Problem 111 - Introductory and Intermediate Algebra for College Students 4th

Question

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. The product of \(\frac{2}{3}\) and a number increased by 6

Step-by-Step Solution

Answer

\((2/3)x + 6\)

Step 1: Identify the operations and numbers involved

The problem states that the product of 2/3 and a number (which we know is represented as x) is increased by 6. Therefore, the operations involved here are multiplication and addition, and the numbers involved are 2/3 and 6.

Step 2: Translate the English sentence into an algebraic expression

In the sentence, 'the product of 2/3 and a number' can be translated into \((2/3)x\), which means 2/3 multiplied by x. 'Increased by 6' can be translated into '+6'. Therefore, the entire sentence 'The product of 2/3 and a number increased by 6' can be translated into the algebraic expression \((2/3)x + 6\).

Key Concepts

Translation to Algebra

Translating verbal phrases into algebraic expressions is like turning words into numbers and symbols. It's all about identifying the operations mentioned and expressing them mathematically. In our given sentence, "The product of \(\frac{2}{3}\) and a number increased by 6," we need to recognize specific keywords:

• "The product of" suggests multiplication is involved.
• "A number" is unknown, so we use a variable, like \(x\), to represent it.
• "Increased by" implies we need to add 6 to whatever result we have from the multiplication.

By interpreting the sentence part by part and organizing it into an equation, we create a mathematical expression: \((2/3)x + 6\). This translation process forms the foundation for solving many word problems in algebra.

Multiplication

Multiplication is one of the basic operations we often use in algebra. It involves finding the product when one number is multiplied by another. Let's explore how this applies to our example:
In the expression "the product of \(\frac{2}{3}\) and a number," you are asked to multiply \(\frac{2}{3}\) by an unknown value, or the variable \(x\). This results in \((2/3)x\). Let's break this down:

• \(\frac{2}{3}\) is a fraction—a small part relative to the whole.
• When multiplied by another value, \(x\), it scales \(x\) by \(\frac{2}{3}\).
• This operation helps in situations where you need a portion or a fraction of a number.

Understanding multiplication involving fractions and variables is crucial to formulating correct algebraic expressions.

Addition

Addition in algebra allows us to combine amounts or increase values systematically. In the phrase "increased by 6," addition means including 6 to what you already have.
In the context of our algebraic expression, which starts as \((2/3)x\), the addition of 6 is represented as: - \((2/3)x + 6\).
Here's why addition is used here:

• "Increased by" naturally suggests adding to the current value or result.
• You're combining two quantities: the result of a multiplication and another fixed number (6).
• Addition is straightforward as it directly conveys the operation of combining or incrementing amounts.

By understanding addition in these expressions, you get closer to figuring out the total value or amount described in word problems.

Variable Representation

Variables are placeholders for unknown numbers in algebra. They help express generality in mathematical expressions, allowing you to solve problems with different values.
In our example, "a number" is something we don't know. So, we use a variable, \(x\), as a symbol to represent this unknown quantity.

• Using \(x\) or any letter is typical in algebra to present numbers without specifying their value.
• It makes expressing mathematical rules or relationships simpler and versatile.
• Variables let you work with general formulas till you solve for their specific values.

Variables in algebra serve as a flexible way to construct equations and expressions that can be solved once more information is available or known.