Problem 46 - Algebra 1

Question

Write an equation or an inequality to model the real-life situation. The simple interest earned on a principal of three hundred dollars at an annual interest rate of \(x\) percent is less than or equal to seventy-two dollars.

Step-by-Step Solution

Answer

The inequality that models the situation is \(3x \leq 72\).

Step 1: Understand the context

In the given problem, a real-world scenario is provided which states that the simple interest earned on a principal of three hundred dollars at an annual interest rate of \(x\) percent is less than or equal to seventy-two dollars.

Step 2: Formulate the problem using simple interest formula

Simple Interest = Principal * Rate * Time. Substituting the given values, we have Interest = $300 * \(x\) (as a decimal) * 1 (since it's an annual interest, time = 1 year). Thus the inequality can be written as: \(300x \leq 72\). Since x is a percentage, we need to convert it to a decimal for this inequality.

Step 3: Getting the final inequality

To convert x from percentage to decimal, we divide by 100. Thus, the final inequality that models the given situation can be expressed as: \(3x\leq 72\)

Key Concepts

Algebraic Inequality

When we face an algebraic inequality, we're dealing with a mathematical statement indicating that two expressions are not necessarily equal, but instead, one is less than, greater than, less than or equal to, or greater than or equal to the other. They are crucial for modeling real-world scenarios where we need to represent a range of possible solutions rather than a single answer.

For instance, the inequality from our exercise, \(300x \leq 72\), signals that the simple interest rate times the principal amount (\(x\) percent of \(300) should not exceed \)72. This is an example of an inequality being used to set a boundary on financial figures, which is common in budgeting, investing, and other financial planning activities.

In learning to solve these inequalities, it's important to understand the properties of inequalities, such as how multiplying or dividing both sides by a negative number reverses the inequality sign. This doesn't apply to our example, as we're dealing with positive numbers, but it's a key point worth noting when tackling algebraic inequalities.

Simple Interest Formula

The simple interest formula is a quick way to calculate the interest earned or paid on a certain principal over a set period at a fixed rate. The formula is given by \(I = P \times r \times t\), where \(I\) is the interest, \(P\) is the principal amount, \(r\) is the rate of interest per period, and \(t\) is the time in years. This formula is essential in finance and helps in understanding how much money one can earn from an investment or, alternatively, how much one will have to pay on a loan.

Why is this important? Well, by using the simple interest formula, individuals can make informed decisions about their loans and investments. For example, by adjusting the rate or the time, one can see how it affects the total interest earned or owed, allowing for better planning of finances. The exercise presents a scenario where the simple interest is capped, which teaches us to work backward from the interest to figure out the maximum possible rate.

Mathematical Modeling

Mathematical modeling is the process of using mathematical language and tools to represent and solve problems from the real world. It allows us to translate situations into equations or inequalities that can be analyzed and solved, giving us insight into the problem at hand.

In the context of our exercise, we modeled a real-life financial situation with an algebraic inequality derived from the simple interest formula. This is a prime example of how mathematical modeling helps in decision-making and predictions based on mathematical calculations. By creating a model, one can experiment with different values and scenarios without impacting the real-world subject, such as testing various interest rates to see their effect on the simple interest earned.