Compound interest

In the equation A is the compound amount, P is the principal, r is the interest rate as a decimal, n is the number of compounding times per year, and t is time in years.

Example 1

If $100 is invested at 5% compounded quarterly, find how long it takes for the investment to double.

Solution .

Using A = 200, P =100, r = 0.05, n =4 we get To solve this problem we must solve this exponential equation for t. First isolate the exponential part by dividing by 100. This gives us Taking the natural log of both sides we get Using property 3 we get and thus = 13.949 or about 14 years.

Example 2

Investor A has $100 invested at 4% compounded annually while investor B has $50 invested at 6% compounded annually. In how many years will B's investment exceed A's?

Solution .

In x years, A will have while B will have We solve Divide both sides by 50. This gives us . Now divide by This yields or Taking the natural log of both sides and using property 3, we get Solving this for x, we get x = 36.389 years. This can be seen in the graph below.