Problem 39
Question
SIMPLE INTEREST The simple interest on an investment is directly proportional to the amount of the investment. By investing \(\$3250\) in a certain bond issue, you obtained an interest payment of \(\$113.75\) after 1 year. Find a mathematical model that gives the interest \(I\) for this bond issue after 1 year in terms of the amount invested \(P\).
Step-by-Step Solution
Verified Answer
The mathematical model that gives the interest \(I\) for this bond issue after 1 year in terms of the amount invested \(P\) is \(I = 0.035P\).
1Step 1: Understand the problem
The problem gives us two paired variables, the interest \(I\) and the amount invested \(P\). It also states that \(I\) and \(P\) are directly proportional. The formula for directly proportional relationships is \(I = kP\), where \(k\) is the proportionality constant.
2Step 2: Finding the constant of proportionality
We use the one data pair provided in the question to find \(k\). By substituting \(I = \$113.75\) and \(P = \$3250\) into \(I = kP\), we can solve for \(k\). We get \(k = \frac{I}{P} = \frac{\$113.75}{\$3250} = 0.035\).
3Step 3: Formulating the mathematical model
Finally, place the proportionality constant \(k\) back into the proportionality formula. The mathematical model that gives the interest \(I\) for this bond issue after 1 year in terms of the amount invested \(P\) is \(I = 0.035P\).
Key Concepts
Directly Proportional RelationshipsProportionality ConstantMathematical ModelsInvestment Math
Directly Proportional Relationships
Understanding directly proportional relationships is crucial when dealing with concepts like simple interest. Direct proportionality means that as one variable increases, the other variable increases at a constant rate. In the context of simple interest, the interest earned (\(I\)) is directly proportional to the amount of money invested (\(P\)).
One key feature of directly proportional relationships is their linear nature. This relationship can be expressed as \(I = kP\), where \(k\) is the constant of proportionality. It shows that these two variables grow together, maintaining a consistent ratio. If you double the amount invested, the interest will also double, given that the time and interest rate remain unchanged.
Grasping this fundamental relationship helps to better understand how small changes in investments can significantly impact the interest earned.
One key feature of directly proportional relationships is their linear nature. This relationship can be expressed as \(I = kP\), where \(k\) is the constant of proportionality. It shows that these two variables grow together, maintaining a consistent ratio. If you double the amount invested, the interest will also double, given that the time and interest rate remain unchanged.
Grasping this fundamental relationship helps to better understand how small changes in investments can significantly impact the interest earned.
Proportionality Constant
The proportionality constant \(k\) is a vital component in the equation of directly proportional relationships. This constant determines how much interest is earned per unit of currency invested. In investment math, \(k\) essentially represents the interest rate expressed as a decimal.
To compute \(k\), you need the known values of interest and the investment amount. For instance, in the exercise problem, we had an interest \(I = \\(113.75\) and an investment \(P = \\)3250\). By solving the equation \(I = kP\) for \(k\), you divide the interest by the principal: \(k = \frac{113.75}{3250} = 0.035\).
This tells us that for each dollar invested in the bond issue, 36.5 cents are earned in interest per year. Understanding the proportionality constant helps in predicting returns on different investment sums, enhancing the decision-making process in financial planning.
To compute \(k\), you need the known values of interest and the investment amount. For instance, in the exercise problem, we had an interest \(I = \\(113.75\) and an investment \(P = \\)3250\). By solving the equation \(I = kP\) for \(k\), you divide the interest by the principal: \(k = \frac{113.75}{3250} = 0.035\).
This tells us that for each dollar invested in the bond issue, 36.5 cents are earned in interest per year. Understanding the proportionality constant helps in predicting returns on different investment sums, enhancing the decision-making process in financial planning.
Mathematical Models
Mathematical models are simplified representations of real-world phenomena using mathematical language. They are extensively used in fields like investment math to predict outcomes and analyze scenarios.
In the case of simple interest, the mathematical model represents the relationship between the interest earned and the amount invested. The simple interest model we derived, \(I = 0.035P\), can be very helpful for investors.
In the case of simple interest, the mathematical model represents the relationship between the interest earned and the amount invested. The simple interest model we derived, \(I = 0.035P\), can be very helpful for investors.
- The model predicts how much interest will be generated for any given amount invested.
- It provides a simple way to understand how bond issues perform under consistent rates.
- This model is linear, suggesting a stable environment for calculating future interest gains.
Investment Math
Investment math involves using mathematical principles to make informed decisions about financial investments. It includes understanding concepts such as simple interest, proportions, and growth predictions. These principles allow investors to estimate returns and evaluate the risks and benefits of various investment options.
In our exercise, computing the simple interest using direct proportionality principles serves as an introduction to investment math. Knowing the constants and applying them accurately provides a foundational skill useful for real-life financial decision-making.
Whether calculating simple interest or more complex financial derivatives, a clear comprehension of investment math enables effective portfolio management, helping investors optimize returns on their investments. It’s fundamental for anyone looking to understand finance, whether for personal savings plans or large-scale investment strategies.
In our exercise, computing the simple interest using direct proportionality principles serves as an introduction to investment math. Knowing the constants and applying them accurately provides a foundational skill useful for real-life financial decision-making.
Whether calculating simple interest or more complex financial derivatives, a clear comprehension of investment math enables effective portfolio management, helping investors optimize returns on their investments. It’s fundamental for anyone looking to understand finance, whether for personal savings plans or large-scale investment strategies.
Other exercises in this chapter
Problem 38
In Exercises 33-40, use the algebraic tests to check for symmetry with respect to both axes and the origin. \( y = \frac{1}{x^2+1} \)
View solution Problem 38
In Exercises 27-38, find the distance between the points. \( (9.5, -2.6) \), \( (-3.9, 8.2) \)
View solution Problem 39
In Exercises 37-40, find (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(g \circ g\). \(f(x) = \sqrt[3]{x-1}\), \(g(x) = x^3 + 1\)
View solution Problem 39
In Exercises 25-54, \(g\) is related to one of the parent functions described in Section 1.6. (a) Identify the parent function \(f\). (b) Describe the sequence
View solution