Part a: find a rational number that is between 7.7 and 7.9. explain why it is rational.

part b: find an irrational number that is between 7.7 and 7.9. explain why it is irrational. include the decimal approximation of the irrational number to the nearest hundredth.

Part A.
A rational number can be expressed as a quotient of two integers.
Therefore 78/10 = 39/5 = 0.78 represents a rational number between 7.7 and 7.9.
39/5

Part B
An irrational number cannot be expressed as a quotient of two integers.
Let π*x = 7.8
Then x = 7.8/π = 2.483
Try 2.5*π = 7.854 which lies between 7.7 and 7.9

7.85 (nearest hundredth) 
Because this number is approximately equal to 2.5π, it is irrational because π is an irrational number.
The actual value of 2.5π = 7.8539816 ...