Problem 59 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. a. 6 miles more than \(15.5 \%\) of the altitude b. \(15.5 \%\) of 6 miles more than the altitude

Step-by-Step Solution

Answer

Part (a): \( 0.155x + 6 \); Part (b): \( x + 0.93 \).

Step 1: Translating Words to Math (Part a)

First, identify the unknown variable. Let's use \( x \) to represent the altitude. The phrase '6 miles more than' translates to adding 6 and '15.5% of the altitude' can be written as \( 0.155x \). Therefore, the algebraic expression for part (a) is: \( 0.155x + 6 \).

Step 2: Translating Words to Math (Part b)

Again, identify the unknown variable. We'll use \( x \) for the altitude. The phrase '15.5% of 6 miles' translates to \( 0.155 imes 6 \), and 'more than the altitude' means we'll be adding to \( x \). Calculate \( 0.155 imes 6 \) which equals \( 0.93 \). Thus, the algebraic expression for part (b) is: \( x + 0.93 \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a fundamental part of algebra. They serve as a way to represent numbers and relationships using variables, numbers, and operations. For students, understanding how to translate words into these expressions is crucial for problem-solving in algebra. An algebraic expression can include constants (fixed numbers), variables (letters representing unknowns), and operators (like addition or multiplication). Consider that when a phrase asks us to add 6 miles more than a certain value, we need to identify what that value is. In this exercise, it might be 15.5% of an altitude 'x', expressed as \(0.155x\). The full expression would then be \(0.155x + 6\). Similarly, a different order of words might require a different expression, as in \(x + 0.93\) when the phrase "15.5% of 6 more than altitude" is considered. This shows that context matters greatly when interpreting expressions.

Working with Percentages in Algebra

Percentages have a critical role in algebra, especially when converting them into numerical values that can be used in expressions. A percentage represents a dimensionless ratio or a fraction of 100. In algebra, the term "percent of" translates to multiplication. For example, "15.5% of the altitude" involves converting the percentage to a decimal (\(0.155\)) and multiplying by the altitude \(x\), shown as \(0.155x\). In our exercise example, understanding percentages allows us to convert the phrase "15.5% of 6 miles" into \(0.155 \times 6\), resulting in 0.93. This conversion is crucial for creating accurate algebraic expressions and ensuring clarity in mathematical statements. It emphasizes the importance of interpreting language precisely to capture the mathematical relationships it describes.

Basic Algebra Concepts for Translating Words Into Math

Being able to translate phrases into algebraic expressions is a pivotal skill in algebra. This process requires a solid grasp of basic algebraic concepts, such as understanding variables, constants, and operations. The variable denotes an unknown quantity, like \(x\) for altitude in the exercises. Constants are numbers with fixed values, such as the 6 miles mentioned. Operations like addition or multiplication are represented symbolically (+, *).
Breaking down each word problem step by step can help:

• Identify the variables and what they represent.
• Understand the constants' role.
• Translate operations such as "more than" or "percent of" into mathematical symbols.

Practicing this technique improves math literacy and equips learners with tools to tackle more complex expressions in the future. Converting narrative problems into algebra may seem challenging at first, but consistent practice enables students to think more critically and solve problems with ease.