Problem 10 - Prealgebra

Question

Translate each sentence into an equation. Then find each number. Ten more than the quotient of a number and \(-2\) is three.

Step-by-Step Solution

Answer

The number is 14.

Step 1: Understand the Phrase

First, identify the mathematical operations described by the words. The phrase "ten more than" indicates an addition, and "the quotient of a number and \(-2\)" means a division operation. Our unknown number can be represented as a variable, say \(x\).

Step 2: Form the Equation

Translate the verbal sentence into a mathematical equation. "The quotient of a number and \(-2\)" becomes \(\frac{x}{-2}\). Therefore, "ten more than the quotient of a number and \(-2\)" can be written as \(10 + \frac{x}{-2}\). This is equal to three, so the equation becomes: \[10 + \frac{x}{-2} = 3\]

Step 3: Solve the Equation

Isolate the variable \(x\) by first subtracting 10 from both sides of the equation: \[ \frac{x}{-2} = -7 \] Now, solve for \(x\) by multiplying both sides by \(-2\): \[ x = (-7) \times (-2) \] Simplify the multiplication: \[ x = 14 \]

Step 4: Verify the Solution

Substitute \(x = 14\) back into the original context to ensure it satisfies the condition. The quotient of 14 and \(-2\) is \(-7\), so ten more than \(-7\) is indeed 3, confirming our solution.

Key Concepts

Algebraic Expressions

Algebraic expressions are the core building blocks in the realm of algebra. They use symbols and numbers to represent real-world problems. In our exercise, the statement "the quotient of a number and \(-2\)" translates into an algebraic expression. The participating elements include:

• A variable, usually represented by letters like \(x\), to stand in for unknown values.
• Operations such as addition, subtraction, multiplication, and division that connect variables and numbers in meaningful ways.

In translating a sentence to an algebraic expression, it is important to recognize keywords:

• "Quotient" indicates division.
• "More than" suggests an addition.

Thus, using these clues aids in forming expressions like \(10 + \frac{x}{-2}\). The goal is clarity and precision, setting the stage for finding solutions.

Problem Solving

Problem solving in algebra involves translating verbal phrases into equations. This process requires an understanding of mathematical language and keen analytical thinking. For instance, in our exercise, we expressed the condition "ten more than the quotient of a number and \(-2\) is three" as the equation:\[10 + \frac{x}{-2} = 3\]Here's how problem solving unfolds in steps:

• Identify the variable: Represent unknown numbers with variables like \(x\).
• Translate words into operations: Understand terms like "quotient" and "more than" to form the equation.
• Perform logical reasoning: Deduce further steps to simplify and solve for the variable.

Understanding the sequence and rationale behind translating, forming, and solving ensures students develop strong problem-solving skills.

Mathematical Operations

Mathematical operations are the processes we use to manipulate numbers and variables to obtain desired results. In this exercise, we used several operations:

• Addition and Subtraction: These inverse operations help in moving terms across an equation. We subtracted 10 to isolate the term with the variable.
• Division and Multiplication: Division often aids in creating expressions for quotients. In contrast, multiplication reverses division, as seen when we multiplied by \(-2\) to solve for \(x\).

To solve equations effectively, one must handle these operations proficiently, always keeping the equation balanced:

• Operations must be consistently applied to both sides of an equation.
• Knowing when to use inverse operations is essential in simplifying and finding unknowns.

Mastering these operations solidifies one's ability to navigate through complex equations with ease.