adam invests $8,000 in an account that earns 1.25% interest, compounded quarterly for 20 years. on the same…

adam invests $8,000 in an account that earns 1.25% interest, compounded quarterly for 20 years. on the same date, jacinta invests $8,000 in an account that earns continuous compounded interest at a rate of 1.25% for 20 years. who do you predict will have more money in their account after 20 years? explain your reasoning.\n\nchoose the correct answer below.\n\na. jacinta will have more money. continuously compounded interest increases at a faster rate.\nb. adam will have more money. interest that is compounded quarterly increases at a faster rate.\nc. they will both have the same amount of money. the compounding periods will not affect the final amount since both have money in accounts for 20 years.

adam invests $8,000 in an account that earns 1.25% interest, compounded quarterly for 20 years. on the same date, jacinta invests $8,000 in an account that earns continuous compounded interest at a rate of 1.25% for 20 years. who do you predict will have more money in their account after 20 years? explain your reasoning.\n\nchoose the correct answer below.\n\na. jacinta will have more money. continuously compounded interest increases at a faster rate.\nb. adam will have more money. interest that is compounded quarterly increases at a faster rate.\nc. they will both have the same amount of money. the compounding periods will not affect the final amount since both have money in accounts for 20 years.

Answer

Explanation:

Step1: Recall compound - interest formulas

The compound - interest formula for compounded $n$ times a year is $A = P(1+\frac{r}{n})^{nt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $n$ is the number of times compounded per year, and $t$ is the number of years. For Adam, $P = 8000$, $r=0.0125$, $n = 4$ (compounded quarterly), and $t = 20$. So $A_{Adam}=8000(1 +\frac{0.0125}{4})^{4\times20}=8000(1 + 0.003125)^{80}$. The formula for continuous compounding is $A=Pe^{rt}$. For Jacinta, $P = 8000$, $r = 0.0125$, and $t=20$. So $A_{Jacinta}=8000e^{0.0125\times20}=8000e^{0.25}$.

Step2: Compare the growth rates

The function $y = e^{x}$ has a faster - growing rate than the function $y=(1 + a)^{x}$ for positive $a$ and $x$. Since the base of the exponential function in continuous compounding ($e\approx2.718$) is larger than the base of the non - continuous compounding function $(1+\frac{0.0125}{4}=1.003125)$, continuously compounded interest grows faster.

Answer:

A. Jacinta will have more money. Continuously compounded interest increases at a faster rate.