If $64 grows to $128 in four years at a constant efective annual interest rate, how much will $10,000 grow to in three years at the same rate of interest?

We are given that $64 grows to $128 in 4 years at a constant effective annual interest rate. This implies that the interest rate, $r$, is constant, and we can use the compound interest formula:

\[
A = P(1 + r)^t
\]

where:
- $A$ is the final amount,
- $P$ is the initial principal,
- $r$ is the annual interest rate,
- $t$ is the time in years.

### Step 1: Calculate the interest rate
From the given information, the initial amount $P = 64$, the final amount $A = 128$, and the time $t = 4$ years. Substituting into the compound interest formula:

\[
128 = 64(1 + r)^4
\]

Dividing both sides by 64:

\[
2 = (1 + r)^4
\]

Taking the fourth root of both sides:

\[
1 + r = \sqrt[4]{2}
\]

\[
r = \sqrt[4]{2} - 1
\]

Now calculate the value of $r$:

\[
r \approx 0.1892 \quad \text{(or 18.92% per year)}
\]

### Step 2: Calculate how much $10,000 will grow to in 3 years
Now that we have the interest rate $r \approx 0.1892$, we can calculate how much $10,000 will grow to in 3 years. Using the same compound interest formula:

\[
A = 10,000(1 + 0.1892)^3
\]

Let's calculate this:

\[
A = 10,000(1.1892)^3
\]

\[
A \approx 10,000 \times 1.682
\]

\[
A \approx 16,820
\]

So, $10,000 will grow to approximately **$16,820** in 3 years.

Would you like more details, or have any questions?

Here are 5 related questions:
1. How does compound interest differ from simple interest?
2. What happens to the final amount if the interest rate increases slightly?
3. How would the growth change if the time period was doubled?
4. What if $64 had grown to $256 in 4 years, what would be the new interest rate?
5. Can you explain the difference between nominal and effective interest rates?

**Tip:** Understanding how to rearrange the compound interest formula can help in solving for any variable, such as time, interest rate, or initial amount.

If $64 grows to $128 in four years at a constant effective annual interest rate, how much will $10,000 grow to in three years at the same rate of interest?

Solve a compound interest problem where $64 grows to $128 in 4 years, and calculate how much $10,000 will grow in 3 years at the same interest rate. Learn key mathematical concepts and use the compound interest formula to find the final amount. Suitable for Grades 10-12.

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