I'm studying and I'm not that sure how to answer this question.
Is $97.1%$ $=$ $650,000,000$?
I was going to find $2.9%$ of $650,000,000$ however this would be wrong as I would finding our the annum of the $2002$ population, if you know what I mean. I know this looks quite easy but thanks.
On
If it is increasing at a rate of $2.9 \%$ per year, and we're looking at one year, the amount it will increase is $0.029*650000000$. But what is the amount it will be after the increase? Well that would be the amount increased plus what we had originally, so $0.029*650000000+650000000 = (0.029+1)*650000000=1.029*650000000$. The single number we multiply the population by is thus $1.029$.
Let's look at what we did from 2002-2003. We multiplied the current population by 1 plus the percentage increase. Now we're told to assume this is constant. Each year, the population is $1.029$ times the previous year's population. As we showed above, in 2003, this would be $1.029*650000000$. Let's look at 2004. It would be $1.029*(1.029*65000000)$. In 2005, it would be $1.029*(1.029*1.029*650000000)$. Do you notice a pattern here? Basically we are multiplying the original population by $1.029$ once for each year that has passed. In 2020, we have passed 18 years, so we need to multiply by $1.029$ 18 times, or $650000000*(1.029)^{18}$. Does that make sense?
If you want to test if you really understand it, try to figure out what the population was in 1998 if each year the population increase was the same, i.e. $2.9\%$.
Let's think about this. Africa has a population of $650,000,000$ and is increasing every year by $2.9\%$. So next year they will have all the $650$ million and another $2.9\%$, this means they will have $$650+2.9\%(650)=650+0.029(650)=668,850,000$$ million people next year. But notice that both the terms above have a $650$ in common! So we can factor that number out. This gives us $$650(1+2.9\%)=650(1+0.029)=650(1.029)$$ So the number we could have multiplied by was $1.029$.
What did we learn from this? If we want to find something if it increases by some percentage $x\%$ after a certain time, we can multiply it by $$1+d$$ where $d$ is the percentage written as a decimal! Of course, if the number is DECREASING by a certain percentage each year, instead we could multiply by $$1-d$$ where $d$ is the decimal representation of our percentage.
Now for the second part. Think after 1 year, the year 2003, the population in Africa was $$650(1+0.029)$$ The next year, it grows by another $2.9\%$. That means the number we just got grows by another $2.9\%$, but from what we just learned that means it's $$\big(650(1+0.029)\big)(1+0.029)=650(1+0.029)^2$$ We can continue this process again and again until we reach the desired year of 2020. This will give us a number of the form $$650(1+0.029)^t$$ I'll let you figure out what the number $t$ will be!