Question

How long will it take for $2500 to double if it is invested at 6% annual interest compounded 6 times a year? Enter exact calculations or round to 3 decimal places.

- It will take ____years to double.
How long will it take if the interest is compounded continuously?
-compounded continuously, it would only take ____years

Answer 1

Part 1)
`t=11.610\ years`

Part 2)
`t=11.552\ years`

Explanation:

Part 1) we know that

The compound interest formula is equal to

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

where

A is the Final Investment Value

P is the Principal amount of money to be invested

r is the rate of interest in decimal

t is Number of Time Periods

n is the number of times interest is compounded per year

in this problem we have

`A=\$5,000\\P=\$2,500\\ r=6\%=6/100=0.06\\n=6`

substitute in the formula above

`5,000=2,500(1+(0.06)/(6))^(6t)`

`2=(1.01)^(6t)`

Apply log both sides

`log(2)=log[(1.01)^(6t)]`

`log(2)=(6t)log(1.01)`

solve for t

`t=log(2)/[6log(1.01)]`

`t=11.610\ years`

Part 2) we know that

The formula to calculate continuously compounded interest is equal to

`A=P(e)^(rt)`

where

A is the Final Investment Value

P is the Principal amount of money to be invested

r is the rate of interest in decimal

t is Number of Time Periods

e is the mathematical constant number

we have

`A=\$5,000\\P=\$2,500\\ r=6\%=6/100=0.06`

substitute in the formula above

`5,000=2,500(e)^(0.06t)`

`2=(e)^(0.06t)`

Apply ln both sides

`ln(2)=ln[(e)^(0.06t)]`

`ln(2)=(0.06t)ln(e)`

`ln(2)=(0.06t)`

`t=ln(2)/(0.06)`

`t=11.552\ years`