Problem 1 - Fundamentals of Mathematics

Question

Translate each phrase or sentence into a mathematical expression or equation. Twelve more than a number.

Step-by-Step Solution

Answer

12 + x

Step 1: Identify the variables

To solve the given phrase, first identify what the unknown quantity is. Here, it is 'a number', which we will represent with the variable \( x \).

Step 2: Analyze phrase 'Twelve more than'

In the phrase 'twelve more than', 'more than' indicates an addition operation. Therefore, we need to add 12 to our unknown number \( x \).

Step 3: Construct the expression

The phrase 'twelve more than a number' translates mathematically to '12 + x', which combines the identified variable and the additional amount.

Key Concepts

Variables

In mathematics, variables are symbols or letters that represent unknown or changeable values. They help us simplify complex statements or expressions and are often used in equations to find a solution for the unknown. For example, in the phrase 'a number', we're dealing with an unknown value. To make things clearer, we assign a letter, called a variable, to represent it. This letter is usually something simple like \( x \), \( y \), or \( z \), but many other options exist.

Using variables:

• They allow us to form general rules or formulas.
• They make it easier to manipulate and solve expressions or equations.
• They aid in creating models for real-world problems.

In our exercise, we use the variable \( x \) to stand in for 'a number' mentioned in the phrase. This substitution is crucial for translating the phrase into a mathematical expression.

Addition Operation

The addition operation is one of the fundamental operations in mathematics, used to calculate the total or sum of two or more numbers or quantities. It's represented by the plus sign \(+\).

Understanding addition:

• Addition combines numbers or variables to form a single new quantity.
• It's commutative, meaning that the order doesn't affect the result (e.g., \( a + b = b + a \)).
• It helps solve problems requiring total amounts or increased values.

In the context of our exercise, when we see the words 'more than', it signals that we need to add one quantity to another. Thus, 'twelve more than a number' means we add 12 to the unknown represented by our variable \( x \), resulting in the expression \( 12 + x \).

Translating Phrases to Expressions

Translating phrases into mathematical expressions involves converting words into a mathematical language. This skill is crucial as it enables us to solve real-world problems by representing them in an easily solvable form.

Key steps in translation:

• Identify quantities, whether known or unknown, and represent unknowns with variables.
• Recognize mathematical operations described by the words.
• Combine the identified elements into a coherent expression or equation.

For example, in the phrase 'twelve more than a number', 'a number' is the unknown, which we represent with the variable \( x \). 'Twelve more than' indicates the use of the addition operation, specifically adding 12 to the number.

Therefore, the phrase translates into the expression \( 12 + x \). This process allows us to understand and solve problems systematically and logically.