Finding the Descending Order of Fractions

To arrange fractions in descending order, we need to compare their values and list them from largest to smallest. We can do this by converting the fractions to decimals or by finding a common denominator.

Method 1: Converting Fractions to Decimals

We can convert each fraction to its decimal equivalent by dividing the numerator by the denominator. This makes comparison easier.

• For \(\frac{2}{5}\): Divide 2 by 5. \(\frac{2}{5} = 2 \div 5 = 0.4\)
• For \(\frac{1}{3}\): Divide 1 by 3. \(\frac{1}{3} = 1 \div 3 = 0.333...\) (approximately 0.33)
• For \(\frac{3}{7}\): Divide 3 by 7. \(\frac{3}{7} = 3 \div 7 \approx 0.428...\) (approximately 0.43)

Now, let's compare the decimal values:

• \(0.4\)
• \(0.333...\)
• \(0.428...\)

Comparing these decimals, we see that \(0.428...\) is the largest, \(0.4\) is the next largest, and \(0.333...\) is the smallest.

So, in descending order, the fractions are \(\frac{3}{7}\), \(\frac{2}{5}\), and \(\frac{1}{3}\).

Method 2: Finding a Common Denominator

Another way to compare fractions is to express them with a common denominator. We find the Least Common Multiple (LCM) of the denominators (5, 3, and 7).

• The denominators are 5, 3, and 7.
• These numbers are prime relative to each other.
• The LCM of 5, 3, and 7 is \(5 \times 3 \times 7 = 105\).

Now, we convert each fraction to an equivalent fraction with a denominator of 105:

• For \(\frac{2}{5}\): Multiply numerator and denominator by \(105 \div 5 = 21\).
  \(\frac{2}{5} = \frac{2 \times 21}{5 \times 21} = \frac{42}{105}\)
• For \(\frac{1}{3}\): Multiply numerator and denominator by \(105 \div 3 = 35\).
  \(\frac{1}{3} = \frac{1 \times 35}{3 \times 35} = \frac{35}{105}\)
• For \(\frac{3}{7}\): Multiply numerator and denominator by \(105 \div 7 = 15\).
  \(\frac{3}{7} = \frac{3 \times 15}{7 \times 15} = \frac{45}{105}\)

Now, we compare the fractions with the same denominator by looking at their numerators:

• \(\frac{42}{105}\) (from \(\frac{2}{5}\))
• \(\frac{35}{105}\) (from \(\frac{1}{3}\))
• \(\frac{45}{105}\) (from \(\frac{3}{7}\))

Comparing the numerators 42, 35, and 45, the descending order is 45 > 42 > 35.

This corresponds to the fractions \(\frac{45}{105}\), \(\frac{42}{105}\), and \(\frac{35}{105}\).

Substituting the original fractions back, the descending order is \(\frac{3}{7}\), \(\frac{2}{5}\), and \(\frac{1}{3}\).

Conclusion on Descending Order

Both methods show that the descending order of the fractions \(\frac{2}{5}\), \(\frac{1}{3}\), and \(\frac{3}{7}\) is \(\frac{3}{7}, \frac{2}{5}, \frac{1}{3}\).

Based on the comparison (either decimals or equivalent fractions), the order from largest to smallest is \(\frac{3}{7}\), \(\frac{2}{5}\), \(\frac{1}{3}\).

Revision Table: Ordering Fractions

Additional Information: Comparing Fractions

Comparing fractions is a fundamental skill in mathematics. When fractions have the same denominator, the one with the larger numerator is greater. When they have the same numerator, the one with the smaller denominator is greater. When both numerator and denominator are different, as in this question with fractions \(\frac{2}{5},\space\frac{1}{3}\) and  \(\frac{3}{7}\), we need a standard way to compare them, such as finding a common ground like equivalent fractions or decimal values. Understanding how to order fractions is essential for various mathematical operations and problem-solving.