Select all the equivalent equations that represent the annual value of an investment of $2,000 at 4% annual interest compounded quarterly.
	A.	A = 2,000(1 + 0.044
)4t
	B.	A = 2,000(1 + 1.01)4t
	C.	A = 2,000(1.01)4t
	D.	A = 2,000(1 + 0.042
)2t
	E.	A = 2,000(1 + 0.01)4t

To determine the correct equivalent equations, we first need to establish the standard formula for compound interest:

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

Where:
- $ A $ is the amount of the investment after time $ t $,
- $ P $ is the principal amount (initial investment),
- $ r $ is the annual interest rate (in decimal form),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the number of years.

For this problem:
- The principal $ P = 2,000 $,
- The annual interest rate $ r = 4\% = 0.04 $,
- The interest is compounded quarterly, so $ n = 4 $.

Substituting these values into the compound interest formula, we get:

\[
A = 2,000\left(1 + \frac{0.04}{4}\right)^{4t} = 2,000\left(1 + 0.01\right)^{4t} = 2,000(1.01)^{4t}
\]

Now, let’s evaluate the given equations:

### Option A: 
\[
A = 2,000(1 + 0.044)^{4t}
\]
This is incorrect because $ 0.044 $ is not the correct quarterly interest rate.

### Option B: 
\[
A = 2,000(1 + 1.01)^{4t}
\]
This is incorrect because $ 1.01 $ should not be added inside the parentheses; it should be directly used as the base.

### Option C: 
\[
A = 2,000(1.01)^{4t}
\]
This is **correct**, as it matches the derived equation from the compound interest formula.

### Option D: 
\[
A = 2,000(1 + 0.042)^{2t}
\]
This is incorrect because $ 0.042 $ is not the correct interest rate, and the compounding frequency is also wrong (should be 4 instead of 2).

### Option E: 
\[
A = 2,000(1 + 0.01)^{4t}
\]
This is **correct**, as $ 1 + 0.01 = 1.01 $, which is the correct base for the equation.

### Correct answers:
- **C**
- **E**

Would you like more details or have any questions?

Here are 5 related questions:
1. What happens to the amount when the interest is compounded monthly instead of quarterly?
2. How would you modify the formula if the interest rate changes to 5%?
3. What is the formula for compound interest if compounded continuously?
4. How would the formula change for a 3-year investment period?
5. What is the difference between compound interest and simple interest?

**Tip:** When dealing with compound interest, the frequency of compounding plays a critical role in determining the final amount!

Select all the equivalent equations that represent the annual value of an investment of $2,000 at 4% annual interest compounded quarterly.

Learn how to identify the correct equations representing the annual value of a $2,000 investment at a 4% interest rate compounded quarterly. This problem explains key concepts in compound interest and exponential growth, suitable for students in Grades 9-12.

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