Understanding and Solving the Mathematical Phrase: Subtract Two-Thirds from Thirty

The phrase 'subtract two-thirds of a number from thirty' can be translated into a mathematical expression. This type of problem requires understanding of basic algebra and arithmetic operations. In this article, we will break down the problem, provide a step-by-step solution, and explain the underlying concepts using clear examples and detailed calculations.

Introduction to the Problem

The problem states: 'Subtract two-thirds of a number from thirty.' This is a straightforward algebraic phrase that can be written as:

Given:

y 30 - (2/3)r

Understand the Variables

In the equation, y represents the result after performing the specified operation. 30 is the constant value we are starting with. r is the variable, representing any real number. The term (2/3)r denotes two-thirds of the variable r.

Step-by-Step Solution

1. Identifying Two-Thirds of a Number

To begin with, let's understand what two-thirds of a number means. If the number is denoted by x, then two-thirds of this number is calculated as:

Calculation:

2/3 * x

2. Subtracting from Thirty

The problem requires us to subtract two-thirds of a number from thirty. Mathematically, this can be represented as:

Expression:

y 30 - (2/3) * x

Alternative Notation and Explanation

Another way to represent the same logic is by directly using the variable r instead of x. This does not change the solution, as the variable is interchangeable:

Given:

y 30 - (2/3)r

Example Calculation

Example 1: Let the number be 15

Using the variable x 15, we calculate two-thirds as:

Step 1: Calculate Two-Thirds

(2/3) * 15 10

Step 2: Subtract from Thirty

y 30 - 10 20

General Case

Generalizing the solution, if we let:

Our number: x

Two-thirds of our number: (2/3)x

Subtract Two-Thirds of our number from Thirty:

30 - (2/3)x

Final Expression:

y 30 - (2/3)x

Conclusion

Understanding and solving the phrase 'subtract two-thirds of a number from thirty' involves a basic application of algebra. By following the outlined steps, we can easily solve such problems and understand the underlying mathematical concepts.

Key Takeaways:

Algebraic Expression: Understanding how to write and solve algebraic expressions is crucial in solving such problems. Basic Operations: Practicing basic arithmetic operations and fractions is essential for these calculations. Variable Interchangeability: Recognizing that variables can be substituted interchangeably does not alter the solution.

For further practice and deeper understanding, consider exploring more similar problems or consulting a comprehensive mathematics textbook.