Question

determine whether the following statement is true or false. if the statement is false, make the necessary change(s) to produce a true statement.
at a given annual interest rate, your money grows faster as the compounding period becomes shorter.

choose the correct answer below.
a. the statement is false. at a given annual interest rate, your money grows in the same manner as the compounding period becomes shorter
b. the statement is false. at a given annual interest rate, your money grows faster as the compounding period becomes longer
c. the statement is false. at a given annual interest rate, your money grows slower as the compounding period becomes shorter
d. the statement is true.

Explanation:

To determine the validity of the statement, we recall the concept of compound interest. The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the amount of money accumulated after $n$ years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (in decimal), $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for in years. When the annual interest rate $r$ and the principal $P$ and time $t$ are fixed, as $n$ (the number of compounding periods per year, which is related to the length of the compounding period - a shorter compounding period means a larger $n$) increases, the value of $(1 + \frac{r}{n})^{nt}$ increases. For example, if we compound interest semi - annually ($n = 2$), quarterly ($n=4$), monthly ($n = 12$), or daily ($n=365$), with the same annual interest rate, the more frequently we compound (shorter compounding period), the more money we will have at the end of the investment period. So, at a given annual interest rate, money grows faster as the compounding period becomes shorter.

Answer:

D. The statement is true.