Two Savings Accounts and a One-Day Loan
A community college student has $5,000 from a summer job and wants to reach $7,500 for a certification program. Two banks advertise savings accounts. Bank A pays 3.6% nominal annual interest compounded quarterly, but requires a $10,000 minimum opening deposit. Bank B pays 3.5% nominal annual interest compounded monthly with no minimum. A friend tells the student, 'Take Bank A because 3.6% is higher than 3.5%.' The student will not need the money for at least five years, but cannot add to the principal. The student asks which account actually earns more, how long it will take to reach the $7,500 goal at the available account, and whether it is worth borrowing $5,000 from a relative for one day to meet Bank A's minimum if the relative charges a $75 fee. Assume no taxes or fees beyond those stated.
1. Compute the effective annual rate for each account and determine which advertised rate actually earns more.
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Bank A: quarterly rate = 0.036 / 4 = 0.009. Effective annual rate = (1.009)^4 - 1. Compute (1.009)^2 = 1.018081, and (1.018081)^2 ≈ 1.036489, so EAR ≈ 3.6489%. Bank B: monthly rate = 0.035 / 12 ≈ 0.00291667. Effective annual rate = (1.00291667)^12 - 1 ≈ 1.035567 - 1 = 3.5567%. Bank A does earn slightly more on the same balance, by about 0.092 percentage points. However, the student cannot open Bank A with only $5,000 because of the $10,000 minimum, so the higher rate is not directly available.
2. At the account the student can actually open, how long will it take $5,000 to reach $7,500? Use logarithms and interpret the result in months and years, including the effect of monthly crediting.
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Use Bank B: 7500 = 5000(1 + 0.035/12)^n, so 1.5 = (1.00291667)^n. Take logs: n = ln(1.5) / ln(1.00291667) ≈ 0.405465 / 0.00291242 ≈ 139.22 months. Interest is credited monthly, so 139 months is slightly short of the goal; the student needs 140 months. 140 ÷ 12 ≈ 11.67 years, or about 11 years 8 months. Check: 5000(1.00291667)^139 ≈ $7,495, just under $7,500, while 5000(1.00291667)^140 ≈ $7,517, above the goal.
3. The student considers borrowing $5,000 from a relative for one day to meet Bank A's minimum, then repaying it. The relative charges a $75 fee. Should the student do this? Consider both the fee and the bank's minimum rule.
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If the loan is for one day and Bank A requires only a $10,000 opening deposit, the student deposits $10,000, then repays the $5,000 the next day, leaving $5,000 in Bank A. Compare five years at A and B on that $5,000. Bank A: 5000(1.009)^20 ≈ 5000 × 1.19625 = $5,981.25. Bank B: 5000(1.00291667)^60 ≈ 5000 × 1.19094 = $5,954.70. The extra interest at A is about $26.55, less than the $75 fee, so the plan loses money. Also, if Bank A requires the $10,000 balance to remain, repaying the loan would violate the minimum and the plan fails outright. The student should stay with Bank B unless the minimum rule is confirmed to apply only at opening and the fee is much lower.