Problem 22 - Elementary Algebra Within Reach

Question

In Exercises \(11-22\), translate the verbal phrase into an algebraic expression. $$ \text { One-half of } y $$

Step-by-Step Solution

Answer

The translation of 'One-half of y' into an algebraic expression is \(0.5*y\) or equivalently \(y/2\).

Step 1: Understanding the verbal phrase

When a verbal phrase states 'one-half of', it means to multiply something by 0.5 or divide it by 2. So, 'one-half of y' means y is being divided by 2.

Step 2: Translating into an algebraic expression

The verbal phrase can be written as \(0.5 * y\) or equivalently as \(y / 2\). Either form is acceptable and they represent the same mathematical operation.

Key Concepts

Translating Verbal Phrases

Translating verbal phrases into algebraic expressions is a fundamental skill in algebra. It involves converting everyday language into a mathematical format. The key is identifying the mathematical operation that corresponds to the words used. For example, when you encounter the phrase 'one-half of y', you need to determine that 'one-half' refers to dividing by 2 or multiplying by 0.5.
This conversion helps in setting up equations that can be solved to find unknown values, making it a valuable tool in both academics and real-life problem solving.
Once you understand the meaning behind the words, translating them becomes much simpler.

Here are some general tips to translate verbal phrases:

• Identify keywords that signal mathematical operations, such as 'of' meaning multiply.
• Translate step-by-step, focusing on one part of the phrase at a time.
• Check your work by reading the algebraic expression back into words to ensure it makes sense.

Mathematical Operations

Mathematical operations are the building blocks of algebra and arithmetic. In the context of translating verbal phrases, knowing these operations allows us to perform the necessary calculations. The core mathematical operations are addition, subtraction, multiplication, and division.
In our exercise, we dealt primarily with the operation of 'one-half', which translates to dividing by 2 or multiplying by 0.5.
This is a division operation, but it can also be viewed as a special kind of multiplication.

Here's a quick overview of how these operations translate from words into algebra:

• **Addition (+):** Words like 'sum', 'plus', or 'increased by'.
• **Subtraction (−):** Words such as 'difference', 'minus', or 'decreased by'.
• **Multiplication (×):** Terms like 'product', 'times', or 'of'.
• **Division (÷):** Phrases including 'quotient', 'divided by', or 'per'.

Understanding these basic operations enables you to break down more complex phrases.

Fractions in Algebra

Fractions often appear in algebra, and it's crucial to become comfortable with them. A fraction represents division, something divided into equal parts, and is especially common when dealing with phrases like 'one-half', 'one-third', etc.
The expression 'one-half of y' can be written as both \( \frac{y}{2} \) and \( 0.5 \times y \), showing how fractions can interchangeably be represented as decimals in algebra.
Grasping fractions is necessary because they frequently emerge in many algebraic problems and solutions.

When working with fractions in algebra:

• Understand the relationship between the divisor and the dividend. Here, y is being divided by 2.
• Remember that fractions imply a part of a whole and are equivalent to multiplying by the reciprocal.
• Practice converting between fractional and decimal forms to improve versatility in problem-solving.

This familiarity with fractions will significantly boost your confidence and competence in tackling algebraic expressions involving them.