Percentages and Copy Machines

Today's question is relatively easy, but very unique. It does not require complex mathematical calculations, but rather a simple mathematical thinking.

A copy machine has 7 buttons to increase or decrease the area of the photocopied area. They are 50%, 75%, 80%, 100%, 120%, 125%, and 150%. Which buttons can be eliminated from the machine in a way that saves the machines original area choices. Notice: when two buttons are pushed consecutively, the effect of the second button is on the area that remains after the first button (for example, two pushes on 50% will photocopy 25%).

It seems easy, and it really is. All you have to do is turn these percents into fractions and see how you can multiply them to get the others.

50% = 1/2
75% = 3/4
80% = 4/5
100% = 1
120% = 6/5
125% = 5/4
150% = 3/2

50% cannot be created by any combination, so it must remain in the machine.
75% = 3/4 = 1/2 * 3/2
80% cannot be created by any combination, so it also stays in the machine.
100% = 1 = 5/4 * 4/5
120% = 4/5 * 3/2
125% is another number that cannot be created from others.
150% is a component in 75% and 120%, so it can't go away.

However, this question has another answer, where 150% = 6/5 * 5/4, and then 120% stays in the machine and 150% goes out.

Easy, isn't it?
Nadav

nadavs