Sure, let's represent the given rational numbers on a number line.
### Part (a): Representing ¾
1. Understand the Rational Number:
- The given rational number is [tex]\( \frac{3}{4} \)[/tex], which translates to 0.75 in decimal form.
2. Placement on the Number Line:
- Locate the area between 0 and 1 on the number line since [tex]\( \frac{3}{4} \)[/tex] is a positive number less than 1.
- Divide this segment into four equal parts, each part representing 0.25 (since [tex]\(\frac{1}{4} = 0.25\)[/tex]).
- Count three parts from 0, reaching 0.75. This point is [tex]\( \frac{3}{4} \)[/tex].
### Part (b): Representing 31/-6
1. Understand the Rational Number:
- The given rational number is [tex]\( \frac{31}{-6} \)[/tex], which simplifies to [tex]\( -\frac{31}{6} \)[/tex].
- Convert this into a mixed number or a decimal for easier placement on the number line:
[tex]\[
\frac{31}{6} \approx 5.1667 \text{, so } -\frac{31}{6} \approx -5.1667
\][/tex]
2. Placement on the Number Line:
- Locate the value between -5 and -6 on the number line since [tex]\( -\frac{31}{6} \)[/tex] is negative.
- Divide this segment into several parts for a more accurate location, or note that [tex]\( -5.1667 \)[/tex] is slightly more than -5 and less than -6.
- Find a point around one-sixth of the way from -6 to -5, which corresponds to [tex]\( -\frac{31}{6} \)[/tex].
### Summary of Steps to Represent on the Number Line:
1. Draw a horizontal line and mark equidistant points for integers, like -6, -5, -4, ..., 0, 1, and so on.
2. For [tex]\( \frac{3}{4} \)[/tex]:
- Focus on the section between 0 and 1, and mark a point at 0.75 within that interval.
3. For [tex]\( -\frac{31}{6} \)[/tex]:
- Focus on the section between -6 and -5, and mark a point at approximately -5.1667 within that interval.
By understanding and plotting these points step-by-step, they can be accurately placed on the number line as described.
