Problem 63 - Algebra 1

Question

Evaluate the expression for the given value of the variable. (Review 1.1) $$\frac{5}{16}-p \text { when } p=\frac{3}{8}$$

Step-by-Step Solution

Answer

The value of the expression \( \frac{5}{16} - p \) when \( p=\frac{3}{8} \) is \( -\frac{1}{16} \).

Step 1: Substitution

Substitute \( \frac{3}{8} \) for 'p' in the expression. Thus, the initial expression \( \frac{5}{16}-p \) becomes \( \frac{5}{16} - \frac{3}{8} \).

Step 2: Convert to the Same Denominator

To simplify the expression, it is necessary to convert the fractions to have the same denominator. Since 16 is an easy multiply of 8, the expression can be rewritten as \( \frac{5}{16} - \frac{6}{16} \).

Step 3: Simplify the Expression

Now, it is simple to subtract the numerators: \( \frac{5 - 6}{16} = \frac{-1}{16} \).

Key Concepts

Variable Substitution

In algebra, variable substitution is a fundamental step in evaluating algebraic expressions. It involves replacing a given variable with a specific value. For example, in the exercise, the expression \( \frac{5}{16} - p \) requires substituting \( p \) with \( \frac{3}{8} \). This substitution transforms the expression into a more numeric form, making it easier to solve.

• Identify the variable in the expression.
• Replace the variable with the given numerical value.
• Simplify the expression if possible.

This step is crucial because it turns abstract algebraic expressions into concrete numerical problems that are easier to manage. By performing substitution, students familiarize themselves with handling variables and expressions, providing a strong base for more complex algebraic operations.

Fractions

Fractions are numerical quantities that are not whole numbers, expressed as \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. In algebra, fractions are often manipulated by standard operations such as addition, subtraction, multiplication, or division.
Specifically in the given exercise, the fractions \( \frac{5}{16} \) and \( \frac{3}{8} \) represent parts of a whole that need to be subtracted.

• When dealing with fractions, ensure the denominators are the same before carrying out operations like addition or subtraction.
• Fractions can represent real-world problems where parts of a whole need to be evaluated.

Understanding how to work with fractions helps students tackle a wide range of mathematical problems with confidence and clarity.

Denominator

The denominator is the bottom part of a fraction, indicating how many parts make up a whole. In the expression \( \frac{5}{16} - \frac{3}{8} \), it is essential to have the same denominator for meaningful subtraction. Different denominators can complicate the process, and thus they need to be converted to a common denominator.
When converting fractions to have the same denominator:

• Identify a common multiple of the denominators.
• Adjust the fractions so that they share the same denominator.
• Once they have the same denominator, the numerators can be directly subtracted or added.

In this exercise, the conversion of \( \frac{3}{8} \) to \( \frac{6}{16} \) allows the subtraction to occur seamlessly. Mastering this concept leads to a better grasp of fraction operations and prepares students for even more complex mathematical challenges.