Problem 40 - Algebra 1: Concepts and Skills

Question

Translate into mathematical symbols "the difference of a number and 4 is 10 " Let \(n\) represent the number. (A) \(n-4=10\) (B) \(4-n=10\) (C) \(10-4=n\) (D) \(10-n=4\)

Step-by-Step Solution

Answer

The correct mathematical translation of the sentence is option (A): \(n - 4 = 10\)

Step 1: Translate the sentence into an equation

The sentence can be translated into the following equation: \(n - 4 = 10\) where n represents the number.

Step 2: Compare the formulated equation with the available options

The formulated equation from the sentence is \(n - 4 = 10\). Comparing this equation with the given options, it matches with option (A): \(n - 4 = 10\). The other options represent different situations and are not accurate translations of the mentioned condition.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical statement made up of numbers, variables, and arithmetic operations. In our exercise, the phrase "the difference of a number and 4 is 10" can be transformed into an algebraic expression before becoming an equation.
Translating a phrase into an algebraic expression involves identifying key components:

• **Variables**: A symbol, usually a letter, represents the unknown value. Here, we use \(n\) to denote "a number."
• **Operations**: Indicates the action to perform, such as addition or subtraction. For our sentence, "the difference" suggests subtraction.
• **Constants**: Fixed numerical values within the expression. Here, 4 and 10 satisfy this role.

Building the expression involves taking these identified elements to form something like \(n - 4\). This sets the stage for defining an equation.

Mathematical Symbols

Mathematical symbols are essential tools that help us communicate complex ideas simply and effectively, especially in solving equations. They provide a universal language that transcends linguistic barriers. Let's break down the symbols involved:

• **Variables**: The letter \(n\) is a placeholder for an unknown number, a versatile way to represent any numeric value we are attempting to find or work with.
• **Arithmetic Operations**: Symbols like \(-\) denote operations, here indicating subtraction, a core operation in our sentence translation.
• **Equality**: The equals sign \(=\) shows that two expressions are equivalent or balanced. In the phrase "is 10," it indicates that the left side equals 10, forming the equation \(n - 4 = 10\).

These symbols make it possible to convert word problems into solvable equations, representing relationships a student must analyze and solve.

Equation Solving

Equation solving is a fundamental skill in algebra, allowing us to find unknown values that satisfy a given condition, expressed mathematically. To solve an equation such as \(n - 4 = 10\), follow these steps:

• **Identify the equation**: Determine from the problem statement that the relationship is represented as \(n - 4 = 10\).
• **Isolate the variable**: Add 4 to both sides to cancel out the subtraction, aiming to get \(n\) alone. This gives \(n = 10 + 4\).
• **Calculate the result**: Compute the right-hand side, resulting in \(n = 14\).

By following these simple steps, we identify that "the number" referred to in our statement is 14. Equation solving allows us to neatly transition from a sentence to a mathematical insight, showing why learning and applying algebra are powerful tools.