Question

Arrange the following fractions in order from largest to the smallest. $$ \frac{3}{16}, \frac{1}{16}, \frac{5}{16}, \frac{14}{16}, \frac{7}{16} $$

Short answer

\(\frac{14}{16}, \frac{7}{16}, \frac{5}{16}, \frac{3}{16}, \frac{1}{16}\)

Step-by-step solution

Analyze Fractions

Observe that all fractions have the same denominator, 16, which means they can be directly compared by their numerators.

List Numerators

Extract the numerators from the fractions. The numerators are: 3, 1, 5, 14, and 7.

Sort Numerators

Arrange the numerators in descending order. The sorted numerators are: 14, 7, 5, 3, 1.

Replace Numerators with Fractions

Replace the sorted numerators back into their respective fractions: \(\frac{14}{16}\), \(\frac{7}{16}\), \(\frac{5}{16}\), \(\frac{3}{16}\), \(\frac{1}{16}\).

Write the Final Order

The fractions ordered from largest to smallest are \(\frac{14}{16}\), \(\frac{7}{16}\), \(\frac{5}{16}\), \(\frac{3}{16}\), \(\frac{1}{16}\).

Key concepts

Understanding the Numerator

In a fraction, the numerator is the number located at the top. It tells you how many parts of the whole are being considered. For example, in the fraction \( \frac{3}{16} \), the number 3 is the numerator. This means that out of 16 equal parts, 3 parts are being counted.

To determine the relative size of fractions with a common denominator, you can compare the numerators directly. This is possible because each fraction represents parts of the same whole. Thus, a larger numerator indicates a larger fraction.

• The bigger the numerator, the larger the fraction.
• In the example, \( \frac{14}{16} \) is larger than \( \frac{3}{16} \) because 14 is greater than 3.

Role of the Denominator

The denominator in a fraction is the number at the bottom. It indicates the total number of equal parts the whole is divided into. For instance, in \( \frac{3}{16} \), 16 is the denominator, meaning the whole is split into 16 parts.

In fractions with the same denominator, like the ones in our exercise, the size relationship is straightforward. You simply look at the numerators, since they indicate how many parts out of the total are being considered. The denominator ensures that each fraction represents parts of an identical whole.

• Common denominators make fraction comparison easier.
• Uniformity in denominators implies direct comparison through numerators.

Origin and Use of Descending Order

Descending order is the arrangement of numbers from the largest to the smallest. This concept is crucial when organizing data, especially in comparison exercises. When ordering fractions with the same denominator, such as \( \frac{14}{16} \), \( \frac{7}{16} \), and \( \frac{5}{16} \), we arrange the numerators in descending order.

The sequence formed after arranging is known as the descending order, which helps in understanding which fractions are larger.

• Useful in lists when the goal is to find the largest or smallest category.
• Facilitates quick visual identification of size differences.

Comprehending Fractions

Fractions are a fundamental part of mathematics, representing parts of a whole. They consist of two numbers: the numerator and the denominator, divided by a line.

These can be used to express amounts less than a whole, like dividing a pizza into equal slices. Understanding how to compare and organize these is vital for solving even the simplest mathematical exercises.

• Fractions can be compared if they share the same denominators.
• Discovering relationships between different fractions enhances problem-solving skills.

Recognizing the basic structure of fractions helps in many problem-solving situations where amounts and comparisons are needed.