Problem 44 - College Algebra

Question

Simplify the rational expression. $$ \frac{\frac{x}{4}-\frac{p}{8}}{p} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2x - p}{8p} \).

Step 1: Identify the Common Denominator

Examine the expression \( \frac{\frac{x}{4} - \frac{p}{8}}{p} \). Inside the numerator, we have two fractions: \( \frac{x}{4} \) and \( \frac{p}{8} \). The common denominator for these fractions is 8.

Step 2: Rewrite Fractions with Common Denominator

Rewrite \( \frac{x}{4} \) as a fraction over 8 to combine it with \( \frac{p}{8} \). Multiplying the numerator and denominator of \( \frac{x}{4} \) by 2 gives us \( \frac{2x}{8} \). Thus, the expression becomes \( \frac{\frac{2x}{8} - \frac{p}{8}}{p} \).

Step 3: Combine Fractions in the Numerator

Combine the fractions in the numerator. We now have \( \frac{2x - p}{8} \) over \( p \). This results in the expression \( \frac{\frac{2x - p}{8}}{p} \).

Step 4: Simplify the Expression

To simplify, divide \( \frac{2x - p}{8} \) by \( p \), which is equivalent to multiplying by the reciprocal of \( p \). This gives \( \frac{2x - p}{8} \times \frac{1}{p} = \frac{2x - p}{8p} \).

Key Concepts

Fractions

Fractions are mathematical expressions that represent a part of a whole. A fraction consists of two parts: a numerator and a denominator. The numerator is the top number and indicates how many parts of the whole we have. The denominator is the bottom number and shows into how many equal parts the whole is divided.

• For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator.
• Fractions can represent quantities less than, equal to, or greater than one. For instance, \( \frac{3}{4} \) represents a part less than a whole, whereas \( \frac{4}{4} \) represents a whole, and \( \frac{5}{4} \) is greater than one.

Understanding fractions is crucial because they are used in many areas of everyday life and in more advanced math, such as rational expressions and algebra.

Common Denominator

When working with fractions, particularly when adding or subtracting them, having a common denominator is essential. A common denominator is a shared multiple of the denominators of two or more fractions.

• For instance, if you have the fractions \( \frac{x}{4} \) and \( \frac{p}{8} \), you need to find a common denominator to combine them.
• In this case, the least common multiple of 4 and 8 is 8. So, each fraction should be adjusted to have 8 as the denominator.
• This involves multiplying the numerator and the denominator of \( \frac{x}{4} \) by 2 to create \( \frac{2x}{8} \), making it compatible with \( \frac{p}{8} \).

Once the fractions share a common denominator, they can be easily added or subtracted by combining their numerators.

Simplifying Rational Expressions

Rational expressions are fractions that have algebraic expressions in the numerator and/or the denominator. Simplifying these expressions involves reducing them to their simplest form while ensuring that the operations and overall value remain unchanged.

• The process often involves factorization and finding common terms to cancel.
• In the original expression \( \frac{\frac{2x - p}{8}}{p} \), simplification occurs by multiplying the numerator by the reciprocal of the denominator \( p \).
• This results in the expression \( \frac{2x - p}{8p} \), where 2x - p is simplified over 8p.

Always check for opportunities to further reduce fractions by canceling common factors in the numerator and denominator, ensuring the expression is as simple as possible.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. They form the building blocks for more complex mathematical concepts.

• For example, \( 2x - p \) is an algebraic expression involving variables \( x \) and \( p \).
• These expressions can be used in a variety of mathematical problems, including the forming of equations and solving rational expressions.
• When handling algebraic expressions, it is essential to follow the proper operation orders, or BODMAS/BIDMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction).

Practicing operations with algebraic expressions helps in understanding how different pieces of algebra fit together to solve equations and simplify expressions.