**EXERCISE 1.1**

**1. Insertion Between 2/9 and 3/8, Descending Order:**

To insert a rational number between 2/9 and 3/8 while arranging them in descending order, we turn to the LCM method to ensure a common denominator for these fractions.

**LCM of 9 and 8:**The LCM of the denominators 9 and 8 is 72. This common denominator guarantees that the fractions share the same base for further calculations.**Equivalent fractions:**We convert both fractions to their equivalent forms with a common denominator of 72:- 2/9 = (2/9) * (8/8) = 16/72
- 3/8 = (3/8) * (9/9) = 27/72

**Insertion calculation:**Since we want to insert a rational number between these two fractions, we consider the average of 16/72 and 27/72: (16/72 + 27/72) / 2 = 43/72.**Descending order:**Arranging them in descending order, we get 27/72, 43/72, and 16/72.

**2. Pairing Between 1/3 and 1/4, Ascending Order:**

Inserting rational numbers between 1/3 and 1/4 while arranging in ascending order requires finding a common denominator through the LCM method.

**LCM of 3 and 4:**The LCM of the denominators 3 and 4 is 12.**Equivalent fractions:**We convert both fractions to equivalent forms with a common denominator of 12:- 1/3 = (1/3) * (4/4) = 4/12
- 1/4 = (1/4) * (3/3) = 3/12

**Insertion calculation:**To insert rational numbers, we use a common difference between fractions. In this case, the difference is 1/12 (12 divided by 12).**Insertion and arrangement:**Adding 1/12 and then 2/12 (simplified as 1/6) to 4/12 gives us the sequence 4/12, 5/12, 6/12 (or 1/3), 7/12, 8/12 (or 2/3), 9/12 (or 3/4).

**3. Inclusion Between -1/3 and -1/2, Ascending Order:**

Incorporating rational numbers between -1/3 and -1/2 while arranging in ascending order calls for the LCM technique to find a common denominator.

**LCM of 3 and 2:**The LCM of the denominators 3 and 2 is 6.**Equivalent fractions:**We convert both fractions to equivalent forms with a common denominator of 6:- -1/3 = (-1/3) * (2/2) = -2/6
- -1/2 = (-1/2) * (3/3) = -3/6

**Insertion calculation:**Similar to the previous question, we insert rational numbers using a common difference, which is 1/6 (6 divided by 6).**Insertion and arrangement:**Adding 1/6 and then 2/6 (simplified as 1/3) to -2/6 gives us the sequence -2/6, -5/6, -8/6 (or -1/2).

**4. Filling Between 1/3 and 4/5, Descending Order:**

Filling rational numbers between 1/3 and 4/5 while arranging them in descending order requires establishing a common denominator through LCM.

**LCM of 3 and 5:**The LCM of the denominators 3 and 5 is 15.**Equivalent fractions:**We convert both fractions to equivalent forms with a common denominator of 15:- 1/3 = (1/3) * (5/5) = 5/15
- 4/5 = (4/5) * (3/3) = 12/15

**Insertion calculation:**Once again, we insert rational numbers using a common difference, which is 1/15 (15 divided by 15).**Insertion and arrangement:**Adding 1/15, then 2/15, and finally 3/15 (or 1/5) successively to 5/15 (or 1/3) gives us the sequence 5/15, 6/15 (or 2/5), 7/15, 8/15, 9/15 (or 3/5), 10/15 (or 2/3), 11/15, 12/15 (or 4/5).

**5. Intervals Between 4 and 4.5:**

The task here is to insert rational numbers between 4 and 4.5 using equal intervals.

**Intervals:**The difference between 4 and 4.5 is 0.5. To divide this range into 3 intervals, we use 0.5 / 3 = 0.166… (repeating decimal).**Insertion and arrangement:**Adding 0.166… successively to 4 gives us the sequence 4, 4.166…, 4.333…, and finally 4.5.

**6. Navigating Between 3 and 4:**

To navigate between 3 and 4 while inserting rational numbers and arranging them, we use the LCM method to establish common intervals.

**Intervals:**The difference between 3 and 4 is 1. To divide this range into 6 intervals, we use 1 / 6 = 1/6.**Insertion and arrangement:**Adding 1/6 successively to 3 gives us the sequence 3, 13/6, 7/3, 5/2, 17/6, and finally 4.

**7. Bridging Between 3/5 and 4/5:**

To bridge rational numbers between 3/5 and 4/5 while arranging them, we once again use the LCM technique for common intervals.

**LCM of 5:**The LCM of the denominator 5 is 5.**Equivalent fractions:**We convert both fractions to equivalent forms with a common denominator of 5:- 3/5 = (3/5) * (1/1) = 3/5
- 4/5 = (4/5) * (1/1) = 4/5

**Intervals:**The difference between 4/5 and 3/5 is 1/5. To divide this range into 5 intervals, we use 1/5 / 5 = 1/25.**Insertion and arrangement:**Adding 1/25 successively to 3/5 gives us the sequence 3/5, 11/25, 13/25, 21/25, and finally 4/5.

**8. Discovery Amidst -2/5 and 1/7:**

In this complex scenario, discovering rational numbers between -2/5 and 1/7 while arranging them requires careful application of the LCM approach.

**LCM of 5 and 7:**The LCM of the denominators 5 and 7 is 35.**Equivalent fractions:**We convert both fractions to equivalent forms with a common denominator of 35:- -2/5 = (-2/5) * (7/7) = -14/35
- 1/7 = (1/7) * (5/5) = 5/35

**Intervals:**The difference between 5/35 and -14/35 is 19/35. To divide this range into 11 intervals, we use 19/35 / 11 = 19/385.**Insertion and arrangement:**Adding 19/385 successively to -14/35 gives us the sequence -14/35, -33/385, -14/77, -7/55, -4/35, -33/385, -14/105, -5/77, -2/35, -19/385, and finally 5/35.

**9. Journeying Between 1/2 and 2/3:**

The final task is to journey between 1/2 and 2/3 by inserting rational numbers and arranging them.

**LCM of 2 and 3:**The LCM of the denominators 2 and 3 is 6.**Equivalent fractions:**We convert both fractions to equivalent forms with a common denominator of 6:- 1/2 = (1/2) * (3/3) = 3/6
- 2/3 = (2/3) * (2/2) = 4/6

**Intervals:**The difference between 4/6 and 3/6 is 1/6. To divide this range into 6 intervals, we use 1/6.**Insertion and arrangement:**Adding 1/6 successively to 3/6 gives us the sequence 3/6, 4/6, 5/6.