Problem 70 - Elementary Algebra Within Reach

Question

In Exercises 69-72, translate the phrase into an algebraic expression. Let \(x\) represent the real number. $$ c \text { divided by } 6 $$

Step-by-Step Solution

Answer

The phrase 'c divided by 6' translates to the algebraic expression \( \frac{c}{6} \)

Step 1: Identify the Variables

The primary variable in this given problem is \(c\). Notice that \(x\) is also mentioned, but it's not part of the phrase we're trying to translate, so we'll ignore it for this problem.

Step 2: Identify the Operations

The operation in this problem with the variable \(c\) is division. In the given phrase, 'divided by' tells us this explicitly.

Step 3: Translate to Algebraic Expression

Now we'll translate the verbal phrase into an algebraic expression. According to the phrase 'c divided by 6', we must write \(c\) atop the division symbol and 6 at the bottom. Alternatively this could be written as a fraction. The fraction form of this instance is \( \frac{c}{6} \) .

Key Concepts

Division in Algebra

When working with algebra, division is one of the essential operations, akin to addition, subtraction, and multiplication. Division can often be represented in different ways, especially in algebraic expressions. Most commonly, you might see this as a fraction.

For example, in the phrase "c divided by 6," division is the primary operation. This results in an expression written as a fraction: \( \frac{c}{6} \). Here, \( c \) is the dividend, or the number being divided, and 6 is the divisor, or the number by which the dividend is divided.

• The division is visually represented with the divisor beneath the dividend in fractional form.
• You can also use the division symbol ("÷"), but fractions are more common in algebra.

Understanding how to translate division into algebraic expressions is crucial as it appears in many different mathematical problems. It helps you manipulate expressions further, especially when solving equations.

Variables in Algebra

Variables play a foundational role in algebra. They serve as symbols that hold the place of numbers and can represent unknown values that need to be determined.

In our specific example, we use \( c \) as a variable to describe a particular number. We insert variables into expressions to:

• Stand in for values that vary or are unknown.
• Allow for the generalization of mathematical principles.
• Enable the solution of equations by isolating and solving for these variables.

Variables can be almost any letter or symbol, though it's common to see lower-case letters like \( x, y, a, b, \) and \( c \). Each variable is powerful in that it provides us a way to form and solve equations and represents real numbers in various contexts within mathematics.

Translating Phrases to Algebraic Expressions

At the heart of many algebra problems is the process of translating words into algebraic expressions. It involves converting a verbal description of a mathematical situation into a formal algebraic equation or expression.

Here's a step-by-step guide to translating phrases:

• Identify the variables: Determine which symbols represent the numbers in question.
• Spot the operations: Look for keywords such as "sum," "product," "difference," and "quotient," which clue you into the operations involved. Words like "plus," "times," "minus," and "divided by" are your indicators.
• Construct the expression: Arrange the variables and numbers to represent the relationships shown in the verbal phrase, using mathematical symbols appropriately.

For instance, the phrase "c divided by 6" directly translates to \( \frac{c}{6} \). Translating phrases effectively allows you to set up problems correctly and solve them using algebraic methods.