Problem 59 - Prealgebra

Question

The number line below extends from 0 to 2, with the segment from 0 to 1 and the segment from 1 to 2 each divided into 8 equal parts. Locate each of the following numbers on this number line. $$\frac{3}{4}$$

Step-by-Step Solution

Answer

\( \frac{3}{4} \) is located at the 6th mark between 0 and 1 on the number line.

Step 1: Understand the Number Line

The number line is divided into two segments: 0 to 1, and 1 to 2. Each of these segments is further divided into 8 equal parts. Therefore, each part or division on the number line represents a fraction of \( \frac{1}{8} \).

Step 2: Convert to Eighths

To locate \( \frac{3}{4} \) on the number line, convert it into a fraction with a denominator of 8. Since each segment is divided into 8 parts, we need to express \( \frac{3}{4} \) as an equivalent fraction with 8 as the denominator. We do this by multiplying both the numerator and the denominator of \( \frac{3}{4} \) by 2, resulting in \( \frac{6}{8} \).

Step 3: Locate on the Number Line

Now that we have \( \frac{3}{4} \) expressed as \( \frac{6}{8} \), we can locate this fraction on the number line. Start at 0, and move 6 segments to the right, since each segment represents \( \frac{1}{8} \). This will place you at \( \frac{6}{8} \), which is the same as \( \frac{3}{4} \), between 0 and 1 on the number line.

Key Concepts

Fractions

Fractions are a way of expressing numbers that are not whole. They consist of two parts: the numerator and the denominator.
- The **numerator** is the top part of the fraction and signifies how many parts we are considering. - The **denominator** is the bottom part and indicates how many equal parts the whole is divided into.
For example, the fraction \( \frac{3}{4} \) means out of 4 equal parts, we have 3. Understanding fractions helps in various mathematical operations, and they're essential in measuring, converting numbers, and understanding quantities.
To visualize fractions, think of them as slices of a pie. If a pie is cut into 4 pieces, each piece is one-fourth of the pie. If you have 3 pieces, that's \( \frac{3}{4} \) of the whole pie.

Equivalent Fractions

Equivalent fractions may look different but they have the same value. If you multiply or divide the numerator and the denominator of a fraction by the same number, you get an equivalent fraction. An important rule is that the operations must not change the value.

• For example, \( \frac{1}{2} \) is equivalent to \( \frac{2}{4} \) because when both the numerator (1) and the denominator (2) are multiplied by 2, the value remains equal.
• Similarly, \( \frac{3}{4} \) can be converted to \( \frac{6}{8} \) by multiplying both the numerator and the denominator by 2.

This is particularly handy when working with number lines or comparing fractions as it makes interpretation intuitive. Finding equivalent fractions can be essential in performing operations like adding and subtracting fractions with different denominators.

Number Line Segments

A number line is a visual representation of numbers in a straight line, making it easier to understand their position and value relative to each other.
On a number line, each segment or division represents a consistent fraction of a number. For instance, consider a number line from 0 to 2, divided into segments of 8 equal parts between each whole number. Here, each segment represents \( \frac{1}{8} \).
Locating a fraction like \( \frac{3}{4} \) involves converting it to an equivalent fraction that matches the number line segments. By converting \( \frac{3}{4} \) to \( \frac{6}{8} \), you can easily identify the correct position by counting 6 segments starting from 0. This approach enhances understanding of scaling and partitioning on a number line and emphasizes the role of fractions as measuring tools.