Lets assume that the weights of five containers are B1, B2, B3, B4 and B5 kg respectively. Also, B1 ≤ B2 ≤ B3 ≤ B4 ≤ B5. It is given that five cartons of oil are weighed two at a time in all possible ways. It means that each of the container is weighted four times. Thus,

4×(B1 + B2 + B3 + B4 + B5) = (165 + 168 + 169.5 + 171 + 172.5 + 174 + 175.5 + 177 + 180 + 181.5)

4×(B1 + B2 + B3 + B4 + B5) = 1734

(B1 + B2 + B3 + B4 + B5) = 433.5 kg……….(1)

Now, B1 and B2 must add to 165 as they are the lightest one.

B1 + B2 = 165……………(2)

Similarly, B4 and B5 must add to 181.5 as they are the heaviest one.

B4 + B5 = 181.5………….(3)

From above three equation, we get B3 = 87 kg

Also, it is obvious that B1 and B3 will add to 168 – the next possible higher value. Similarly, B3 and B5 will add to 180 – the next possible lower value.

B1 + B3 = 168………….(4)

B3 + B5 = 180………….(5)

Substituting B3 = 87, we get B1 = 81 and B5 = 93

From 2 & 3 equations, we get B2 = 84 and B4 = 88.5

Hence, the weights of five containers are 81, 84, 87, 88.5 and 93 kg.