Problem 69 - College Algebra and Calculus: An Applied Approach

Question

Find the distance between \(a\) and \(b\).\(a=\frac{16}{5}, b=\frac{112}{75}\)

Step-by-Step Solution

Answer

So, the distance between \(a=\frac{16}{5}\) and \(b= \frac{112}{75}\) is \(\frac{128}{75}\).

Step 1: Identify the given points

The points that are given are \(a\) and \(b\). They are \(a=\frac{16}{5}\) and \(b=\frac{112}{75}\) respectively. These will be the two points we will use to calculate the distance.

Step 2: Conversion to common denominator

To facilitate the subtraction, convert both fractions to have the same denominator. From the given values, the least common multiple of 5 and 75 is 75. So, we convert \(\frac{16}{5}\) to \(\frac{240}{75}\).

Step 3: Apply the distance formula

The distance between any two points \(a\) and \(b\) on the number line is given by the absolute difference between them. So the distance between \(a\) and \(b\) can be found using the formula \(|a - b|\). Hence the distance = \(|\frac{240}{75} - \frac{112}{75}|\).

Step 4: Evaluate the Expression

Subtract the values in the absolute value resulting in: \(|\frac{128}{75}|\). The absolute value of a positive number is the number itself, so the distance is \(\frac{128}{75}\).

Key Concepts

Fractions

Fractions represent parts of a whole or a way of expressing numbers that are not whole, using a numerator and a denominator. The fraction's numerator is the top part, indicating how many parts we have. The denominator is the bottom part, showing how many equal parts the whole is divided into.
For example, in the fraction \(\frac{16}{5}\), 16 is the numerator, and 5 is the denominator. This tells us that we have 16 parts, each of which is a fifth of something.

• Fractions can be simplified or converted into equivalent forms to make calculations easier.
• They can also be added, subtracted, multiplied, or divided, but often we'll need to prepare by finding a common denominator first.

Understanding how to work with fractions is crucial when dealing with various math problems, especially when calculating distances or converting between units.

Common Denominator

Finding a common denominator is a technique used when you need to perform operations such as addition or subtraction on two or more fractions. For fractions to be directly added or subtracted, they need to have the same denominator. This ensures that the fractions are part of the same whole.

To find a common denominator:

• Identify the denominators of the fractions you are working with.
• Determine the least common multiple (LCM) of these denominators.
• Convert each fraction to have this common denominator by adjusting the numerator accordingly.

In our example, we had the fractions \(\frac{16}{5}\) and \(\frac{112}{75}\). By finding the LCM of 5 and 75, which is 75, we convert \(\frac{16}{5}\) to \(\frac{240}{75}\). This step prepares the fractions for direct subtraction to find the distance between two points.

Distance Formula

The distance between two points on a number line can easily be calculated using the distance formula. This formula is all about finding the absolute difference between two numbers. The formula is expressed as \(|a - b|\), where \(a\) and \(b\) are the points you are measuring the distance between.

The steps to use the distance formula are straightforward:

• Ensure that both points are expressed as fractions with a common denominator, enabling easy subtraction.
• Subtract the second point from the first to find the difference.
• Take the absolute value of this difference. The absolute value ensures the distance is always a non-negative number, as distance cannot be negative.

In our current exercise, after converting to a common denominator, the distance is calculated as \(|\frac{240}{75} - \frac{112}{75}| = \frac{128}{75}|\). This value represents the distance between \(a\) and \(b\) on a number line.