David H. answered • 05/21/20
5th Year High School Math Teacher & Tutor
Hi Michael,
This problem requires knowledge of the compound interest formula, percentages and decimals, exponents, and the relationship between exponential and logarithmic equations. If you have any questions about any of these concepts (like why the formula works, or what a logarithm is), please let me know.
To begin, you need to use the formula for compound interest:
A = P(1 + r/n)^nt
A: Final Amount (3)
P: Principal Amount (1)
r: interest rate (0.06)
n: number of times the interest compounds per time period (1)
t: number of time periods that have elapsed (variable)
It doesn't technically matter what we set A equal to, as long as it is 3 times the value of P. To make our lives easier, we will choose 1 for P and 3 for A (we are pretending that we started with a principal amount of 1 dollar, and are curious when that 1 dollar turns into a final amount of 3 dollars). The interest rate, r, must be expressed as a decimal, and 6% translates to 0.06. Since the time period is one year, and the interest compounds annually (once per year), n must equal 1. We will leave the number of elapsed years as a variable, and solve for that variable.
Substituting, we get:
3 = 1(1.06)^1t
Which simplifies to:
3 = 1.06^t
To isolate the variable, we will perform a logarithm of both sides. Since the base of the exponential equation is 1.06, we will use this for the logarithmic base as well.
log1.06(3) = log1.06(1.06^t)
Which simplifies to:
log1.06(3) = t
Some calculators allow you to type in any base. If this is the case with your calculator, you're ready to solve. If not, then you must use the change of base formula, using either the log or ln button on your calculator, which have bases of 10 and e, respectively. I will use the log button (which has a base of 10 that is left out of the written expression).
log(3)/log(1.06) = t
Your calculator should tell you that your answer is roughly 18.85. This means that at the rate your money is increasing, it would triple at exactly 18.85 years. However, the interest only compounds once every year, so it won't actually reach that amount until year 19. Thus, the answer is 19.
Hope that helps!
