Problem 69 - Algebra and Trigonometry

Question

\(67-70\) . Find the distance between the given numbers. (a) 2 and 17 (b) \(-3\) and 21 (c) \(\frac{11}{8}\) and \(-\frac{3}{10}\)

Step-by-Step Solution

Answer

(a) 15, (b) 24, (c) \( \frac{67}{40} \).

Step 1: Understand the Concept of Distance

The distance between two numbers on a number line is calculated by finding the absolute value of their difference. Absolute value is used because distance is always positive.

Step 2: Calculate the Distance for (a)

To find the distance between 2 and 17, use the formula: \[ \text{Distance} = |2 - 17| = |-15| = 15 \] Thus, the distance between 2 and 17 is 15.

Step 3: Calculate the Distance for (b)

To find the distance between -3 and 21, use the formula: \[ \text{Distance} = |-3 - 21| = |-24| = 24 \] Thus, the distance between -3 and 21 is 24.

Step 4: Calculate the Distance for (c)

To find the distance between \( \frac{11}{8} \) and \( -\frac{3}{10} \), first find their difference: \[ \text{Difference} = \frac{11}{8} - \left(-\frac{3}{10}\right) = \frac{11}{8} + \frac{3}{10} \]Find a common denominator to perform the addition. The least common multiple of 8 and 10 is 40, so:\[ \frac{11}{8} = \frac{55}{40} \quad \text{and} \quad \frac{3}{10} = \frac{12}{40} \]Now, add these fractions:\[ \frac{55}{40} + \frac{12}{40} = \frac{67}{40} \]Finally, take the absolute value of the result to find the distance:\[ \text{Distance} = \left| \frac{67}{40} \right| = \frac{67}{40} \] Thus, the distance between \( \frac{11}{8} \) and \( -\frac{3}{10} \) is \( \frac{67}{40} \).

Key Concepts

Absolute Value

The absolute value of a number is a way to describe how far that number is from zero on the number line, regardless of direction. In simpler terms, it is the non-negative value of a number without regard to its sign. For example, both 5 and -5 have the same absolute value, which is 5, because they are both five units away from zero.

• Absolute value is denoted by two vertical lines around the number, such as \(|x|\).
• The concept is crucial when calculating distance, as distance is never negative.

Hence, when you're finding the distance between two numbers, you always calculate the absolute value of their difference: \[ \text{Distance} = |a - b| \] where \( a \) and \( b \) are any two numbers.

Number Line

A number line is a straight line that visually represents numbers at spaced intervals. It is a simple yet powerful tool to understand the order and distance between numbers. Each number on the line is represented by a point, and the distance between points can be measured to find how far apart the numbers are.

• All numbers increase to the right and decrease to the left, with zero generally placed somewhere in the middle.
• Using a number line facilitates understanding operations like addition, subtraction, and identifying negative and positive numbers.

A number line helps in visualizing concepts like:

• Understanding absolute value as the distance from zero.
• Finding the distance between two numbers, which translates to finding how many units or spaces lie between two points.

Try placing numbers like 2 and 17 or \(-3\) and 21 on a number line to see how far apart they are from each other and from zero.

Fractions

Fractions represent parts of a whole and can be sometimes tricky to work with, especially when finding distances. A fraction consists of a numerator and a denominator, where the numerator represents how many parts we have, and the denominator indicates the total number of equal parts the whole is divided into. For example, \( \frac{3}{4} \) denotes three parts of a whole divided into four equal parts.
To calculate the distance between two fractions, you must first determine their difference and then find the absolute value of that difference. Here's how:

• Ensure the fractions have a common denominator so that you can easily perform subtraction or addition.
• Convert fractions to have the same denominator if necessary.
• Perform the arithmetic operation (addition or subtraction).
• Find the absolute value of the result to get the distance.

This approach ensures a straightforward calculation and increases accuracy when working with fractional numbers.