Compound interest

Question

Determine the time necessary for $1000 to double if it is invested at interest rate r compounded (a) annually, (b) monthly, (c) daily, and (d) continuously.

r = 10%

Answer 1

Step 1:

The compound investment  , annual rate .

The annual rate .

The amount after maturity period is double to the investment is .

The formula for n compounding per year: .

Step 2:

(a)

Annually compounding means .

Substitute , , and .

Take on each side.

Apply inverse property: .

Apply change-of-base formula : .

.

The required time to double the investment is  years.

Solution :

The required time to double the investment is  years when compounded annually.

Answer 2

(b)

Step 1:

The compound investment  , rate of interest .

The annual rate .

The amount after maturity period is double to the investment is .

The formula for n compounding per year: .

Step 2:

Monthly compounding means .

Substitute , , and .

Take of each side.

Apply inverse property: .

Apply change-of-base formula : .

The required time to double the investment is  years.

Solution :

The required time to double the investment is  years when compounded monthly.

Answer 3

(c)

Step 1:

The compound investment  , annual rate .

The annual rate .

The amount after maturity period is double to the investment is .

The formula for n compounding per year: .

Step 2:

daily compounding means (one year has 365 days).

Substitute  , , and .

Take on each side.

Apply inverse property: .

Apply change-of-base formula : .

 

             

.

The required time to double the investment is  years.

Solution :

The required time to double the investment is  years when compounded daily.

Answer 4

(d)

Step 1:

The formula for continuous compounding: .

Substitute , and .

 

Take on each side.

Apply inverse property:

  

.

The required time to double the investment is years.

Solution :

The required time to double the investment is when compounded continuously.