Question

Multiple Choice John invests \(\$ 200\) at 4.5\(\%\) compounded annually. About how long will it take for John's investment to double in value? (A) 6 yrs (B) 9 yrs (C) 12 yrs (D) 16 yrs (E) 20 yrs

Short answer

It will take approximately 16 years for John's investment to double in value. So, the correct answer is (D) 16 yrs.

Step-by-step solution

Set up the formula

We first set up our equation using the formula for compound interest: \( A = P (1 + r/n)^{nt} \). Here, A is the final amount which is double the original investment, so \( A = \$ 400 \). P is the principal amount, or the initial investment, which is \( \$ 200 \), the interest rate r is 4.5/100 = 0.045 (converted from percentage to decimal), and n equals to 1 because the interest is compounded annually. We are solving for t, the time in years, so our equation becomes: \( \$ 400 = \$ 200 (1 + 0.045/1)^{1t} \)

Simplify the equation

Divide both sides of the equation by \( \$ 200 \), the initial investment, to isolate the formula on the right. This gives us: \( 2 = (1 + 0.045)^t \)

Solve for t

This is now an exponential equation, which can be solved using the natural logarithm (ln). We can take the natural log of both sides to get rid of the exponent on the right side. This gives us: \( ln(2) = t*ln(1 + 0.045) \). We then divide both sides by \( ln(1 + 0.045) \) to solve for t: \( t = ln(2) / ln(1 + 0.045) \)

Calculate the value of t

When calculated, it gives \( t \approx 15.401 \) years. Because we can't have a fraction of a year in this context, we round it up to the nearest year, which is 16 years.

Key concepts

Exponential Equations

Exponential equations are fundamental in understanding growth processes, including financial scenarios like compound interest. When dealing with these equations, the variable we're looking to solve for is located in the exponent. In the context of compound interest, the formula used is typically represented as
\[ A = P (1 + \frac{r}{n})^{nt} \]
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 (in decimal form), n represents the number of times that interest is compounded per year, and t is the time in years. To solve an exponential equation like the one in our exercise, we must employ logarithms to isolate the variable in the exponent (t in this case). Exponential equations often describe processes where something is growing at a constant rate per period, such as money in a bank account or a population of bacteria in a lab culture.

Natural Logarithms

A natural logarithm is the logarithm to the base e (Euler's number), where e is an irrational and transcendental number approximately equal to 2.71828. In financial mathematics, natural logarithms are crucial when we want to 'reverse' the process of exponentiation. That is, they let us solve for the time variable in the compound interest formula.

Using the formula from our exercise:
\[ 2 = (1 + 0.045)^t \]
We can apply the natural logarithm to both sides because ln(a^b) = b*ln(a). This yields:
\[ \ln(2) = t * \ln(1 + 0.045) \]
By then dividing both sides by \(\ln(1 + 0.045)\), we isolate t, enabling us to calculate how many years it will take for an investment to double. This application of natural logarithms is a powerful tool in unraveling exponential growth scenarios and finding specific points in time for financial plans.

Financial Mathematics

Financial mathematics is an application of advanced mathematical methods to financial problems. Using algebra, calculus, and probability, it allows individuals and businesses to make informed decisions based on the potential outcomes of their financial investments.

In our compound interest problem, financial mathematics enables us to determine the time required for an investment to double. The key formula for compound interest
\[ A = P (1 + \frac{r}{n})^{nt} \]
is a cornerstone in financial mathematics, helping investors understand the effects of interest rates and time on their initial investments. By calculating when the investment will double, a concept known as the 'Rule of 72' often emerges as a shortcut, which states that dividing 72 by the interest rate gives an approximation of the time needed to double. However, for precise calculations, especially for higher interest rates or other compounding periods, the algebraic method with natural logarithms is preferred.