Find the compound amount for the deposit and the amount of interest earned.
​$6400 at 4​% compounded quarterly for 3 years
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Part 1
The compound amount after 3 years is ​$
  
enter your response here.

To find the compound amount for a deposit and the amount of interest earned, we can use the formula for compound interest:

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

where:
- $ A $ is the compound amount (the total amount after interest),
- $ P $ is the principal (initial deposit),
- $ r $ is the annual nominal interest rate (expressed as a decimal),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time the money is invested for, in years.

### Given:
- $ P = 6400 $ (initial deposit),
- $ r = 0.04 $ (4% annual interest rate),
- $ n = 4 $ (compounded quarterly),
- $ t = 3 $ years.

Now, let's plug these values into the formula:

\[
A = 6400 \left(1 + \frac{0.04}{4}\right)^{4 \times 3}
\]

### Step-by-step Calculation:

1. **Calculate the quarterly interest rate**:
   \[
   \frac{r}{n} = \frac{0.04}{4} = 0.01
   \]

2. **Calculate the total number of compounding periods**:
   \[
   nt = 4 \times 3 = 12
   \]

3. **Plug these values into the formula and solve for $ A $**:
   \[
   A = 6400 \left(1 + 0.01\right)^{12}
   \]
   \[
   A = 6400 \left(1.01\right)^{12}
   \]

Let's compute this value:

\[
A = 6400 \times 1.126825 \approx 7203.68
\]

### Compound Amount:
The compound amount after 3 years is **$7203.68**.

### Amount of Interest Earned:
The amount of interest earned can be found by subtracting the initial deposit from the compound amount:

\[
\text{Interest Earned} = A - P = 7203.68 - 6400 = 803.68
\]

### Summary:
- The compound amount after 3 years is **$7203.68**.
- The amount of interest earned is **$803.68**.

Would you like more details or have any questions?

### Related Questions:
1. How would the compound amount change if the interest rate was compounded monthly instead of quarterly?
2. What would be the compound amount for a deposit of $6400 at a 5% annual interest rate compounded quarterly for 3 years?
3. How much interest would be earned if the deposit was for 5 years instead of 3 years, keeping all other variables the same?
4. What is the formula for continuously compounded interest, and how does it differ from the formula used here?
5. How would the total amount change if the interest was compounded annually instead of quarterly?

### Tip:
When calculating compound interest, always make sure to adjust the rate and number of periods to match the compounding frequency (annually, semi-annually, quarterly, monthly, etc.).

Learn how to calculate compound interest for a deposit of $6400 at 4% interest compounded quarterly over 3 years. Discover the compound amount and interest earned using step-by-step calculations. Suitable for high school students and anyone interested in financial mathematics.

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