Problem 95 - Beginning and Intermediate Algebra

Question

In \(2006,\) Gap Inc operated a total of 3054 stores worldwide. The following chart shows the store breakdown by brand. $$ \begin{array}{|l|l|} \hline \text { Brand } & {\text { Number of Stores }} \\ \hline \text { Gap(\mathrm{ Domestic) } } & {1335} \\ \hline \text { Gap (International) } & {256} \\ \hline \text { Banana Republic } & {498} \\ \hline \text { Old Navy } & {960} \\ \hline \text { Forth \& Towne } & {5} \\ \hline \text { Total } & {3054} \\ \hline \end{array} $$ What fraction of Gap-brand stores were Old Navy stores? Simplify this fraction.

Step-by-Step Solution

Answer

The fraction is \( \frac{160}{509} \).

Step 1: Identify Total Gap-brand Stores

The total Gap-brand stores include all the stores listed, which is 3054.

Step 2: Identify Old Navy Stores

From the chart, the number of Old Navy stores is given as 960.

Step 3: Calculate the Fraction

The fraction of Gap-brand stores that were Old Navy stores is given by the ratio of Old Navy stores to the total Gap-brand stores:\[ \text{Fraction} = \frac{960}{3054} \]

Step 4: Simplify the Fraction

To simplify the fraction \( \frac{960}{3054} \), we need to find the Greatest Common Divisor (GCD) of 960 and 3054:- The GCD of 960 and 3054 is 6.- Divide both the numerator and the denominator by their GCD:\[ \frac{960 \div 6}{3054 \div 6} = \frac{160}{509} \]

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and rules for manipulating those symbols. It forms the basis for nearly all other branches of mathematics and is essential for understanding and solving equations. In the context of this exercise, we use basic algebra to represent and solve the problem of finding what fraction of the Gap Inc stores were Old Navy stores. Algebra here involves understanding how to create a ratio using numeric representations of the number of stores. By placing the number of Old Navy stores as the numerator and the total number of Gap-brand stores as the denominator, we establish a fraction which is essentially an algebraic expression without unknowns. The basic tools of algebra assist in structuring this information logically and help us in simplifying the problem further.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a critical concept in mathematics for reducing fractions. It helps us find the largest number that can divide two numbers without a remainder. In our exercise, the GCD of 960 and 3054 is used to simplify the original fraction. Understanding the GCD is important for working with fractions, as simplifying them requires dividing both the numerator and the denominator by their GCD.

• To find the GCD, you can use several methods:
• Prime factorization: break down both numbers into their prime factors and multiply the common factors.
• Euclidean algorithm: repeatedly subtract the smaller number from the larger number or use division until a remainder of zero is reached; the last non-zero remainder is the GCD.

In our example, after determining that the GCD of 960 and 3054 is 6, we divide both numbers by this GCD to achieve a simpler fraction.

Simplifying Fractions

Simplifying fractions means expressing them in their simplest form. A fraction is simplified when the numerator and the denominator are only divisible by 1 and not by any other common numbers. Simplifying is important because it makes fractions easier to understand and compare.

• The steps to simplify a fraction are:
• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

For example, with the fraction \( \frac{960}{3054} \), we use our previously determined GCD of 6. Dividing the numerator and denominator by 6, we get the simplified fraction \( \frac{160}{509} \). This process not only makes the fraction simpler but also ensures we are dealing with the most reduced form possible, making further arithmetic operations easier.