Austin invested $13,000 in an account paying an interest rate of 6 3/4% compounded monthly. Easton invested $13,000 in an account paying an interest rate of 7 1/4% compounded daily. After 19 years, how much more money would Easton have in his account than Austin, to the nearest dollar?

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

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

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

### Step 1: Calculate Austin's Investment

- Principal ($ P $): $13,000
- Annual interest rate ($ r $): $ 6 \frac{3}{4}\% = 6.75\% = 0.0675 $
- Compounding frequency ($ n $): 12 times per year (monthly)
- Time ($ t $): 19 years

\[
A_{\text{Austin}} = 13000 \left(1 + \frac{0.0675}{12}\right)^{12 \times 19}
\]

### Step 2: Calculate Easton's Investment

- Principal ($ P $): $13,000
- Annual interest rate ($ r $): $ 7 \frac{1}{4}\% = 7.25\% = 0.0725 $
- Compounding frequency ($ n $): 365 times per year (daily)
- Time ($ t $): 19 years

\[
A_{\text{Easton}} = 13000 \left(1 + \frac{0.0725}{365}\right)^{365 \times 19}
\]

### Step 3: Calculate the Difference

Finally, we'll subtract Austin's accumulated amount from Easton's accumulated amount to find the difference. Let's calculate this now.After 19 years, Easton would have approximately $4,832 more in his account than Austin, to the nearest dollar.

Would you like more details or have any questions?

Here are some related questions:
1. How does the frequency of compounding affect the final amount in a savings account?
2. What would the difference be if both investments were compounded annually?
3. How would the difference change if the investment period was 25 years instead of 19 years?
4. What if the interest rates were reversed between Austin and Easton?
5. How much would Austin have after 19 years if his interest rate was increased to 7.25%?

**Tip:** The more frequently interest is compounded, the greater the amount of interest accrued.

Explore the impact of different compounding frequencies on long-term investments. Learn how Austin's and Easton's investments grow over 19 years with monthly and daily compounding. This detailed solution demonstrates key concepts in compound interest and exponential growth. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation