Problem 111 - Introductory and Intermediate Algebra for College Students 4th

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 subtraction \(\frac{9}{17}-\frac{5}{17}\) is \(\frac{4}{17}\)

Step 1: Identify the Fractions

The two fractions that are intended to be subtracted are \( \frac{9}{17} \) and \( \frac{5}{17} \). Here, 9 and 5 are the numerators while 17 is the common denominator.

Step 2: Apply the Formula

A formula is supplied in the problem, telling us how to subtract fractions that have the same denominator: \(\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}\). Replace a, b and c with the corresponding values from our fractions. In this case, a is 9, b is 17 and c is 5. This results in \(\frac{9-5}{17}\).

Step 3: Subtract the Numerators

By subtracting the numerators, we find the answer to the fraction. So, \(\frac{9-5}{17} = \frac{4}{17}\).

Key Concepts

Common Denominator

When it comes to fraction subtraction, having a common denominator is vital. The denominator is the bottom number in a fraction, which shows how many equal parts make up a whole. To subtract two fractions, they must have the same denominator. This common denominator ensures that we are working with fractions that describe parts of the same size.

Imagine you have two pizzas. One has 9 slices, and another has 5, all cut from the same pizza with 17 slices in total. Here, 17 is the common denominator. Both fractions, \(\frac{9}{17}\) and \(\frac{5}{17}\), are parts of pizzas cut into the same number of slices. It allows us to directly subtract \(9 - 5\) slices because each slice, or part, is equivalent.

Numerators

The numerator is the top number in a fraction. It indicates how many parts of the whole we have. Understanding the role of numerators is crucial in performing fraction operations like addition and subtraction.

In the exercise, \(\frac{9}{17}\) and \(\frac{5}{17}\) have numerators of 9 and 5, respectively. These numerators tell us that, with a common denominator of 17, one fraction represents 9 parts of the 17, and the other represents 5 parts.

• The numerator is essential because it indicates quantity in relation to the denominator.
• In the given formula for fraction subtraction, the operation occurs in the numerators: \(a - c\).

So, the focus when subtracting fractions is on those numerators to find the difference while the common denominator stays the same.

Subtracting Fractions

Subtracting fractions may seem daunting, but it boils down to simple steps when the denominators are common.

First, ensure the fractions have the same denominator. Once confirmed, keep that common denominator and look directly at the numerators. By applying the subtraction operation to the numerators, you can quickly find the difference between the two fractions.

Using the formula \(\frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}\), we highlight the critical step:

• Subtract the numerators: In \(\frac{9}{17} - \frac{5}{17}\), calculate \(9 - 5 = 4\).
• Keep the denominator the same: The common denominator remains 17.

So, the result is \(\frac{4}{17}\). The method simplifies because the fractions share a common denominator, enabling direct subtraction of numerators.