Question

In how many years will a certain amount double, if it attracts 5%

interest rate compounded annually?

Answer 1

In about 14.21 years a certain amount will double if it attracts 5% interest rate compounded annually

Explanation:

The formula for compound interest, including principal sum, is

`A=P(1+(r)/(n))^(nt)` , where

• A is the future value of the investment/loan, including interest
• P is the principal investment amount
• r is the annual interest rate (decimal)
• n is the number of times that interest is compounded per unit t
• t is the time the money is invested or borrowed for

∵ A certain amount is double in t years

- That means A is double P

∴ A = 2 P

∵ It attracts 5% interest rate compounded annually

∴ r = 5% =
`(5)/(100)` = 0.05

∴ n = 1 ⇒ compounded annually

- Substitute all of these values in the formula above

∵
`2P=P(1+(0.05)/(1))^((1)t)`

∴
`2P=P(1.05)^(t)`

- Divide both sides by P

∴
`2=(1.05)^(t)`

- Insert ㏑ for both sides

∴
`ln(2)=ln(1.05)^(t)`

- Remember
`ln(1.05)^(t)` = t . ㏑(1.05)

∴ ln(2) = t . ㏑(1.05)

- Divide both sides by ㏑(1.05)

∴ 14.2067 = t

- Round it to the nearest hundredth

∴ t = 14.21 years

In about 14.21 years a certain amount will double if it attracts 5% interest rate compounded annually