Problem 25 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. Six plus five times an unknown number.

Step-by-Step Solution

Answer

The expression is \( 6 + 5x \).

Step 1: Identify the Operations

Start by breaking down the phrase into the arithmetic operations involved. The phrase 'six plus five times an unknown number' contains addition and multiplication.

Step 2: Assign a Variable for the Unknown

Let the unknown number be represented by the variable \( x \). This variable will be used in constructing the mathematical expression.

Step 3: Translate Multiplication

The phrase 'five times an unknown number' translates to the arithmetic operation \( 5 \times x \), or simply \( 5x \) in mathematical notation.

Step 4: Translate Addition

With the multiplication translated, now look at the phrase 'six plus.' This indicates that 6 should be added to the result of the previous multiplication, forming the expression \( 6 + 5x \).

Key Concepts

Arithmetic Operations

In mathematics, understanding arithmetic operations is essential for performing calculations and solving problems. When translating phrases into mathematical expressions, recognizing these fundamental operations in a sentence helps greatly. The four primary operations are:

• Addition (+): This operation combines two numbers to yield a sum. For example, adding 6 to 5 equals 11, which is expressed as \(6 + 5\).
• Subtraction (−): This operation indicates removing a quantity from another. For instance, subtracting 3 from 10 gives 7, represented as \(10 - 3\).
• Multiplication (×): Used to find the total in groups of a number. If you multiply 4 by 3, the result is 12, shown as \(4 \times 3\) or simply \(4 \cdot 3\).
• Division (÷): It involves splitting a number into equal parts. For example, 15 divided by 5 is 3, which can be written as \(15 \div 5\) or \(\frac{15}{5}\).

In the given exercise, the phrase 'six plus five times an unknown number' involves addition and multiplication. Recognizing these operations is the first step in converting the phrase into a mathematical expression.

Variables in Math

Variables play a crucial role in mathematics, especially when dealing with expressions and equations. They are symbols, often letters, used to represent unknown or variable quantities. When translating phrases into math expressions, variables come handy to denote numbers that are not yet known.

In the problem 'Six plus five times an unknown number,' the unknown number can be represented by a variable like \(x\). This use of variables allows us to write and manipulate expressions without having the exact value. Here’s a quick guide to the concept of variables:

• Purpose of Variables: They simplify expressions and make equations flexible and adaptable to different scenarios.
• Naming Variables: Although letters like \(x, y,\) and \(z\) are common, any symbol can be used as a variable.
• Role in Expressions: By using a variable, we can create expressions that are solvable, adaptable, and extensible.

Understanding variables helps in creating clear and concise mathematical expressions, as seen in the translation process of the exercise.

Translation of Phrases to Expressions

Translating phrases into mathematical expressions is a skill that connects language with math. It involves identifying verbal cues in a sentence that signal arithmetic operations and realizing their mathematical counterparts. Here's how you can translate phrases like 'six plus five times an unknown number':

• Identify Keywords: Terms like 'plus,' 'times,' 'difference,' and 'quotient' indicate specific operations.
• Break Down the Phrase: Focus on each part separately. The exercise contains 'six plus' and 'five times an unknown number.'
• Convert to Math: 'Five times an unknown number' translates to \(5x\), with \(x\) as the variable. The word 'plus' signals addition, leading to the final expression \(6 + 5x\).

This translation process not only clarifies the meaning but also demonstrates how languages and math can coexist to articulate mathematical ideas.