One Less Than Twice A Number

One Less Than Twice a Number: Exploring Mathematical Concepts and Applications

The seemingly simple phrase "one less than twice a number" hides a wealth of mathematical concepts and applications. This seemingly straightforward expression forms the basis for numerous algebraic equations, word problems, and even more advanced mathematical explorations. Let's delve into this concept, exploring its various interpretations and practical uses.

Understanding the Core Concept

At its heart, "one less than twice a number" is a concise way of expressing an algebraic relationship. Let's break it down:

• A number: This represents an unknown quantity, typically denoted by a variable like x, y, or n.
• Twice a number: This means multiplying the number by two (2 * x, 2 * y, or 2 * n).
• One less than twice a number: This implies subtracting one from the result of doubling the number (2x - 1, 2y - 1, or 2*n - 1).

Therefore, the phrase translates directly into the algebraic expression: 2x - 1 (assuming 'x' represents the unknown number). This simple expression is the cornerstone for solving many mathematical problems.

Translating Words into Equations: Word Problems

A significant application of "one less than twice a number" lies in solving word problems. These problems often present real-world scenarios that require translating verbal descriptions into mathematical equations. Let's look at a few examples:

Example 1: The Age Problem

• Problem: John's age is one less than twice his sister's age. If John is 17 years old, how old is his sister?

• Solution:

  • Let's represent his sister's age with the variable x.
  • "Twice his sister's age" translates to 2x.
  • "One less than twice his sister's age" becomes 2x - 1.
  • We know John's age is 17, so we set up the equation: 2x - 1 = 17.
  • Solving for x:
    • Add 1 to both sides: 2x = 18
    • Divide both sides by 2: x = 9
  • Therefore, John's sister is 9 years old.

Example 2: The Geometry Problem

• Problem: The length of a rectangle is one less than twice its width. If the perimeter of the rectangle is 34 cm, find the length and width.

• Solution:

  • Let's represent the width with the variable w.
  • The length is "one less than twice its width," which translates to 2w - 1.
  • The perimeter of a rectangle is given by the formula: P = 2(length + width).
  • Substituting the expressions for length and width, we get: 34 = 2((2w - 1) + w)
  • Simplifying the equation: 34 = 2(3w - 1)
  • Dividing both sides by 2: 17 = 3w - 1
  • Adding 1 to both sides: 18 = 3w
  • Dividing both sides by 3: w = 6
  • Therefore, the width is 6 cm.
  • The length is 2w - 1 = 2(6) - 1 = 11 cm.

Example 3: The Profit Problem

• Problem: A company's profit is one less than twice its revenue. If the profit is $10,000, what is the revenue?

• Solution:

  • Let's represent the revenue with the variable r.
  • The profit is "one less than twice its revenue," which translates to 2r - 1.
  • We know the profit is $10,000, so we set up the equation: 2r - 1 = 10000
  • Solving for r:
    • Add 1 to both sides: 2r = 10001
    • Divide both sides by 2: r = 5000.5
  • Therefore, the revenue is $5000.50.

These examples demonstrate the practical application of translating verbal descriptions into algebraic equations, highlighting the importance of understanding the meaning of "one less than twice a number."

Beyond Basic Algebra: Advanced Applications

The expression "one less than twice a number" isn't limited to simple algebraic equations. It can appear in more complex scenarios:

Sequences and Series

The expression could define a term in a sequence. For instance, a sequence could be defined as a_n = 2n - 1, where a_n represents the nth term. This would generate the sequence 1, 3, 5, 7… (odd numbers).

Functions and Graphs

The expression can be represented as a function, f(x) = 2x - 1. This function is a linear function with a slope of 2 and a y-intercept of -1. Graphing this function would reveal a straight line. Analyzing the slope and intercept provides valuable insights into the function's behavior.

Inequalities

Instead of an equation, the expression might form part of an inequality. For example: 2x - 1 > 5. Solving this inequality would yield x > 3, meaning any value of x greater than 3 would satisfy the inequality.

Quadratic Equations and Beyond

While not directly apparent, the expression can be embedded within more complex equations. For instance, a quadratic equation might involve (2x - 1)² = 9. Solving this requires expanding the expression and applying quadratic equation solving techniques.

Practical Applications in Real Life

Beyond textbook problems, "one less than twice a number" finds its way into various real-world scenarios:

• Pricing Strategies: Businesses might use this type of calculation to determine pricing, considering factors like cost and desired profit margin.

• Resource Allocation: In project management or resource allocation, similar calculations can help distribute resources effectively.

• Scientific Modeling: Simple linear relationships like this can be components of larger scientific models, approximating real-world phenomena.

• Computer Programming: The concept is fundamental in programming logic and algorithm design.

Strengthening Mathematical Skills: Practice Problems

To solidify your understanding, consider the following practice problems:

1. A mother is one less than twice her daughter's age. If the mother is 37, how old is the daughter?

2. The area of a triangle is one less than twice the base. If the area is 29 square units and the height is 10 units, what is the base of the triangle? (Remember: Area of a triangle = (1/2) * base * height)

3. The cost of a laptop is one less than twice the cost of a tablet. If the total cost of the laptop and tablet is $1200, what is the individual cost of each item?

4. Solve the inequality: 2x - 1 < 7

5. Solve the equation: (2x - 1)² = 25

These exercises will enhance your ability to translate word problems into algebraic expressions and solve for unknowns, demonstrating a practical grasp of "one less than twice a number."

Conclusion: A Foundation for Mathematical Understanding

The expression "one less than twice a number" serves as a fundamental building block in algebra and beyond. Its seemingly simple nature belies its importance in solving various mathematical problems and its relevance in real-world applications. By understanding this concept thoroughly, you build a strong foundation for tackling more complex mathematical challenges and applying mathematical thinking to diverse situations. Mastering this basic algebraic concept unlocks a broader understanding of mathematical relationships and their practical uses.