Investment: Rule of 70

Verify that the time necessary for an investment to double in value is approximately 10/r, where r is the annual interest rate entered as a percent.

Verification given below.

Step-by-step explanation:

This question requires us to calculate the rate of interest that will double the investment amount in 10 years. Let suppose that 100 dollars is the amount to be invested. The double of it is 200 dollars, so by using dicount factor formula we can easily find interest rate r

FV = PV (1+r) ^n

200 = 100 (1+r) ^10

Log 2 = 10 log (1+r)

R = 7.2%

n = ln (2) / ln (1+r) ... 3

But ln2 ~=0.7 and ln (1+r) ~ = r

Therefore; substituting into eqn 3

n = 0.7/r

Multiplying both denominator and numerator by 100.

n = (0.7) (100) / (r) (100)

n = 70/r%

Where r% = rate in percentage.

Here is the correct question:

Verify that the time necessary for an investment to double in value is approximately 70/r, where r is the annual interest rate entered as a percent.

Step-by-step explanation:

The compound interest formula can be written as;

A = P (1+r) ^n ... 1

n is the number of years of investment.

A is the final value of investment

P is the principal (initial investment)

r is the rate in fraction

When the investment doubles, the final value is twice the principal;

A = 2P ... 2

Substituting eqn 2 into eqn 1

2P = P (1+r) ^n

dividing both sides by P

2 = (1+r) ^n

Finding the natural log of both sides.

ln (2) = nln (1+r)

Making n the subject of formula.

n = ln (2) / ln (1+r) ... 3

But ln2 ~=0.7 and ln (1+r) ~ = r

Therefore; substituting into eqn 3

n = 0.7/r

Multiplying both denominator and numerator by 100.

n = (0.7) (100) / (r) (100)

n = 70/r%

Where r% = rate in percentage.