Certainly! Let's represent each of the given fractions on the number line. We'll provide a detailed explanation for each fraction's placement on the number line.
### (i) [tex]\(\frac{1}{3}\)[/tex]
- Explanation: The fraction [tex]\(\frac{1}{3}\)[/tex] means one part out of three equal parts of a whole.
- Decimal form: [tex]\(0.333...\)[/tex] (which is approximately equal to 0.3333333333333333)
- Position on number line: This fraction is slightly less than [tex]\(\frac{1}{2}\)[/tex] (which is 0.5). So, [tex]\(\frac{1}{3}\)[/tex] lies between 0 and 0.5, closer to 0.33.
### (ii) [tex]\(\frac{1}{4}\)[/tex]
- Explanation: The fraction [tex]\(\frac{1}{4}\)[/tex] means one part out of four equal parts of a whole.
- Decimal form: [tex]\(0.25\)[/tex]
- Position on number line: This fraction is exactly one-fourth and lies at the quarter mark between 0 and 0.5, at 0.25.
### (iii) [tex]\(\frac{5}{7}\)[/tex]
- Explanation: The fraction [tex]\(\frac{5}{7}\)[/tex] means five parts out of seven equal parts of a whole.
- Decimal form: Approximately [tex]\(0.714...\)[/tex] (which is 0.7142857142857143)
- Position on number line: This fraction is more than [tex]\(\frac{1}{2}\)[/tex] (which is 0.5) and slightly more than [tex]\(\frac{2}{3}\)[/tex] (which is approximately 0.666). So, [tex]\(\frac{5}{7}\)[/tex] lies between 0.5 and 1, closer to 0.71.
### (iv) [tex]\(\frac{4}{9}\)[/tex]
- Explanation: The fraction [tex]\(\frac{4}{9}\)[/tex] means four parts out of nine equal parts of a whole.
- Decimal form: Approximately [tex]\(0.444...\)[/tex] (which is 0.4444444444444444)
- Position on number line: This fraction is less than [tex]\(\frac{1}{2}\)[/tex] (which is 0.5). So, [tex]\(\frac{4}{9}\)[/tex] lies between 0 and 0.5, closer to 0.44.
### (v) [tex]\(\frac{3}{4}\)[/tex]
- Explanation: The fraction [tex]\(\frac{3}{4}\)[/tex] means three parts out of four equal parts of a whole.
- Decimal form: [tex]\(0.75\)[/tex]
- Position on number line: This fraction is exactly three-fourths and lies at the three-quarter mark between 0.5 and 1, at 0.75.
### Visualization on Number Line:
```
0 0.25 0.33 0.44 0.5 0.66 0.71 0.75 1
|-------|--------|--------|--------|--------|--------|--------|--------|
0 1/4 1/3 4/9 1/2 2/3 5/7 3/4 1
```
- [tex]\(\frac{1}{4}\)[/tex] at 0.25
- [tex]\(\frac{1}{3}\)[/tex] at approximately 0.33
- [tex]\(\frac{4}{9}\)[/tex] at approximately 0.44
- [tex]\(\frac{5}{7}\)[/tex] at approximately 0.71
- [tex]\(\frac{3}{4}\)[/tex] at 0.75
By following these steps, we can accurately represent each of the given fractions on the number line.
