Given the following two investments:

Investment X: $2,000 invested at 6% simple interest rate.

Investment Y: $1,000 invested at 5% interest rate with interest compounded annually.

Assume no money is ever withdrawn from either investment. Which of the following is true?

Group of answer choices

X will always have a higher balance because it has a higher principal and higher interest rate.

Y will eventually have a higher balance than X but it will take at least 50 years.

Y will have a higher balance than X after 39 years.

We need to compare the future value of **Investment X** (simple interest) and **Investment Y** (compound interest) over time.

### Investment X (Simple Interest):
The formula for simple interest is:

\[
A_X = P(1 + rt)
\]
Where:
- $ P = 2000 $
- $ r = 6\% = 0.06 $
- $ t = \text{time in years} $

Thus, the future value of Investment X at time $ t $ is:

\[
A_X = 2000(1 + 0.06t)
\]

### Investment Y (Compound Interest):
The formula for compound interest is:

\[
A_Y = P\left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $ P = 1000 $
- $ r = 5\% = 0.05 $
- $ n = 1 $ (compounded annually)
- $ t = \text{time in years} $

Thus, the future value of Investment Y at time $ t $ is:

\[
A_Y = 1000\left(1 + \frac{0.05}{1}\right)^{1 \cdot t} = 1000(1.05)^t
\]

### Step 1: Find the time when Investment Y surpasses Investment X
We need to find the time $ t $ when $ A_Y > A_X $.

\[
1000(1.05)^t > 2000(1 + 0.06t)
\]

We'll solve this inequality to find the time $ t $ when Investment Y surpasses Investment X.

### Step 2: Solve numerically

Let me calculate the approximate value of $ t $.It seems solving this equation directly requires an approximation method. Instead, I will numerically calculate when Investment Y surpasses Investment X by testing values of $ t $. Let me compute that.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you’d like, I can guide you on how to compute this manually or use a calculator to find the time when Investment Y surpasses Investment X.

Would you like further assistance with this or another topic?

Here are some related questions:
1. How does simple interest compare to compound interest in general?
2. What are the advantages of compounding more frequently (quarterly, monthly, etc.)?
3. How can you solve similar financial equations using logarithms?
4. What other factors affect the growth of investments besides interest rates?
5. How can one calculate the effective annual rate (EAR) for compound interest?

**Tip**: Compound interest grows faster because the interest earned itself generates interest, leading to exponential growth over time.

Given the following two investments: Investment X: $2,000 invested at 6% simple interest rate. Investment Y: $1,000 invested at 5% interest rate with interest compounded annually. Assume no money is ever withdrawn from either investment. Which of the following is true? Group of answer choices: X will always have a higher balance because it has a higher principal and higher interest rate. Y will eventually have a higher balance than X but it will take at least 50 years. Y will have a higher balance than X after 39 years.

Explore how two different investments grow over time with simple and compound interest. Investment X ($2,000 at 6% simple interest) and Investment Y ($1,000 at 5% compound interest) are compared to determine when Y surpasses X. Learn the relevant formulas for simple and compound interest and how to solve for the time when Y overtakes X.

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