Question

continuous compounding means that interest is compounded to an account infinitely many times each year.

• until now, interest was compounded a specific number of times: annually, semi - annually, or quarterly.

continuous compound interest formula
$b = pe^{rt}$ where $b=$ ending balance
$p = $ principal
$e = $ exponential base
$r = $ interest rate expressed as a decimal
$t = $ number of years
example 1: if you deposited $1,000 at 100% interest, compounded continuously, what would you ending balance be after 1 year?
example 2: if you deposit $1,000 at 2.3% interest compounded continuously, what would your ending balance be to the nearest cent after 3 years?
example 3: craig deposits $5,000 at 1.12% interest, compounded continuously for 4 years. what would his ending balance be to the nearest cent?

Explanation:

Since the problem involves calculating continuous compound interest, we'll use the Step - by - Step Format.

The formula for continuous compound interest is $B = Pe^{rt}$, where $B$ is the ending balance, $P$ is the principal amount, $r$ is the annual interest rate (in decimal), and $t$ is the number of years.

Example 1:

Step 1: Identify the values

We have $P=\$1000$, $r = 100\%=1.00$ (since $100\%$ as a decimal is $1$), and $t = 1$ year.

Step 2: Substitute into the formula

Substitute the values into $B = Pe^{rt}$. So we get $B=1000\times e^{1\times1}$. Since $e\approx2.71828$, then $B = 1000\times2.71828=\$2718.28$.

Example 2:

Step 1: Identify the values

Here, $P = \$1000$, $r=2.3\% = 0.023$ (because to convert a percentage to a decimal, we divide by 100), and $t = 5$ years.

Step 2: Substitute into the formula

Using the formula $B = Pe^{rt}$, we substitute the values: $B=1000\times e^{0.023\times5}$. First, calculate the exponent: $0.023\times5 = 0.115$. Then, $e^{0.115}\approx1.12187$. Multiply this by $1000$: $B = 1000\times1.12187=\$1121.87$ (to the nearest cent, it can also be rounded to $\$1121.88$ depending on the level of precision in the calculation of $e^{0.115}$).

Example 3:

Step 1: Identify the values

We have $P=\$5000$, $r = 1.12\%=0.0112$ (dividing $1.12$ by $100$ to get the decimal), and $t = 4$ years.

Step 2: Substitute into the formula

Substitute into $B = Pe^{rt}$: $B = 5000\times e^{0.0112\times4}$. First, calculate the exponent: $0.0112\times4=0.0448$. Then, $e^{0.0448}\approx1.04585$. Multiply this by $5000$: $B = 5000\times1.04585=\$5229.25$ (to the nearest cent).

Answer:

• Example 1: The ending balance is $\boldsymbol{\$2718.28}$.
• Example 2: The ending balance (to the nearest cent) is $\boldsymbol{\$1121.87}$ (or $\boldsymbol{\$1121.88}$).
• Example 3: The ending balance (to the nearest cent) is $\boldsymbol{\$5229.25}$.