Question

Please explain #2 for me please! The question deals with proportions.

Answer 1

We claim that the ratio of the areas is (3/2)^3=27/8, not just for these lengths, but also for every possible set of side-lengths x, y, and z.

To find the volume of the area of the original suitcase, multiply together x, y, and z to get xyz.  The modified suitcase will have all these dimensions multiplied by 1 1/2, or 3/2, for a result of 3x/2*3y/x*3z/2=27xyz/8.  The ratio is (27xyz/8)/(xyz)=27/8, as desired.

Note that we made no assumptions about the value of x, y, and z throughout the whole solution!  Therefore, we can plug ANY side-lengths, including the problem's set (28, 16, 8) into this problem to achieve the same ratio.