Problem 81 - Prealgebra

Question

Which of the following fractions is closest to the number 0? a. \(\frac{1}{8}\) b. \(\frac{3}{8}\) c. \(\frac{5}{8}\) d. \(\frac{7}{8}\)

Step-by-Step Solution

Answer

The fraction closest to zero is \(\frac{1}{8}\).

Step 1: Identify the Fractions

List each of the given fractions: \( \frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8} \). We need to determine which of these is closest to 0.

Step 2: Understanding Proximity to Zero

A fraction will be closer to zero if its numerator is much smaller in comparison to its denominator. Generally, the smaller the numerator, the closer the fraction will be to zero.

Step 3: Compare the Fractions

Comparing the numerators of the fractions since they have the same denominators: \(1, 3, 5, 7\). The fraction with the smallest numerator will be the closest to zero.

Step 4: Identify the Closest Fraction

Among 1, 3, 5, and 7, the smallest number is 1, making \(\frac{1}{8}\) the fraction where the numerator is least, hence closest to zero.

Key Concepts

Understanding Fractions

Fractions represent parts of a whole. When you look at a fraction like \(\frac{1}{8}\), it consists of two parts:

• The numerator, which is the top number (1 in this case), tells you how many parts you have.
• The denominator, which is the bottom number (8), tells you into how many parts the whole is divided.

Each fraction indicates a division; thus, \(\frac{1}{8}\) means "1 divided by 8" or one part of eight equal sections. This means whenever you're dealing with fractions, you’re dealing with divisions of a whole, where the numerator signifies the particular segments included and the denominator shows how many such segments create a complete item.

How to Compare Fractions

Comparing fractions is crucial for understanding their sizes relative to each other. When comparing, it’s often easier when fractions have like denominators, which is the case here: \(\frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}\).

• If the denominators are the same, compare the numerators directly. The fraction with the smaller numerator will be smaller of the two fractions.
• For our set of fractions, since all share the denominator 8, simply look at the numerators: 1, 3, 5, and 7.

Here, \(\frac{1}{8}\) with the smallest numerator 1, is the smallest, hence closest to zero. This represents how understanding the relationship between numerators helps in identifying the size of fractions quickly.

The Role of Numerators and Denominators

When working with fractions, the numerator and denominator play different roles.

• The numerator shows how many parts of the fraction you are considering. If we have a fraction \(\frac{a}{b}\), 'a' is what you have.
• The denominator tells you into how many parts the total is divided. In \(\frac{a}{b}\), 'b' establishes the size of those parts.

When the denominators of the fractions are the same, the numerator becomes the deciding factor for determining which is smaller or closer to a certain value like zero. The fraction \(\frac{1}{8}\), as an example, is small. This is due to its numerator '1' being small compared to its denominator '8', making it closer to zero compared to fractions with larger numerators.