A. Investment Account Comparison
To determine which account the business owner should choose for investing $10,000 for 10 years, we will calculate the amount earned from both Account A and Account B.
Account A: 7.5% simple interest per year
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Formula: \[ A = P(1 + rt) \] where:
- \( P = 10,000 \)
- \( r = 0.075 \)
- \( t = 10 \)
\[ A = 10,000(1 + 0.075 \times 10) = 10,000(1 + 0.75) = 10,000(1.75) = 17,500 \]
Account B: 5% compounded annually
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Formula: \[ A = P(1 + r)^t \] where:
- \( P = 10,000 \)
- \( r = 0.05 \)
- \( t = 10 \)
\[ A = 10,000(1 + 0.05)^{10} = 10,000(1.05)^{10} \]
Using \( (1.05)^{10} \approx 1.62889 \):
\[ A \approx 10,000 \times 1.62889 \approx 16,288.90 \]
Comparison:
- Account A will yield $17,500 after 10 years, while Account B will yield approximately $16,288.90. Thus, the business owner should choose Account A for higher returns.
B. Earning Equal Amounts
You deposit $275 in a savings account that earns 4% simple interest. Your friend deposits $275 in an account that earns 3% compounded yearly. We need to find out when the amounts will be equal.
Your Account (Simple Interest):
-
Formula: \[ A = P(1 + rt) \]
- \( P = 275 \)
- \( r = 0.04 \)
\[ A_Y = 275(1 + 0.04t) = 275 + 11t \]
Your Friend's Account (Compounded Yearly):
-
Formula: \[ A = P(1 + r)^t \]
- \( P = 275 \)
- \( r = 0.03 \)
\[ A_F = 275(1 + 0.03)^t \]
Setting these equal: \[ 275 + 11t = 275(1.03^t) \]
To solve this equation, either use numerical methods or test values. Let's try a few values for \( t \):
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For \( t = 1 \):
- \( A_Y = 275 + 11(1) = 286 \)
- \( A_F = 275(1.03) \approx 283.25 \)
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For \( t = 2 \):
- \( A_Y = 275 + 11(2) = 297 \)
- \( A_F = 275(1.03^2) \approx 291.98 \)
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For \( t = 5 \):
- \( A_Y = 275 + 11(5) = 330 \)
- \( A_F = 275(1.03^5) \approx 319.29 \)
Eventually, after checking:
- At approximately \( t = 10 \):
- \( A_Y = 275 + 110 = 385 \)
- \( A_F \approx 275(1.3439) \approx 369.60\)
You can continue until \( t = 11 \) where they would be close enough, suggesting around 11 years for equal amounts.
Who Has More After 5 and 10 Years?
-
After 5 years:
- Your Account: \( A_Y = 275 + 55 = 330 \)
- Friend's Account: \( A_F \approx 275(1.159274) \approx 319.30 \)
You have more.
-
After 10 years:
- Your Account: \( A_Y = 275 + 110 = 385 \)
- Friend's Account: \( A_F \approx 275(1.3439) \approx 369.30 \)
You still have more.
C. Time to Reach $7000
We need to find how long it will take for $2000 to grow to $7000 at 5% compounded annually.
Using the formula: \[ A = P(1 + r)^t \] where
- \( A = 7000 \)
- \( P = 2000 \)
- \( r = 0.05 \)
Setting it up: \[ 7000 = 2000(1.05)^t \] \[ 3.5 = (1.05)^t \]
Taking the logarithm: \[ \log(3.5) = t \cdot \log(1.05) \]
Calculating:
- \( \log(3.5) \approx 0.544 \)
- \( \log(1.05) \approx 0.02119 \)
Thus: \[ t \approx \frac{0.544}{0.02119} \approx 25.7 \]
It will take approximately 26 years.
D. Total Amount Owed
Stan is late for 2 months (August 1 to October 1), with monthly interest of 3% on a $300 payment.
Calculating: Total owed after 2 months: \[ Amount = 300(1 + 0.03)^2 = 300(1.0609) \approx 318.27 \]
Total owed to the company is approximately $318.27 $ in late fees.
E. Simple vs. Compounded Interest
In general, an account that earns compound interest will always earn more than an account that earns simple interest over the same period, given the same interest rate.
This is because, in a compounded interest account, interest is calculated on both the initial principal and the interest that has already been added to that principal. Thus, the overall amount earning interest continually increases, while simple interest is only calculated on the original amount. After one year, compounded interest will result in a higher balance than simple interest.