Asked by o
Suppose you invest $10,000 in a savings account with 2.5% compound interest. Which account has the greatest value after 10 years?
Responses
A an account compounded annuallyan account compounded annually
B an account compounded semiannuallyan account compounded semiannually
C an account compounded quarterlyan account compounded quarterly
D an account compounded monthlyan
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Answered by
GPT-5 mini
AI
D — an account compounded monthly.
Reason: For principal P, nominal rate r and time t, the balance is P(1 + r/n)^{nt} where n is compounding frequency. For fixed r and t this increases as n increases, so monthly (n = 12) gives the largest value.
Numeric comparison (P = $10,000, r = 0.025, t = 10):
- Annual (n=1): 10,000(1.025)^{10} ≈ $12,805.06
- Semiannual (n=2): 10,000(1.0125)^{20} ≈ $12,827.70
- Quarterly (n=4): 10,000(1.00625)^{40} ≈ $12,835.97
- Monthly (n=12): 10,000(1+0.025/12)^{120} ≈ $12,839.93
Thus the monthly-compounded account has the greatest value.
Reason: For principal P, nominal rate r and time t, the balance is P(1 + r/n)^{nt} where n is compounding frequency. For fixed r and t this increases as n increases, so monthly (n = 12) gives the largest value.
Numeric comparison (P = $10,000, r = 0.025, t = 10):
- Annual (n=1): 10,000(1.025)^{10} ≈ $12,805.06
- Semiannual (n=2): 10,000(1.0125)^{20} ≈ $12,827.70
- Quarterly (n=4): 10,000(1.00625)^{40} ≈ $12,835.97
- Monthly (n=12): 10,000(1+0.025/12)^{120} ≈ $12,839.93
Thus the monthly-compounded account has the greatest value.
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