Final answer:
Bert's earnings would not be twice as large if he switched to a bank offering a doubled interest rate of 6%. This is because compound interest grows exponentially, so the increase in earnings is not linearly proportional to the increase in the interest rate.
Step-by-step explanation:
Understanding Compound Interest
Bert deposited $2000 in an account earning with 3% annually compounded interest. He's considering switching to another bank offering 6% annually compounded interest. Would Bert's earnings be twice as large if the interest rate doubles? The answer is no. The reason is that compound interest grows exponentially, not linearly.
To understand this, let's take a closer look at compound interest. The formula for compound interest is A = P(1 + r/n)^(nt), where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (decimal).
n is the number of times that interest is compounded per year.
t is the time the money is invested or borrowed for, in years.
Using this formula, if Bert kept his money at the 3% interest rate for 1 year, he would have:
$2000(1 + 0.03/1)^(1*1) = $2060
With a 6% interest rate, it would be:
$2000(1 + 0.06/1)^(1*1) = $2120
As we can see, doubling the interest rate does not result in doubling the earnings because of the exponential nature of compound interest.