In 1963, an investor opened a savings account with $LaTeX: \text{K} K earning simple interest at annual rate of LaTeX: 2.5\% 2.5 % . Four years later, the investor closed the account and invested the accumulated amount in a savings account earning LaTeX: 5\% 5 % compound interest. Determine the number of years (since 1963) necessary for the balance to reach $LaTeX: 3K 3 K .

The number of years necessary for the balance to turn from K to 3K (since 1963) in the given situation = 24.5636 years rounded off to 25 years

Explanation:

The simple interest earned is at the rate of 2.5%. The formula for simple interest per year is,

Simple interest per year = Investment * interest rate

Simple interest per year = 1K * 2.5%  => $0.025K

Simple interest for 4 years = 0.025 * 4 = $0.1K

So, total investment at the after 4 years = 1K + 0.1K = $1.1K

The formula for future value of a sum of amount will be used to calculate the value of investment at a future date. The formula is as follows,

Future value = Present value * (1+r)^t

Where,

So, accumulated earnings ($1.1K) are invested at 5% compound interest. The value of t necessary for 1.1K to turn into 3K can be found as follows,

3 = 1.1 * (1.05)^t

3 / 1.1 = 1.05^t

2.727272727 = 1.05^t

ln(2.727272727) / ln(1.05) = t

t = 20.5636 years rounded off to 21 years

The number of years necessary for the balance to turn from K to 3K in the given situation = 4 + 20.5636 = 24.5636 rounded off to 25 years

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