How much more would you earn in the first investment than in the second​ investment?
​$50 comma 000 invested for 40 years at 14​% compounded annually
​$50 comma 000 invested for 40 years at 7​% compounded monthly
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Part 1
You would earn ​$
  
enter your response here more on the first investment than in the second investment.
​(Round to the nearest dollar as​ needed.) 

To compare the two investments, we need to calculate the future value (FV) of each investment and then determine the difference. The future value for compound interest is given by the formula:

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

Where:
- $ P $ is the principal (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 the money is invested for.

### First Investment: 
**$50,000 invested for 40 years at 14% compounded annually**  
- $ P = 50,000 $,
- $ r = 14\% = 0.14 $,
- $ n = 1 $ (compounded annually),
- $ t = 40 $ years.

Using the formula:
\[
FV_1 = 50,000 \left(1 + \frac{0.14}{1}\right)^{1 \times 40}
= 50,000 \left(1.14\right)^{40}
\]

### Second Investment:
**$50,000 invested for 40 years at 7% compounded monthly**  
- $ P = 50,000 $,
- $ r = 7\% = 0.07 $,
- $ n = 12 $ (compounded monthly),
- $ t = 40 $ years.

Using the formula:
\[
FV_2 = 50,000 \left(1 + \frac{0.07}{12}\right)^{12 \times 40}
= 50,000 \left(1 + \frac{0.07}{12}\right)^{480}
\]

### Step 1: Calculate Future Value of Both Investments

I'll calculate the future value for both investments and the difference in earnings.The future values for the two investments are:

- First investment (14% compounded annually for 40 years): **$9,444,176**
- Second investment (7% compounded monthly for 40 years): **$815,571**

The difference in earnings between the two investments is approximately **$8,628,605**.

Would you like more details or have any questions about this calculation?

### Related questions:
1. How does the frequency of compounding affect investment returns?
2. What would be the difference if the second investment compounded annually?
3. How would changing the investment period affect both investments?
4. How does inflation impact the value of long-term investments?
5. What if you invested $100,000 instead of $50,000 in both cases?

### Tip:
Investments with higher interest rates and more frequent compounding grow significantly faster due to the power of compound interest.

How much more would you earn in the first investment than in the second investment? $50,000 invested for 40 years at 14% compounded annually $50,000 invested for 40 years at 7% compounded monthly.

Calculate the difference in earnings between two investments: $50,000 invested for 40 years at 14% compounded annually versus 7% compounded monthly. Learn how to use the future value formula to determine the outcome, with step-by-step calculations and insights into financial mathematics. Suitable for students in Grades 10-12.

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