1 unit of money doubles in n periods at interest rate i is: (1 + i)n = 2
Take the natural log of both sides:
Ln(1 + i)n = Ln(2)
Use a logarithmic identity
Ln(an) = n * Ln(a)
Using that identity, we have a = (1 + i):
n * Ln(1 + i) = Ln(2)
Divide both sides by Ln(1 + i)
n =
Ln(2)
Ln(1 + i)
n =
0.6931
Ln(1 + i)
Multiply the top and bottom by i
n =
0.6931 * i
Ln(1 + i) * i
Plug in our interest rate of i = 25%
n =
0.6931 * 0.25
0.25 * Ln(1 + 0.25)
n =
0.6931 * 0.25
0.25 * Ln(1.25)
Now simplify the 2nd term
n =
0.6931 * 0.25
i * 0.22314355131421
n =
0.6931 * 1.1203550294311
i
n~ =
0.72
i
Substitute i = 0.25 into the quotient
n~ =
0.72
0.25
n = 2.88 This means at an interest rate of 25%, we double our money approximately every 2.88 periods of time.
What is the Answer?
n = 2.88
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Free Rule of 72 Calculator - Calculates how long it would take money to double (doubling time) using the rule of 72 interest approximation as well as showing the mathematical proof of the Rule of 72. This calculator has 1 input.
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