Rule of 72 at 5%

Publish date: 2024-06-26
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Using the Rule of 72...

How long for money to double at 5%?

Doubling Time Definition:

(1 + i)n = 2

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 = 5%
n  =  0.6931 * 0.05
  0.05 * Ln(1 + 0.05)

n  =  0.6931 * 0.05
  0.05 * Ln(1.05)

Now simplify the 2nd term
n  =  0.6931 * 0.05
  i * 0.048790164169432

n  =  0.6931 * 1.0247967157144
  i

n~  =  0.72
  i

Substitute i = 0.05 into the quotient
n~  =  0.72
  0.05

n = 14.4
This means at an interest rate of 5%, we double our money approximately every 14.4 periods of time.


What is the Answer?

How does the Rule of 72 Calculator work?

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.
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What 3 formulas are used for the Rule of 72 Calculator?

Doubling Money Time (t) = 0.72/i
(1 + i)t = 2

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What 6 concepts are covered in the Rule of 72 Calculator?

approximationanything that is intentionally similar but not exactly equal to something else.compound interestthe interest you earn on principal and interest
A = (1 + r/n)ntinterest ratethe proportion of a loan that is charged as interest to the borrower or proportion of principal credit given to a depositorlogarithmthe exponent or power to which a base must be raised to yield a given numberrule of 72a simplified formula that calculates how long it will take for an investment to double in value, based on its rate of return.
t ~ 72/iyieldHow much an investment returns in terms of interest rate

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