Simple interest at a rate of 7.5% per annum is paid on an investment of $1200.The term of the investment is chosen so that the total interest earned will be at least $460.The minimum term for this investment, to the nearest month, is:

Question

BlackTom AI · Accepted Answer

1. Relevant knowledge points and concepts
Simple interest is computed by I = P · r · t, where I is interest, P is principal, r is annual rate (as a decimal), and t is time in years. To express time in months, convert months to years by dividing by 12.

2. Complete reasoning process step by step
Apply I = P r t with P = 1200 and r = 7.5% = 0.075. Solve for t (years) that yields at least I = 460. Convert the resulting t in years to months and round to the nearest month as requested.

3. Calculations with intermediate steps
I = P r t → 460 = 1200 × 0.075 × t
Compute 1200 × 0.075 = 90, so 460 = 90 t
t = 460 / 90 = 5.111111... years
Convert to months: months = 5.111111... × 12 = 61.333333... months
Rounding 61.333... to the nearest whole month gives 61 months.

4. Why the correct answer is correct
The algebra yields 61.333... months as the exact minimum duration in months. The instruction asks for the minimum term "to the nearest month," so rounding 61.333... to the nearest month produces 61 months. That is why option d (61 months) is the correct choice under the phrasing "to the nearest month."

5. Why the other options are incorrect
a. 50 months: 50/12 years = 4.1667 years → interest = 90 × 4.1667 = $375 (well below $460).
b. 62 months: rounded from 61.333 up to 62 would give interest = 90 × (62/12) = 90 × 5.1667 = $465 (satisfies the at-least condition) but is not the nearest-month rounding of 61.333.
c. 58 months: 58/12 years = 4.8333 years → interest = 90 × 4.8333 = $435 (below $460).
Thus only 61 months matches the nearest-month rounding of the exact solution.

6. Conclusion
Rounding the exact 61.333... months to the nearest month gives 61 months, so the correct answer is d. 61 months, matching the provided answer.

Simple interest at a rate of 7.5% per annum is paid on an investment of $1200.The term of the investment is chosen so that the total interest earned will be at least $460.The minimum term for this investment, to the nearest month, is:单项选择题

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