Problem 17 - Intermediate Algebra

Question

In \(2009,\) there was a combined total of \(4,046\) Gap and Aéropostale clothing stores worldwide. The number of Gap stores was \(3 \frac{1}{4}\) times more than the number of Aéropostale stores. How many Gap stores and how many Aéropostale stores were there that year'? (Source: wikinvest.com)

Step-by-Step Solution

Answer

There were 3094 Gap stores and 952 Aéropostale stores in 2009.

Step 1: Define Variables

Let's denote the number of Aéropostale stores as \( x \) and the number of Gap stores as \( y \). We need to find the values of \( x \) and \( y \).

Step 2: Formulate Equations

We know the combined number of Gap and Aéropostale stores is 4046. This gives us the equation: \[ x + y = 4046 \]Additionally, the problem states the number of Gap stores was \(3\frac{1}{4}\) times more than the number of Aéropostale stores. This can be translated to an equation as: \[ y = \left(3\frac{1}{4}\right) x = \frac{13}{4}x \]

Step 3: Solve the Equations

Substitute \( y = \frac{13}{4}x \) into the first equation: \[ x + \frac{13}{4}x = 4046 \]Combine like terms: \[ \left(1 + \frac{13}{4}\right)x = 4046 \]\[ \frac{17}{4}x = 4046 \] To find \( x \), multiply both sides by \( \frac{4}{17} \):\[ x = 4046 \times \frac{4}{17} \]

Step 4: Calculate Number of Aéropostale Stores

Perform the multiplication to solve for \( x \):\[ x = 4046 \times \frac{4}{17} = 952 \]Thus, there were 952 Aéropostale stores.

Step 5: Calculate Number of Gap Stores

Using the value of \( x \), substitute back into the equation for \( y \):\[ y = \frac{13}{4} \times 952 = 3094 \]So, there were 3094 Gap stores.

Key Concepts

Variables in Algebra

In algebra, variables are letters or symbols that stand in for unknown values we want to find. In this exercise, we have two variables: \( x \) and \( y \). These variables represent the number of Aéropostale stores and Gap stores, respectively. Variables are crucial in algebra because they allow us to create equations that model real-world situations. For instance:

• Let \( x \) be the number of Aéropostale stores.
• Let \( y \) be the number of Gap stores.

By assigning these variables, we can express the relationships described in the problem using algebraic equations. This allows us to solve for the unknowns and understand the situation better.

Solving Equations

Solving equations involves finding the value of unknown variables that make the equation true. In this exercise, two equations are formed based on the problem description:

• The total number of stores is given by \( x + y = 4046 \).
• The number of Gap stores is \(3\frac{1}{4}\) times the number of Aéropostale stores: \( y = \frac{13}{4}x \).

To solve these equations, substitute the expression for \( y \) from the second equation into the first equation:
\[ x + \frac{13}{4}x = 4046 \]
This combined equation helps to eliminate one of the variables, making it easier to solve for the other. By simplifying it, we find the value of \( x \), which tells us the number of Aéropostale stores. Then, we use this value to find \( y \), the number of Gap stores, completing the solution.

Fraction Multiplication

Fraction multiplication plays a vital role in this problem. Understanding how to handle fractions is essential in algebra. Here, we multiply a variable by a fraction.
For the Gap stores, we have the equation \( y = \frac{13}{4}x \). This equation means that the number of Gap stores is \( \frac{13}{4} \) times the number of Aéropostale stores. To work with this:

• We multiply the variable \( x \) by the fraction \( \frac{13}{4} \).
• When isolating \( x \), use the reciprocal of \( \frac{17}{4} \) to solve \( \frac{17}{4}x = 4046 \) by multiplying both sides by \( \frac{4}{17} \).

These steps show how to effectively use fractions to transform equations and solve for unknowns. Understanding fraction multiplication allows us to translate complex statements into solvable equations efficiently.