Problem 33 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. One tenth of a number is that number less one.

Step-by-Step Solution

Answer

Question: Translate the phrase "one-tenth of a number is equal to that number less one" into a mathematical equation. Answer: \[\frac{1}{10}x = x - 1\]

Step 1: Identify the unknown number

Let x be the unknown number.

Step 2: Express one-tenth of the number

One-tenth of the number x can be written as: \[\frac{1}{10}x\]

Step 3: Express the number less one

The number x less one can be written as: \[x - 1\]

Step 4: Set up the equation

According to the phrase, one-tenth of a number is equal to that number less one. So, we can set up the equation as: \[\frac{1}{10}x = x - 1\] So this is the final equation translated from the given phrase: \[\frac{1}{10}x = x - 1\]

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is crucial for solving algebra problems. An algebraic expression is a mathematical phrase that can include numbers, variables (representing numbers), and operation symbols. For instance, when the problem states 'one tenth of a number', we think of dividing a number by ten, which in algebraic terms is expressed as \( \frac{1}{10}x \). There's no equal sign in an algebraic expression, which distinguishes it from an equation. A firm grasp of how to create and manipulate these expressions is essential for effective problem solving in algebra.

Mathematical Equations

Moving from expressions to mathematical equations, an equation is a statement that two things are equal. It consists of two expressions separated by an equal sign (\( = \)). A simple algebraic equation, as seen in the exercise, \( \frac{1}{10}x = x - 1 \) suggests a balance; whatever is on one side of the equal sign has the same value as what's on the other side. Understanding how to structure an equation from a verbal statement is a key skill in algebra that bridges the gap between real-world scenarios and mathematical analysis.

Algebra Problem Solving

The process of algebra problem solving often involves translating words into mathematical symbols, much like a new language. The step-by-step solution provided demonstrates this approach. Firstly, an unknown quantity is identified, which we often denote with a variable such as 'x'. After that, mathematical operations described in words are converted into symbols (e.g., 'less one' becomes '-1'). Setting up the equation from these pieces then allows mathematicians to solve for the unknown, revealing the underlying relationships between the variables and constants involved. Cultivating the skill to interpret and solve these problems is at the heart of elementary algebra.

Elementary Algebra

Finally, elementary algebra lays the groundwork for all subsequent mathematics education. It introduces the basic concepts of variables and functions, and it teaches students how to manipulate these variables to solve equations. From the problem presented, understanding the basic operations (addition, subtraction, multiplication, division) and their corresponding algebraic expressions is fundamental. Elementary algebra also emphasizes the importance of both sides of an equation being balanced, a concept that extends into more advanced areas of mathematics. Through practice and patience, students will not only learn to solve equations but also to appreciate the logical structure and beauty of mathematics.