Problem 31 - Algebra 1: Concepts and Skills

Question

One share of ABC stock is worth three times as much as XYZ stock. An investor has 100 shares of each. If the total value of the stocks is \(\$4500\), how much money is invested in each stock?

Step-by-Step Solution

Answer

The investor has invested $1125 in XYZ stock and $3375 in ABC stock.

Step 1: Define the Variables

Let's denote the value of each share of XYZ stock as \(x\) and the value of each share of ABC stock as \(3x\) (since ABC share is worth three times as much as XYZ share). Now we can set up equations based on the information given.

Step 2: Formulate the Equation

We know that the total value of the stocks is $4500, which means 100 shares of XYZ and 100 shares of ABC together sum up to $4500. Hence, the equation would be \(100x + 100*3x = 4500\). Simplifying this gives us \(400x = 4500\).

Step 3: Solve the Equation

The next step is to solve the equation for \(x\). In order to do that, divide 4500 by 400, which simplifies to \(x = 11.25\). This indicates that each share of XYZ stock is worth $11.25.

Step 4: Find the Value of ABC Stock

It was stated that each ABC share is worth three times as much as XYZ. Therefore ABC = 3 * 11.25 = $33.75. Therefore, each ABC share is worth $33.75.

Step 5: Calculate the Total Investment in Each Stock

Given that the investor has 100 shares of each stock, the total investment in XYZ = 100 * 11.25 = $1125. And the total investment in ABC = 100 * 33.75 = $3375.

Key Concepts

Variable Definition

Understanding the role of variables is a foundational step in solving algebraic equations. In this exercise, the problem revolves around two stocks, ABC and XYZ. To find their individual values, we designate a variable for the unknown quantity. For instance:

• Let \( x \) represent the value of one share of XYZ stock.
• Since ABC is worth three times more, a share of ABC stock is represented as \( 3x \).

By defining these variables, we provide a clear representation of the relationships between the different values in the problem. This method allows us to convert a word problem into a mathematical one, paving the way for further steps in solving the equation.

Equation Formulation

Once the variables are defined, the next step is to formulate an equation. We know from the problem that the combined value of the stocks is \( \$4500 \). With 100 shares each, we compute:

• 100 shares of XYZ: \( 100x \)
• 100 shares of ABC: \( 100 \times 3x = 300x \)

Thus, the equation reflecting the total value is \( 100x + 300x = 4500 \). The primary goal in this step is to establish a clear mathematical relationship that represents the stated problem, which funnels us toward solving for \( x \). By simplifying, we get \( 400x = 4500 \).

Equation Solving

Solving the equation involves isolating the variable. Here, our simplified equation is \( 400x = 4500 \). To find \( x \), we need to divide both sides by 400:\[x = \frac{4500}{400} = 11.25\]This solution tells us that each share of XYZ stock is valued at \( \$11.25 \). At this point, we have used basic algebraic techniques to find the value of the shared variable, allowing us to determine the price of each category of stock based on the given condition.

Investment Calculation

With the value of \( x \) found, the next step is to compute the total investment in each type of stock. We previously established the price of each stock:

• XYZ stock: \( \\(11.25 \) per share
• ABC stock: \( 3 \times 11.25 = \\)33.75 \) per share

To find the total money invested:

• Investment in XYZ: \( 100 \times 11.25 = \\(1125 \)
• Investment in ABC: \( 100 \times 33.75 = \\)3375 \)

Summing these investments confirms our original total, ensuring that the calculations align with the problem's parameters. By following these structured steps, we've calculated the individual contributions to a portfolio's total value.