Question

How long does it take, approximately, when the interest generated by principal equals the interest generated by interest for a particular year (i.e., it refers to the interest generated by interest for a particular year, not the interest generated by interest that has cumulated so far), when the interest rate = 9%?

Answer 1

Let n be the number of years

let principal be 1

interest rate = 9%

interest generated by principal = 9%*1 = 0.09 ----------------(1)

Interest in nth year = compounded value in nth year - compounded value in (n-1)th year

= 1*(1+9%)ⁿ - 1*(1+9%)^n-1

= (1+9%)ⁿ - (1+9%)^n-1

interest generated by interest = 9% * [(1+9%)ⁿ - (1+9%)^n-1] ------------(2)

Equating (1) and (2)

9% * [(1+9%)ⁿ - (1+9%)^n-1] = 0.09

[(1+9%)ⁿ - (1+9%)^n-1] = 1

(1+9%)^n-1*[(1+9%)-1] = 1

(1+9%)^n-1*9% = 1

(1+9%)^n-1 = 1/9%

(1+9%)^n-1 = 11.111111

taking natural logarithm on both sides

ln[(1+9%)^n-1] = ln(11.111111)

(n-1)*ln(1+9%) = ln(11.111111)

n-1 = ln(11.111111)/ln(1.09) = 2.407946/0.086178

n-1 = 27.94163355

n = 27.94163355 + 1 = 28.94163355 = 29 years approximately

Hence interest generated in a year equals the principal value in approximately 29 years

but the interest generated on that interest happens in the next year, hence the answer is 30 years