Problem 72 - College Algebra and Calculus: An Applied Approach

Question

Stock Mix You invest \(\$ 4000\) in two stocks. In one year, the value of stock A increases by \(5.4 \%\) and the value of stock B increases by \(12.8 \%\). The total value of the stocks is now \(\$ 4401\). How much did you originally invest in each stock?

Step-by-Step Solution

Answer

The original investment in stock A was approximately \$2147.06 and in stock B was approximately \$1852.94.

Step 1: Setting Up the Equations

Let's denote the amount originally invested in stock A as \(x\) and in stock B as \(y\). We can set up the following equations: 1) \(x + y = \$4000\) because the total original investment is \$40002) \(x \cdot (1 + 0.054) + y \cdot (1 + 0.128) = \$4401\), because the value of each investment increases by their respective percentage rates (5.4% for A and 12.8% for B) and summing to the total increased value.

Step 2: Solve the System of Equations

A little bit of algebra is needed in this step. First, we could simplify the second equation:2.1) \(x \cdot 1.054 + y \cdot 1.128 = \$4401\), we multiply out the right hand sideAfterwards, we can multiply the first equation by 1.054, which should give:1.1) \(x \cdot 1.054 + y \cdot 1.054 = \$4216\)Finally, subtracting equation 1.1 from equation 2.1, the new equation will only have \(y\) in it: \(y \cdot (1.128 - 1.054) = \$4401 - \$4216\)

Step 3: Calculate the Investment for Stock B

Solving the equation from step 2 for \(y\) gives: \(y = \frac{\$4401 - \$4216}{1.128 - 1.054}\), or \(y \approx \$1852.94\) which is the initial investment for stock B.

Step 4: Calculate the Investment for Stock A

Substitute \(y\) back into the first equation: \(x = \$4000 - y\), or \(x \approx \$2147.06\), which is the initial investment for stock A.

Key Concepts

System Of Equations

Understanding how to set up and solve a system of equations is crucial for investment problems like this one. You have two equations in this problem that describe the relationship between your investments in stocks A and B.

• The first equation is based on the total amount initially invested: \( x + y = \\(4000 \). This tells us that the sum of the initial amounts invested in stock A and stock B is \)4000.
• The second equation considers the value increase: \( x \cdot 1.054 + y \cdot 1.128 = \$4401 \). This equation incorporates the percentage increase for each stock and sums their final values after these increases.

Understanding how these equations are derived is key. The variables \(x\) and \(y\) represent the unknown original investments in stocks A and B respectively. By solving these equations simultaneously, we can determine the values of \(x\) and \(y\). This involves using methods like substitution or elimination to find the solution to the system, which reveals the initial investments.

Algebra

Algebra helps us work with the equations we derive in investment scenarios. In this exercise, algebra is used to simplify and manipulate equations to find unknown values.
By using algebraic techniques, we started with two complex equations and broke them down step by step. Initially, we simplified the equation related to the increase in value:

• Expanding the equation: \( x \cdot 1.054 + y \cdot 1.128 = \\(4401 \).
• Manipulating one of the equations by multiplying both sides, \( x \cdot 1.054 + y \cdot 1.054 = \\)4216 \), facilitates the elimination of one variable.
• The goal is to subtract the second equation from the first to isolate a single variable.

This approach reduces the complexity and number of variables, allowing us to solve for one variable at a time. Mastering algebra for investment problems aids in understanding how different elements of an investment interact and affect each other.

Percentage Increase

Percentage increase is a common concept in finance that describes how much a value has risen relative to its original amount. In this problem, we deal with both stocks appreciating by different percentages.

• Stock A increased by 5.4%, so its increased value is represented as \( x \cdot (1 + 0.054) \) or \( x \cdot 1.054 \).
• Similarly, stock B's value increased by 12.8%, represented as \( y \cdot (1 + 0.128) \) or \( y \cdot 1.128 \).

The percentage increases are crucial because they help calculate the new values and thus set the second equation. Understanding percentage increase ensures accurate calculations, as it directly affects the total value and how much each component contributes to it. Knowing how to convert percentages into decimals and apply them to original values is key to solving these problems efficiently.