Let's go through the problems step by step.

1. Converting Mixed Numbers to Improper Fractions

This section asks you to convert the given mixed numbers into improper fractions.

• The first mixed number is $2 \frac{7}{9}$.

  • Formula: $\text{Improper Fraction} = \text{Whole Number} \times \text{Denominator} + \text{Numerator}$
  • Calculation: $2 \times 9 + 7 = 18 + 7 = 25$
  • So, $2 \frac{7}{9} = \frac{25}{9}$.

• The second mixed number is $3 \frac{1}{5}$.

  • Calculation: $3 \times 5 + 1 = 15 + 1 = 16$
  • So, $3 \frac{1}{5} = \frac{16}{5}$.

• The third mixed number is $1 \frac{4}{8}$.

  • Calculation: $1 \times 8 + 4 = 8 + 4 = 12$
  • So, $1 \frac{4}{8} = \frac{12}{8}$, which simplifies to $\frac{3}{2}$.

• The fourth mixed number is $4 \frac{3}{5}$.

  • Calculation: $4 \times 5 + 3 = 20 + 3 = 23$
  • So, $4 \frac{3}{5} = \frac{23}{5}$.

2. Completing the Table of Improper Fractions and Mixed Numbers

This section asks you to fill out the table by converting improper fractions to mixed numbers.

• For $\frac{13}{4}$:

  • Divide $13$ by $4$: $13 \div 4 = 3$ remainder $1$.
  • So, $\frac{13}{4} = 3 \frac{1}{4}$.

• For $\frac{19}{6}$:

  • Divide $19$ by $6$: $19 \div 6 = 3$ remainder $1$.
  • So, $\frac{19}{6} = 3 \frac{1}{6}$.

• For $\frac{11}{3}$:

  • Divide $11$ by $3$: $11 \div 3 = 3$ remainder $2$.
  • So, $\frac{11}{3} = 3 \frac{2}{3}$.

• For $\frac{15}{8}$:

  • Divide $15$ by $8$: $15 \div 8 = 1$ remainder $7$.
  • So, $\frac{15}{8} = 1 \frac{7}{8}$.

Lastly, place the mixed numbers on the number line between the integers 1, 2, 3, and 4.

3. Comparing Lengths

In this section, you are to compare the given pairs of fractions and select the longer length.

• $\frac{3}{4} \text{ metre}$ and $\frac{2}{3} \text{ metre}$:

  • $\frac{3}{4}$ is greater than $\frac{2}{3}$, so tick $\frac{3}{4} \text{ metre}$.

• $\frac{2}{5} \text{ metre}$ and $\frac{1}{2} \text{ metre}$:

  • $\frac{1}{2}$ is greater than $\frac{2}{5}$, so tick $\frac{1}{2} \text{ metre}$.

• $\frac{5}{8} \text{ metre}$ and $\frac{3}{4} \text{ metre}$:

  • $\frac{3}{4}$ is greater than $\frac{5}{8}$, so tick $\frac{3}{4} \text{ metre}$.

Do you have any questions or need further details on any of these steps?

Here are five related questions to further your understanding:

1. How do you convert a mixed number to an improper fraction in general?
2. Can improper fractions be greater than 1?
3. What is the process to simplify fractions?
4. How do you compare two fractions without converting them to decimals?
5. Why might it be important to place fractions on a number line?

Tip:

When comparing fractions, it can be helpful to find a common denominator or convert them to decimals to make the comparison easier.