My growth rate is 1.41. My starting point is 1359033. I am trying to calculate my growth over 5 years

Question

I am trying to calculate population growth over 5 years, and my rate is 1.41%. My starting point is 1,359,033. I should be ending up with a number of 1,486,521 after that 5 year period, but for some reason I keep getting 11 thousand something or 81. I am also in Algebra 1 and am in 8th grade. What would the formula for my population growth be, using the y=ab^x method?

Answer 1

For an initial population of $P_0$ and a $\color{blue}{\text{growth}}$ rate of $r\%$ per time period, the size of the population after $t$ time periods have passed will be

$$P(t) = P_0\cdot (1\color{blue}{+}\frac{r}{100})^t$$

Again, remembering that $r$ in the above was written as a percentage.

For your example, after five years we have $1359033\cdot (1+0.0141)^5 \approx 1457585$ as we expected.

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Had it been a $\color{red}{\text{decay}}$ rate instead, we would be subtracting instead of adding.

Note also that some authors will prefer not to talk in terms of percentages, but will instead talk about raw rates. "Grows by a factor of $2$" for instance in which case the formula will need to be adjusted to accommodate.

In order to accommodate such changes in wording, I find it most helpful to spend the time to understand why the formula looks the way it does and not just memorize the formula itself. Honestly, half of the time I don't remember the formulas for annuities or growth rates and such exactly or doubt my memory on them and so just come up with the formulas again on the spot by recognizing what they are meant to represent. In your case of an exponential growth, we start with a value and after some period of time it has increased by some factor. After another period of time, the new amount increases again by that factor, and so on... leading to the general form of $a\cdot b^t$ for some appropriate choices of $a,b,t$

Answer 2

Let $P_{\alpha,\gamma}^\beta$ the population from $\gamma$ over $\alpha$ years and with rate $\beta$. I have: $P_{\alpha,\gamma}^\beta=\gamma\cdot(1+ \alpha/100)^\beta$. With your inputs: $$P_{5,13590033}^{0.41}=1359033\cdot(1+0.41/100)^5=1486521$$

Answer 3

If the starting population is $P_0$ and the annual growth rate, in percent, is $r$, then the appropriate formula for the population after $n$ years is

$$P_n=P_0\left(1+{r\over100}\right)^n$$

However, something is a bit funky here, because $1{,}359{,}033(1.0141)^5\approx1{,}457{,}585$, not $1{,}486{,}521$, as asserted in the OP.