Problem 301 - Intermediate Algebra

Question

In the following exercises, locate the numbers on a number line. $$ \frac{3}{10}, \frac{7}{2}, \frac{11}{6}, 4 $$

Step-by-Step Solution

Answer

Locate 0.3 between 0 and 1, 3.5 between 3 and 4, 1.8333 between 1 and 2, and 4 at 4.

Step 1: Convert Fractions to Decimal Form

To make it easier to locate the numbers on a number line, convert each fraction to its decimal form:\[\frac{3}{10} = 0.3,\frac{7}{2} = 3.5,\frac{11}{6} \approx 1.8333,4 = 4.0\]

Step 2: Draw a Number Line

Draw a horizontal line and mark evenly spaced intervals. Label these intervals with whole numbers to represent the scale of the number line, such as 0, 1, 2, 3, 4, 5, etc.

Step 3: Locate Each Number on the Number Line

Place each decimal representation of the numbers at the appropriate location on the number line. For example:\(0.3i s b e t w e e n 0\an d 1,3.5i s b e t w e e n 3a n d 4,1.8333i s b e t we en1a n da n da n d1.8 33a n40.4.0 o n t h e m. e s i ta\)

Step 4: Verify the Locations

Ensure that the converted decimal numbers are correctly positioned between the whole numbers on the number line. This verifies the accuracy of their positions.

Key Concepts

fractions to decimals

Understanding how to convert fractions to decimals is vital in solving many math problems. It helps simplify the process of locating numbers on a number line. To convert a fraction to a decimal, divide the numerator (the top number) by the denominator (the bottom number). For instance, to convert \(\frac{3}{10}\) to a decimal, you would divide 3 by 10, which equals 0.3.
Here are the conversions for our exercise:
\(\frac{3}{10} = 0.3\)
\(\frac{7}{2} = 3.5\)
\(\frac{11}{6} \approx 1.8333\)
and 4 remains 4. These decimals make it much easier to work with the numbers on a number line.

locating numbers

Once we have our numbers in decimal form, putting them on a number line becomes straightforward. Start by drawing a horizontal line and marking evenly spaced intervals. Label these intervals with whole numbers to represent the scale, for example, 0, 1, 2, 3...
With our converted decimals, we locate them as follows:
- 0.3 is between 0 and 1.
- 3.5 is between 3 and 4.
- 1.8333 is between 1 and 2.
- 4.0 exactly on 4.
Placing these values accurately will help visualize their positions relative to each other.

step-by-step solutions

Using a step-by-step approach ensures clarity and accuracy. Here’s how to achieve it:

1. **Convert Fractions to Decimals**: As explained earlier, divide the numerator by the denominator to get the decimal form.
2. **Draw a Number Line**: Mark evenly spaced intervals and label them with whole numbers.
3. **Locate Each Number**: Place the decimal values at the correct spots on the number line. For instance, for 0.3, find the point between 0 and 1.
4. **Verify the Locations**: Double-check each number’s placement to ensure they are correctly positioned.

Following these steps allows you to systematically and accurately locate numbers, making for a clean and understandable solution.