Question

Understanding Interest Practice
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Question
Serenity wants to invest $25,000 of her savings for her retirement fund. Determine the difference in interest earnings between a 10% simple interest rate and a 7% compound interest rate with interest being compounded annually if Serenity invests her money for 25 years. Round the answer to two decimal places as needed.(1 point)
The difference in interest earnings is $
.
Remaining Attempts : 3

Answer 1

To find the difference in interest earnings between a 10% simple interest rate and a 7% compound interest rate after 25 years, we can calculate each separately.

### Step 1: Calculate Simple Interest

The formula for simple interest is given by:

\[
\text{Simple Interest} = P \times r \times t
\]

Where:
- \( P \) is the principal amount (initial investment)
- \( r \) is the rate of interest (as a decimal)
- \( t \) is the time (in years)

For Serenity's investment at 10%:

\[
P = 25000, \quad r = 0.10, \quad t = 25
\]

Substituting the values:

\[
\text{Simple Interest} = 25000 \times 0.10 \times 25 = 25000 \times 2.5 = 62500
\]

So, the total simple interest earned at 10% is **$62,500**.

### Step 2: Calculate Compound Interest

The formula for compound interest is given by:

\[
A = P \times (1 + r)^t
\]

Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial investment)
- \( r \) is the annual interest rate (decimal)
- \( t \) is the number of years the money is invested for

For Serenity's investment at 7%:

\[
P = 25000, \quad r = 0.07, \quad t = 25
\]

Now, calculate the total amount:

\[
A = 25000 \times (1 + 0.07)^{25}
\]

Calculating \( (1 + 0.07)^{25} \):

\[
(1.07)^{25} \approx 5.427
\]

Now, substitute back:

\[
A \approx 25000 \times 5.427 \approx 135675
\]

To find the compound interest earned, we use:

\[
\text{Compound Interest} = A - P = 135675 - 25000 = 110675
\]

So, the total compound interest earned at 7% is **$110,675**.

### Step 3: Calculate the Difference in Interest Earnings

Now we find the difference between the two interest amounts:

\[
\text{Difference} = \text{Compound Interest} - \text{Simple Interest}
\]

Calculating:

\[
\text{Difference} = 110675 - 62500 = 48175
\]

### Final Answer

The difference in interest earnings is **$48,175.00**.