Problem 31 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. When a number is divided by four, the result is sixty-eight.

Step-by-Step Solution

Answer

Question: Translate the statement "When a number is divided by four, the result is sixty-eight" into an equation. Answer: \(\frac{x}{4} = 68\)

Step 1: Understand the problem

We are given a statement describing a mathematical situation. The statement tells us that when a number is divided by four, we get sixty-eight as the result. We need to translate this statement into an equation.

Step 2: Identify the variables

Let's use the variable x to represent the unknown number. Our task is to create an equation involving x that represents the given situation.

Step 3: Translate the statement into an equation

We know that when a number (x) is divided by four, the result is sixty-eight. So, we can represent this situation as an equation: \frac{x}{4} = 68

Step 4: Final equation

The final equation that represents the given statement is: \frac{x}{4} = 68

Key Concepts

Translating Sentences to Equations

Turning words into mathematical expressions is a skill that makes solving problems much easier. We start by reading the sentence carefully.
Next, we identify the mathematical operations involved in the problem.
In the given sentence: "When a number is divided by four, the result is sixty-eight," we notice the division operation.
The key is to look for signal words:

• "Is divided by" suggests division.
• "The result is" indicates equality.

Here, these words help us see that the problem revolves around division and equality, shaping them into one neat equation. This forms the backbone of turning verbal statements into equations. The context clues are critical in bridging the gap between words and math.

Variable Identification

Identifying variables is a crucial part of formulating equations. In our example, the sentence talks about an unknown number.
This unknown quantity needs a placeholder—a "variable" that will represent it throughout the equation. Choosing a Variable:

• A variable is typically a letter from the alphabet; common choices include x, y, or z.
• The choice of letter does not affect the outcome; we commonly use x for simplicity.

In this exercise, we choose 'x' to stand for the unknown number. By doing this, we translate ambiguous language into something concrete and mathematical.
This allows the sentence to become more logical and structured, almost like a math puzzle awaiting a solution.

Equation Formation

Forming an equation is the culmination of translating a statement and identifying variables. You now merge these elements into a mathematical equation.
In this exercise, you've already identified 'x' as the unknown number. The division in the sentence "When a number is divided by four" becomes the fraction \( \frac{x}{4} \).Steps for Creating the Equation:

• Identify operations: Here, division suggests a fraction.
• Focus on the outcome: The phrase "the result is sixty-eight" indicates an equation format where \( \frac{x}{4} = 68 \).
• Ensure balance: An equation must maintain equilibrium between its two sides.

You've now formed the complete equation \( \frac{x}{4} = 68 \). It's like completing a math sentence, transforming words into a mathematical relationship that's understandable and countable.