Problem 77 - Elementary Algebra

Question

Translate the following sentences into linear equations and then solve. A number \(n\) divided by 8 is 5 .

Step-by-Step Solution

Answer

The number is 40.

Step 1: Translate the Sentence into an Equation

The problem states that a number \( n \) divided by 8 equals 5. We translate this into a linear equation:\[\frac{n}{8} = 5\].

Step 2: Solve for the Number \( n \)

To solve for \( n \), we need to eliminate the fraction. Multiply both sides of the equation by 8:\[\frac{n}{8} \times 8 = 5 \times 8\]This simplifies to:\[n = 40\]

Key Concepts

Solving Equations

Solving equations is a fundamental part of algebra that involves finding the unknown value that makes the equation true. When you approach solving an equation, here are a few steps to keep in mind:

• Understand the Equation: Begin by examining the equation carefully. Identify what you are solving for, which variables are present, and what operations are being used.
• Isolate the Variable: The goal is to get the unknown variable by itself on one side of the equation. This often involves undoing operations such as addition, subtraction, multiplication, and division.
• Check Your Solution: Once you solve the equation, it's always a good idea to plug the solution back into the original equation to check its accuracy.

These basic steps will help you tackle any linear equation with confidence. It's all about reversing operations and strategically isolating the variable that represents the unknown.

Algebraic Translation

Algebraic translation is converting a word problem or sentence into a mathematical equation. This skill is crucial for bridging real-world problems and mathematical solutions. Here's how you can become adept at this:

• Identify Keywords: Look for words that indicate mathematical operations. For instance, words like "divided by" point to division, while "equals" suggests an equation.
• Turn Sentences into Symbols: Replace parts of the sentence with mathematical symbols. In the given example, the sentence "a number \(n\) divided by 8 is 5" translates to \(\frac{n}{8} = 5\).
• Follow a Logical Structure: Ensure the translated equation makes logical sense. Each part of the sentence should correspond to a part of the equation.

Practicing these skills will allow you to solve algebra problems presented in a variety of contexts effortlessly.

Fraction Elimination

Fraction elimination is a technique used to simplify equations that contain fractions, making them easier to solve. Here is how you can master this technique:

• Find a Common Denominator: If an equation has multiple fractions, identify a common denominator or focus on one fraction at a time.
• Clear the Fraction: Multiply every term in the equation by the denominator of the fraction you are dealing with. This helps you remove the fraction and simplify the equation.
• Solve the Resulting Equation: Once the fractions are gone, you can proceed to solve the linear equation using basic algebra techniques.

In the example, multiplying both sides by 8 cleared the fraction \(\frac{n}{8}\), resulting in the simple equation \(n = 40\). This method streamlines the solving process and ensures accuracy.