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Sagot :
Let's solve the problem by calculating the total interest Carlos would earn from each bank over the 3-year period.
First, we will consider the offer from the Bank of Peace:
### Bank of Peace:
- Annual interest rate: 3.2% per year
- Compound interest calculation for 3 years
1. Convert the percentage rate into a decimal:
[tex]\( \text{Rate} = \frac{3.2}{100} = 0.032 \)[/tex]
2. Use the compound interest formula to calculate the future value:
[tex]\[ \text{Future Value} = P \left(1 + r\right)^t \][/tex]
Here, [tex]\( P = 18000 \)[/tex], [tex]\( r = 0.032 \)[/tex], and [tex]\( t = 3 \)[/tex].
3. Calculate:
[tex]\[ \text{Future Value} = 18000 \times \left(1 + 0.032\right)^3 \][/tex]
[tex]\[ \text{Future Value} = 18000 \times 1.032^3 \][/tex]
[tex]\[ \text{Future Value} = 18000 \times 1.099168 \][/tex]
[tex]\[ \text{Future Value} = 19783.885824 \][/tex]
So, the future value with the Bank of Peace is [tex]\( \$19783.885824 \)[/tex].
4. Calculate the interest earned:
[tex]\[ \text{Interest Earned} = \text{Future Value} - \text{Initial Amount} \][/tex]
[tex]\[ \text{Interest Earned} = 19783.885824 - 18000 \][/tex]
[tex]\[ \text{Interest Earned} = 1783.885824 \][/tex]
Now, let's consider the offer from the Bank of Trust:
### Bank of Trust:
- 3.6% interest for the 1st year
- 1.4% interest for the next 2 years
- Compound interest calculation
1. Convert the percentage rates into decimals:
[tex]\[ \text{Rate for 1st year} = \frac{3.6}{100} = 0.036 \][/tex]
[tex]\[ \text{Rate for 2nd and 3rd years} = \frac{1.4}{100} = 0.014 \][/tex]
2. Calculate the future value after each year.
- After the 1st year:
[tex]\[ \text{Amount after Year 1} = P \times \left(1 + \text{Rate for 1st year}\right) \][/tex]
[tex]\[ \text{Amount after Year 1} = 18000 \times \left(1 + 0.036\right) \][/tex]
[tex]\[ \text{Amount after Year 1} = 18000 \times 1.036 \][/tex]
[tex]\[ \text{Amount after Year 1} = 18664.8 \][/tex]
- After the 2nd year:
[tex]\[ \text{Amount after Year 2} = \text{Amount after Year 1} \times \left(1 + \text{Rate for extra years}\right) \][/tex]
[tex]\[ \text{Amount after Year 2} = 18664.8 \times \left(1 + 0.014\right) \][/tex]
[tex]\[ \text{Amount after Year 2} = 18664.8 \times 1.014 \][/tex]
[tex]\[ \text{Amount after Year 2} = 18926.092800000001 \][/tex]
- After the 3rd year:
[tex]\[ \text{Amount after Year 3} = \text{Amount after Year 2} \times \left(1 + \text{Rate for extra years}\right) \][/tex]
[tex]\[ \text{Amount after Year 3} = 18926.092800000001 \times \left(1 + 0.014\right) \][/tex]
[tex]\[ \text{Amount after Year 3} = 18926.092800000001 \times 1.014 \][/tex]
[tex]\[ \text{Amount after Year 3} = 19173.799008 \][/tex]
So, the future value with the Bank of Trust is [tex]\( \$19173.799008 \)[/tex].
3. Calculate the interest earned:
[tex]\[ \text{Interest Earned} = \text{Future Value} - \text{Initial Amount} \][/tex]
[tex]\[ \text{Interest Earned} = 19173.799008 - 18000 \][/tex]
[tex]\[ \text{Interest Earned} = 1173.799008 \][/tex]
### Conclusion
By comparing the interest earned from both banks over 3 years:
- Bank of Peace: [tex]\( \$1783.885824\)[/tex]
- Bank of Trust: [tex]\( \$1173.799008\)[/tex]
Carlos should choose the Bank of Peace to get the most interest over the 3-year period.
