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

The Riddle of Four

Yesterday I saw an amazing riddle on Yahoo Answers, and I had to share it, but it was after I published the calculus question, and I didn't want to post too much. Even though I answered it with detail, the asker promised to give the best answer to the first one who answers, and my long answer took away my 10 points.

Without further explaining, here is the riddle:

A number has six digits, and the last one (the units digit) is 4. When the 4 is removed from the units place and placed on the beginning of the number (the hundred thousands), the number is multiplied by 4. What is the number?

It sounds difficult and tempting to solve with complex algebra and calculus methods, but the solution is much simpler: use your head.

First, we know that when the 4 is moved to the beginning of the number, it's multiplied by 4. Since it was and remains 6 digits, the first digit must be 1. Now we have two digits:
1 _ _ _ _ 4

When we multiply 4 by 4, we get 16. Since the tens digit becomes the units digit and the bigger number must end in 6, the tens digit is 6. Now we're half way there!
1 _ _ _ 6 4

Now look at the end of this number: 64. When multiplied by 4, we get 4 * 64 = 256. Notice it ends in 6, which is perfect for us, because it'll fit right in the bigger number. That means we have two more digits, and only one is missing!
1 _ 2 5 6 4

To find the last digit, we need some more thinking. The bigger number looks like 4 1 _ 2 5 6. Notice that the 1 is not changed, and so does the 2. That means we need a digit that does not change when multiplied by 4, and the winner is zero.

The number is 102564. When you multiply it by 4, you get 410246. Riddle solved!

Nadav

nadavs