Problem 62 - 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. $$\frac{5}{4}$$

Step-by-Step Solution

Answer

Locate \( \frac{5}{4} \) as the second mark after 1 on the number line divided into eighths.

Step 1: Understanding the Number Line

The number line is divided into the range from 0 to 2. The segment from 0 to 1 and from 1 to 2 are both divided into 8 equal parts. Each part represents \( \frac{1}{8} \) of a whole number, meaning the entire number is divided into 16 equal parts.

Step 2: Placing Whole Numbers and Fractions

Given the information that the number line is divided into 8 parts per whole number, knowing \( 1 = \frac{8}{8} \) can help us locate fractions. Since \( \frac{5}{4} = 1 + \frac{1}{4} \), we can simplify finding its exact position.

Step 3: Converting to Number Line Index

The fraction \( \frac{5}{4} \) is equivalent to \( 1.25 \), which can be written as \( 1 + \frac{2}{8} \), referring to the position just after reaching 1, and then moving two parts further on the scale divided into eighths.

Step 4: Identifying the Location

On the number line, since \( 1 = \frac{8}{8} \) and each eighth equals \( \frac{1}{8} \), we need to move 2 more parts to reach \( \frac{5}{4} \). Starting from 1 move right by two eighths (i.e., \( \frac{2}{8} = \frac{1}{4} \)). This positions \( \frac{5}{4} \) or 1.25 exactly on the second mark past 1.

Key Concepts

Fractions

Fractions are a crucial part of mathematics, representing parts of a whole. When a whole is divided into equal parts, a fraction helps express the number of such parts you have. For example, \( \frac{1}{4} \) means one part out of four equal parts. Understanding fractions can be simplified by visualizing them on a number line, which gives a clear illustration of their size relative to whole numbers.

• Numerator: The top number in a fraction, indicating how many parts are being considered.
• Denominator: The bottom number, showing the total number of equal parts the whole is divided into.

For example, the fraction \( \frac{5}{4} \) indicates that we have five parts out of four, suggesting we have more than one whole unit (specifically, one whole and one extra quarter). This visualization on a number line helps us understand more complex concepts, such as how fractions build upon whole numbers.

Whole Numbers

Whole numbers are the set of numbers without fractions or decimals. They start from zero and increase incrementally by one: 0, 1, 2, 3, and so on. Essentially, whole numbers are basic counting numbers.They are fundamental in mathematics as building blocks that students use before learning about more complex numbers such as fractions and decimals. Recognizing whole numbers on a number line is straightforward, as they are neatly represented as separate, distinct points.Using whole numbers as reference points helps in understanding where fractions lie. For example, on a number line from 0 to 2 divided into eight parts, knowing where 1 is helps us place fractions like \( \frac{5}{4} \) between whole numbers.Whole numbers help in adding fractions. For \( \frac{5}{4} \), the whole number 1 acts as an anchor, from which we step further into more detailed parts (eighths) to precisely locate its position.

Equivalent Fractions

Equivalent fractions are those that represent the same portion of a whole, even though they have different numerators and denominators. For example, \( \frac{2}{4} \) and \( \frac{1}{2} \) are equivalent because they both equal the same part of a whole.Finding equivalent fractions is useful in comparing, adding, or subtracting fractions. On a number line, understanding equivalent fractions helps identify values accurately, since one can transform complicated fractions into simpler forms.To find equivalent fractions, you multiply or divide the numerator and denominator by the same number. For instance, multiplying the numerator and denominator of \( \frac{1}{4} \) by 2 results in \( \frac{2}{8} \), which is equivalent to the original fraction.On a number line, this becomes particularly helpful in locating points. For \( \frac{5}{4} \), breaking it down to \( 1 + \frac{1}{4} \) or \( 1.25 \) helps find its exact spot, a step past the whole number 1 on a scale of eighths. This division ensures precise placement and comprehension.