Problem 304 - Intermediate Algebra

Question

In the following exercises, locate the numbers on a number line. $$ \frac{2}{5},-\frac{2}{5}, 1 \frac{3}{4},-1 \frac{3}{4}, \frac{8}{3},-\frac{8}{3} $$

Step-by-Step Solution

Answer

Place \(\frac{2}{5}\) at 0.4, \(-\frac{2}{5}\) at -0.4, \(\frac{7}{4}\) at 1.75, \(-\frac{7}{4}\) at -1.75, \(\frac{8}{3}\) at 2.67, and \(-\frac{8}{3}\) at -2.67 on the number line.

Step 1: Understand the Numbers

The provided numbers are \(\frac{2}{5}\), \(-\frac{2}{5}\), \(1 \frac{3}{4}\), \(-1 \frac{3}{4}\), \(\frac{8}{3}\), and \(-\frac{8}{3}\). Some are fractions and some are mixed numbers. All need to be placed on a number line.

Step 2: Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers \(1 \frac{3}{4}\) and \(-1 \frac{3}{4}\) into improper fractions. \(1 \frac{3}{4} = \frac{7}{4}\) and \(-1 \frac{3}{4} = -\frac{7}{4}\).

Step 3: Identify the Position of Each Number

Identify and simplify each number so they can be located on the number line. \(\frac{2}{5}\) remains \(\frac{2}{5}\) which is 0.4. \(-\frac{2}{5}\) remains \(-\frac{2}{5}\) which is -0.4. \(\frac{7}{4} = 1.75\) and \(-\frac{7}{4} = -1.75\). \(\frac{8}{3} \text{ remains as } \frac{8}{3} = 2.67\) and \(-\frac{8}{3} \text{ remains as } -\frac{8}{3} = -2.67\).

Step 4: Draw the Number Line

Draw a straight horizontal line and mark the origin (0). Then, mark equal intervals to represent integers and fractional parts both positive and negative.

Step 5: Locate and Place Numbers on the Number Line

Place each number on the number line based on their identified decimal or fractional value from Step 3. \(\frac{2}{5}\) at 0.4, \(-\frac{2}{5}\) at -0.4, \(\frac{7}{4}\) at 1.75, \(-\frac{7}{4}\) at -1.75, \(\frac{8}{3}\) at 2.67, and \(-\frac{8}{3}\) at -2.67.

Key Concepts

fractions

Fractions are a fundamental concept in mathematics, representing parts of a whole. For example, \(\frac{2}{5}\) indicates that you have 2 out of 5 equal parts of something. Proper fractions have numerators smaller than their denominators. This means their value is between 0 and 1. Improper fractions, such as \( \frac{8}{3} \), have numerators larger than denominators, meaning their value is greater than 1. These fractions can be converted into mixed numbers to simplify reading and locating them on number lines. When locating fractions on a number line, knowing the decimal equivalent helps. For instance, \( \frac{2}{5} \) is equivalent to 0.4 and \( \frac{8}{3} \) to approximately 2.67.

mixed numbers

Mixed numbers combine whole numbers and fractions. For example, \( 1 \frac{3}{4} \) includes a whole part (1) and a fractional part (\( \frac{3}{4} \)). This form makes it easier to understand and visualize amounts greater than one. To work with mixed numbers on a number line, converting them to improper fractions is useful. This conversion is done by multiplying the whole number by the fraction's denominator and adding the numerator: \( 1 \frac{3}{4} = \frac{7}{4} \). Such conversions simplify the process of locating these numbers precisely on a number line.

number line

A number line is a visual tool to represent numbers in an ordered manner. It consists of a horizontal line marked with equally spaced intervals, including positive and negative integers, fractions, and decimals. To display numbers like \( \frac{2}{5},- \frac{2}{5} \), and \( 1 \frac{3}{4} \) correctly, follow these steps:

• Draw a horizontal line and mark the origin (0).
• Divide the line into equal intervals corresponding to integers.
• Identify the decimal or fractional equivalents of each number.
• Carefully place each number at the correct position by measuring the intervals.

This approach ensures that all numbers, whether fractions, mixed numbers, or whole, are accurately represented, making it easier to comprehend their size and relationship to each other.