Problem 45 - Precalculus Mathematics for Calculus

Question

Stock Portfolio 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. $$\begin{array}{|l|c|c|c|} \hline & \text { Stock A } & \text { Stock B } & \text { Stock C } \\ \hline \text { Monday } & \$ 10 & \$ 25 & \$ 29 \\ \text { Tuesday } & \$ 12 & \$ 20 & \$ 32 \\ \text { Wednesday } & \$ 16 & \$ 15 & \$ 32 \\ \hline \end{array}$$ 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?

Step-by-Step Solution

Answer

1,500 shares of A, 1,200 shares of B, 1,000 shares of C.

Step 1: Set Up Equations for Daily Totals

For each day, the total value of stocks is given to be \( \$74,000 \). We need to form equations for this total value in terms of the number of shares of each stock.Let \( x \) be the number of shares of Stock A, \( y \) be the number of shares of Stock B, and \( z \) be the number of shares of Stock C.On Monday, the equation for the total value is:\[ 10x + 25y + 29z = 74,000 \]

Step 2: Formulate Tuesday's Equation

Using the same logic as Step 1, write the equation for Tuesday:Given Tuesday's prices, we have:\[ 12x + 20y + 32z = 74,000 \]

Step 3: Create Wednesday's Equation

Following the same pattern, write the equation for Wednesday:Using Wednesday's prices, the equation is:\[ 16x + 15y + 32z = 74,000 \]

Step 4: Solve the System of Equations

Now that we have three equations: 1. \( 10x + 25y + 29z = 74,000 \)2. \( 12x + 20y + 32z = 74,000 \)3. \( 16x + 15y + 32z = 74,000 \)We will solve them step-by-step. To eliminate one variable, subtract the second equation from the first:\[ (10x + 25y + 29z) - (12x + 20y + 32z) = 0 \]Which simplifies to:\[ -2x + 5y - 3z = 0 \] Next, subtract the third equation from the second:\[ (12x + 20y + 32z) - (16x + 15y + 32z) = 0 \]Which simplifies to:\[ -4x + 5y = 0 \] From \( -4x + 5y = 0 \), we can express \( y \) in terms of \( x \):\[ 5y = 4x \]\[ y = \frac{4}{5}x \]

Step 5: Solve for the Number of Shares

Substitute \( y = \frac{4}{5}x \) back into \( -2x + 5y - 3z = 0 \):\[ -2x + 5\left(\frac{4}{5}x\right) - 3z = 0 \]Simplifying gives:\[ -2x + 4x - 3z = 0 \]\[ 2x = 3z \]\[ z = \frac{2}{3}x \]Substitute both \( y = \frac{4}{5}x \) and \( z = \frac{2}{3}x \) into one of the original equations (e.g., \( 10x + 25y + 29z = 74,000 \)):\[ 10x + 25\left(\frac{4}{5}x\right) + 29\left(\frac{2}{3}x\right) = 74,000 \]\[ 10x + 20x + \frac{58}{3}x = 74,000 \]\[ 30x + \frac{58}{3}x = 74,000 \]Multiply through by 3 to clear the fraction:\[ 90x + 58x = 222,000 \]\[ 148x = 222,000 \]\[ x = 1,500 \]Using this value of \( x \), find \( y \) and \( z \):\[ y = \frac{4}{5} \times 1,500 = 1,200 \]\[ z = \frac{2}{3} \times 1,500 = 1,000 \]

Step 6: Conclusion

The investor owns 1,500 shares of Stock A, 1,200 shares of Stock B, and 1,000 shares of Stock C.

Key Concepts

Linear Algebra

Linear Algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations. It provides tools for modeling and solving problems related to multi-dimensional spaces, which is exactly what we encounter in systems of linear equations.
At the core, a system of linear equations consists of multiple equations that share the same variables. In the context of this exercise, we have a system of three equations, each corresponding to a different trading day, and they together form a linear system.

• The equations are linear because they involve variables that aren't raised to any power other than one.
• By solving the system, we aim to find the values of the variables (in this case, shares of stocks) that satisfy all equations simultaneously.
• Methods often used in Linear Algebra for solving such systems include substitution, elimination, and matrix operations.

In this problem, we employ the elimination method to reduce the system step by step until we solve for each individual variable.

Stock Portfolio

A stock portfolio is a collection of stocks owned by an investor or individual. It represents the total value of the stocks, which can fluctuate daily based on market prices.
Understanding a stock portfolio allows investors to make informed decisions to maximize returns while managing risks.

• Stock prices change daily based on market demand and supply.
• A balanced portfolio may include stocks of various sectors to spread risk.
• Investors aim to maintain stability, as seen in this exercise where the total portfolio value remains constant despite daily price changes.

The problem demonstrates a real-world application of Linear Algebra—analyzing how an investor’s portfolio holds its total value constant over three days, while different stocks have varying prices, illustrating the importance of diversification and strategy in stock investment.

Problem Solving in Mathematics

Problem solving in mathematics involves a structured approach to find solutions to complex problems. This exercise exemplifies a classic mathematical problem-solving technique where logical reasoning and mathematical calculations combine to find the answer. To solve the given exercise:

• We began by translating the problem into mathematical equations. Each day's stock prices created one equation, giving us a system of three linear equations.
• We used the elimination method to simplify the system. This involved removing variables step by step, reducing the complexity of the equations one by one.
• Using logical steps, we derived relationships between the variables (number of shares of each stock).
• Once relationships were clear, back-substitution allowed us to find the exact number of shares for each stock.

Structured problem solving like this is crucial in mathematics, developing not only analytical skills but also enhancing one's ability to tackle real-world problems with a methodical approach.