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Chapter 1: Problem 86

Simplify the given expression. $$ \frac{7+1}{8-4} $$

Short Answer

Expert verified
The simplified expression is 2.

Step by step solution

01

Simplify the Numerator

The numerator of the fraction is \( 7 + 1 \). First, add the two numbers: \( 7 + 1 = 8 \). Replace the numerator with 8.
02

Simplify the Denominator

The denominator of the fraction is \( 8 - 4 \). Subtract the two numbers: \( 8 - 4 = 4 \). Replace the denominator with 4.
03

Simplify the Fraction

The expression now reads \( \frac{8}{4} \). Simplify by dividing the numerator (8) by the denominator (4). \( 8 \div 4 = 2 \). So, the simplified expression is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Numerator and Denominator
Fractions are made up of two parts: a numerator and a denominator. These concepts form the foundation of fraction operations.
The **numerator** is the top part of the fraction, indicating how many parts you have. In the fraction \( \frac{a}{b} \), "a" is the numerator.
The **denominator** is the bottom part of the fraction, showing the total number of equal parts in the whole. Here, "b" is the denominator. Recognizing these components is crucial. When simplifying fractions or performing any operation involving fractions, always identify the numerator and denominator. This understanding helps to accurately manipulate and simplify the fraction.
Applying Basic Arithmetic Operations
To simplify fractions, you often need to use basic arithmetic operations: addition, subtraction, multiplication, and division.
For instance, simplifying \( \frac{7+1}{8-4} \) requires addition in the numerator and subtraction in the denominator.
  • **Addition** is used to simplify the numerator: \( 7 + 1 = 8 \).
  • **Subtraction** is used in the denominator: \( 8 - 4 = 4 \).
  • Finally, **division** simplifies the entire fraction: \( \frac{8}{4} = 2 \).
These straightforward operations simplify the expression effectively. Understanding how to perform each operation is key for not just math classes, but also real-world problem solving.
Building on Prealgebra Concepts
Prealgebra concepts lay the groundwork for understanding and simplifying fractions. They introduce important skills needed for more complex mathematics later on.**Key Prealgebra Skills Include:**
  • **Recognizing Patterns**: Understanding the way numbers relate strengthens simplification processes.
  • **Order of Operations**: Ensures calculations, like in \( \frac{7+1}{8-4} \), are handled correctly.
  • **Basic Computation Skills**: Essential for adding, subtracting, or dividing as seen throughout fraction simplification.
These skills prepare you to handle not just simple expressions but also set you on the path toward solving more advanced mathematical challenges.

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