Understanding Fraction Comparison

To determine which of the given options correctly orders the fractions, we need to compare the fractions \(\frac{2}{3}\), \(\frac{3}{5}\), and \(\frac{11}{15}\). The easiest way to compare fractions is to find a common denominator.

Finding a Common Denominator for Fractions

The denominators of the fractions are \(3\), \(5\), and \(15\). We need to find the least common multiple (LCM) of these denominators.

• Multiples of 3: 3, 6, 9, 12, 15, 18, ...
• Multiples of 5: 5, 10, 15, 20, ...
• Multiples of 15: 15, 30, ...

The smallest number that is a multiple of \(3\), \(5\), and \(15\) is \(15\). So, the LCM of \(3\), \(5\), and \(15\) is \(15\). This will be our common denominator.

Converting Fractions to the Common Denominator

Now, we convert each fraction to an equivalent fraction with a denominator of \(15\).

For \(\frac{2}{3}\): We need to multiply the denominator \(3\) by \(5\) to get \(15\). We must do the same to the numerator to keep the fraction equivalent.

\(\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}\)

For \(\frac{3}{5}\): We need to multiply the denominator \(5\) by \(3\) to get \(15\). We must do the same to the numerator.

\(\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}\)

For \(\frac{11}{15}\): The denominator is already \(15\).

\(\frac{11}{15}\)

Ordering Fractions by Comparing Numerators

Now that all fractions have the same denominator (\(15\)), we can compare their numerators. The fractions are now \(\frac{10}{15}\), \(\frac{9}{15}\), and \(\frac{11}{15}\).

Let's compare the numerators: \(10\), \(9\), and \(11\).

In ascending order (smallest to largest), the numerators are \(9 < 10 < 11\).

Therefore, the fractions in ascending order are:

\(\frac{9}{15} < \frac{10}{15} < \frac{11}{15}\)

Relating Back to Original Fractions

Substituting the original fractions back into the inequality:

• \(\frac{9}{15}\) is equivalent to \(\frac{3}{5}\)
• \(\frac{10}{15}\) is equivalent to \(\frac{2}{3}\)
• \(\frac{11}{15}\) is \(\frac{11}{15}\)

So, the correct ascending order is:

\(\frac{3}{5} < \frac{2}{3} < \frac{11}{15}\)

Analyzing the Options for Correct Fraction Order

Let's compare our derived correct order with the given options:

Based on the comparison of equivalent fractions, the correct order is \(\frac{3}{5} < \frac{2}{3} < \frac{11}{15}\).

Revision Table: Key Concepts for Comparing Fractions

Additional Information: Other Ways to Compare Fractions

While finding a common denominator is a standard method, here are other ways to compare fractions:

• Cross-Multiplication: To compare \(\frac{a}{b}\) and \(\frac{c}{d}\), compare \(ad\) and \(bc\). If \(ad < bc\), then \(\frac{a}{b} < \frac{c}{d}\). If \(ad > bc\), then \(\frac{a}{b} > \frac{c}{d}\). This is useful for comparing two fractions at a time.
• Converting to Decimals: Divide the numerator by the denominator for each fraction to get a decimal value. Then compare the decimal values. For example, \(\frac{2}{3} \approx 0.667\), \(\frac{3}{5} = 0.6\), \(\frac{11}{15} \approx 0.733\). Comparing these decimals: \(0.6 < 0.667 < 0.733\), which gives \(\frac{3}{5} < \frac{2}{3} < \frac{11}{15}\).
• Comparing to a Benchmark (like 1/2 or 1): Compare fractions to a known value. For example, \(\frac{3}{5}\) is slightly more than \(\frac{1}{2}\) (\(0.5\)), \(\frac{2}{3}\) is also more than \(\frac{1}{2}\), and \(\frac{11}{15}\) is closer to \(1\) than the others. This method is less precise but can help eliminate options quickly.