Understanding the Problem: Ordering Fractions

The question asks us to arrange a given set of fractions in ascending order. Ascending order means arranging numbers from the smallest value to the largest value. To compare fractions and put them in order, we need a way to see their values clearly. One common method is converting the fractions to decimal numbers.

Step-by-Step Solution: Converting Fractions to Decimals

Let's convert each fraction into its decimal equivalent by dividing the numerator by the denominator:

• Fraction 1: $\frac{4}{5}$
• Calculation: $4 \div 5 = 0.8$
• Decimal value: $0.8$

• Fraction 2: $\frac{2}{3}$
• Calculation: $2 \div 3$
• Decimal value: $0.666...$ (a repeating decimal)

• Fraction 3: $\frac{1}{11}$
• Calculation: $1 \div 11$
• Decimal value: $0.0909...$ (a repeating decimal)

• Fraction 4: $\frac{2}{9}$
• Calculation: $2 \div 9$
• Decimal value: $0.222...$ (a repeating decimal)

Comparing Decimal Values for Ascending Order

Now that we have the decimal values, we can easily compare them:

• $0.8$ (from $\frac{4}{5}$)
• $0.666...$ (from $\frac{2}{3}$)
• $0.0909...$ (from $\frac{1}{11}$)
• $0.222...$ (from $\frac{2}{9}$)

Arranging these decimal values from smallest to largest (ascending order):

$0.0909... < 0.222... < 0.666... < 0.8$

Final Ascending Order of Fractions

Mapping the decimal values back to their original fractions, the ascending order is:

$\frac{1}{11}, \frac{2}{9}, \frac{2}{3}, \frac{4}{5}$

Checking the Options

Let's look at the provided options and see which one matches the ascending order we found:

• Option 1: $\frac{4}{5}, \frac{2}{3}, \frac{1}{11}, \frac{2}{9}$ (This is $0.8, 0.666..., 0.0909..., 0.222...$ - Incorrect order)
• Option 2: $\frac{1}{11}, \frac{2}{9}, \frac{2}{3}, \frac{4}{5}$ (This is $0.0909..., 0.222..., 0.666..., 0.8$ - Correct order)
• Option 3: $\frac{2}{9}, \frac{1}{11}, \frac{4}{5}, \frac{2}{3}$ (This is $0.222..., 0.0909..., 0.8, 0.666...$ - Incorrect order)
• Option 4: $\frac{2}{3}, \frac{4}{5}, \frac{1}{11}, \frac{2}{9}$ (This is $0.666..., 0.8, 0.0909..., 0.222...$ - Incorrect order)

Option 2 presents the fractions in the correct ascending order.

Ordered decimal values: $0.0909... < 0.222... < 0.666... < 0.8$

Corresponding fractions in ascending order: $\frac{1}{11}, \frac{2}{9}, \frac{2}{3}, \frac{4}{5}$

Revision Table: Key Concepts

Additional Information: Comparing Fractions Methods

Besides converting fractions to decimals, another common method to compare fractions is finding a common denominator.

Method 2: Finding a Common Denominator

To compare $\frac{4}{5}, \frac{2}{3}, \frac{1}{11}, \frac{2}{9}$, we could find the Least Common Multiple (LCM) of the denominators $5, 3, 11, 9$.

• Prime factorization: $5=5$, $3=3$, $11=11$, $9=3^2$
• LCM is $3^2 \times 5 \times 11 = 9 \times 5 \times 11 = 45 \times 11 = 495$.

Now, convert each fraction to have a denominator of 495:

• $\frac{4}{5} = \frac{4 \times 99}{5 \times 99} = \frac{396}{495}$
• $\frac{2}{3} = \frac{2 \times 165}{3 \times 165} = \frac{330}{495}$
• $\frac{1}{11} = \frac{1 \times 45}{11 \times 45} = \frac{45}{495}$
• $\frac{2}{9} = \frac{2 \times 55}{9 \times 55} = \frac{110}{495}$

Comparing the numerators: $45 < 110 < 330 < 396$.

This corresponds to the fractions: $\frac{45}{495}, \frac{110}{495}, \frac{330}{495}, \frac{396}{495}$.

Substituting back the original fractions:

$\frac{1}{11}, \frac{2}{9}, \frac{2}{3}, \frac{4}{5}$

This confirms the order found using the decimal method. Both methods are valid for ordering fractions.