To solve this problem, we need to calculate the amount of interest earned in a simple interest account versus an account with compound interest. Let's go through each calculation step by step.
**Simple Interest Calculation:**
Simple interest is calculated using the formula:
\[I = P \times r \times t\]
Where:
- \(I\) represents the interest earned,
- \(P\) is the principal amount ($3000 in this case),
- \(r\) is the annual interest rate (7% or 0.07 as a decimal), and
- \(t\) is the time the money is invested for (1 year).
Let's calculate the simple interest:
\[I = 3000 \times 0.07 \times 1\]
\[I = 3000 \times 0.07\]
\[I = 210\]
The simple interest earned in one year would be $210.
**Compound Interest Calculation:**
Compound interest is calculated using the formula:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
- \(A\) represents the amount in the account after interest,
- \(P\) is the principal amount ($3000),
- \(r\) is the annual interest rate (6.9% or 0.069 as a decimal),
- \(t\) is the time the money is invested for (1 year), and
- \(n\) is the number of times that interest is compounded per year (daily compounding means \(n = 365\)).
Let's calculate the total account balance using compound interest:
\[A = 3000 \left(1 + \frac{0.069}{365}\right)^{365 \times 1}\]
Now, let's determine what \(A\) is:
\[A = 3000 \left(1 + \frac{0.069}{365}\right)^{365}\]
\[A \approx 3000 \left(1 + 0.000189041\right)^{365}\]
\[A \approx 3000 \times (1.000189041)^{365}\]
\[A \approx 3000 \times 1.0715112\] (Use a calculator for this step)
\[A \approx 3214.53\]
The total account balance after compound interest is approximately $3214.53.
To find out how much compound interest was earned, we subtract the principal from this total:
\[Compound\ Interest = A - P\]
\[Compound\ Interest \approx 3214.53 - 3000\]
\[Compound\ Interest \approx 214.53\]
The compound interest earned in one year would be approximately $214.53.
**Determining Which is Greater and by How Much:**
Now we compare the simple interest $210 to the compound interest approximately $214.53:
- Simple interest earned: $210
- Compound interest earned: approximately $214.53
It's clear that the compound interest amount is greater. To find out how much more interest is earned in the compound account, subtract the simple interest from the compound interest.
\[Difference = Compound\ Interest - Simple\ Interest\]
\[Difference \approx 214.53 - 210\]
\[Difference \approx 4.53\]
Rounded to the nearest dollar, the difference is approximately $5.
So, the answer is:
B. You will earn approximately $5 more interest in the compound interest account than in the simple interest account.
