The Bank Account
Task
Most savings accounts advertise an annual interest rate, but they actually compound that interest at regular intervals during the year. That means that, if you own an account, you’ll be paid a portion of the interest before the year is up, and, if you keep that payment in the account, you’ll start earning interest on the interest you’ve already earned.
For example, suppose you put \$500 in a savings account that advertises 5% annual interest. If that interest is paid once per year, then your balance $B$ after $t$ years could be computed using the equation $B = 500(1.05)^t$, since you’ll end each year with 100% + 5% of the amount you began the year with.
On the other hand, if that same interest rate is compounded monthly, then you would compute your balance after $t$ years using the equation $$B=500\left(1+\frac{.05}{12}\right)^{12t}$$
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Why does it make sense that the equation includes the term $\frac{.05}{12}$? That is, why are we dividing .05 by 12?
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How does this equation reflect the fact that you opened the account with \$500?
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What do the numbers 1 and $\frac{.05}{12}$ represent in the expression $\left(1+\frac{.05}{12}\right)$?
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What does the “$12t$” in the equation represent?
Solution
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The 5% is the annual interest rate. Since this interest is compounded monthly (12 times per year), the rate needs to be divided by 12 to figure out the monthly interest rate.
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Looking at the structure of the expression on the right side of the equation, you can see that the \$500 starting value is multiplied by a factor that depends on the interest rate and the amount of time that has passed. If you let $t = 0$, you will find the amount in the account after 0 years have passed: $$B=500\left(1+\frac{.05}{12}\right)^{12(0)}=500(1)=500.$$ In other words, the coefficient of the exponential expression corresponds to the initial amount in the account.
Each month the value of the account is multiplied by $\left(1+\frac{.05}{12}\right)$, so if we begin a month with $D$ dollars, we end the month with $$ \left(1+\frac{.05}{12}\right)D=1\cdot D+\frac{.05}{12}D. $$ Now it's clear that the 1 represents the (100% of the) money in the account at the start of the month, and the $.05/12$ represents the percentage of $D$ that gets added in at the end of the month, i.e., the montly interest rate.
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Interest is compounded each month, and $12t$ tells the number of months that have passed in $t$ years. This quantity becomes an exponent since we multiply the account by $\left(1+\frac{.05}{12}\right)$ each month.
The Bank Account
Most savings accounts advertise an annual interest rate, but they actually compound that interest at regular intervals during the year. That means that, if you own an account, you’ll be paid a portion of the interest before the year is up, and, if you keep that payment in the account, you’ll start earning interest on the interest you’ve already earned.
For example, suppose you put \$500 in a savings account that advertises 5% annual interest. If that interest is paid once per year, then your balance $B$ after $t$ years could be computed using the equation $B = 500(1.05)^t$, since you’ll end each year with 100% + 5% of the amount you began the year with.
On the other hand, if that same interest rate is compounded monthly, then you would compute your balance after $t$ years using the equation $$B=500\left(1+\frac{.05}{12}\right)^{12t}$$
-
Why does it make sense that the equation includes the term $\frac{.05}{12}$? That is, why are we dividing .05 by 12?
-
How does this equation reflect the fact that you opened the account with \$500?
-
What do the numbers 1 and $\frac{.05}{12}$ represent in the expression $\left(1+\frac{.05}{12}\right)$?
-
What does the “$12t$” in the equation represent?