Problem 4 - Fundamentals of Mathematics

Question

Translate each phrase or sentence into a mathematical expression or equation. Six times a number is fifty-four.

Step-by-Step Solution

Answer

The equation is \( 6x = 54 \).

Step 1: Identify the Unknown

In the sentence 'Six times a number is fifty-four,' we need to identify the unknown number. We will denote this unknown number by a variable, commonly chosen as \( x \).

Step 2: Translate the Phrase into a Mathematical Expression

The phrase 'six times a number' means that the unknown number, represented by \( x \), is multiplied by six. This can be written as \( 6x \).

Step 3: Form the Equation

The complete sentence states 'Six times a number is fifty-four.' The word 'is' signifies equality, so we set the expression \( 6x \) equal to the number fifty-four. This translates to the equation: \[ 6x = 54 \].

Key Concepts

Mathematical Expressions

In mathematics, expressions are combinations of numbers, variables, and operations. They help us represent real-world problems in a mathematical form that can be manipulated to find solutions. When translating a phrase like "six times a number," we're tasked with turning words into a mathematical expression.

To do this, we identify operations such as addition, subtraction, multiplication, or division that correspond to the words. Here, the word "times" suggests multiplication. So, when you see "six times a number," it means you multiply 6 by an unknown number. In mathematical terms, this becomes the expression:

• \( 6x \)

Remember, expressions do not have an equal sign, as they are not complete equations. They're simply expressions of operations and quantities put together.

Variable Identification

Recognizing and defining variables is a fundamental skill in algebra. A variable symbolizes a quantity that can change or is unknown, represented commonly by letters such as \( x \) or \( y \). In our exercise, the sentence "six times a number is fifty-four" indicates that we're dealing with an unknown number.

You can choose any letter to represent this unknown, but \( x \) is often used for simplicity. Identifying this variable is essential as it serves as a placeholder for the value we aim to find.

• Choose a letter to represent the unknown quantity.
• Acknowledge that this letter is the variable in the expression.
• Use this variable to form meaningful mathematical expressions or equations.

Once a variable is chosen, it becomes the building block for translating the sentence into mathematical language.

Forming Equations

Forming equations is the step where expressions transform into equations by introducing a relationship between two expressions using an equality. In our example, we took the expression "six times a number" and acknowledged the association of equality through the word "is."

In mathematics, "is" typically signals an equation, and serves to equate the expression "six times a number" with fifty-four. Thus, we write:

• \( 6x = 54 \)

Equations are powerful because they create an assertion of equality, allowing us to calculate the unknown. In this instance, solving the equation involves finding the value of \( x \) that makes the equation true.

It's essential to learn that translating words into equations often relies on identifying key phrases of equality and understanding the mathematical operations implied by those phrases. This forms the basis for solving many algebraic problems.