Introduction:
Finding rational numbers between two given numbers involves looking for numbers that can be expressed as a fraction with a numerator and denominator. In this case, we need to find five rational numbers between 2/3 and 4/5.

Step 1: Finding the common denominator:
To compare the fractions 2/3 and 4/5, we need to find a common denominator. The least common multiple (LCM) of 3 and 5 is 15. Therefore, we can rewrite the fractions as 10/15 and 12/15.

Step 2: Dividing the range:
We now have a range of numbers from 10/15 to 12/15 (inclusive) between which we need to find five rational numbers. To divide this range equally, we can calculate the difference between the two fractions and divide it into 6 equal parts.

Step 3: Dividing the difference:
The difference between 12/15 and 10/15 is 2/15. Dividing this difference into 6 equal parts gives us 1/15.

Step 4: Finding the rational numbers:
We can now find five rational numbers by adding multiples of 1/15 to the initial fraction 10/15. Adding 1/15, 2/15, 3/15, 4/15, and 5/15 to 10/15, we get the following rational numbers:

1. 11/15: Adding 1/15 to 10/15 gives us 11/15.
2. 12/15: Adding 2/15 to 10/15 gives us 12/15.
3. 13/15: Adding 3/15 to 10/15 gives us 13/15.
4. 14/15: Adding 4/15 to 10/15 gives us 14/15.
5. 15/15: Adding 5/15 to 10/15 gives us 15/15, which simplifies to 1.

Conclusion:
In conclusion, we have found five rational numbers between 2/3 and 4/5 by dividing the difference between the two fractions into six equal parts and adding them to the initial fraction. The five rational numbers are 11/15, 12/15, 13/15, 14/15, and 15/15.

LCM of 2/3 and 4/5 :
LCM = 15
2/3 × 5/5 = 10/15
4/5 × 5/5 = 20/15

10/15 × 4/4 = 40/60
20/15 × 4/4 = 80/60
Five rational numbers between 40/60 and 80/60 are:
45/60 , 50/60 , 55/60 , 67/60 , 78/80 etc.