To solve this exercise, let's break it down step by step, considering the given operations for different fractions:
### Part a) Counting in halves
First, we'll start with a number (3) and add [tex]\(\frac{1}{2}\)[/tex] step by step.
1. Start at 3.
2. Add [tex]\(\frac{1}{2}\)[/tex] to 3:
[tex]\[
3 + \frac{1}{2} = 3.5
\][/tex]
3. Add [tex]\(\frac{1}{2}\)[/tex] to 3.5:
[tex]\[
3.5 + \frac{1}{2} = 4
\][/tex]
4. Add [tex]\(\frac{1}{2}\)[/tex] to 4:
[tex]\[
4 + \frac{1}{2} = 4.5
\][/tex]
5. Add [tex]\(\frac{1}{2}\)[/tex] to 4.5:
[tex]\[
4.5 + \frac{1}{2} = 5
\][/tex]
So, the sequence in halves is: [tex]\(3, 3.5, 4, 4.5, 5\)[/tex].
### Part b) Counting in thirds
Now, we start again from the number 3 but add [tex]\(\frac{1}{3}\)[/tex] each step.
1. Start at 3.
2. Add [tex]\(\frac{1}{3}\)[/tex] to 3:
[tex]\[
3 + \frac{1}{3} = 3.333\ldots
\][/tex]
3. Add [tex]\(\frac{1}{3}\)[/tex] to 3.333...:
[tex]\[
3.333\ldots + \frac{1}{3} = 3.666\ldots
\][/tex]
4. Add [tex]\(\frac{1}{3}\)[/tex] to 3.666...:
[tex]\[
3.666\ldots + \frac{1}{3} = 4
\][/tex]
5. Add [tex]\(\frac{1}{3}\)[/tex] to 4:
[tex]\[
4 + \frac{1}{3} = 4.333\ldots
\][/tex]
So, the sequence in thirds is: [tex]\(3, 3.333\ldots, 3.666\ldots, 4, 4.333\ldots\)[/tex].
### Part c) Counting in quarters
Finally, we add [tex]\(\frac{1}{4}\)[/tex] each step starting from 3.
1. Start at 3.
2. Add [tex]\(\frac{1}{4}\)[/tex] to 3:
[tex]\[
3 + \frac{1}{4} = 3.25
\][/tex]
3. Add [tex]\(\frac{1}{4}\)[/tex] to 3.25:
[tex]\[
3.25 + \frac{1}{4} = 3.5
\][/tex]
4. Add [tex]\(\frac{1}{4}\)[/tex] to 3.5:
[tex]\[
3.5 + \frac{1}{4} = 3.75
\][/tex]
5. Add [tex]\(\frac{1}{4}\)[/tex] to 3.75:
[tex]\[
3.75 + \frac{1}{4} = 4
\][/tex]
So, the sequence in quarters is: [tex]\(3, 3.25, 3.5, 3.75, 4\)[/tex].
In summary:
- The sequence in halves: [tex]\(3, 3.5, 4, 4.5, 5\)[/tex]
- The sequence in thirds: [tex]\(3, 3.333\ldots, 3.666\ldots, 4, 4.333\ldots\)[/tex]
- The sequence in quarters: [tex]\(3, 3.25, 3.5, 3.75, 4\)[/tex]
This detailed step-by-step guide should help you understand how counting in different fractions works.
