Problem 60 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. a. twice the sum of the tax and 200 b. the sum of twice the tax and 200

Step-by-Step Solution

Answer

a. \( 2(t + 200) \), b. \( 2t + 200 \)

Step 1: Identify Variables

First, identify appropriate variables to use in the expressions. In this case, let's use \( t \) to represent the tax.

Step 2: Construct the First Expression

For part (a), you need to translate 'twice the sum of the tax and 200'. Start inside the parentheses: the sum of the tax and 200 is \( t + 200 \). Now, multiply this entire sum by 2 (which is twice): \( 2(t + 200) \).

Step 3: Construct the Second Expression

For part (b), the phrase is 'the sum of twice the tax and 200'. Start by finding twice the tax first: \( 2t \). Add 200 to this result to get the expression: \( 2t + 200 \).

Key Concepts

Translating Phrases to Expressions

Turning a verbal phrase into an algebraic expression might seem a bit tricky at first, but breaking it down makes it manageable. This process involves identifying keywords that often indicate certain operations. For instance:

• "Sum" usually means addition.
• "Twice" or "double" suggests multiplication by 2.

Take, for instance, the exercise of translating phrases like "twice the sum of the tax and 200." In situations like this, it's important to first address what's happening inside the phrase. The sum in this case is the combination of "the tax" and 200, leading to the expression: \( t + 200 \). After forming this initial sum, multiply it by 2, as "twice" indicates, resulting in \( 2(t + 200) \). By following a step-by-step approach, translation becomes a straightforward task.

Variables in Algebra

In algebra, letters are often used to stand in for unknown quantities, these letters are called variables. Choosing the right variable for your algebraic expressions is crucial.

• Variables help in creating general formulas.
• They allow expressions to represent a range of possible numbers.

In the given exercise, we use the letter \( t \) to represent an unknown tax amount. This is a common practice, as it simplifies communication and calculation processes across different algebraic problems. Once you assign a variable to stand in for a value, it becomes easier to perform operations and write expressions.

Mathematical Operations

Mathematical operations such as addition, multiplication, and subtraction form the backbone of algebraic expressions. Understanding how to apply these operations correctly is essential to success in translating these expressions from phrases.

• Addition (+): Combines two values together.
• Multiplication (\(\times\)): Represents repeated addition.
• Subtraction (-): Involves taking away from a total.

In the provided exercise, the operation of addition is demonstrated by phrases like "the sum of," indicating that two numbers should be added together. Multiplication is highlighted through words such as "twice," which requires multiplying a number by 2. By understanding these operations and their indicators, transforming verbal statements into mathematical expressions becomes far simpler and more logical.