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Chapter 2: Problem 393

In the following exercises, translate each English sentence into an algebraic equation. The product of 4 and 8 is equal to \(32 .\)

Short Answer

Expert verified
4 x 8 = 32

Step by step solution

01

Identify Terms

Recognize key terms in the English sentence: 'product', '4', '8', 'is equal to', and '32'.
02

Understand 'Product'

The term 'product' indicates multiplication. In this context, it means to multiply 4 and 8.
03

Set Up the Multiplication

Write the multiplication of 4 and 8 as: \[4 \times 8\]
04

Translate 'Is Equal To'

The phrase 'is equal to' corresponds to the equals sign '=' in an algebraic equation.
05

Combine and Translate

Combine the translation of 'product' and 'is equal to' into one algebraic equation: \[4 \times 8 = 32\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Algebraic expressions are a way to represent numbers and operations using symbols and variables. For example, instead of saying 'the product of 4 and 8', we can write \[4 \times 8\]. This makes it easier to work with and understand mathematical relationships.
In any algebraic expression, each number, symbol, or variable represents a specific quantity or operation. So, when you convert words into symbols, you create a concise and precise mathematical expression.
Remember, learning to translate sentences into algebraic expressions is a fundamental skill in algebra. It helps solve real-world problems with ease.
Basic Multiplication
Multiplication is one of the four basic operations in arithmetic. When you multiply, you are essentially adding a number to itself a certain number of times. For example, \[4 \times 8\] means adding 4 together eight times:
4 + 4 + 4 + 4 + 4 + 4 + 4 + 4.
Using the multiplication symbol (\[ \times \]), we can quickly represent and calculate problems involving repeated additions. Multiplying two numbers is often called finding the 'product'.
In our example, the product of 4 and 8 is 32. This means that \[4 \times 8 = 32\].
Equation Translation
Equation translation is the process of converting a written sentence into a mathematical equation. This involves recognizing keywords and substituting them with appropriate symbols.
Let's break down the steps:
  • Identify key terms: Focus on words like 'product', 'sum', 'difference', and 'is equal to'.
  • Understand operations: The term 'product' means multiplication, 'sum' means addition, 'difference' means subtraction, etc.
  • Set up the equation: Arrange numbers and symbols to reflect the sentence. For example, 'the product of 4 and 8 is equal to 32' becomes \[4 \times 8 = 32\].
This skill is crucial, as it allows you to solve real-world problems using algebra.
Mathematical Terminology
In algebra, understanding terminology is key. Here are some essential terms:
  • Product: The result of multiplying two numbers. In our example, 32 is the product of 4 and 8.
  • Is Equal To: Represented by the equals sign (\[ = \]). It shows that two values on either side of the sign are the same.
  • Multiply: To combine groups of equal sizes. Represented by symbols like \[ \times \] or \[ \bullet \].

    • Grasping these terms helps you translate and solve equations easily. With practice, recognizing these keywords will become second nature, allowing you to quickly set up and solve algebraic equations.

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