Problem 43 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. Twenty divided by eight times a number added to one is nine.

Step-by-Step Solution

Answer

Question: Translate the phrase "Twenty divided by eight times a number added to one is nine" into a mathematical equation and simplify it. Answer: (5/2)x + 1 = 9

Step 1: Identify the operations and variables

The phrase describes the following operations: 1. "Twenty divided by eight" 2. "Times a number" 3. "Added to one" 4. "Is nine" A number is represented by a variable, let's call it "x".

Step 2: Translate the operations into a mathematical equation

Now we will translate the operations into a mathematical equation: 1. "Twenty divided by eight" can be written as \(\frac{20}{8}\). 2. "Times a number" can be represented as \((\frac{20}{8})x\). 3. "Added to one" can be written as \((\frac{20}{8})x + 1\). 4. "Is nine" can be written as \((\frac{20}{8})x + 1 = 9\).

Step 3: Simplify the equation

The given equation is \((\frac{20}{8})x + 1 = 9\). We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4. So, the fraction becomes \(\frac{5}{2}\). Thus, the simplified equation is \((\frac{5}{2})x + 1 = 9\). That's the final equation representing the given phrase.

Key Concepts

Variable Representation

In mathematics, translating verbal expressions into equations often requires representing unknown values with variables. A variable serves as a placeholder for numbers we do not know yet. In the exercise, the phrase "a number" refers to this unknown value. We use a variable, commonly represented by "x" in algebra, to indicate this number. By doing this, we can create an equation that embodies the entire verbal expression.

• Choosing a Variable: Most of the time, algebraic problems prefer using letters like "x," "y," or "z". You can choose any letter, but it is common to use "x."
• Role of a Variable: It acts as an important component, allowing you to solve and manipulate equations easily.

Understanding how to consistently represent numbers with variables is a foundational skill essential for creating and solving equations in algebra.

Mathematical Operations

In the transition from words to math, identifying mathematical operations in the sentence is crucial. These operations serve as the building blocks for forming an equation. Each word or phrase in the original sentence correlates to a particular operation:

• "Divided by" indicates division.
• "Times" suggests multiplication.
• "Added to" involves addition.
• "Is" translates to the equals sign, connecting two parts of the equation.

For instance, in this exercise, "Twenty divided by eight" translates to the fraction \(\frac{20}{8}\). Similarly, "Times a number" suggests multiplying this fraction by a variable \(x\).
Recognizing these operations correctly transforms the verbal narrative into a workable equation. It's akin to decoding a recipe, where each word corresponds to a step in cooking the equation.

Simplification of Equations

Simplifying equations is a critical process that makes them easier to solve or understand. Once we've translated a verbal expression into a mathematical equation, we can often refine it further by simplifying complex fractions or expressions.

In the original task, the equation \((\frac{20}{8})x + 1 = 9\) was simplified by dividing the numerator and denominator of the fraction by their greatest common divisor, 4. This simplifies \(\frac{20}{8}\) to \(\frac{5}{2}\), resulting in the equation \((\frac{5}{2})x + 1 = 9\).

• Steps in Simplifying:
  • Locate fractions or complex expressions in your equation.
  • Determine if any numbers in the fraction can be reduced by finding a common factor.
  • Rewrite the equation with the simplified components.
• Benefits: Simplified equations are easier to manage and solve, reducing the risk of errors during calculations.

Practicing simplification not only streamlines your equations but also enhances your overall understanding of algebraic manipulations.