Difficulty: Medium
Correct Answer: 6.3 years
Explanation:
Introduction / Context:
This question involves growth of an investment with quarterly compounding. You are given the initial deposit, the nominal annual rate and the target amount. The task is to determine approximately how long it takes to reach that target, which requires solving the compound interest equation for time.
Given Data / Assumptions:
Concept / Approach:
For nominal annual rate r, compounded m times per year, periodic rate i is r / m. The amount after n periods is: A = P * (1 + i)^n Here, i = 0.07 / 4 and n is the number of quarters. We set A = 12,400 and solve for n. Then, number of years = n / 4.
Step-by-Step Solution:
Step 1: Compute the quarterly rate. i = 7% / 4 = 1.75% per quarter = 0.0175. Step 2: Set up the compound interest equation. 12,400 = 8,000 * (1.0175)^n. (1.0175)^n = 12,400 / 8,000 = 1.55. Step 3: Solve for n using logarithms. n = log(1.55) / log(1.0175) ≈ 25.26 quarters. Step 4: Convert quarters to years. Years = 25.26 / 4 ≈ 6.3 years.
Verification / Alternative check:
We expect a little less than double, and at roughly 7% per year with quarterly compounding, the doubling time will be near 10 years. Reaching 1.55 times (not full doubling) should take somewhat over half of that, around 6 to 7 years. The computed value of about 6.3 years therefore makes sense.
Why Other Options Are Wrong:
2.3, 3.3, and 4.3 years are too short. At 7% nominal, there is not enough time in those options for $8,000 to grow to $12,400. Calculations show the balance would still be well below the target in those time frames.
Common Pitfalls:
Typical mistakes include using 7% as the quarterly rate instead of dividing by 4, or using simple interest instead of compound interest. Others forget to convert the number of quarters back into years, leading to misinterpretation of the numeric answer.
Final Answer:
It will take approximately 6.3 years for $8,000 to grow to $12,400 at 7% compounded quarterly.
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