In the pairs, each number is mentioned $4$ times. Since the sum of the sums of the pairs is $216$, it follows that the sum of our $5$ numbers is $54$.
The sum of the two smallest is $16$, the sum of the two biggest is $27$, so the middle person is $11$.
But then the smallest plus $11$ is $18$, making the smallest $7$. You can take it the rest of the way.
Remark: With basic "figuring" we can find the only $5$ numbers that could possibly work. However, the calculation by no means uses all the information provided. So after we have found the only possible $5$ candidate numbers, we need to check that the $10$ sums are as given: we might have been lied to. They do work.