Problem 31 - Algebra 1

Question

You invest \(\$ 80\) at a simple annual interest rate of \(2 \%\). How much simple interest would you earn in 1.5 years? Use unit analysis to check the units in your response.

Step-by-Step Solution

Answer

The simple interest earned would be $2.4.

Step 1: Understand Simple Interest

First, understand what simple interest is: it is interest calculated on the original principal only. You can calculate simple interest using this formula: \(I = PRT\), where: \(I\) is the interest, \(P\) is the principal (the initial amount of money), \(R\) is the rate of interest per period, and \(T\) is the time the money is invested for, in years.

Step 2: Substitute Values into the Formula

Substitute the given values into the formula. Here, \(P = $80\), \(R = 2% = 0.02\) (as percent means per hundred, we divide the given number by 100 to convert it into decimal form), \(T = 1.5\) years. We find that \(I = 80 * 0.02 * 1.5\).

Step 3: Calculate the Result

Perform the multiplication to calculate the interest. This gives us \(I = $2.4\).

Step 4: Perform Unit Analysis

For unit analysis, ensure that your final answer is in the correct units. In this case, the question asks for the interest in dollars, so you should present your answer in dollars. The result from step 3 is already in the correct unit.

Key Concepts

Unit Analysis

Unit analysis is a critical step in ensuring that your calculations are correct and meaningful. It involves carefully checking the units that you are using in the problem to ensure consistency. If mismatched units are used, the result will not make sense.

Here's how to apply unit analysis in interest calculation:

• Make sure that all the quantities are in compatible units before performing any calculations. For example, if your interest rate is given annually, ensure that your time period is also in years.
• Convert percentages into decimals for calculations as percentages are not directly usable in equations. For example, converting a rate of 2% to 0.02 by dividing by 100.
• Check that the final answer's units match what the problem specifies. In our example, calculating interest should result in an amount of money, expressed in dollars.

This method helps avoid common errors in mathematical education and ensures that students grasp the importance of consistent units in their calculations.

Interest Calculation

Interest calculation, especially in the context of simple interest, revolves around a straightforward formula: \[ I = PRT \] where:

• \( I \) is the interest earned,
  
• \( P \) is the principal amount or initial investment,
  
• \( R \) is the annual interest rate, and
  
• \( T \) is the time in years.
  

In our example, you start by replacing the variables with the given values: an initial amount (\( P \) = \(80), an interest rate (\( R \) = 0.02), and the time period (\( T \) = 1.5 years). The structure of the formula allows for easy substitution and solving.

After plugging in these values, you perform the simple multiplication operation: \( I = 80 \times 0.02 \times 1.5 \).

The result, calculated as \( I = \)2.4 \), represents the interest earned over the specified time. This approach to simple interest is foundational in mathematics education, paving the way for more complex financial calculations and concepts.

Mathematics Education

Mathematics education often emphasizes understanding over memorization. Concepts like simple interest encourage this form of learning, as they apply mathematical formulas to real-world scenarios. Here are a few educational insights:

• Understanding formulas: Dissect formulas to understand each component. For instance, recognize \( I = PRT \) encapsulates how interest grows over a set period, affecting the initial investment.
• Relating to everyday life: Simple interest is not just theoretical. It's used in bank savings, loans, and investments—providing a real context for abstract numbers.
• Developing problem-solving skills: Encountering diverse problems encourages students to think critically, adapt strategies, and apply learned knowledge effectively.
• Connecting to other mathematical areas: The principles behind interest calculation connect to algebra, decimals, and even use basic arithmetic, demonstrating the interconnectedness of mathematical learning.

By focusing on these aspects, mathematics education becomes more interactive and engaging, which can lead to a better understanding and appreciation of mathematical concepts in students' daily lives.