Problem 41 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. six more than the quoticnt of a number and 30

Step-by-Step Solution

Answer

The final algebraic expression for the phrase 'six more than the quotient of a number and 30' is \(6 + \frac{x}{30}\).

Step 1: Identify the Variable

The first step here is to understand what our variable represents. We are given in the problem that \(x\) represents the number. So wherever we read 'the number' in our English phrasing, we will substitute it with \(x\).

Step 2: Translating 'Quotient of a Number and 30'

Now the second part to tackle is 'the quotient of a number and 30'. In mathematical terms, this phrase can be translated to a division operation between 'the number', which is \(x\), and 30. Therefore, 'the quotient of a number and 30' becomes \(\frac{x}{30}\).

Step 3: Adding Six

The statement 'six more than' tells us that we are adding six to our previously translated expression. Hence, 'six more than the quotient of a number and 30' transforms into \(6 + \frac{x}{30}\). This is our final algebraic expression.

Key Concepts

Variables in Algebra

Variables are fundamental in algebra. They are symbols, often letters like \( x \), that represent unknown quantities. This allows us to create expressions and equations that model real-world situations.

• Think of a variable as a placeholder used to work with numbers that can change or are not defined yet.
• In our example, \( x \) represents "a number"—it's a way to handle unknown values flexibly.
• Knowing how to use variables is key to forming and solving algebraic expressions.

This ability to symbolize unknowns makes algebra a powerful tool in solving problems.

Translation of Phrases into Algebra

Translating phrases into algebra requires understanding how everyday language translates to mathematical operations. It's like learning a new language, where words become equations. Here are some basic tips:

• Phrases like "the sum of" or "more than" typically indicate addition.
• Words such as "minus" or "less than" suggest subtraction.
• "Product of" refers to multiplication, while "quotient of" signals division.

In our example, "the quotient of a number and 30" directs us to divide \( x \) by 30, turning the phrase into the expression \( \frac{x}{30} \). Adding "six more than" cues us to add 6 to this expression, leading to the final translation \( 6 + \frac{x}{30} \). Understanding this translation is essential in transforming word problems into solvable algebraic form.

Quotients in Algebra

When dealing with quotients in algebra, we're looking at division expressions. Understanding quotients lets you solve problems that involve dividing one term by another.

• The phrase "quotient of a number and 30" in our example simplifies to the algebraic expression \( \frac{x}{30} \).
• The numerator (\( x \)) is the number being divided, and the denominator (30) is the number dividing it.
• Recognizing and working with quotients helps students tackle division-based word problems with ease.

Learning how to express and solve these division relationships is crucial in developing robust algebra skills.