Using a term of
50
years in all cases, calculate the account balance in each case to see
which variable has the most impact on the amount.
You should use your calculator for these calculations. a=P(1+r/n)^nt
P=15,000
r=5
n=12
To find the account balance using the formula a=P(1+r/n)^nt, considering a term of 50 years, we need to calculate the value of a in each case.
Case 1: r = 5%, n = 12 times per year
a = 15000(1 + 0.05/12)^(12*50)
a ≈ 15000(1.0041666667)^(600)
a ≈ 15000(5.52684455)
a ≈ 82,902.67
Case 2: r = 5%, n = 1 time per year (compounded annually)
a = 15000(1 + 0.05/1)^(1*50)
a ≈ 15000(1.05)^(50)
a ≈ 15000(11.467456)
a ≈ 172,011.84
Comparing the account balances, the variable that has the most impact on the amount is the compounding frequency (n). In Case 1, where the compounding frequency is 12 times per year, the account balance after 50 years is approximately $82,902.67. In Case 2, where the compounding frequency is 1 time per year, the account balance after 50 years is approximately $172,011.84. This shows that compounding more frequently leads to a higher account balance over time.
thats incorrect
Apologies for the confusion. Let me recalculate the account balances using the correct formula.
Case 1: r = 5%, n = 12 times per year
a = 15000(1 + 0.05/12)^(12*50)
a ≈ 15000(1.0041666667)^(600)
a ≈ 15000(8.138428767)
a ≈ 122,076.43
Case 2: r = 5%, n = 1 time per year (compounded annually)
a = 15000(1 + 0.05/1)^(1*50)
a ≈ 15000(1.05)^(50)
a ≈ 15000(11.467456)
a ≈ 172,011.84
Upon recalculating, we find that the correct account balance in Case 1 (with the compounding frequency of 12 times per year) after 50 years is approximately $122,076.43. In Case 2 (with the compounding frequency of 1 time per year), the account balance after 50 years remains the same at approximately $172,011.84.
Therefore, the correct conclusion is that the variable that has the most impact on the amount is the interest rate (r). In both cases, the compounding frequency (n) does not affect the final account balance after 50 years, but the interest rate significantly affects the amount.
thats still not right
Apologies for the mistakes. Let me recalculate the account balances using the correct formula.
Case 1: r = 5%, n = 12 times per year
a = 15000(1 + 0.05/12)^(12*50)
a ≈ 15000(1.0041666667)^(600)
a ≈ 15000(3.19661642) ≈ 47,949.50
Case 2: r = 5%, n = 1 time per year (compounded annually)
a = 15000(1 + 0.05/1)^(1*50)
a ≈ 15000(1.05)^(50)
a ≈ 15000(11.467456)
a ≈ 172,011.84
Upon recalculating, we find that the correct account balance in Case 1 (with the compounding frequency of 12 times per year) after 50 years is approximately $47,949.50. In Case 2 (with the compounding frequency of 1 time per year), the account balance after 50 years remains the same at approximately $172,011.84.
Therefore, the correct conclusion is that the variable that has the most impact on the amount is the compounding frequency (n). In Case 2, where the compounding frequency is 1 time per year, the account balance after 50 years is significantly higher than Case 1, where the compounding frequency is 12 times per year.