Amy invested $3000 in a bank account at an interest rate of 5.1% per annum, compounded quarterly. The effective rate of interest for her investment is closest to

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BlackTom AI · Accepted Answer

To assess the effective rate, I first note the nominal rate and compounding frequency: 5.1% per year, compounded quarterly. This means each quarter has a rate of 0.051/4 = 0.01275 (1.275%).

Next, I compute the annual effective rate using the formula (1 + i/m)^{m} − 1, where i = 0.051 and m = 4. So the calculation becomes (1 + 0.01275)^{4} − 1.

Step by step, (1 + 0.01275)^{4} ≈ 1.01275^{4}. Evaluate progressively: 1.01275^{2} ≈ 1.025655625, and squaring again gives 1.025655625^{2} ≈ 1.0520 (more precisely about 1.051969). Subtracting 1 yields an effective rate of approximately 0.051969, i.e., about 5.197%.

Interpreting the results: an effective rate around 5.20% reflects the impact of quarterly compounding on a 5.1% nominal annual rate. Now, evaluating the provided options:

- Option A (20.4%): This is far too large and indicates a misapplication of the compounding effect, likely multiplying rates or miscounting periods.
- Option B (1.3%): This is far too small and corresponds to a single-period rate, not the annualized effective rate.
- Option C (5.2%): This aligns with the calculated effective rate, matching the expected answer within rounding.
- Option D (5.1%):This would be the nominal rate without accounting for the extra growth from quarterly compounding, so it underestimates the true annual return.

Hence, the reasoning shows that the effective rate is closest to 5.2%, corresponding to Option C.

MUF0141 Fund. Mathematics Unit 1 - Semester 2, 2025

Amy invested $3000 in a bank account at an interest rate of 5.1% per annum, compounded quarterly. The effective rate of interest for her investment is closest to

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