Problem 22 - Algebra A Combined Function

Question

Saralee Rose has investments in Google and Nintendo stock. During a particular day in 2009 , Google stock was at \(\$ 596\) per share, and Nintendo stock was at \(\$ 30\) per share. Saralee's portfolio made up of these two stocks was worth \(\$ 22,682\) at that time. If Saralee owns 16 more shares of Google stock than she owns of Nintendo stock, how many shares of each type of stock does she own?

Step-by-Step Solution

Answer

Saralee owns 21 shares of Nintendo stock and 37 shares of Google stock.

Step 1: Define the Variables

Let's define two variables to represent the unknowns in this problem. Let \( x \) be the number of Nintendo shares Saralee owns, and \( y \) be the number of Google shares Saralee owns.

Step 2: Set Up Equations from Problem Statements

We have two main pieces of information: 1. The total value of the stocks: \[ 30x + 596y = 22,682 \] 2. There are 16 more Google shares than Nintendo shares: \[ y = x + 16 \]

Step 3: Substitute and Simplify

Use the second equation to express \( y \) in terms of \( x \) and substitute into the first equation: \[ 30x + 596(x + 16) = 22,682 \] Expand and simplify: \[ 30x + 596x + 9536 = 22,682 \] Combine like terms: \[ 626x + 9536 = 22,682 \]

Step 4: Solve for \( x \)

Subtract 9536 from both sides to isolate terms with \( x \): \[ 626x = 13,146 \] Now, divide both sides by 626 to solve for \( x \): \[ x = \frac{13,146}{626} \] \[ x = 21 \]

Step 5: Solve for \( y \)

Use the expression for \( y \) in terms of \( x \) from Step 2: \[ y = x + 16 \] Substitute \( x = 21 \) into the equation: \[ y = 21 + 16 \] \[ y = 37 \]

Step 6: Conclusion

Therefore, Saralee owns 21 shares of Nintendo stock and 37 shares of Google stock.

Key Concepts

Linear Equations

Linear equations are mathematical expressions used to describe relationships between variables. In the context of Saralee's investment problem, linear equations help represent the value of her stocks.

A linear equation in one variable looks like this: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. However, the problem with Saralee's stocks requires a system of two linear equations.

The first equation involves the total value of the stocks:

• \( 30x + 596y = 22,682 \) represents the combined value of Nintendo and Google stocks.
• \( y = x + 16 \) describes the relationship between the number of Google and Nintendo shares.

These combine to form a system of equations we need to solve to find the number of shares Saralee owns.

Variable Substitution

Variable substitution is a method to solve systems of equations by replacing one variable with an expression involving the other variable. This technique simplifies the system, making it easier to find a solution.

In Saralee's investment problem:

• We start by expressing one variable, \( y \), in terms of another variable, \( x \), using the second equation: \( y = x + 16 \).
• This expression for \( y \) is then substituted into the first equation: \( 30x + 596(x + 16) = 22,682 \).

Substituting and simplifying allows us to solve for \( x \) first by removing \( y \) from the equation. This makes the math more straightforward and moves us closer to the solution.

Problem-Solving

Problem-solving, especially in mathematics, involves a strategic approach to find solutions. In the context of Saralee's stock problem, it involves identifying relationships, setting equations, and step-by-step simplification.

Here is the breakdown:

• Understanding: Interpret key information like stock values and the additional shares.
• Equation Setup: Formulate equations based on the relationships between the stocks.
• Solving: Utilize methods like substitution to find solutions systematically.
• Verification: Cross-check with the original problem to ensure the solution fits.

Each step relies heavily on logical thinking and understanding the components of the situation to fulfill the problem's requirements efficiently.

Investment Portfolio Analysis

Investment portfolio analysis deals with evaluating an individual's investments to understand their value and performance. It's a crucial skill for managing stocks, as seen in Saralee's situation.

Key aspects of portfolio analysis include:

• Value assessment: Calculate the total worth of stocks using their current market price.
• Diversification assessment: Check the distribution of shares across different companies, like Google and Nintendo.
• Risk and return: Evaluate the potential risk of stock fluctuations versus their expected return.

Portfolio analysis helps investors like Saralee make informed decisions and manage their investments to meet their financial goals. Analyzing stocks as simple equations makes the complex task of investment tracking more manageable.