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Sagot :
To find the savings plan balance after 9 months with an Annual Percentage Rate (APR) of 7% and monthly payments of [tex]$200, follow these steps:
1. Determine the monthly interest rate:
- APR is 7% annually.
- Monthly interest rate is calculated by dividing the annual rate by 12.
\[
\text{Monthly Interest Rate} = \frac{7\%}{12} = \frac{0.07}{12} = 0.005833333333333334
\]
2. Initialize the balance:
- Start with an initial balance of $[/tex]0.
3. Calculate the balance for each month:
- For each of the 9 months, add the monthly payment to the balance and then apply the monthly interest rate.
- The formula for the new balance each month is:
[tex]\[ \text{New Balance} = (\text{Previous Balance} + \text{Monthly Payment}) \times (1 + \text{Monthly Interest Rate}) \][/tex]
Let's compute this month by month:
1. Month 1:
[tex]\[ \text{Balance} = (0 + 200) \times (1 + 0.005833333333333334) = 200 \times 1.0058333333333334 = 201.16666666666669 \][/tex]
2. Month 2:
[tex]\[ \text{Balance} = (201.16666666666669 + 200) \times (1 + 0.005833333333333334) = 401.1666666666667 \times 1.0058333333333334 = 403.5079861111111 \][/tex]
3. Month 3:
[tex]\[ \text{Balance} = (403.5079861111111 + 200) \times (1 + 0.005833333333333334) = 603.5079861111112 \times 1.0058333333333334 = 606.0277361111111 \][/tex]
4. Month 4:
[tex]\[ \text{Balance} = (606.0277361111111 + 200) \times (1 + 0.005833333333333334) = 806.0277361111112 \times 1.0058333333333334 = 810.7304604166667 \][/tex]
5. Month 5:
[tex]\[ \text{Balance} = (810.7304604166667 + 200) \times (1 + 0.005833333333333334) = 1010.7304604166668 \times 1.0058333333333334 = 1016.6207020833336 \][/tex]
6. Month 6:
[tex]\[ \text{Balance} = (1016.6207020833336 + 200) \times (1 + 0.005833333333333334) = 1216.6207020833336 \times 1.0058333333333334 = 1223.7030038194445 \][/tex]
7. Month 7:
[tex]\[ \text{Balance} = (1223.7030038194445 + 200) \times (1 + 0.005833333333333334) = 1423.7030038194445 \times 1.0058333333333334 = 1431.9819088875142 \][/tex]
8. Month 8:
[tex]\[ \text{Balance} = (1431.9819088875142 + 200) \times (1 + 0.005833333333333334) = 1631.9819088875142 \times 1.0058333333333334 = 1641.4619604313864 \][/tex]
9. Month 9:
[tex]\[ \text{Balance} = (1641.4619604313864 + 200) \times (1 + 0.005833333333333334) = 1841.4619604313864 \times 1.0058333333333334 = 1853.332693869144 \][/tex]
4. Round the final balance to the nearest cent:
- The final balance after 9 months is approximately [tex]$1853.33. Thus, the savings plan balance after 9 months with an APR of 7% and monthly payments of $[/tex]200 is $1853.33.
3. Calculate the balance for each month:
- For each of the 9 months, add the monthly payment to the balance and then apply the monthly interest rate.
- The formula for the new balance each month is:
[tex]\[ \text{New Balance} = (\text{Previous Balance} + \text{Monthly Payment}) \times (1 + \text{Monthly Interest Rate}) \][/tex]
Let's compute this month by month:
1. Month 1:
[tex]\[ \text{Balance} = (0 + 200) \times (1 + 0.005833333333333334) = 200 \times 1.0058333333333334 = 201.16666666666669 \][/tex]
2. Month 2:
[tex]\[ \text{Balance} = (201.16666666666669 + 200) \times (1 + 0.005833333333333334) = 401.1666666666667 \times 1.0058333333333334 = 403.5079861111111 \][/tex]
3. Month 3:
[tex]\[ \text{Balance} = (403.5079861111111 + 200) \times (1 + 0.005833333333333334) = 603.5079861111112 \times 1.0058333333333334 = 606.0277361111111 \][/tex]
4. Month 4:
[tex]\[ \text{Balance} = (606.0277361111111 + 200) \times (1 + 0.005833333333333334) = 806.0277361111112 \times 1.0058333333333334 = 810.7304604166667 \][/tex]
5. Month 5:
[tex]\[ \text{Balance} = (810.7304604166667 + 200) \times (1 + 0.005833333333333334) = 1010.7304604166668 \times 1.0058333333333334 = 1016.6207020833336 \][/tex]
6. Month 6:
[tex]\[ \text{Balance} = (1016.6207020833336 + 200) \times (1 + 0.005833333333333334) = 1216.6207020833336 \times 1.0058333333333334 = 1223.7030038194445 \][/tex]
7. Month 7:
[tex]\[ \text{Balance} = (1223.7030038194445 + 200) \times (1 + 0.005833333333333334) = 1423.7030038194445 \times 1.0058333333333334 = 1431.9819088875142 \][/tex]
8. Month 8:
[tex]\[ \text{Balance} = (1431.9819088875142 + 200) \times (1 + 0.005833333333333334) = 1631.9819088875142 \times 1.0058333333333334 = 1641.4619604313864 \][/tex]
9. Month 9:
[tex]\[ \text{Balance} = (1641.4619604313864 + 200) \times (1 + 0.005833333333333334) = 1841.4619604313864 \times 1.0058333333333334 = 1853.332693869144 \][/tex]
4. Round the final balance to the nearest cent:
- The final balance after 9 months is approximately [tex]$1853.33. Thus, the savings plan balance after 9 months with an APR of 7% and monthly payments of $[/tex]200 is $1853.33.
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