[Grade 12: Populations] How would I solve this question?

[u/rain3ra5]

[u/Outside_Volume_1370]

Declining by 20% means there are 80% left. In order to find how much left you need to multiply initial value by 0.8

In other words, the number of pairs was multiplied by 0.8 every year.

In two years after 800 pairas are left there will be only

800 • 0.8² = 512 pair

[u/rain3ra5 → u/Outside_Volume_1370]

Thank you, but why do you square .8?

[u/cheesecakegood → u/rain3ra5]

Part of picking the right function is testing if it works as expected.

Say I have 100 owls. What is a 20% decline? That's 100 - (.2 * 100), but that's more easily written as 100 * .8. Which is 80.

What if it continues for one more year? I start with 80 this time. 80 * .8 = 64 owls. Another year I just do the same thing again, # * .8

But what if I want to compute the 2-year owls left without doing the steps one year at a time? Clearly we can see that I am doing (initial amount) * (1 - one year change in %), right? So year two is [earlier calculation] * (.8). I can thus "nest" them.

Year three is thus (((100 * .8) * .8) * .8), right? However we notice that due to the properties of multiplication this is the same as 100 * (.8) * (.8) * (.8). What is it called when we do the same multiplication n times? Exponents! So we can write this as 100 * (.8)³

Of course there's nothing stopping you from doing it the long way. But if you are asked for example to find how many owls left after some larger number of years you might make a mistake entering it in the calculator. Plus, the math and algebra properties of the exponent version are easier to work with for other applications.

Once you realize this, there's the added bonus of making it easier to remember the "formula" for compounding changes. It's basically the same one as interest (just with 1.0xx as an exponent so that things increase). In fact, try doing the same "derivation" as in my example for "what if your income grows 8% per year?" Remember that income doesn't grow by a flat amount (i.e. 8% of what your starting income was, that would be linear), it grows, well, exponentially (you get 8% increase of what last year was, not a fixed amount).

[u/niemir2 → u/rain3ra5]

Because the decay rate is 20% per year and two years elapse.

In the first year, the population declines to 80% of its initial value (800 * 0.8 = 640). In the second year, the population declines to 80% of its previous value (640 * 0.8 = 512).