Problem 302 - Intermediate Algebra

Question

In the following exercises, locate the numbers on a number line. $$ \frac{7}{10}, \frac{5}{2}, \frac{13}{8}, 3 $$

Step-by-Step Solution

Answer

Locate \(\frac{7}{10}\) at 0.7, \(\frac{5}{2}\) at 2.5, \(\frac{13}{8}\) at 1.625, and 3 on the number line.

Step 1: Understand the Number Line

A number line is a straight line where numbers are placed at equal intervals or segments along its length. Both positive and negative numbers can be represented on a number line, but for this exercise, focus only on positive values because all given numbers are positive.

Step 2: Identify Fractions and Convert if Necessary

The given numbers are \(\frac{7}{10}, \frac{5}{2}, \frac{13}{8}, 3\). Understand that \(\frac{7}{10}\) and \(\frac{13}{8}\) are already in simplest form, but it might be helpful to convert \(\frac{5}{2}\) into a mixed number. Since \(\frac{5}{2} = 2 \frac{1}{2}\), this helps in placing it on the number line.

Step 3: Determine Decimal Equivalents

Convert fractions to decimal form to better understand their position on the number line: \(\frac{7}{10} = 0.7\), \(\frac{5}{2} = 2.5\), and \(\frac{13}{8} \approx 1.625\). The number 3 remains as is.

Step 4: Draw the Number Line

Draw a horizontal line with evenly spaced marks and label it with integers. For example: 0, 1, 2, 3, 4. Make sure to include fractional and decimal positions.

Step 5: Locate the Numbers on the Number Line

Place each number on the number line: \(\frac{7}{10}\) goes to the right of 0.5 and before 1. Place \(\frac{5}{2}\) between 2 and 3, closer to 2.5. \(\frac{13}{8}\) should be placed between 1.5 and 2, closer to 1.62. Finally, place 3 exactly on the mark for 3.

Key Concepts

Fractions

Fractions represent parts of a whole number and are written as \(\frac{a}{b}\), where \(a\) is the numerator (the top number) and \(b\) is the denominator (the bottom number).
For example, in \( \frac{7}{10} \), 7 is the numerator and 10 is the denominator. This means that \( \frac{7}{10} \) represents 7 parts out of 10.
Understanding fractions is crucial for accurately placing them on a number line.
When you have fractions like \( \frac{13}{8} \), if the numerator is larger than the denominator, it's an improper fraction. Improper fractions can be converted into mixed numbers for easier visualization.

• Example: \( \frac{5}{2} = 2 \frac{1}{2} \)

This process helps in understanding where the fraction fits on the number line as it breaks down into whole numbers and parts of a whole.

Decimal Conversion

Converting fractions to decimals can make it easier to plot them on a number line. To convert a fraction to a decimal, divide the numerator by the denominator.

• For instance, \( \frac{7}{10} = 0.7 \)
• Similarly, \( \frac{5}{2} = 2.5 \)
• And, \( \frac{13}{8} \approx 1.625 \)

Decimal conversion provides a clear understanding of where a number lies between whole numbers, which is very helpful for placement.
In our exercise, after converting \( \frac{7}{10} \), \( \frac{5}{2} \), and \( \frac{13}{8} \) to 0.7, 2.5, and 1.625 respectively, you can see where they should be located between the whole number intervals on a number line.
This step simplifies the process, especially when dealing with non-standard fractions.

Locating Numbers on the Number Line

A number line is a visual representation of numbers laid out in a straight, horizontal line. It helps in understanding the position and comparison of numbers. Here's how you can locate decimal and whole numbers on a number line:

• Start by drawing a horizontal line and marking it at regular intervals with whole numbers (e.g., 0, 1, 2, 3).
• Between each pair of whole numbers, mark the positions for fractions and decimals.

For our exercise:
- 0.7 lies between 0 and 1, closer to 1.
- 1.625 lies between 1 and 2, closer to 1.5.
- 2.5 lies between 2 and 3, exactly halfway.
- 3 lies exactly on the mark for 3.

Placing these numbers correctly helps in visualizing their relative sizes and learning how to compare different types of numbers.

Understanding how to locate decimal and fractional values on a number line builds a solid foundation for more advanced math concepts.