Principal $575, time is 1 year, rate of compounded interest 4%, compounded quarterly. What’s the periods, rate, total amount, and total interest?

To solve this problem, we'll use the compound interest formula:

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

Where:
- $A$ = Total amount after interest
- $P$ = Principal amount ($575)
- $r$ = Annual interest rate (4% or 0.04)
- $n$ = Number of times the interest is compounded per year (quarterly means $n = 4$)
- $t$ = Time the money is invested or borrowed for (1 year)

Let's break it down:

### 1. **Periods (total number of compounding periods)**
- Since the interest is compounded quarterly and the time is 1 year:
\[
nt = 4 \times 1 = 4 \text{ periods}
\]

### 2. **Rate per period**
- The interest rate is compounded quarterly, so the rate per period is:
\[
\frac{r}{n} = \frac{0.04}{4} = 0.01 = 1\%
\]

### 3. **Total amount (A)**
Now we can calculate the total amount after 1 year:

\[
A = 575 \left(1 + \frac{0.04}{4}\right)^{4 \times 1}
\]
\[
A = 575 \left(1 + 0.01\right)^4
\]
\[
A = 575 \times (1.01)^4
\]
\[
A = 575 \times 1.04060401
\]
\[
A \approx 598.35
\]

So, the total amount after 1 year is approximately **$598.35**.

### 4. **Total interest**
The total interest earned is the difference between the final amount and the principal:
\[
\text{Interest} = A - P = 598.35 - 575 = 23.35
\]

So, the total interest earned is approximately **$23.35**.

### Summary:
- **Periods:** 4
- **Rate per period:** 1%
- **Total amount (A):** $598.35
- **Total interest:** $23.35

Would you like more details or have any questions? 

Here are 5 related questions:
1. What if the time was 2 years instead of 1 year?
2. How would the calculations change if interest was compounded monthly?
3. How does simple interest differ from compound interest in this scenario?
4. How would the total interest change if the rate were 5% instead of 4%?
5. What is the effect of increasing the compounding frequency (e.g., semi-annually, monthly)?

**Tip:** The more frequently interest is compounded, the higher the total amount, as the interest is calculated on previously accumulated interest as well.

Learn how to calculate the total amount and interest for a principal of $575 with a 4% interest rate compounded quarterly over 1 year. This problem demonstrates the use of the compound interest formula and covers concepts such as the number of compounding periods and rate per period. Suitable for students in Grades 9-12.

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