Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Find the distance between \(a\) and \(b\).\(a=\frac{1}{4}, b=\frac{11}{4}\)

Step-by-Step Solution

Answer

The distance between \( a \) and \( b \) is \( 2.5 \).

Step 1: Identify the larger and smaller numbers

Here, \( b = \frac{11}{4} \) is larger than \( a = \frac{1}{4} \).

Step 2: Subtract the smaller number from the larger

The distance between \(a\) and \(b\) is found by subtracting \( a = \frac{1}{4} \) from \( b = \frac{11}{4} \), giving us \( \frac{11}{4} - \frac{1}{4} \).

Step 3: Simplify the result

The operation simplifies to \( \frac{10}{4} \). The fraction simplifies further to \( 2.5 \)

Key Concepts

Fractions

When dealing with fractions, it is important to understand that a fraction represents a part of a whole. A fraction is composed of two parts:

• The numerator, which is the number above the line, indicates how many parts we have.
• The denominator, which is the number below the line, shows into how many parts the whole is divided.

For example, in the fraction \( \frac{1}{4} \), the numerator 1 indicates one part, and the denominator 4 implies that a whole is divided into four parts. This fraction expresses one part out of four equal parts of a whole.

Understanding the concept of fractions helps when comparing amounts and performing operations such as addition, subtraction, and especially when finding the distances between numbers in the form of fractions. Comparing fractions like \( \frac{1}{4} \) and \( \frac{11}{4} \) first involves looking at the numerators, given that the denominators are the same.

Subtraction

Subtraction is one of four basic arithmetic operations, and it represents taking one quantity away from another. For fractions, subtraction involves subtracting the numerators while keeping the denominator the same. For example, to subtract \( \frac{1}{4} \) from \( \frac{11}{4} \), you need to:

• Subtract the numerators: 11 - 1 = 10
• Keep the denominator: 4

This results in the fraction \( \frac{10}{4} \).

Before subtracting fractions, it is essential to ensure they have a common denominator. In our example, both fractions \( \frac{1}{4} \) and \( \frac{11}{4} \) share the same denominator, making it straightforward to perform the subtraction. Remember, if the denominators were different, you would need to find a common denominator before subtracting.

Simplification

Simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and work with.

To simplify \( \frac{10}{4} \), you need to find the greatest common factor (GCF) of the numerator and the denominator, which in this case is 2:

• Divide the numerator (10) by 2, resulting in 5
• Divide the denominator (4) by 2, resulting in 2

So, \( \frac{10}{4} \) simplifies to \( \frac{5}{2} \), which can be expressed as a mixed number, 2.5. This final step of simplification is important because it gives a clear, reduced representation of the original calculation, making results easier to interpret and compute further.