Problem 111 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. In each exercise, use the given formula to perform the indicated operation with the two fractions. $$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b} ; \frac{9}{17}-\frac{5}{17}$$

Step-by-Step Solution

Answer

The result of the operation \(\frac{9}{17}-\frac{5}{17}\) is \(\frac{4}{17}\)

Step 1: Identify 'a', 'b', and 'c'

In this problem, 'a' refers to the numerator of the first fraction which is 9, 'b' denotes the common denominator of the two fractions which is 17, and 'c' stands for the numerator of the second fraction which is 5.

Step 2: Subtract 'c' from 'a'

According to the formula, we subtract 'c' from 'a'. So \(a - c = 9 - 5 = 4\). This forms the numerator of our resulting fraction.

Step 3: Form the Answering Fraction

After subtraction, the numerator becomes 4 and 'b', the denominator remains 17. So we have \(\frac{4}{17}\) as the resulting fraction after the subtraction of the given fractions.

Key Concepts

Understanding Numerators in Fractions

In the world of fractions, the numerators are the numbers above the fraction line. They represent the number of parts we have out of a whole when that whole is divided into equal parts. When subtracting fractions, the numerators play a crucial role as they are the values that we manipulate directly.

• For example, in the given fractions, the numerators are 9 and 5.
• To subtract these fractions, we focus on subtracting these two numbers, 9 - 5.
• This operation only affects the numerators since a common denominator is shared.

It's essential to distinguish the numerators from the denominators, as they have different roles and aren't treated the same way in operations such as addition or subtraction of fractions.
This leads us to the next important concept—how these fractions share something that allows for straightforward subtraction: the common denominator.

The Role of a Common Denominator

A common denominator is a shared base for fractions that simplifies their comparison. Think of the denominator as the total number of equal parts a whole is divided into. When two or more fractions have the same denominator, they are easy to work with because each fraction represents parts of the same sized whole.

• In our subtraction problem, both fractions have a denominator of 17. This is our common denominator.
• Since they share this common denominator, we can directly subtract the numerators, simplifying our job significantly.
• The denominator remains unchanged in the operation; only the numerators are adjusted.

So, having a common denominator facilitates direct subtraction and is key to performing fraction operations like addition and subtraction efficiently. Understanding how to spot and use a common denominator helps in carrying out these operations correctly.

Fraction Operations: Subtraction

Fraction operations may seem tricky, but they become straightforward once you understand the process. In subtraction, the similarity in denominators is paramount, as it allows us to focus solely on the numerators. This process requires a few simple steps:

• Verify that the fractions share the same denominator. This makes the subtraction permissible.
• Subtract the numerator of the second fraction (5) from that of the first fraction (9).
• The result provides the new numerator for the difference, which is written over the shared denominator (17).

This simple approach yields the fraction \(\frac{4}{17}\). The denominator, representing the identical division of the whole, remains the same throughout.
Mastering these operations is all about executing each step carefully and understanding the role of each component, particularly the numerators and common denominators.