Problem 116 - Elementary and Intermediate Algebra

Question

For 2010 , the net income of Rite Aid Corporation (the drugstore chain) was about \(-\$ 508\) million. The previous year, the company's net income was even worse, by a factor of about \(5.75 .\) What signed number represents Rite Aid's net income in \(2009 ?\)

Step-by-Step Solution

Answer

The net income for Rite Aid in 2009 was \(-2919\) million dollars.

Step 1: Understand the Problem

We need to find the net income for Rite Aid Corporation in 2009. We are given that in 2010, the net income was $508 million, and the net income in 2009 was worse by a factor of 5.75.

Step 2: Set Up the Mathematical Expression

We know that the 2009 net income is 5.75 times worse than the 2010 net income. If we denote the 2009 income as \( x \), then we can write the equation: \[ x = -5.75 imes (-508) \].

Step 3: Solve the Equation

Substitute the net income value for 2010 into the equation:\[ x = 5.75 imes 508 \].Calculate the product to find the income for 2009:\[ x = 5.75 imes 508 = 2919 .\]

Step 4: Determine the Sign of the Result

Since the problem states the net income was worse by 5.75 (meaning more negative), the sign of the expression remains negative. Therefore, the net income for 2009 is \[ -2919 .\]

Step 5: Interpret the Result

The net income for Rite Aid Corporation in 2009 was a loss, which corresponds to the negative value we calculated. Thus, the signed number that represents Rite Aid's net income in 2009 is \[ -2919 ext{ million dollars}.\]

Key Concepts

Algebraic Expressions

In the world of mathematics, algebraic expressions are essential as they help simplify, model, and analyze real-world situations. An algebraic expression is a combination of numbers, variables, and at least one arithmetic operation. In our exercise about Rite Aid's net income, the expression helps us understand the change in revenue between 2009 and 2010.
A crucial aspect of this concept is setting up the correct expression to solve the problem. In our case, we started with the information that the net income in 2009 was worse by a factor of 5.75 compared to 2010. Using the variable \( x \,\), the expression \[ x = 5.75 \times 508 \] was formed to calculate the net income for 2009. This equation reflects how such expressions can encapsulate relationships in a straightforward manner, allowing us to isolate unknowns, like the 2009 income, with ease.

Negative Numbers

Negative numbers are integral to understanding financial situations, particularly losses and debts. In our exercise, Rite Aid Corporation experienced net losses, represented by negative numbers in both 2009 and 2010.
When dealing with negative numbers, remember that they behave differently from positive numbers in arithmetic operations. A crucial rule is that multiplying or dividing two negative numbers results in a positive number. Conversely, the product or quotient of a positive and a negative number is negative. This concept is essential when calculating Rite Aid's income change, as we initially multiplied using positive values to simplify calculations but ultimately accounted for the negative nature of results by affixing a negative sign to denote losses.

Problem-Solving Steps

Approaching a math problem systematically ensures accuracy. Follow these simple problem-solving steps to tackle similar tasks:

• Understand the Problem: Clearly define what is given and what needs to be found. Identify the key information and the relationships between them. In this case, we started with known figures for net incomes and the factor of change.


• Set Up the Mathematical Expression: Translate the problem's word description into a mathematical equation or expression. Here, we set up the expression \[ x = 5.75 \times 508 \] to represent the 2009 income.


• Solve the Equation: Proceed to calculate the value of the expression. Be precise with arithmetic to avoid mistakes, resulting in our solution of \[ x = 2919 \].


• Determine the Sign of the Result: Ensure the resulting calculations align with the context of the problem, adjusting signs accordingly to reflect reality. Since the income was worse, the final sign was negative, giving us \[ -2919 \].


• Interpret the Result: Relate the answer back to the original scenario and confirm it makes sense. In our case, the negativity of \[ -2919 \] signifies the deeper loss in 2009, consistent with the problem statement.

By adopting these steps, you will find navigating similar mathematical problems much more manageable.