Problem 61 - 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. $$\frac{3}{2}$$

Step-by-Step Solution

Answer

\( \frac{3}{2} \) is located at the 4th subdivision between 1 and 2, midway on the line.

Step 1: Understand the Number Line Segments

The number line extends from 0 to 2 and each of the segments from 0 to 1 and from 1 to 2 is divided into 8 equal parts. This means that each part represents a fraction of \( \frac{1}{8} \) of the entire segment between two whole numbers.

Step 2: Convert the Fraction

The number \( \frac{3}{2} \) needs to be located on the number line. First, convert \( \frac{3}{2} \) into an equivalent decimal or mixed number to better visualize its placement.\[ \frac{3}{2} = 1.5 \] which is equivalent to 1 and \( \frac{1}{2} \). This lies between 1 and 2 on the number line.

Step 3: Calculate the Subdivision for 1.5 on the Number Line

Since the segment from 1 to 2 is divided into 8 parts, we need to find which part corresponds to \( 1.5 \). \( \frac{1}{2} \) in the segment from 1 to 2 corresponds to the midpoint, which is the 4th part out of 8 (as \( \frac{4}{8} = \frac{1}{2} \)).

Step 4: Locate \( \frac{3}{2} \) on the Number Line

Starting at 1, count 4 steps to the right, each representing \( \frac{1}{8} \). The position after 4 steps marks \( \frac{3}{2} \) or 1.5 on the number line.

Key Concepts

Understanding Fractions

Fractions represent parts of a whole. They comprise two numbers: the numerator (top number) and the denominator (bottom number). The numerator indicates how many parts you have, while the denominator shows the total number of equal parts the whole is divided into.
For example, in the fraction \( \frac{3}{2} \), the numerator is 3, and the denominator is 2. This means you have 3 parts out of 2 equal sections of a whole, or simply put, one whole and one half. Fractions can represent values greater than 1, as in this case, where \( \frac{3}{2} \) equals 1.5.
Fractions are commonly used in everyday measurements such as recipes, where precision in the amount of ingredients is essential. Understanding fractions helps in tasks like dividing items equally, adjusting recipe quantities, and calculating portions.

Decimal Conversion Explained

Converting fractions to decimals can make it easier to understand and visualize their value. To convert a fraction like \( \frac{3}{2} \) to a decimal, you divide the numerator by the denominator: \( 3 \div 2 = 1.5 \).
Decimals are another way to express fractions and are commonly used in financial transactions, measurements, and scientific data. They offer an easier means of adding, subtracting, and comparing values than working with fractions directly.

• To convert any fraction to a decimal, simply perform the division indicated by the fraction.
• Some fractions convert into terminating decimals (like \( \frac{1}{2} \) = 0.5) while others convert into repeating decimals (like \( \frac{1}{3} \) = 0.333...).

Understanding both representations allows you to choose the most convenient method for solving problems and communicating numerical values.

Exploring Number Line Segments

A number line is a visual representation of numbers laid out on a straight path. It helps us visualize the size, order, and value of numbers. When a number line is divided into segments, it aids in identifying fractions and decimals.
In this exercise, the number line from 0 to 2 is split into segments, with each whole number further divided into 8 equal parts. This effectively highlights intervals of \( \frac{1}{8} \).

• For a fraction like \( \frac{3}{2} \), knowing it equals 1.5 allows us to find it on the number line.
• We identify 1 first, then move rightward along the segments to the midpoint, each step representing \( \frac{1}{8} \), reaching 1.5 or the equivalent \( \frac{4}{8} \) in the 1 to 2 segment.

Number lines offer an intuitive way of comparing numbers and understanding their relationship to one another, enhancing numerical literacy and comprehension.