A few days ago
Wonders

# How do I use the Rule of 70?

Rule of 70= 70/percentage of growth rate=doubling time in years..

my percentage growth is 2%. Now do i put that in for percentage growth rate so that it shows up as

70/2=doubling time in years?

I have to use the rule of 70 to calculate how many years it would take for the population of a country to double if it was growing at 2% per year.

A few days ago
RU11

The rule of 70 as distinct from the rule of 72 is derived based on continuous compounding of interest at lower rates.

When the interest is compounded k times per year, with i / 100 replaced by r and A/P = 2, the formula you have becomes:

2 = (1 + r/k)^(kt)

Expanding the RHS using the binomial theorem gives:

2 = 1 + (kt)(r/k) + (kt)(kt – 1) (r/k)^2 / 2!

+ (kt)(kt – 1)(kt – 2) (r/k) / 3! + …

2 = 1 + tr + t(t – 1/k)(r^2 / 2!) + t(t – 1/k)(t – 2/k)(r^3 / 3!) + …

As k tends to infinity, 1/k tends to 0, and this becomes:

2 = 1 + tr + (tr)^2 / 2! + (tr)^3 / 3! + …

The RHS of this formula is the power series for e^(rt).

Hence:

2 = e^(rt)

rt = log(base e) 2

t = [ log(base e)2 ] / r

t = 0.6913 / r

Substituting 100i back for r:

t = 69.13 / i

but I don’t really see the justification for 70.

However this is the basis of the derivation in the link below, where at the end the article just says ’70 is close enough’

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A few days ago
alicias7768
http://mmcconeghy.com/students/supscruleof70.html
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