Question

The simple interest received from an investment is directly proportional to the amount of the investment. By investing $$\$ 2500$$ in a bond issue, you obtain an interest payment of $$\$ 187.50$$ at the end of 1 year. Find a mathematical model that gives the interest \(I\) at the end of 1 year in terms of the amount invested \(P\).

Short answer

The mathematical model that gives the interest \(I\) at the end of 1 year in terms of the amount invested \(P\) is \(I = 0.075P\).

Step-by-step solution

Understand the Problem

From the problem, we know that the interest \(I\) is directly proportional to the amount \(P\) invested. This means that as \(P\) changes, \(I\) will change in the same manner. The relationship between \(I\) and \(P\) can be expressed as \(I = kP\), where \(k\) is the proportionality constant.

Find the Proportionality Constant

To find the constant \(k\), we can use the information provided by the problem that when \(P = \$2500\), \(I = \$187.50\). Substituting these values into the equation \(I = kP\), we obtain: \(\$187.50 = k \cdot \$2500\). Solving for \(k\) gives us \(k = \$187.50 / \$2500 = 0.075\).

Write down the Mathematical Model

Now, we can substitute the obtained value for \(k\) back into the formula to get the mathematical model: \(I = 0.075P\).

Key concepts

Directly Proportional Relationships

Understanding directly proportional relationships is crucial when delving into various aspects of mathematics, including finance. In simple terms, when two quantities are directly proportional, as one increases, the other does so too, at a constant rate. This consistent rate is known as the proportionality constant, often represented by the symbol 'k'.

In the context of financial mathematics, if we consider the interest earned on an investment, we can see this direct relationship at work. For example, the more money you invest, the more interest you will receive, assuming a fixed interest rate. This direct proportionality is what forms the foundation of simple interest calculations.

When students encounter problems involving proportions, it's important to identify what the constant of proportionality represents. In our exercise, it was determined that the interest earned was directly proportional to the investment made. This linear relationship allows us to create a simple mathematical model to predict future interest earnings based on varying investment amounts.

Interest Calculation

Interest calculation is a key concept in financial mathematics, especially when it comes to savings, loans, and investments. Simple interest, which is the topic at hand, is calculated based on the original principal amount of a loan or investment. Unlike compound interest, simple interest does not take into account the accumulated interest in previous periods.

The formula for simple interest is straightforward:
\[\begin{equation}I = Prt\text{, where:}\begin{itemize}\item {I} is the interest.\item {P} is the principal amount (initial investment).\item {r} is the annual interest rate (in decimal form).\item {t} is the time in years.\begin{itemize}\end{equation}\]

However, as seen in our initial problem, if we know the interest and the principal for a given time period, we can rearrange the formula to find the rate. That's what was done to find the proportionality constant 'k', which in this case is equivalent to the interest rate when 't' is one year. This kind of understanding allows for greater flexibility in financial planning and analysis.

Mathematical Modeling in Finance

Mathematical models are invaluable tools in the world of finance. They provide a systematic and quantitative framework for representing financial scenarios to make predictions or decisions. In our textbook example, we've seen a simple yet powerful mathematical model for predicting the interest earned from an investment over a year.

The model's formula, derived from understanding directly proportional relationships and interest calculations, is clear and concise:
\[\begin{equation}I = 0.075P\begin{itemize}\end{equation}\]

This equation is a result of the mathematical modeling process which involves identifying variables, establishing relationships between them, and using given data to find constants that can be applied across a range of scenarios. When creating or using mathematical models in finance, it's essential to understand their limitations and assumptions, such as the constancy of the interest rate in this case.

In real-world finance, models can become much more complex, taking into account multiple factors and variable rates. Nonetheless, the basic principles behind them remain the same as those illustrated in this simple interest exercise.