Define the relation M @ N as M @ N = M^{2} + 2M/N. What is the value of 9 @ 4? Express your answer as a common fraction.

*Substituting 9 for M and 4 for N in the given relation, we get 9 @ 4 = 9 ^{2} + 2 × 9 ÷ 4 = 81 + 18/4 = 324/4 + 18/4 = *

**171/2***.*

Define the relation A # B as A # B = (A^{2} – B^{2} + AB)/(2B). What is the value of 5 # 4? Express your answer as a common fraction.

*Substituting 5 for A and 4 for B in the given relation, we get (5 ^{2} – 4^{2} + 5 × 4)/(2 × 4) = (25 – 16 + 20)/8 = *

*29/8**.*

Using the two relations defined above, what is the value of (4 @ 2) # 10?

*We’ll start with the relation in parentheses first. So, substituting 4 for M and 2 for N in the relation provided in problem 1, we get 4 @ 2 = 4 ^{2} + 2 × 4 ÷ 2 = 16 + 8/2 = 20. Now, we can substitute 20 for A and 10 for B in the relation provided in problem 2 to get 20 # 10 = (20^{2} – 10^{2} + 20 × 10)/(2 × 10) = (400 – 100 + 200)/20 = 500/20 = 25.*

**♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.**

**Page 2 contains ONLY PROBLEMS. ♦**