Problem 109 - Introductory and Intermediate Algebra for College Students 4th

Question

Translate from English to an algebraic expression or equation, whichever is appropriate. Let the variable \(x\) represent the number. The sum of \(\frac{1}{7}\) of a number and \(\frac{1}{8}\) of that number gives 12.

Step-by-Step Solution

Answer

The algebraic equation that represents the given English statement is: \(\frac{1}{7}x + \frac{1}{8}x = 12\).

Step 1: Identify the Unknown

Let \(x\) be the number we are looking for. In algebra, it's common to use a letter to represent an unknown quantity we are trying to solve for.

Step 2: Convert the words to Algebra

The sum of two quantities can be represented by an addition operation between these quantities. So, the algebraic expression for '\(\frac{1}{7}\) of a number and \(\frac{1}{8}\) of that number' is '\(\frac{1}{7}x + \frac{1}{8}x\)'.

Step 3: Translate the complete sentence into an Equation

Now, the full statement 'The sum of \(\frac{1}{7}\) of a number and \(\frac{1}{8}\) of that number gives 12' can be translated to '\(\frac{1}{7}x + \frac{1}{8}x = 12\)'.

Key Concepts

Understanding Equations

Equations are like scales trying to stay balanced. They consist of two sides, connected by an equals sign, symbolizing that both sides have the same value. In the context of our exercise, we create an equation to show the relationship between two expressions that sum up to 12.
When translating a word problem into an equation, look for keywords like 'sum,' 'product,' 'difference,' or 'quotient.' These suggest operations like addition, multiplication, subtraction, and division. Here, the keyword was 'sum,' indicating addition.
To form the equation from the given problem, we equate the known result (12) to the algebraic expressions derived from the fractions, which are parts of the unknown number. This forms the equation \( \frac{1}{7}x + \frac{1}{8}x = 12 \). It's like saying one part of the number plus another part adds up to a total sum of 12.

Grasping Variable Representation

Variable representation involves choosing a letter to denote an unknown value in an equation or expression. Often, the letter chosen is \(x\), but any letter can work. This technique allows flexibility in dealing with unknowns and provides a way to abstractly discuss numbers without specifying them.
In our problem, we've used \(x\) as the variable to represent 'a number' mentioned in the sentence. Using a variable turns a word phrase into an algebraic language format, making it easier to manipulate and solve mathematically.
The phrase 'of a number' in our problem suggests multiplication with the variable \(x\). So, "\(\frac{1}{7}\) of a number" converts to \(\frac{1}{7}x\), and likewise for "\(\frac{1}{8}\) of a number," becoming \(\frac{1}{8}x\). This symbolic representation lays the foundation for forming and solving equations.

Fractional Coefficients in Equations

Fractional coefficients are simply fractions used in multiplying a variable within an algebraic expression. These coefficients provide specific parts or proportions of the variable's value.
In our equation \( \frac{1}{7}x + \frac{1}{8}x = 12 \), the coefficients \(\frac{1}{7}\) and \(\frac{1}{8}\) show that we're dealing with fractions of the variable \(x\). These fractions indicate specific portions of the unknown number.
When you encounter fractional coefficients, the first step is often to find a common denominator to simplify solving the equation. This can make addition or subtraction straightforward as these terms can then be combined easily. Fractional coefficients are a nifty algebraic tool for dealing with proportional relationships in equations.