Problem 34 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$\frac{9-5}{5-9}$$

Step-by-Step Solution

Answer

The expression simplifies to -1.

Step 1: Evaluate the Numerator

Find the value of the expression in the numerator of the fraction. The numerator is given as \( 9 - 5 \). Calculate \( 9 - 5 = 4 \). So, the numerator simplifies to 4.

Step 2: Evaluate the Denominator

Find the value of the expression in the denominator of the fraction. The denominator is given as \( 5 - 9 \). Calculate \( 5 - 9 = -4 \). So, the denominator simplifies to -4.

Step 3: Simplify the Fraction

Now that we have the simplified numerator and denominator (4 and -4, respectively), simplify the fraction \( \frac{4}{-4} \). This fraction evaluates to -1 because dividing a positive number by its negative counterpart yields -1.

Key Concepts

Fractions

A fraction is a way to express numbers that are not whole. It represents a part of a whole and is written in the form \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. Fractions are important in mathematics because they allow us to work with values that fall between integers.
Understanding fractions is key to making sense of many problems. A fraction essentially tells you how many parts of a certain size you have. For example, if you have \( \frac{1}{2} \) of a cake, this means you have one of two equal parts of the cake.
When working with fractions, it's crucial to remember:

• The numerator signifies how many parts you have.
• The denominator signifies the total number of equal parts the whole is divided into.

When simplifying or operating with fractions, one must pay close attention to these two parts to ensure accurate calculations.

Numerators and Denominators

The numerator and the denominator each play a crucial role in the form and function of a fraction. The numerator, which is the top number, indicates how many pieces of the whole or set are being considered. On the other hand, the denominator, the bottom number, specifies the total number of pieces the whole is divided into.
It's quite useful to analyze numerators and denominators separately when dealing with fractions, like in mathematical operations or simplifications. For instance, in the exercise given, the numerator \(9 - 5\) results in \(4\), while the denominator \(5 - 9\) results in \(-4\).
When evaluating or manipulating fractions:

• Always follow the order of operations for expressions in the numerator or denominator individually first.
• Re-evaluate the fraction based on these simplified values.
• For any division by negative or positive values, consider the sign for your final answer.

This approach not only ensures accuracy but also aids in developing a deeper understanding of fractions.

Simplifying Expressions

Simplifying expressions, particularly those involving fractions, is a fundamental skill in mathematics. It involves reducing an expression to its simplest form, making it easier to understand and use. To simplify a fraction, you generally divide both the numerator and the denominator by their greatest common factor.
In the exercise provided, after determining the simplified values of the numerator and the denominator (as 4 and -4 respectively), you then create the simplified fraction \( \frac{4}{-4} \). This fraction simplifies to -1 because when a positive number is divided by its negative counterpart, the result is negative.
Here are a few tips when simplifying expressions:

• Break down all components separately before combining them.
• Consider the sign of the fraction, especially when the denominator or numerator is negative.
• Simplify fractions by canceling out matching factors in the numerator and the denominator.

By mastering the art of simplification, you'll not only find solutions more efficiently but also gain a clearer understanding of underlying mathematical structures.