A person invests 5000 dollars in a bank. The bank pays 5% interest compounded monthly. To the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 10500 dollars?
A, equals, P, left parenthesis, 1, plus, start fraction, r, divided by, n, end fraction, right parenthesis, start superscript, n, t, end superscript
A=P(1+ 
n
r
​
 ) 
nt

The formula for compound interest is given by:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $A$ is the final amount (10500 dollars),
- $P$ is the initial principal (5000 dollars),
- $r$ is the annual interest rate (5% = 0.05),
- $n$ is the number of times the interest is compounded per year (monthly, so $n = 12$),
- $t$ is the time in years (which we need to find).

We need to solve for $t$, given that:

\[
10500 = 5000 \left(1 + \frac{0.05}{12}\right)^{12t}
\]

Let’s solve this equation step by step.

### Step 1: Simplify the equation
First, divide both sides of the equation by 5000:

\[
\frac{10500}{5000} = \left(1 + \frac{0.05}{12}\right)^{12t}
\]

\[
2.1 = \left(1 + \frac{0.05}{12}\right)^{12t}
\]

Now, simplify the term inside the parentheses:

\[
1 + \frac{0.05}{12} = 1 + 0.004167 = 1.004167
\]

So the equation becomes:

\[
2.1 = (1.004167)^{12t}
\]

### Step 2: Take the natural logarithm of both sides
To solve for $t$, take the natural logarithm (ln) of both sides:

\[
\ln(2.1) = \ln\left((1.004167)^{12t}\right)
\]

Using the logarithmic property $\ln(a^b) = b \cdot \ln(a)$:

\[
\ln(2.1) = 12t \cdot \ln(1.004167)
\]

### Step 3: Solve for $t$
Now, divide both sides by $12 \cdot \ln(1.004167)$ to isolate $t$:

\[
t = \frac{\ln(2.1)}{12 \cdot \ln(1.004167)}
\]

Let me calculate this value for you.

A person invests 5000 dollars in a bank. The bank pays 5% interest compounded monthly. To the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 10500 dollars?

Learn how to calculate the time required for a $5000 investment to grow to $10500 with 5% interest compounded monthly. This problem involves the compound interest formula, logarithmic calculations, and exponential growth. Suitable for students in Grades 10-12.

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