Problem 70 - Algebra for College Students

Question

Set up an equation and solve each problem. Suppose that \(\$ 10,000\) is invested at a certain rate of interest compounded annually for 2 years. If the accumulated value at the end of 2 years is \(\$ 12,544\), find the rate of interest.

Step-by-Step Solution

Answer

The interest rate is 12%.

Step 1: Understand the Problem

We need to find the interest rate given an initial investment, the time period, and the final accumulated amount. The formula for compound interest is: \[ A = P(1 + r)^t \] where \( A \) is the accumulated amount, \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time in years.

Step 2: Identify Known Values

From the problem, we know: - \( P = 10,000 \) (the initial investment),- \( A = 12,544 \) (the accumulated amount),- \( t = 2 \) years (the time period). We need to find \( r \), the rate of interest.

Step 3: Set Up the Equation

Substitute the known values into the compound interest formula: \[ 12,544 = 10,000(1 + r)^2 \] This equation will help us solve for \( r \).

Step 4: Simplify the Equation

First, divide both sides by 10,000 to isolate the expression involving \( r \): \[ \frac{12,544}{10,000} = (1 + r)^2 \] This simplifies to: \[ 1.2544 = (1 + r)^2 \]

Step 5: Solve for \( 1 + r \)

Take the square root of both sides to solve for \( 1 + r \): \[ \sqrt{1.2544} = 1 + r \] Calculate the square root: \[ 1.12 = 1 + r \]

Step 6: Solve for the Interest Rate \( r \)

Subtract 1 from both sides to solve for \( r \): \[ r = 1.12 - 1 \] Thus, \( r = 0.12 \) or 12%.

Key Concepts

Interest Rate Calculation

Interest rate calculation is the process of determining the rate at which your principal investment will grow over time. This rate is represented as a percentage and helps in calculating how much extra money, besides your initial investment, you will earn. To find this rate, specific values are required:

• The principal amount (e.g., $10,000).


• The time period for which the money is invested (e.g., 2 years).


• The final accumulated value (e.g., $12,544).

Once these values are available, you substitute them into the compound interest formula to calculate the interest rate. In financial mathematics, understanding the interest rate is crucial, as it can affect decisions like whether to save or invest money.

Compound Interest Formula

The compound interest formula is fundamental in financial mathematics. This formula helps calculate the total amount of money that will be in your account after a certain period, given a specific interest rate applied to your initial investment.The formula is:\[ A = P(1 + r)^t \]where:

• \( A \) is the accumulated value (e.g., \(12,544).


• \( P \) is the principal amount (e.g., \)10,000).


• \( r \) is the annual interest rate (expressed as a decimal).


• \( t \) is the time the money is invested (in years).

Using the formula, you can plug in the values you know and solve for the unknown, which in many cases is the interest rate. This formula is especially powerful because it reflects the concept of interest being earned on interest, hence the term "compound."

Investment Growth

Investment growth refers to the increase in value of an investment over time due to factors like compounding interest. This growth is crucial for understanding how your money can work over an extended period to yield significant returns. The main elements contributing to investment growth include:

• The original sum of money invested (the principal).


• The interest rate applied.


• The length of the investment period (time).

The compound interest formula captures this growth by showing how the original investment increases not just by earning interest, but by earning interest on the previously earned interest as well. This exponential growth is why people are often encouraged to invest early and let their investments grow over time.

Financial Mathematics

Financial mathematics involves using math equations and models to solve problems related to finance, such as determining the best investments or savings strategies. It combines areas like algebra, statistics, and calculus to help forecast financial scenarios and make informed decisions. Core areas of financial mathematics include:

• Interest calculations, both simple and compound.


• Investment valuations, determining the present and future values.


• Risk assessment, evaluating the potential upsides and downsides of investments.

Understanding financial mathematics allows individuals to take charge of their financial future by using mathematical tools to optimize their personal and business finances.