Weights - Objects into bags puzzle

Question

I found a maths puzzle somewhere and a part of it as below:

• Kelly wants to place n objects $a_1 , a_2 , ··· , a_n$ into $k > 1$ bags.
• For each $i = 1 , 2 , ··· , n $, the weight of $a_i$ is $w_i$ kilograms.

• The capacity of each bag is not a problem as it is not limited.

  Kelly wants to minimize the weight of the most heavy bag. So this is what she does:

• She ordered in an arbitrary order.

• She always places the object under consideration into the least heavy bag.

At the end when all objects are in the bags, Kelly finds that the weight of the most heavy bag is at most 2 times the optimum.

How? ? ? How can she get to know that?

Answer 1

Denote by $M$ the optimal weight of the most heavy bag.

Hint:

• At all times, the weigh of the least heavy bag has to be smaller than $M$ (consider the total weight of items).
• The weigh of any item has to be smaller or equal than $M$ (the most heavy item has to go somewhere).

I hope this helps $\ddot\smile$