$$ \$ 500 $$ at $$ 4.7 \% $$ for $$ 2 \frac{1}{2} $$ years

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UpStudy AI · Accepted Answer

Certainly! It looks like you're interested in calculating the growth of a **\$500** investment at an annual interest rate of **4.7%** over **2.5 years**. Depending on how the interest is applied, there are two primary methods to calculate this:

1. **Simple Interest**
2. **Compound Interest**

Let’s explore both:

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### 1. Simple Interest

**Simple Interest** calculates interest only on the **principal amount** (the original sum of money).

**Formula:**
$$ \text{Simple Interest (SI)} = P \times r \times t $$

Where:
- $$ P $$ = Principal amount (\$500)
- $$ r $$ = Annual interest rate (4.7% or 0.047)
- $$ t $$ = Time in years (2.5)

**Calculation:**
$$ \text{SI} = 500 \times 0.047 \times 2.5 $$
$$ \text{SI} = 500 \times 0.1175 $$
$$ \text{SI} = \$58.75 $$

**Total Amount After 2.5 Years:**
$$ \text{Total} = P + \text{SI} = 500 + 58.75 = \$558.75 $$

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### 2. Compound Interest

**Compound Interest** calculates interest on the principal amount **and** on the accumulated interest from previous periods. The frequency of compounding (e.g., annually, semi-annually, quarterly, monthly) affects the total interest earned.

Since the compounding frequency isn't specified, I'll demonstrate **annual compounding**. If you have a different compounding frequency in mind, let me know!

**Formula:**
$$ A = P \times \left(1 + \frac{r}{n}\right)^{n \times t} $$

Where:
- $$ A $$ = Amount of money accumulated after n years, including interest.
- $$ P $$ = Principal amount (\$500)
- $$ r $$ = Annual interest rate (4.7% or 0.047)
- $$ n $$ = Number of times interest is compounded per year
- $$ t $$ = Time the money is invested for (2.5 years)

**Assuming Annual Compounding ($$ n = 1 $$):**
$$ A = 500 \times \left(1 + \frac{0.047}{1}\right)^{1 \times 2.5} $$
$$ A = 500 \times (1.047)^{2.5} $$

**Calculating $$ (1.047)^{2.5} $$:**
First, calculate the natural logarithm to handle the exponent:
$$ \ln(1.047) \approx 0.046 $$
$$ 2.5 \times \ln(1.047) \approx 2.5 \times 0.046 = 0.115 $$
$$ e^{0.115} \approx 1.122 $$

So,
$$ A \approx 500 \times 1.122 = \$561 $$

**Total Interest Earned:**
$$ \text{Interest} = A - P = 561 - 500 = \$61 $$

*Note:* The exact amount may vary slightly based on more precise calculations or different compounding frequencies.

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### **Summary**

- **Simple Interest:** \$58.75
- **Compound Interest (Annually):** Approximately \$61

If you have a specific compounding frequency (like monthly or quarterly) or require more precise calculations, please provide those details, and I can adjust the calculations accordingly!
After 2.5 years, \$500 at 4.7% interest grows to approximately \$558.75 with simple interest and about \$561 with annual compound interest.

at for years

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