Problem 47 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. A number plus six, divided by two, plus five, is forty-three.

Step-by-Step Solution

Answer

Answer: The unknown number is 70.

Step 1: Assign a variable to represent the unknown number

Let x be the unknown number.

Step 2: Break down the phrase into a mathematical equation

We need to translate the phrase "A number plus six, divided by two, plus five, is forty-three" into a mathematical equation. We can do this by following the operations step by step: 1. A number plus six -> x + 6 2. Divided by two -> (x + 6) / 2 3. Plus five -> (x + 6) / 2 + 5 4. Equals forty-three -> (x + 6) / 2 + 5 = 43 So, the resulting equation is: (x + 6) / 2 + 5 = 43

Step 3: Solve the equation for x

To find the value of x, we will isolate x by following these steps: 1. Subtract 5 from both sides of the equation: (x + 6) / 2 = 38 2. Multiply both sides by 2 to remove the fraction: x + 6 = 76 3. Subtract 6 from both sides to isolate x: x = 70 The unknown number, x, is 70.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are like puzzles that use mathematical expressions to relate quantities. They consist of numbers, variables (which represent unknown values), and operations like addition and division. In this problem, we translate a verbal statement into an algebraic equation.
For example, the phrase "a number plus six, divided by two, plus five, is forty-three" is transformed into the equation \( \frac{(x + 6)}{2} + 5 = 43 \). This transformation is crucial in solving problems because it helps us understand the relationship among different quantities.
Once written, the equation provides a clear path to solve for the unknown variable.

The Role of Variable Assignment

In mathematical expressions, variable assignment is the method of choosing a letter to represent an unknown quantity—usually using symbols like \( x \), \( y \), or \( z \). This not only simplifies the problem-solving process but also helps us clearly denote these unknowns in the equation.
In the given exercise, we use \( x \) to represent the unknown number mentioned in the phrase. Assigning a variable is the first step in translating a word problem into an algebraic format.
This makes it easier to manipulate and solve the equation systematically.

Techniques in Problem Solving

Problem-solving in mathematics often requires systematic techniques that can be learned and practiced. Let's break down the approach used here:

• Translation: Convert the verbal problem into a mathematical equation, as discussed earlier.
• Isolation: The goal is to isolate the variable \( x \) to solve for its value. This involves performing operations that balance both sides of the equation.
• Simplification: By performing arithmetical operations step by step, we simplify the equation to make \( x \) clear.

The logical progression from understanding the phrase to writing and solving the equation is a key technique in mathematical problem-solving.

Performing Arithmetical Operations

Arithmetical operations are the basic building blocks of solving equations. In the context of our problem, these operations include adding, subtracting, multiplying, and dividing numbers.

These steps need careful execution:1. Subtraction: First, subtract 5 from both sides of the equation, resulting in \( \frac{(x + 6)}{2} = 38 \).2. Multiplication: Then, multiply each side by 2 to eliminate the fraction, giving us \( x + 6 = 76 \).3. Subtraction again: Finally, subtract 6 from both sides to isolate \( x \), which reveals that \( x = 70 \).
By mastering these operations, students can efficiently navigate through more complex algebraic equations.