Problem 38 - 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 \(\$ 2000\) increased to \(\$ 2420\) in 2 years. Write your answer as a percent.

Step-by-Step Solution

Answer

The interest rate is 10%.

Step 1: Understand the given formula

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

Step 2: Substitute the known values

We are given that \( A = 2420 \) and \( P = 2000 \). Substitute these values into the equation: \( 2420 = 2000(1+r)^2 \).

Step 3: Solve for \( (1 + r)^2 \)

Divide both sides of the equation by 2000 to isolate \((1 + r)^2\):\[ \frac{2420}{2000} = (1 + r)^2 \]\[ 1.21 = (1 + r)^2 \]

Step 4: Take the square root of both sides

To find \( 1 + r \), take the square root of both sides:\[ \sqrt{1.21} = 1 + r \]\[ 1.1 = 1 + r \]

Step 5: Solve for \( r \)

Subtract 1 from both sides to solve for \( r \):\[ r = 1.1 - 1 \]\[ r = 0.1 \]

Step 6: Convert \( r \) to a percentage

To express \( r \) as a percentage, multiply by 100:\[ r = 0.1 \times 100\% = 10\% \].

Key Concepts

Interest Rate

The interest rate is a critical factor in determining how your investments grow over time. In our exercise, we are dealing with the simple yet powerful process of compounding. The interest rate, denoted by \( r \), is the proportion of the principal amount, \( P \), that is charged as interest over a set period. It is generally expressed as a decimal or percentage.

Consider a hypothetical situation where your initial investment of \( \\(2000 \) grows to \( \\)2420 \). Here, you could find the interest rate to determine just how much interest was gained annually. To begin with, convert the interest rate from a percentage to a decimal for computational purposes.

Key concepts about interest rates:

• It indicates how much you earn on your investment.
• It is crucial in calculating both simple and compound interest.
• Results can widely vary based on the interest rate's frequency and duration.

In our exercise, calculating the interest rate helped identify a \(10\%\) annual increase.

Investment Growth

Investment growth measures how much your money increases over time due to interest. It's an important concept for anyone looking to save or invest money effectively. In this particular exercise, we're exploring how an investment grows with compound interest over time.

By understanding the formula \( A = P(1 + r)^2 \), it's clear that we're describing exponential growth. Initially, your investment amount, or principal, is \( P = 2000 \). After two years and with interest factored in, your investment grows to \( A = 2420 \).

Consider these factors impacting investment growth:

• The principal amount.
• The interest rate applied.
• The compounding frequency.

Calculating the difference between the final amount and the principal gives a concrete view of how much your initial investment has grown.

Compounded Annually

When interest is compounded annually, it means that the interest is calculated and added to the principal once per year. This process can significantly impact the growth of investments over time. For our exercise, compounding annually means every year the interest earned is based on the total amount at the end of the previous year.

To break it down simply:

• At the end of year one, interest is added to your principal.
• In the next period, interest is calculated on the new total.
• This creates a cycle where each year builds upon the last.

This method of compounding accelerates the growth of an investment compared to simple interest, which does not take the accrued interest into account when calculating for the next period. In the example provided, compounding annually helps the initial \( \$2000 \) grow over two years, influenced further by the discovered \(10\%\) interest rate.