Problem 38 - Precalculus: Mathematical for Calculus

Question

An investor owns three stocks: \(A, B\) and C. The closing prices of the stocks on three successive trading days are given in the table. Despite the volatility in the stock prices, the total value of the investor's stocks remained unchanged at \(\$ 74,000\) at the end of each of these three days. How many shares of each stock does the investor own? $$\begin{array}{|l|c|c|c|} \hline & \text { Stock } A & \text { Stock B } & \text { Stock } C \\ \hline \text { Monday } & \$ 10 & \$ 25 & \$ 29 \\ \text { Tuesday } & \$ 12 & \$ 20 & \$ 32 \\ \text { Wednesday } & \$ 16 & \$ 15 & \$ 32 \end{array}$$

Step-by-Step Solution

Answer

1500 shares of A, 1200 shares of B, 1000 shares of C.

Step 1: Define Variables

Let \(x\), \(y\), and \(z\) represent the number of shares owned for Stocks A, B, and C respectively. Our objective is to formulate equations using these variables to eventually find their values.

Step 2: Create Equations for Each Day

Using the stock prices given for Monday, Tuesday, and Wednesday, we create the following equations based on the total stock value being $74,000 each day:\[10x + 25y + 29z = 74000\] (Monday) \[12x + 20y + 32z = 74000\] (Tuesday) \[16x + 15y + 32z = 74000\] (Wednesday)

Step 3: Subtract Equations to Simplify

Subtract the first equation from the second to eliminate terms and find a new equation:\((12x + 20y + 32z) - (10x + 25y + 29z) = 0\).This simplifies to:\[2x - 5y + 3z = 0\]Now, subtract the second equation from the third:\((16x + 15y + 32z) - (12x + 20y + 32z) = 0\).This simplifies to:\[4x - 5y = 0\]

Step 4: Solve the System of Equations

From the equation \[4x - 5y = 0\], solve for \(y\):\[ y = \frac{4}{5}x \]Substitute \(y = \frac{4}{5}x\) into the equation \[2x - 5y + 3z = 0\]:\[ 2x - 5\left(\frac{4}{5}x\right) + 3z = 0\] simplifies to:\[ 2x - 4x + 3z = 0\]\[ -2x + 3z = 0\]\[ 3z = 2x \]\[ z = \frac{2}{3}x \]

Step 5: Substitute Back to Find Shares

Substitute \(y = \frac{4}{5}x\) and \(z = \frac{2}{3}x\) into one of the original equations, say the first one, to solve for \(x\):\[10x + 25\left(\frac{4}{5}x\right) + 29\left(\frac{2}{3}x\right) = 74000\]Simplify and solve for \(x\):\[10x + 20x + 19.33x = 74000\]\[49.33x = 74000\]\[x \approx 1500\]

Step 6: Calculate y and z

Using \(x = 1500\), calculate \(y\) and \(z\):\[y = \frac{4}{5} \times 1500 = 1200\]\[z = \frac{2}{3} \times 1500 = 1000\]

Step 7: Final Step: Write Final Solution

The number of shares owned for Stock A is 1500, for Stock B is 1200, and for Stock C is 1000.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. When we have multiple relationships that need to be satisfied simultaneously, we turn to systems of equations for solutions. In this exercise, the investor's stock portfolio provides us with three key equations—one for each day—based on the total value of their stocks being $74,000.
These equations link the number of shares and the stock prices:

• Monday: \(10x + 25y + 29z = 74000\)
• Tuesday: \(12x + 20y + 32z = 74000\)
• Wednesday: \(16x + 15y + 32z = 74000\)

While setting these equations looks straightforward, solving them requires strategic elimination and substitution of variables. By interpreting and solving these, we align with the financial condition provided, aiming to break down the simultaneous impact of stock prices on each day.
This requires rearranging terms to understand how one equation influences another, leading to a practical resolution of the investor's situation.

Matrix Method

The matrix method is a streamlined approach for handling systems of equations. It involves representing these systems numerically, allowing us to exploit mathematical efficiencies and computational tools. To use the matrix method here, we express the system of equations in terms of matrices.
First, place the coefficients of variables into a matrix, known as the coefficient matrix, alongside a column matrix for the variables (shares of stocks in this case). Thus, the equations:

• Matrix \[\begin{bmatrix}10 & 25 & 29 \12 & 20 & 32 \16 & 15 & 32\end{bmatrix}\]
• Variable matrix \[\begin{bmatrix} x \ y \ z \end{bmatrix}\]
• Result matrix \[\begin{bmatrix}74000 \74000 \74000\end{bmatrix}\]

This transforms the original problem into the matrix equation:\[A \mathbf{x} = \mathbf{b}\]where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the variable matrix, and \(\mathbf{b}\) is the results matrix.
The goal here is to find \(\mathbf{x}\) by operations like row reduction or applying the inverse, though in practical scenarios, techniques like these facilitate the use of computer software for solutions.

Problem Solving in Finance

In the realm of finance, problem-solving often involves analyzing data and making decisions amidst fluctuations. This exercise highlights real-world applications of mathematics in finance through calculating shares amidst varying stock prices. The objective is to deduce precise financial conditions, such as stock holdings, given constraints on value across several days.
Financial problem-solving requires both analytical skills to process numerical data and a strategic mindset to foresee market movements. The input—stock prices over three days—acts as the volatile factor, while our fixed output is the total portfolio value. By leveraging math and logic, one deciphers actionable insights into an investor’s strategy or situation.
Having a sound grasp of systems like these, particularly involving matrices and algebra, enhances our capability to make logical inferences about investments. Solutions potentially guide decision-making processes, resource allocation, or portfolio adjustments, enhancing both personal and professional financial health.