santishahi280
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CLASSWORK EXAMPLES

1. [tex]\(\begin{tabular}{l|r|}
+\quad 9 & 63785 \\
\hline 77458 & 63749
\end{tabular}\)[/tex]

Similarly, to locate:

### Example 1
Locate [tex]\(\frac{3}{5}\)[/tex] on the number line.

Solution:
1. Construct a number line.
2. Construct two parallel lines [tex]\(\ell_1\)[/tex] and [tex]\(\ell_2\)[/tex] from 0 and respectively.

### Example 2
Locate point [tex]\(X\)[/tex].

Solution:
1. Insert 6 rational numbers between 2 and 3.
2. One of the rational numbers is the arithmetic average of 2 and 3, which is [tex]\(\frac{1}{2}(2 + 3) = 2.5\)[/tex].
3. Therefore, one rational number is 2.5.
4. The other rational numbers are calculated similarly by taking the arithmetic average of the subsequent intervals.

Construct 5 equal divisions:
1. Using a pencil compass, divide the interval into 5 equal parts.
2. Example: [tex]\(\frac{1}{2}(2 + \frac{1}{2}) + \ldots\)[/tex]

This process will help you locate the desired points and rational numbers on the number line accurately.



Answer :

GinnyAnswer
Sure, let's tackle these various math exercises step-by-step. We'll aim to understand what's required and explain the reasoning clearly.

### Classwork Examples Breakdown

#### Example 1: Locate [tex]\(\frac{3}{5}\)[/tex] on the Number Line
To locate [tex]\(\frac{3}{5}\)[/tex] on the number line:

1. Construct a Number Line: Draw a straight horizontal line and mark the points [tex]\(0\)[/tex] and [tex]\(1\)[/tex].

2. Divide the Interval: Divide the segment between [tex]\(0\)[/tex] and [tex]\(1\)[/tex] into 5 equal parts, because the denominator is 5.

3. Locate the Point: Count 3 parts from 0 towards 1 to locate [tex]\(\frac{3}{5}\)[/tex]. Mark this point on the number line.

So, [tex]\(\frac{3}{5}\)[/tex] is 3 units to the right of 0 if 1 unit is split into 5 equal parts.

#### Example 2: Insert 6 Rational Numbers

To insert 6 rational numbers between two given numbers:

1. Identify the Given Numbers: Assume the two numbers are [tex]\(a\)[/tex] and [tex]\(b\)[/tex].

2. Find the Arithmetic Sequence: To insert numbers between [tex]\(a\)[/tex] and [tex]\(b\)[/tex], find the common difference [tex]\(d\)[/tex].

3. Calculate the Rational Numbers: Use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence:
[tex]\[ a_n = a + (n-1)d \][/tex]
For six rational numbers in between, create the sequence [tex]\(a\)[/tex], [tex]\(a+d\)[/tex], [tex]\(a+2d\)[/tex], ..., [tex]\(a+6d\)[/tex], [tex]\(b\)[/tex].

4. Determine the Correct Placement: Ensure that each number follows the arithmetic sequence logic and place them evenly between [tex]\(a\)[/tex] and [tex]\(b\)[/tex].

#### Additional Number Line Example (Constructing Divisions)

To further understand constructing divisions on a number line equitably:

1. Draw the Interval: Draw a straight line representing numerical value from 0 to another specific number, say 2.

2. Mark Points Equidistantly: If tasked to divide into 5 equal parts, use a compass or ruler to mark them with equal spacing.

3. Verify Equal Divisions: Check that the segments are indeed equal, ensuring accuracy by perhaps using a compass or a ruler.

By placing segments accurately, you ensure that the divisions such as [tex]\(\frac{1}{2}\left[2 \frac{1}{2}\right]\)[/tex] are correctly represented.

### Summary

- Locating Fractions on a Number Line: Split the interval as per the denominator and locate the specified numerator.
- Inserting Rational Numbers: Utilize arithmetic sequences to evenly distribute numbers between two given values.
- Constructing Equal Divisions: Use tools like compasses or rulers to ensure precision when dividing intervals.

This exercise strengthens our understanding of the number line concept, arithmetic sequences, and accurate placement of rational numbers.

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