Problem 21 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. the ratio of the number of games won and games played

Step-by-Step Solution

Answer

The algebraic expression is \( \frac{w}{p} \).

Step 1: Understand the Phrase

The phrase is 'the ratio of the number of games won and games played'. We need to translate this into an algebraic expression.

Step 2: Identify Variables

Choose variables to represent the quantities in the phrase. Let \( w \) represent the number of games won, and \( p \) represent the number of games played.

Step 3: Express the Ratio

A ratio is represented as a division between two quantities. Thus, the ratio of the number of games won to the number of games played is expressed as \( \frac{w}{p} \).

Key Concepts

Ratios in Algebra

In everyday language, a ratio is used to compare two quantities. It tells us how much of one thing there is relative to another. In algebra, ratios become more powerful because they can involve variables, allowing us to explore a wide range of mathematical relationships and problems.

To translate a phrase involving a ratio into an algebraic expression, follow these key steps:

• Identify the two quantities being compared.
• Choose appropriate variables to represent these quantities (for example, let \( x \) be the first quantity and \( y \) be the second).
• Write the ratio as a fraction. For our variables, this would be \( \frac{x}{y} \).

Ratios are crucial in solving problems because they help express relationships in a clear and concise way. You might encounter ratios when dealing with situations like comparing speeds, prices, or—as in the given exercise—the number of games won to the number of games played.

Variables in Algebra

Variables are the letters that you see in algebraic expressions and equations. They act like placeholders, representing numbers we don’t know yet, or that could change depending on the context.

Variables make algebra incredibly flexible and powerful:

• They allow us to form general expressions that apply to many different situations by simply substituting different numbers for the variables.
• By representing unknown or varying quantities with variables, we can create equations and inequalities to find solutions or describe relationships.
• They facilitate communication of mathematical ideas in a universal way, allowing for the sharing of formulas and solutions worldwide.

In the original exercise, \( w \) and \( p \) are the variables used to represent the number of games won and the number of games played, respectively. Choosing the right variables is important—they should make sense in the context of the problem and make calculations straightforward.

Translating Phrases to Expressions

Turning words into algebraic expressions is a key skill in understanding and solving word problems. This process involves analyzing the language used and converting it into mathematical symbols. Here’s a simple process you can follow:

• Read the phrase carefully to understand what quantities and operations are involved.
• Select variables to represent the quantities mentioned. This could be anything from total cost, distance, time, or—as in our exercise—the number of games won or played.
• Use mathematical operations (like addition, subtraction, multiplication, or division) to link these variables, based on the language of the phrase. Words like "sum," "difference," "product," and "ratio" can give clues about which operations to use.

Understanding how to translate a phrase like "the ratio of games won to games played" into \( \frac{w}{p} \) requires both an understanding of ratios and the ability to choose and use variables effectively. Practice makes perfect, and the more you translate these phrases, the more intuitive it will become.