the $1.6 billion] and the payment is p [in this case $52,560,000]. You want the number of payments [years] n until the balance is less than or equal to B [in this case $620,000,000]. The formula you need to solve for n is

\(\displaystyle A\, x^n\, -\, p\, \frac{x^n\, -\, 1}{i}\, =\, B\)

where x=1+i. Note that if B is zero, this is just the formula for computing the payment on a loan. Solving for n we have

n = \(\displaystyle \frac{log(\frac{B\, i\, -\, p}{A\, i\, -\, P})}{log(1\, +\, i)}\)