Understanding Ordering Fractions

Ordering fractions involves comparing their values to arrange them from largest to smallest (descending order) or smallest to largest (ascending order). For this question, we need to find the correct descending order of the given fractions: \(\frac{1}{2}, \frac{2}{5}, \frac{3}{4}, \frac{2}{3}\). Finding the correct ordering fractions is a fundamental skill in mathematics.

Methods for Comparing Fractions

There are several ways to compare fractions and determine their descending order:

• Finding a Common Denominator: Convert all fractions to equivalent fractions with the same denominator. Then, compare the numerators.
• Converting to Decimals: Convert each fraction to its decimal equivalent. Then, compare the decimal numbers.
• Cross-Multiplication: This is useful for comparing two fractions at a time.

Let's use the method of converting to decimals as it often makes the comparison straightforward when ordering multiple fractions.

Converting Fractions to Decimals

We will convert each given fraction into its decimal form:

• For \(\frac{1}{2}\): \(1 \div 2 = 0.5\)
• For \(\frac{2}{5}\): \(2 \div 5 = 0.4\)
• For \(\frac{3}{4}\): \(3 \div 4 = 0.75\)
• For \(\frac{2}{3}\): \(2 \div 3 \approx 0.666...\)

So, the decimal equivalents are \(0.5, 0.4, 0.75, 0.666...\)

Arranging Decimals in Descending Order

Descending order means arranging from the largest value to the smallest value. Let's look at the decimal values:

\(0.75, 0.666..., 0.5, 0.4\)

Arranging these decimals from largest to smallest gives us:

\(0.75 > 0.666... > 0.5 > 0.4\)

Identifying the Descending Order of Fraction

Now, we convert the ordered decimals back to their original fraction forms:

• \(0.75\) corresponds to \(\frac{3}{4}\)
• \(0.666...\) corresponds to \(\frac{2}{3}\)
• \(0.5\) corresponds to \(\frac{1}{2}\)
• \(0.4\) corresponds to \(\frac{2}{5}\)

Therefore, the correct descending order of the fractions is \(\frac{3}{4}, \frac{2}{3}, \frac{1}{2}, \frac{2}{5}\).

Comparing with Options

Let's compare our calculated descending order of fraction with the given options:

The ordering fractions \(\frac{3}{4}, \frac{2}{3}, \frac{1}{2}, \frac{2}{5}\) is the correct descending order based on our calculations. Mastering fraction comparison techniques is key to solving such problems correctly.