Problem 43 - Elementary Algebra

Question

Applications involving variation. The price of a share of common stock in a company is directly proportional to the earnings per share (EPS) of the previous 12 months. If the price of a share of common stock in a company is \(\$ 22.55\) and the EPS is published to be \(\$ 1.10,\) then determine the value of the stock if the EPS increases by \(\$ 0.20\).

Step-by-Step Solution

Answer

The new stock price is $26.65.

Step 1: Understand the direct variation formula

Direct variation means that the price of the stock, denoted by \( P \), is directly proportional to the earnings per share (EPS), denoted by \( E \). This can be described by the equation \( P = kE \), where \( k \) is the constant of proportionality.

Step 2: Determine the constant of proportionality

We know that when \( P = \\(22.55 \), \( E = \\)1.10 \). Substitute these values into the direct variation formula to find \( k \): \( 22.55 = k \times 1.10 \). Solving for \( k \), we divide both sides by \( 1.10 \): \( k = \frac{22.55}{1.10} = 20.5 \).

Step 3: Calculate the new EPS

The new EPS after an increase of \( \\(0.20 \) from the old EPS of \( \\)1.10 \) is \( 1.10 + 0.20 = \$1.30 \).

Step 4: Determine the new stock price

Use the constant \( k = 20.5 \) and the new EPS \( E = \\(1.30 \) in the formula \( P = kE \) to find the new stock price: \( P = 20.5 \times 1.30 = \\)26.65 \).

Key Concepts

Direct Proportionality

In algebra, direct proportionality is a relationship where one quantity increases or decreases consistently with another. If one entity, say the stock price (\(P\)), is directly proportional to another, such as earnings per share (EPS, \(E\)), the relationship can be expressed mathematically. The formula used is \(P = kE\).
In this equation, \(k\) represents the constant of proportionality, which ensures the relationship remains constant as \(E\) changes.
Direct proportionality is straightforward: if the EPS doubles, the stock price doubles, assuming \(k\) remains the same.

• This creates a clear, predictable connection between stock price and EPS.
• Understanding direct proportionality assists in predicting financial outcomes based on measurable changes in financial metrics.

Constant of Proportionality

The constant of proportionality, represented as \(k\), is critical in expressing direct relationships in algebraic variation. In the equation \(P = kE\), \(k\) quantifies how many units the stock price \(P\) changes for every single unit change in the EPS \(E\).
Calculating \(k\) involves taking known values of \(P\) and \(E\) and rearranging the direct variation formula to solve for \(k\): \(k = \frac{P}{E}\).
This constant is significant for several reasons:

• It allows you to predict future stock prices when there are changes in EPS.
• Once determined, all subsequent calculations regarding stock price become easier and reliable.

In our exercise, \(k\) was determined to be 20.5, meaning every dollar increase in EPS will increase the stock price by \(20.5\). Understanding and calculating \(k\) is an essential competency when dealing with stocks and similar financial calculations.

Stock Price Calculation

Calculating the stock price based on EPS is a practical application of algebraic variation. Given the formula for direct proportionality \(P = kE\), once the constant of proportionality \(k\) is known, determining the stock price becomes a straightforward task.
Following the correct steps, you:

• Substitute the current EPS into the formula.
• Multiply it by the constant \(k\) to find the new price.

For instance, with an EPS increase from \\(1.10 to \\)1.30, and \(k = 20.5\), the formula \(P = 20.5 \times 1.30\) yields a new stock price of \$26.65.
Stock price calculations using direct variation are valuable for financial forecasting and decision-making. Investors can gauge potential stock value changes with EPS variations, aiding strategic planning and investment insights.