1. A very magical teacher had a student select a two-digit number between 50 and 100 and write it on the board out of view of the instructor. Next, the student was asked to add 76 to the number, producing a three-digit sum. If the digit in the hundred’s place is added to the remaining two-digit number and this result is subtracted from the original number, the answer is 23, which was predicted by the instructor. How did the instructor know that the answer would be 23?
Take the word problem and convert it to a number problem. Let x = the number that you choose between 50 and 100. You add 76 to x or x+76 then you take the hundreds place digit away from its place in the sum of the 2 numbers and add it to the x+76. This is the same as subtracting 99 from the x+76 Why? The hundreds place number was moved out of its place by you. Its no longer there and you add it as a 1 to x+76. That is the same as adding (-100+1) or -99 to the x+76 so, knowing this, lets see how it would look in purely numeric form. Choose a number x between 50 and a hundred. Add 76 to it x+76 then x+76-99 =x-23. Subtract this number from your original number x or x-(x-23) or x-x+23 then the x's cancel each other out and the only thing remaining is 23. How much you add to your original number is the key factor. if instead of adding 76 I added 89 then the result would always be 10 assuming you followed the rest of the guidelines.
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