Problem 59 - Elementary Algebra

Question

Translate the phrases or sentences to mathematical expressions or equations. Two fifths of a number minus five.

Step-by-Step Solution

Answer

Question: Translate the phrase "two fifths of a number minus five" into a mathematical expression. Answer: (2/5)x - 5

Step 1: Identify the components of the phrase

Identify the main components of the phrase: - Two fifths of a number: We will represent this number as x, so this portion becomes (2/5)x - Minus five: This portion can be directly translated to -5.

Step 2: Translate the entire phrase into a mathematical expression

Combine the components found in step 1 to create the mathematical expression: (2/5)x - 5

Key Concepts

Mathematical Expressions

Translating English phrases or sentences into mathematical expressions is a fundamental skill in algebra. A mathematical expression is a combination of numbers, variables, and arithmetic operations. This is crucial for solving word problems in mathematics.

For example, in the given problem "two fifths of a number minus five," we need to represent "a number" with a variable, typically denoted as \(x\). The phrase "two fifths of a number" translates to \(\frac{2}{5}x\), where "of" indicates multiplication. The expression "minus five" simply translates to \(-5\). Putting these parts together gives us the expression \(\frac{2}{5}x - 5\).

Creating mathematical expressions helps simplify complex problems, making them easier to analyze and solve. This skill allows students to transition from words to symbols, bridging the gap between language and mathematics.

Algebraic Translation

Algebraic translation involves converting verbal statements into algebraic expressions or equations. This skill is pivotal in understanding and solving real-world problems. The process can initially be challenging due to the need to accurately interpret language specifics.

In our example, the phrase "two fifths of a number minus five" requires breaking down into parts: "two fifths," "of a number," and "minus five." Each component has a corresponding mathematical operation or symbol. "Two fifths" is \(\frac{2}{5}\), "of" implies multiplication, and "a number" is represented as \(x\). Together, they form \(\frac{2}{5}x\). The "minus five" translates directly, leading to the final expression \(\frac{2}{5}x - 5\).

Mastering algebraic translation helps in forming a strong mathematical foundation, essential for higher mathematics and various applications in science and engineering. It transforms verbal problems into solvable mathematics formats, facilitating easier problem-solving.

Educational Exercises

Educational exercises like translating word problems into mathematical expressions boost comprehension and analytical skills in students. These exercises are designed to enhance logical reasoning, symbol manipulation, and application of concepts learned.

The task of breaking down sentences and translating them into mathematics encourages critical thinking. When students engage in exercises such as our example problem, they learn to identify keywords, recognize mathematical operations and apply them correctly. Such consistency in practice builds confidence and improves performance in not just mathematics but in logical reasoning across various disciplines.

By repeatedly practicing similar problems, students become more adept at understanding the language of mathematics, which is crucial for success in future math courses and real-life problem-solving situations. Educational exercises serve as stepping stones in a student’s learning journey, laying foundational skills for lifelong learning.