Problem 109 - Introductory Algebra for College Students

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 number `x` which satisfies the algebraic equation derived from the given English phrase is x = 44.8

Step 1: Understand the statement

The problem shortly translates to: if the number is denoted by `x`, then the sum of \(\frac{1}{7}\) of `x` and \(\frac{1}{8}\) of `x` gives us 12.

Step 2: Translate to algebraic equation

The English statement can be translated into an algebraic equation as follows: \(\frac{1}{7}x + \frac{1}{8}x = 12\)

Step 3: Simplify the equation

We can simplify the left side of the equation by finding a common denominator for \(\frac{1}{7}\) and \(\frac{1}{8}\), which is 56. This simplifies our equation to: \(\frac{8}{56}x + \frac{7}{56}x = 12 \Rightarrow \frac{15}{56}x = 12\)

Step 4: Solve for x

Next, determine `x` by dividing 12 by \(\frac{15}{56}\) to get: `x = \frac{12 * 56}{15} = 44.8`

Key Concepts

Solving Equations

Solving equations is like finding the value for a missing piece in a puzzle, where the puzzle is the equation. An equation is a mathematical statement that asserts the equality of two expressions. In this context, it involves understanding how different parts of an equation can be manipulated to find the value of the unknown variable. Often, the variable is denoted by letters like \( x \), \( y \), or \( z \).

When solving equations, we follow these general steps:

• Isolate the variable: Work to get the variable by itself on one side of the equation. This often involves adding, subtracting, multiplying, or dividing both sides by the same number.
• Simplify: Use mathematical operations to simplify both sides of the equation if needed.
• Check: Substitute your solution back into the original equation to verify that it results in a true statement.

Understanding these steps will make finding a solution easier. In our example, we solved the equation \( \frac{15}{56}x = 12 \) by firstly figuring out \( x \) in terms of constants.Once the equation was simplified, solving involved dividing both sides of the equation to retrieve the value for \( x \). This value, \( x = 44.8 \), is then the solution to the equation.

Fractions

Fractions represent a part of a whole and consist of a numerator (top number) and a denominator (bottom number). Understanding fractions is vital, as they appear frequently in algebraic expressions and equations. In the context of this exercise, they express parts of the unknown number, \( x \).

When working with fractions, being able to translate them into common terms is important:

• Equivalent Fractions: Fractions with different numerators and denominators that express the same value. For instance, \( \frac{2}{4} = \frac{1}{2} \).
• Common Denominator: A shared multiple of the denominators of two or more fractions. In this exercise, we used 56 to combine \( \frac{1}{7} \) and \( \frac{1}{8} \) into a single fraction.
• Addition/Subtraction of Fractions: First change each fraction to the common denominator, then add or subtract the numerators.

In this exercise, upon finding a common denominator of 56, the fractions were rewritten as \( \frac{8}{56}x \) and \( \frac{7}{56}x \), making addition straightforward.

Translation from Words to Algebra

Translation from words to algebra refers to converting verbal statements into mathematical expressions or equations. This skill is essential for solving many real-life mathematical problems, as it involves identifying keywords and understanding their mathematical significance.

To successfully translate words into algebra:

• Identify the Unknown: Determine what the unknown quantity is in the problem and assign it a variable, often denoted as \( x \).
• Recognize Mathematical Operations: Words like "sum," "difference," "product," and "quotient" correspond to operations like addition, subtraction, multiplication, and division.
• Write the Expression or Equation: Using the identified operations and the unknowns, write a matching mathematical statement. For example, "the sum of \( \frac{1}{7} \) of a number and \( \frac{1}{8} \) of that number" becomes \( \frac{1}{7}x + \frac{1}{8}x \).

In the given exercise, the statement "The sum of \( \frac{1}{7} \) of a number and \( \frac{1}{8} \) of that number gives 12" translates to the algebraic equation \( \frac{1}{7}x + \frac{1}{8}x = 12 \). Recognizing this translation is crucial in setting the stage for solving the equation.