Problem 37 - Algebra A Combined Function

Question

At the end of 2 years, \(P\) dollars invested at an interest rate \(r\) compounded annually increases to an amount, \(A\) dollars, given by $$ A=P(1+r)^{2} $$ Find the interest rate if \(\$ 100\) increased to \(\$ 144\) in 2 years. Write your answer as a percent.

Step-by-Step Solution

Answer

The interest rate is 20%.

Step 1: Write Down the Formula

The given formula for compound interest is \( A = P(1+r)^2 \). Here, \( A \) is the final amount after 2 years, \( P \) is the principal amount, and \( r \) is the annual interest rate.

Step 2: Substitute Known Values

We know that \( P = 100 \) and \( A = 144 \). Substitute these values into the formula: \( 144 = 100(1+r)^2 \).

Step 3: Isolate the Compound Term

Divide both sides by 100 to isolate the compound term: \( 1.44 = (1+r)^2 \).

Step 4: Solve for \((1+r)\)

Take the square root of both sides to solve for \((1+r)\): \( \sqrt{1.44} = 1+r \). This gives \( 1.2 = 1 + r \).

Step 5: Solve for \(r\)

Subtract 1 from both sides to find \( r \): \( r = 1.2 - 1 = 0.2 \).

Step 6: Convert to Percentage

Convert the decimal interest rate to a percentage by multiplying by 100: \( 0.2 \times 100 = 20\% \).

Key Concepts

Interest Rate Calculation

Calculating interest rates is a foundational aspect of understanding financial growth and investment potential. In the context of compound interest, the interest is calculated on the initial principal, which also includes all of the accumulated interest from previous periods.

The formula given for compound interest, with annual compounding, is:

• \[ A = P(1+r)^n \]

where:

• \( A \) is the final amount after the investment has grown over time.
• \( P \) is the principal amount initially invested.
• \( r \) is the annual interest rate.
• \( n \) is the number of compound periods.

For our specific problem, we know all these values except for the interest rate \( r \), which we need to calculate. With this, the fundamental step is substituting the given numbers into the formula and rearranging to solve for \( r \).

Remember, interest rate calculations are vital because they help us understand growth over time, allowing for better financial planning and decision-making.

Algebraic Formulas

Algebraic formulas are essential tools when solving mathematical problems. They allow for the manipulation of equations to isolate and solve for unknown variables.

In this problem, we use an algebraic approach to isolate the term containing our unknown, \( r \):

• Starting with \( 144 = 100(1+r)^2 \).
• We divide both sides by 100: \( 1.44 = (1+r)^2 \).
• Next, take the square root of both sides to further isolate \( 1+r \): \( \sqrt{1.44} = 1+r \).
• Finally, we find that \( r \) by solving \( 1.2 = 1 + r \). Thus, \( r = 0.2 \).

Algebraic manipulation is a powerful skill that allows us to solve for unknowns in any formulaic context, not just in finance.

This methodical approach helps simplify complex equations, making it easier to find solutions and understand the underlying relationships between different mathematical elements.

Mathematical Problem Solving

Mathematical problem-solving involves breaking down complex problems into manageable steps. It's a skill that combines logic, mathematical theory, and practical application.

In our compound interest problem, identifying what is given, what is unknown, and how the mathematical relationship unfolds is key:

• We begin by understanding the problem requirements: determining the interest rate with given values over a specified period.
• The next step involves applying the compound interest formula, substituting known values, and simplifying the equation logically.
• Finally, convert the interest rate from a decimal to a percentage, ensuring the answer is in a practical format used in everyday financial discussions.

This systematic approach not only provides a clear path to the solution but also enhances critical thinking and analytical skills. Understanding the logical progression of a solution helps to build confidence in solving similar problems, whether they are in academic settings or real-world financial scenarios.