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.
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