Find two rational numbers between the following using average method:

• a) -2 and 3
• b) ⅔ and 13/14

To find two rational numbers between given pairs using the average method, we can follow a straightforward approach. The average of two numbers is calculated by adding them together and then dividing by two. This average will give us a rational number that lies between the two. We can then find another rational number by taking the average of one of the original numbers and the average we just calculated. Let’s break this down for both pairs you provided.

Finding Rational Numbers Between -2 and 3

First, let’s calculate the average of -2 and 3:

• Step 1: Add the two numbers: -2 + 3 = 1
• Step 2: Divide by 2: 1 / 2 = 0.5

So, the first rational number we found is 0.5. Now, let’s find another rational number by averaging -2 and 0.5:

• Step 1: Add -2 and 0.5: -2 + 0.5 = -1.5
• Step 2: Divide by 2: -1.5 / 2 = -0.75

Thus, the two rational numbers between -2 and 3 are -0.75 and 0.5.

Finding Rational Numbers Between ⅔ and 13/14

Next, let’s apply the same method to the second pair, ⅔ and 13/14:

• Step 1: Convert both fractions to a common denominator. The least common multiple of 3 and 14 is 42.
• Step 2: Convert ⅔: (⅔) × (14/14) = 28/42
• Step 3: Convert 13/14: (13/14) × (3/3) = 39/42

Now, we can find the average of 28/42 and 39/42:

• Step 4: Add the two fractions: 28/42 + 39/42 = 67/42
• Step 5: Divide by 2: (67/42) / 2 = 67/84

The first rational number we found is 67/84. Now, let’s find another rational number by averaging ⅔ (28/42) and 67/84:

• Step 6: Convert ⅔ to the common denominator of 84: (28/42) × (2/2) = 56/84
• Step 7: Add 56/84 and 67/84: 56/84 + 67/84 = 123/84
• Step 8: Divide by 2: (123/84) / 2 = 123/168

Thus, the two rational numbers between ⅔ and 13/14 are 67/84 and 123/168.

Summary of Results

In summary, the two rational numbers between -2 and 3 are -0.75 and 0.5, while the two rational numbers between ⅔ and 13/14 are 67/84 and 123/168. This method of finding averages is a reliable way to identify rational numbers between any two given values.