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Chapter 1: Problem 691

Locate the numbers on a number line. $$ \frac{3}{4}, \frac{8}{5}, \frac{10}{3} $$

Short Answer

Expert verified
Convert to decimals: 0.75, 1.6, 3.33. Plot and label these points on the number line.

Step by step solution

01

Understand the problem

First, identify the fractions that need to be located on the number line: \ \( \frac{3}{4}, \frac{8}{5}, \frac{10}{3} \ \).
02

Convert fractions to decimals

Convert each fraction to a decimal to make it easier to locate them on the number line. \ \( \frac{3}{4} = 0.75 \ \), \ \( \frac{8}{5} = 1.6 \ \), \ \( \frac{10}{3} \approx 3.33 \ \).
03

Draw a number line

Draw a horizontal line and mark points for at least 0, 1, 2, 3, and 4, as the largest decimal value is around 3.33.
04

Locate the decimals

Plot the decimal values on the number line: 0.75 between 0 and 1, 1.6 between 1 and 2 (closer to 2), and 3.33 a bit after 3.
05

Label the fractions

Label the points on the number line with their corresponding fractions: \ \( 0.75 \rightarrow \frac{3}{4} \ \), \ \( 1.6 \rightarrow \frac{8}{5} \ \), \ \( 3.33 \rightarrow \frac{10}{3} \ \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convert Fractions to Decimals
To locate fractions like \(\frac{3}{4}\), \(\frac{8}{5}\), and \(\frac{10}{3}\) on a number line, converting them to decimals is often helpful. This makes it easier to understand their exact positions on the line.

Let's break down the conversion process:
  • Divide the numerator (top number) by the denominator (bottom number).
  • For \(\frac{3}{4}\), divide 3 by 4 to get 0.75.
  • For \(\frac{8}{5}\), divide 8 by 5 to get 1.6.
  • For \(\frac{10}{3}\), divide 10 by 3 to get approximately 3.33.

Remember, you might encounter repeating decimals or approximation signs (鈮�). In such cases, decide an appropriate level of precision. Once you have these decimals, plotting them becomes straightforward.
Plotting Decimals
After you have converted the fractions to decimals, the next step is to plot them on the number line. This requires understanding the position of each decimal between whole numbers.

Here's how to plot them:
  • Identify the main intervals on the number line, for example 0, 1, 2, 3, and so on.
  • For 0.75, it falls between 0 and 1. It鈥檚 closer to 1 as \(\frac{3}{4}\) is more than half.
  • For 1.6, it鈥檚 between 1 and 2. It鈥檚 closer to 2 since 1.6 is well past the halfway mark of these two values.
  • For 3.33, it鈥檚 just over 3 and falls between 3 and 4. Since 3.33 is slightly more than 3, it鈥檒l be just past 3.

This way, you can easily place the decimals at their correct locations.
Number Line Representation
A number line helps visualize the relative positions of numbers, whether they are whole numbers, fractions, or decimals. Here's a stepwise guide to effectively represent these:

  • Draw a horizontal line and mark the major points (0, 1, 2, 3, etc.).
  • Identify the ranges where the decimals fall and mark these points.
  • For \(\frac{3}{4}\) or 0.75, mark it three-fourths of the way from 0 to 1.
  • For \(\frac{8}{5}\) or 1.6, place it a little past halfway between 1 and 2.
  • For \(\frac{10}{3}\) or 3.33, locate it just a bit over 3 on the number line.

Label each decimal point with its corresponding fraction. This helps in visualizing and understanding where each fraction lies relative to whole numbers and between other fractions or decimals. Using a number line is a great way to see the comparative size of numbers at a glance.

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