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The world's population has grown at an average rate of $1.9$ percent per year since $1945$. There were approximately $4$ billion people in the world in $1975$. Which of the following functions represents the world's population $P$, in the billions of people, $t$ years since $1975$?

A) $P(t) = 4(1.019)^t$

B) $P(t) = 4(1.9)$

C) $P(t) = 1.19t + 4$

D) $P(t) = 1.019t + 4$

The answer is A, but I want to know the solution method.

I understand the idea of the question, but I want to know why we used $1.019$ and not $1.9$ in the answer

Ahmad Alnajjar
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    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Aug 20 '23 at 11:49
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    If there was $4$ billion people, since the solution is in the billions, you will take $4$ and multiply it by the percentage of increase plus one. Since 100% is considered as 1, then 1.9% is 0.019. So to get increase after first year you calculate $4\cdot 1.019$, for second year $4\cdot 1.019\cdot 1.019$, for t year $4\cdot (1.019)^t$. – bb_823 Aug 20 '23 at 11:51
  • @JoséCarlosSantos thanks, give me 1 minute – Ahmad Alnajjar Aug 20 '23 at 11:51

1 Answers1

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To say that quantity $P$ increased by $R$ percent over a year means that the difference between the new and old value is $r$ times the old value, where $r = \frac R{100}$ is the percentage converted to a ratio. So $$\begin{align}P(0) &= P\\ P(1) - P(0) &= rP(0)\\P(1) &= P(0) + rP(0)\\&= (1+r)P(0)\\&= (1+r)P\end{align}$$ If the following years all can be treated as being at the same rate of increase, then $$\begin{align} P(2) &= (1+r)P(1) = (1+r)(1+r)P=(1+r)^2P\\ P(3) &= (1+r)P(2) = (1+r)(1+r)^2P=(1+r)^3P\\ P(4) &= (1+r)P(3) = (1+r)(1+r)^3P=(1+r)^4P\\ &\ \ \vdots\\ P(t) &= (1+r)^tP = P(1+r)^t \end{align}$$

Paul Sinclair
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