Problem 36 - Algebra A Combined Function

Question

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve these compound interest problems. Round to the nearest tenth. How long does it take for \(\$ 600\) to double if it is invested at \(12 \%\) interest compounded monthly?

Step-by-Step Solution

Answer

It takes approximately 6.0 years for $600 to double at 12% interest compounded monthly.

Step 1: Identify Variables

Let's identify the variables in the formula. You have an initial principal amount \(P\) of \(600\), a final amount \(A\) of \(1200\) (since it's doubling), an annual interest rate \(r\) of \(0.12\), and \(n\) compounding periods per year of \(12\) (since it's compounded monthly). We need to find \(t\).

Step 2: Substitute Known Values

Substitute the known values into the compound interest formula: \[ 1200 = 600\left(1 + \frac{0.12}{12}\right)^{12t} \]

Step 3: Simplify the Equation

First, simplify the term inside the parentheses: \[ 1 + \frac{0.12}{12} = 1 + 0.01 = 1.01 \] The equation now looks like: \[ 1200 = 600(1.01)^{12t} \]

Step 4: Isolate the Exponential Component

Divide both sides of the equation by 600 to isolate the exponential part:\[ 2 = (1.01)^{12t} \]

Step 5: Solve for \(t\) using Logarithms

To solve for \(t\), use logarithms to get:\[ \log(2) = \log((1.01)^{12t}) \]Using the power rule of logarithms, rewrite as:\[ \log(2) = 12t \cdot \log(1.01) \]

Step 6: Calculate \(t\)

Solve for \(t\):\[ t = \frac{\log(2)}{12 \cdot \log(1.01)} \]Calculate the values using a calculator:\[ t = \frac{0.3010}{12 \cdot 0.0043} \approx 5.95 \]

Step 7: Round the Result

Since we need to round to the nearest tenth, \(t\) is approximately 6.0 years.

Key Concepts

exponential functions

Exponential functions are an essential concept in mathematics, particularly when dealing with growth and decay problems like compound interest. In an exponential function, the variable appears in the exponent, and the base is a constant. For example, in the equation \( A = P(1 + r)^t \), the term \( (1 + r) \) is the base, and \( t \) is the exponent.
Exponential functions describe situations where a quantity grows or shrinks at a rate proportional to its current value. This forms the basis for understanding compound interest calculations. In the context of finance, money grows depending on the interest rate applied over time, exemplifying how exponential growth can significantly increase initial investments.
The hallmark of exponential functions is their rapid growth after a certain point, making them powerful tools in financial planning and investing, especially for calculating future investment values.

logarithms

Logarithms are the inverse operation of exponentiation, meaning they allow us to work backwards from exponential equations. In financial mathematics, logarithms are particularly useful when solving for time periods in compound interest problems. For instance, if we need to find how long it takes for an investment to double, we can express the problem exponentially and use a logarithm to solve for time.
The properties of logarithms that come in handy include the power rule, \( \log_b(x^y) = y \cdot \log_b(x) \), which helps in isolating variables in the exponent. Furthermore, the ability to convert complex multiplications and divisions into additions and subtractions via logarithms simplifies many calculations.
Understanding how to manipulate logarithms is crucial for solving various equations in financial mathematics, especially when the equation involves an unknown time variable, as demonstrated in the compound interest equation.

compounded interest calculation

The compounded interest calculation allows us to determine the future value of an investment given certain parameters. The formula \( A = P\left(1+\frac{r}{n}\right)^{nt} \) represents the compound interest equation, where \( A \) is the amount of money accumulated after n years, including interest.

• \( P \) is the principal amount (initial investment).
• \( r \) is the annual interest rate (as a decimal).
• \( n \) is the number of times the interest is compounded per year.
• \( t \) is the time in years.

This formula takes into account the frequency with which interest is applied to the principal, making it more accurate than simple interest calculations. By plugging in the known values and solving for the unknowns, one can effectively plan financial growth over time.
Accurate compounded interest calculations are vital for personal finance decisions, helping individuals to assess investment potential and eventual returns based on different compounding periods and interest rates.

financial mathematics

Financial mathematics involves applying mathematical methods to solve problems related to finance, such as calculating interest, understanding loan amortizations, and evaluating investment returns. It draws heavily from various fields of mathematics like algebra, calculus, and probability.
In the realm of compound interest, financial mathematics provides tools to determine how investments grow over time, considering various factors like interest rates and compounding frequency. It involves using exponential growth models to predict future financial obligations and opportunities.
By understanding the principles of financial mathematics, individuals can make informed decisions about saving, investing, and managing debt. It empowers individuals to evaluate financial products more critically, ensuring they maximize returns while minimizing risks.
Financial mathematics is not just about numbers but the analysis and interpretation of financial scenarios to support personal and professional financial planning.