The enrollment capacity of a certain college is 22,200 students. If the current enrollment is 20,000, but the enrollement increases by 5% per year, how many years will it take before the enrollment limit is reached?

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We can approach this question with simple percents. Every year, the enrollment will increase by 5%. $$20{,}000 \tag*{\tiny current enrollment}$$ $$20{,}000+20{,}000(0.05) \tag*{\tiny after 1 year}$$ We can quickly calculate 5% by first taking 10% of the number, and halving it. In this case, 10% of 20,000 is 2,000, so 5% would be 1,000.

Therefore, $$20{,}000(1.05)=20{,}000+1{,}000$$ $$=21{,}000$$ Repeat the process for the second year. $$21{,}000(1.05)$$ $$=21{,}000+1{,}050$$ $$=22{,}050$$ We can estimate that another 5% increase the third year will certainly bring the number above the 22,200 threshold.

Use the compound interest equation. $$A=P(1+\frac{r}{n})^{nt}$$ $A=\text{final amount}$
$P=\text{initial amount}$
$r=\text{interest rate}$
$n=\text{number of times interest applied for time period}$
$t=\text{number of time periods elapsed}$

In our situation, we start with 20,000 and have an increase of 5% ($r$=0.05) once every year ($n=1$). We're trying to figure out what year $t$ results in an amount greater than 22,200. $$22{,}200 \lt 20{,}000\left(1+\frac{0.05}{1}\right)^{(1)t}$$ $$22{,}200 \lt 20{,}000(1.05)^t$$ From here we have two options for how to proceed.