Problem 11 - Fundamentals of Mathematics

Question

Translate each phrase or sentence into a mathematical expression or equation. Five less than some quantity is eight.

Step-by-Step Solution

Answer

The quantity is 13.

Step 1: Identify the Unknown Quantity

First, let's identify the unknown number that we need to find. We can use a variable, such as \( x \), to represent 'some quantity.'

Step 2: Translate 'Five Less Than' into an Expression

The phrase 'five less than some quantity' implies that five is being subtracted from that quantity. Therefore, this translates to the expression \( x - 5 \).

Step 3: Set Up the Equation

According to the statement, 'five less than some quantity is eight.' This suggests that the expression we formed is equal to 8. Hence, the equation is: \[ x - 5 = 8 \]

Step 4: Solve the Equation for the Unknown

To find \( x \), add 5 to both sides of the equation to isolate \( x \): \[ x - 5 + 5 = 8 + 5 \] Simplifying gives: \[ x = 13 \]

Key Concepts

Understanding Mathematical Expressions

A mathematical expression is a combination of numbers, symbols, and operations that represent a particular value. In essence, it is any mathematical phrase that can include:

• Numbers
• Variables
• Operators (such as +, -, *, /)

Expressions do not have an equality sign, differentiating them from equations. In this exercise, we have the expression \( x - 5 \), where \( x \) is a variable representing an unknown quantity, and the operation performed is subtraction. This simple expression indicates that 5 is subtracted from some unknown quantity \( x \). By mastering how to form expressions, you become better equipped to tackle more complex mathematical problems efficiently.

Translating Phrases to Equations

Translating everyday language into mathematical language is a valuable skill in solving problems. In this step, we learn to convert phrases into equations. Let's break down the phrase 'Five less than some quantity is eight.'First, "some quantity" is an unknown number, commonly represented by a variable, such as \( x \). "Five less than" indicates a subtraction, so the phrase translates to \( x - 5 \).Finally, "is eight" transforms the expression into an equation by equating it to 8, forming \( x - 5 = 8 \). Equations are crucial as they allow you to find unknown values. By understanding how to translate phrases, you can interpret real-world problems mathematically.

The Basics of Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It enables us to describe relationships and solve for unknowns. In algebra, an equation like \( x - 5 = 8 \) is a way to express a relationship between the variable \( x \) and the numbers 5 and 8.To solve such equations, you need to isolate the variable. This involves performing the same mathematical operation on both sides of the equation to maintain the balance. By adding 5 to both sides of \( x - 5 = 8 \), you get \( x = 13 \). This solution process reflects the elegance and logic that algebra brings to mathematical reasoning. Understanding these principles is fundamental to mastering algebra and other advanced math topics.

Variable Representation in Algebra

Variables are symbols used to represent unknown values in mathematical expressions and equations. They function as placeholders that can assume various numerical values. In our exercise, \( x \) is the variable representing "some quantity."Variables allow flexibility in math, letting equations stand for real-world problems without specifying exact numbers. Choosing the right symbol, commonly starting with \( x, y, z \), simplifies communication in math.By using variables, we can model situations where numbers are unknown. This capability is powerful, allowing for generalization and the ability to solve complex problems. Embracing variables is a key step toward math proficiency, opening doors to endless possibilities in calculations and analysis.