To find the number of periods (n) for an investment with a deposit of $29,385, an interest rate of 14.05% per year compounded half-yearly, we can use the formula for compound interest. The number of periods (n) is approximately 0.0387 years or 0.0774 half-years.
Solution:
To find the number of periods (n) for the investment, we can use the formula for compound interest:
Future Value = Present Value × (1 + Interest Rate / Number of Periods)^(Number of Periods × Time)
Given:
- Present Value (P) = $29,385
- Interest Rate (r) = 14.05% = 0.1405
- Compounding is done half-yearly, so Number of Periods (n) will be 20 × 2 = 40
Plugging in the values into the formula:
$10,000 = $29,385 × (1 + 0.1405 / 40)^(40 × Time)
Simplifying the equation:
1.35144275 = (1 + 0.0035125)^(40 × Time)
Taking the natural logarithm (ln) of both sides:
ln(1.35144275) = ln(1.0035125) × (40 × Time)
Using the logarithmic property, we can simplify the equation:
40 × Time = ln(1.35144275) / ln(1.0035125)
Solving for Time:
Time = (ln(1.35144275) / ln(1.0035125)) / 40
Using a calculator to evaluate the right-hand side of the equation:
Time ≈ 0.0387 years (rounded to 4 decimal places)
Therefore, the number of periods (n) for the investment is approximately 0.0387 years or 0.0387 × 2 = 0.0774 half-years.