# How to calculate rate of future population decline with loss and gain parameters?

#### PistolSlap

##### New member
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!

#### stapel

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

(also could you please show how you did it?)
Sorry but, as you saw in the "Read Before Posting" thread that you read before posting (right?), we don't "do" students' work for them, or give out complete solutions at the drop of a hat.

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?
"The problem is this" suggests that you've already completed the modelling without the "in addition" parameter. How did you do this? Where are you stuck in the computations with the additional constraint?

Please be complete, so we can see where things are going sideways. Thank you!

#### PistolSlap

##### New member
Sorry but we don't "do" students' work for them, or give out complete solutions at the drop of a hat.

"The problem is this" suggests that you've already completed the modelling without the "in addition" parameter. How did you do this? Where are you stuck in the computations with the additional constraint?

Please be complete, so we can see where things are going sideways. Thank you!

I'm not a student. I'm an author. I'm creating a science fiction universe and I have to have internal consistency while determining the rate of decline of the inhabitants of my extremely overpopulated nation-state as they go into the future with some very unethical measures of drastically culling the population. They exterminate a set number of individuals per year, with an aim to get down to a particular population number. I want to know how long it would take them, but I can't just take the current population, subtract the desired population, then divide that number by the constant number eliminated per year, because that doesn't take into account population gain and loss parameters of birth and mortality rates. Since these gains and losses are based on percentages, as the population changes each year by the constant extermination number, the resulting numbers of gains and losses must change dynamically as they reflect that updated total each and every year. Therefore, unless I can determine all three of these factors together, I can't get an accurate estimate on how long it will take my population to be culled to its desired size.

The reason I am asking if you can show me how you did it is because I might have to do it again in the future, if aspects of the plot shift which affect the population of the state. (for example, to begin with, the nation occupied the whole continent of Australia = 8.5 million km^2, then I decided to devote 40% of that continent to wastelands, so the area dropped to 6.4 million km^2. To keep the population density proportionate, the entire starting population also has to drop by 40%. Therefore, I have to do all of these prospective population calculations again. And I'd rather know how, otherwise a one-off answer "at the drop of a hat" as you put it, is pretty useless for me.

Thanks! I really do appreciate the help. As is the case with many authors, ath is not my strong suit. ;-)

#### Ishuda

##### Elite Member
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 = $$\displaystyle \frac{log(\frac{B\, i\, -\, p}{A\, i\, -\, P})}{log(1\, +\, i)}$$

#### Ishuda

##### Elite Member
Ishuda, that's exactly what I suggested here:
http://mathhelpforum.com/algebra/254010-how-calculate-rate-future-population-decline-loss-gain-parameters.html

...except I used 1.4%; why are you using 1.388%?

Btw, seems our esteemed author posted this problem all over the net!
brain _art - just multiplied both percentages which actually would give, for example, a growth per period of [(Initial_Value * mortality rate) * growth rate] instead of a growth of [Initial_Value * growth rate - Initial_Value * mortality rate].

Off to the corner I go

#### PistolSlap

##### New member
Ishuda, that's exactly what I suggested here:
http://mathhelpforum.com/algebra/254010-how-calculate-rate-future-population-decline-loss-gain-parameters.html

...except I used 1.4%; why are you using 1.388%?

Btw, seems our esteemed author posted this problem all over the net!
Haha, yes, I did post in multiple forums, the old "eggs in multiple baskets" trick.
And yes, the happenings would work in the same way, as you have outlined, except your numbers are highly reduced (1.6 billion became 160,000?)

To Ishuda, I am trying to understand the formulae you posted, but failing. lol.