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?

1.Select a two-digit number between 50 and 100. Add 83 to your number. From this number form a new number by adding the digit in the hundreds place to the number formed by the other two digits (the digits in the tens place and the ones place). Now subtract this newly formed number from your original number. Your final result is 16. Use a deductive approach to show that the final result is always 16 regardless of which number you start with.

i don't get it

ITS 88

To figure out how the instructor knew that the answer would be 23, let's break down the given information step by step:

1. The student selected a two-digit number between 50 and 100 and added 76 to it, resulting in a three-digit sum.
Let's call the original two-digit number "xy". So, the three-digit sum would be "xy + 76".

2. The digit in the hundred's place is added to the remaining two-digit number.
The digit in the hundred's place is 'x', and the remaining two-digit number is 'y'. So, the sum of 'x' and 'y' can be written as "x + y".

3. Subtracting this sum from the original number produces the answer of 23.
The original number is "xy", and subtracting the sum "x + y" from it gives us "xy - (x + y)".

Now, since the instructor predicted that the answer would be 23, we can set up an equation with the given information:

xy - (x + y) = 23

To solve this equation step by step, we'll need to express the two-digit number "xy" in terms of its individual digits 'x' and 'y':

xy = 10x + y

Substituting this expression into the equation:

(10x + y) - (x + y) = 23

Now, we can simplify the equation:

10x + y - x - y = 23
9x = 23
x = 23/9

However, since 'x' must be an integer, this equation has no solution. This means that the given problem has no valid solution within the given constraints.

Therefore, it seems like the instructor couldn't have predicted that the answer would be 23 based on the information provided. It's possible that there might be a mistake or missing information in the problem.

Just convert the word problem 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? Because you moved the hundreds place number out of its place. 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. The key factor is how much you add to your original number. if instead of adding 76 I added 89 then the result would always be 10 assuming you followed the rest of the guidelines.