First, we will consider the offer from the Bank of Peace:
### Bank of Peace:
- Annual interest rate: 3.2% per year
- Compound interest calculation for 3 years
1. Convert the percentage rate into a decimal:
[tex]\( \text{Rate} = \frac{3.2}{100} = 0.032 \)[/tex]
2. Use the compound interest formula to calculate the future value:
[tex]\[ \text{Future Value} = P \left(1 + r\right)^t \][/tex]
Here, [tex]\( P = 18000 \)[/tex], [tex]\( r = 0.032 \)[/tex], and [tex]\( t = 3 \)[/tex].
3. Calculate:
[tex]\[ \text{Future Value} = 18000 \times \left(1 + 0.032\right)^3 \][/tex]
[tex]\[ \text{Future Value} = 18000 \times 1.032^3 \][/tex]
[tex]\[ \text{Future Value} = 18000 \times 1.099168 \][/tex]
[tex]\[ \text{Future Value} = 19783.885824 \][/tex]
So, the future value with the Bank of Peace is [tex]\( \$19783.885824 \)[/tex].
4. Calculate the interest earned:
[tex]\[ \text{Interest Earned} = \text{Future Value} - \text{Initial Amount} \][/tex]
[tex]\[ \text{Interest Earned} = 19783.885824 - 18000 \][/tex]
[tex]\[ \text{Interest Earned} = 1783.885824 \][/tex]
Now, let's consider the offer from the Bank of Trust:
### Bank of Trust:
- 3.6% interest for the 1st year
- 1.4% interest for the next 2 years
- Compound interest calculation
1. Convert the percentage rates into decimals:
[tex]\[ \text{Rate for 1st year} = \frac{3.6}{100} = 0.036 \][/tex]
[tex]\[ \text{Rate for 2nd and 3rd years} = \frac{1.4}{100} = 0.014 \][/tex]
2. Calculate the future value after each year.
- After the 1st year:
[tex]\[ \text{Amount after Year 1} = P \times \left(1 + \text{Rate for 1st year}\right) \][/tex]
[tex]\[ \text{Amount after Year 1} = 18000 \times \left(1 + 0.036\right) \][/tex]
[tex]\[ \text{Amount after Year 1} = 18000 \times 1.036 \][/tex]
[tex]\[ \text{Amount after Year 1} = 18664.8 \][/tex]
- After the 2nd year:
[tex]\[ \text{Amount after Year 2} = \text{Amount after Year 1} \times \left(1 + \text{Rate for extra years}\right) \][/tex]
[tex]\[ \text{Amount after Year 2} = 18664.8 \times \left(1 + 0.014\right) \][/tex]
[tex]\[ \text{Amount after Year 2} = 18664.8 \times 1.014 \][/tex]
[tex]\[ \text{Amount after Year 2} = 18926.092800000001 \][/tex]
- After the 3rd year:
[tex]\[ \text{Amount after Year 3} = \text{Amount after Year 2} \times \left(1 + \text{Rate for extra years}\right) \][/tex]
[tex]\[ \text{Amount after Year 3} = 18926.092800000001 \times \left(1 + 0.014\right) \][/tex]
[tex]\[ \text{Amount after Year 3} = 18926.092800000001 \times 1.014 \][/tex]
[tex]\[ \text{Amount after Year 3} = 19173.799008 \][/tex]
So, the future value with the Bank of Trust is [tex]\( \$19173.799008 \)[/tex].
3. Calculate the interest earned:
[tex]\[ \text{Interest Earned} = \text{Future Value} - \text{Initial Amount} \][/tex]
[tex]\[ \text{Interest Earned} = 19173.799008 - 18000 \][/tex]
[tex]\[ \text{Interest Earned} = 1173.799008 \][/tex]
### Conclusion
By comparing the interest earned from both banks over 3 years:
- Bank of Peace: [tex]\( \$1783.885824\)[/tex]
- Bank of Trust: [tex]\( \$1173.799008\)[/tex]
Carlos should choose the Bank of Peace to get the most interest over the 3-year period.
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