Question 303·Hard·Math

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A town’s population increased by $15\%$ in 2019, decreased by $12\%$ in 2020, and then increased by $10\%$ in 2021. At the end of 2021, the population was $22,264$ . What was the town’s population at the beginning of 2019?

Your answer

A string of yearly changes signals successive percent changes: each year's percent is taken from the population left by the year before. Turn each change into a multiplier, chain the multipliers in one equation, and use a Desmos regression to find the missing starting population. Adding the signed percents would treat every change as though it used the same starting population.

Hints

Hint 1
A multiplier is the number that takes an amount to its new value. An increase keeps the whole amount and adds a percent of it; a decrease removes a percent. What multiplier belongs to each year?
Hint 2
With successive percent changes, the next year acts on what the previous year left. If the beginning population is $x$ , the 2019 population becomes $1.15x$ . What does the 2020 decrease do to that new amount?

Step-by-step

Chain the yearly multipliers
Step 1
Find the 2019 multiplier
A multiplier is what you multiply a population by to get its new value. The 2019 increase keeps the whole population and adds $15\%=0.15$ of it. Add that to $1$ , which represents the whole population:
$1+0.15=1.15.$
Step 2
Find the 2020 multiplier
The 2020 decrease removes $12\%=0.12$ of the population left after 2019. Subtract that from the whole $1$ :
$1-0.12=0.88.$
Step 3
Find the 2021 multiplier
The 2021 increase adds $10\%=0.10$ to the whole population left after 2020. Add it to $1$ :
$1+0.10=1.10.$
Step 4
Connect the starting and ending populations
Let $x$ be the population at the beginning of 2019. The 2019 multiplier makes it $1.15x$ . Apply the 2020 multiplier to that result, then the 2021 multiplier to the next result. Set what you get equal to the given ending population:
$1.10\bigl(0.88(1.15x)\bigr)=22{,}264.$
Each percent acts on the population left by the previous year. Adding the percents would use the starting population as the base for all three changes.
Step 5
Solve for the original population
Type $1.10(0.88(1.15x_1))\sim22264$ in Desmos. Use $x_1$ instead of $x$ and $\sim$ instead of $=$ so Desmos finds the value that makes both sides match. Under PARAMETERS, it shows $x_1=20{,}000$ . Since $x_1$ stands for the beginning-of-2019 population, the town started with $20{,}000$ people. Grid in 20000.

Question 303·Hard·Math

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