Question

Arrange the following fractions in order from largest to the smallest. $$ \frac{5}{12}, \frac{5}{32}, \frac{5}{8}, \frac{5}{6}, \frac{5}{64} $$

Short answer

Fractions ordered largest to smallest: \(\frac{5}{6}, \frac{5}{8}, \frac{5}{12}, \frac{5}{32}, \frac{5}{64}\).

Step-by-step solution

Identify the Common Elements

Notice that all the fractions have the same numerator, which is 5. This means that the size of the fractions depends solely on the denominators.

Understand Fraction Comparison with Same Numerator

For fractions with the same numerator, the fraction with the smaller denominator is larger. This is because the numerator is split into fewer parts, making each part larger.

List the Denominators

List the denominators of the given fractions: 12, 32, 8, 6, 64. This will help in comparing the sizes of the fractions.

Order the Denominators from Smallest to Largest

Reorder the denominators from smallest to largest: 6, 8, 12, 32, 64. This helps in identifying which fraction is smallest and which is largest.

Arrange Fractions by Denominator

Using the order of denominators from Step 4, arrange the fractions from the largest to the smallest: \(\frac{5}{6}, \frac{5}{8}, \frac{5}{12}, \frac{5}{32}, \frac{5}{64}\).

Key concepts

Fractions with Common Numerator

When dealing with fractions that share the same numerator, like \(\frac{5}{12}, \frac{5}{32}, \frac{5}{8}, \frac{5}{6}, \frac{5}{64}\), there is an interesting property that makes comparing their sizes straightforward. The numerator is the same across all fractions, which means we focus on the denominators to compare them.

In any fraction with a common numerator, the part you should focus on is the denominator. Since they share the same numerator, the value of the fraction is determined by how many parts this numerator is split into. If the numerator is split into fewer parts, each part is larger, making the entire fraction larger. In contrast, more parts mean smaller parts, hence a smaller fraction.

To illustrate, between \(\frac{5}{6}\) and \(\frac{5}{12}\), the numerator 5 is divided by 6 in the first fraction and by 12 in the second. Therefore, \(\frac{5}{6}\) is larger than \(\frac{5}{12}\) because each part is larger with a smaller denominator.

Comparing Fractions

Comparing fractions with the same numerator is a much simpler task than when the numerators differ. Here, you only need to compare the denominators to determine which fraction is larger or smaller.

For example:

• If you have fractions like \(\frac{5}{8}\) and \(\frac{5}{32}\), simply look at the denominators. The smaller the denominator, the larger the fraction.
• Thus, \(\frac{5}{8}\) is larger than \(\frac{5}{32}\) because the denominator 8 is smaller than 32.

This method makes ordering fractions quite efficient, especially when solving problems that involve multiple fractions with common numerators. Simply write down the denominators, rearrange them in increasing order, and this will help you arrange the fractions from largest to smallest or vice versa if needed.

Fraction Comparison Rules

There are a few rules or steps you can follow to make comparing fractions easier. When fractions have the same numerator, these rules are particularly useful as you can solely focus on the denominators.

Here are some simple steps to follow:

• List the denominators of the fractions you need to compare.
• Reorder the denominators from smallest to largest.
• The fraction with the smallest denominator is the largest fraction, and the one with the largest denominator is the smallest fraction.

By following these steps, you can quickly determine the order of fractions. For example, with the fractions \(\frac{5}{12}, \frac{5}{32}, \frac{5}{8}, \frac{5}{6}, \frac{5}{64}\), the denominators are ordered as 6, 8, 12, 32, 64. Thus, \(\frac{5}{6}\) is the largest, and \(\frac{5}{64}\) is the smallest.
Understanding these rules not only helps in academic exercises but also in real-life applications where comparing portions or shares is necessary.