Problem 31 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. triple the number of waiters

Step-by-Step Solution

Answer

The algebraic expression is \( 3w \).

Step 1: Identify Key Elements

The phrase 'triple the number of waiters' involves multiplying the number of waiters by 3. Identify the key operation in the phrase, which is 'triple', meaning to multiply by 3.

Step 2: Choose a Variable

Choose a variable to represent the number of waiters. Common choices include letters such as \( w \), \( x \), or \( n \). For this solution, we will use \( w \).

Step 3: Formulate the Algebraic Expression

Write the algebraic expression by multiplying the chosen variable by 3, since 'triple' indicates this operation. Therefore, the expression is \( 3w \).

Key Concepts

Multiplication in Algebra

Multiplication in algebra is just like multiplication in arithmetic, but it involves letters (variables) as well as numbers. When you multiply in algebra, you use the same rules as in arithmetic, except now you're often working with variables. Multiplication is a way to combine equal groups, and in algebra, it can be represented by placing the number and the variable next to each other. For example, if you're asked to "triple" something, you multiply it by 3. In algebra, you would write this as \( 3n \) if \( n \) is your variable.

• Example without variables: Triple 2 becomes \( 3 \times 2 = 6 \)
• Example with variables: Triple \( n \) becomes \( 3n \)

Multiplication between a number and a variable is as straightforward as writing them together. So, \( 4 \times n \) is written as \( 4n \). You can think of it as four groups of \( n \). Likewise, when you see \( 3w \), it means there are three groups of \( w \). This operation is a fundamental building block in algebra that helps us express and solve many real-world problems.

Variables in Algebra

Variables are an essential part of algebra. They are symbols, often letters like \( x \), \( y \), or \( w \), that stand in for unknown values or quantities. A variable allows you to create expressions and equations that can change depending on the problem or context. For instance, in our phrase 'triple the number of waiters,' a variable can represent the number of waiters.

• A variable is flexible: It can take on different values depending on what you're trying to solve.
• Choosing a variable: It helps to select a letter that reminds you of what it represents, like \( w \) for waiters.

Variables are not just placeholders; they allow for generality and can simplify complex problems. When you define a variable, you can perform operations with it, use it to set up equations, and solve for unknowns. In our example with the waiters, using the variable \( w \) lets us easily represent situations where the number of waiters could change. This is a powerful tool for turning words into mathematical sentences.

Translating Phrases to Expressions

Translating phrases into algebraic expressions is a fundamental skill in algebra that involves identifying the mathematical operations described by words. This process allows us to convert everyday situations into mathematical form, which can then be analyzed and solved.
Here's how to translate the phrase 'triple the number of waiters' into an expression.
1. **Identify the operation:** "Triple" means to use multiplication.2. **Choose a variable:** Pick a letter, such as \( w \), to represent the number of waiters.3. **Combine them into an expression:** Based on the operation, multiplying the variable by 3 gives us \( 3w \).In this example:

• "Triple" directly translates to the number 3 and the operation of multiplication.
• The phrase "the number of waiters" suggests the use of a variable to represent that unknown quantity.

Translating phrases accurately is crucial because it allows mathematical techniques to be applied to real-world situations. Once in expression form, you can do more with the problem — solve for unknowns or evaluate expressions given specific values. Mastering this translation skill will make algebra simpler and more connected to everyday experiences.