Question

Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75\(\%\) compounded annually.

Short answer

To find \(n\), plug the values into the equation, \(n = \ln(3) / \ln(1.0575)\), and calculate. Thus, it will take approximately 20.22 years for the investment to triple its value.

Step-by-step solution

Understanding the problem

The problem is to find out how many years it would take an investment to triple if there's an annual interest rate of 5.75\%. So let's use the compound interest formula: \(A = P (1 + i)^n\) where \(A\) is the amount of money accumulated after \(n\) years, \(i\) is the annual interest rate, \(n\) is the number of years and \(P\) is the initial principal.

Set up the equation

Since it's stated that the final amount \(A\) is triple the initial investment \(P\), the equation becomes: \(3P = P (1 + 0.0575)^n\). Here, 0.0575 is the decimal equivalent of 5.75%. After simplifying, we get \(3 = (1 + 0.0575)^n\).

Solve for \(n\)

To solve the equation \(3 = (1 + 0.0575)^n\) for \(n\), we take the natural logarithm on both sides. That'll cancel the exponent on the right and we can solve the equation for \(n\): \(n = \ln(3) / \ln(1.0575)\).

Compute the value of \(n\)

Now compute \(n\) using a calculator to get the value of \(n\).

Key concepts

Understanding the Compound Interest Formula

The compound interest formula is a critical concept for understanding how investments grow over time. It's described by the equation \(A = P(1 + r)^n\), where \(A\) represents the future value of the investment, including interest, \(P\) is the principal amount (the initial sum of money), \(r\) is the annual interest rate (expressed as a decimal), and \(n\) represents the number of times the interest is compounded per year multiplied by the number of years the money is invested.

The power of compound interest comes from the fact that interest is calculated not only on the initial principal but also on the accumulated interest from previous periods. This creates exponential growth, significantly increasing the value of the investment over time.

When solving problems involving compound interest, it's important to convert the interest rate to a decimal by dividing by 100 and to make sure the rate and number of compounding periods align if they are not annual.

Deciphering Natural Logarithm

The natural logarithm is a mathematical function that is denoted by \(\ln\). It is the inverse of the exponential function with base \(e\), where \(e\) approximates to 2.71828. Essentially, the natural logarithm of a number answers the question: 'To what power must we raise \(e\) to obtain this number?'

Using the natural logarithm is particularly helpful in solving for variables in exponential equations, as it allows us to 'bring down' exponents and make the equations solvable by conventional means. For example, in the equation \(e^x = b\), taking the natural logarithm of both sides yields \(x = \ln(b)\), thereby solving for \(x\).

In the context of compound interest, if we have an equation like \(A = P(1 + r)^n\), by taking the natural logarithm of both sides, we can begin to solve for \(n\) when \(A\), \(P\), and \(r\) are known.

Solving Exponential Equations

Exponential equations feature variables in the exponent, such as \(a^x = b\) and are fundamental to understanding growth processes, including compound interest. Solving such equations is often carried out by utilizing logarithms.

There's a quick process to solve these equations: First, isolate the exponential part if necessary. Then, apply the logarithm on both sides of the equation to eliminate the exponents. After applying logarithms, use algebra to solve for the variable.

For the compound interest problem, once we have the equation \(3 = (1 + 0.0575)^n\), we apply the natural logarithm, yielding \(\ln(3) = n \ln(1.0575)\). To solve for \(n\), we then divide both sides by \(\ln(1.0575)\), giving us the formula \(n = \frac{\ln(3)}{\ln(1.0575)}\). This equation allows us to calculate the precise number of years needed for the investment to triple, considering the compounded annual interest.