Question 1194539
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Answer: <font color=red>425 is the original number</font>


See below for the work shown and explanation.


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x = hundreds digit
y = tens digit
z = units or ones digit
These variables represent nonnegative single-digit whole numbers from the set {0,1,2,3,4,5,6,7,8,9}
Though I should be more careful to point out that x = 0 is not possible since something like 012 = 12 is a two digit number. 
Luckily, zero is allowed for the other digits.


Any three-digit number is of the form 100x+10y+z
For example, if we had the following three values: 
x = 1
y = 2
z = 3
then,
100x+10y+z = 100*1+10*2+3 = 123


We're told that "the units digit is 3 more than the tens digit", which means symbolically we would say z = y+3
Whatever y is, add on 3 to get z.


We also know that "the sum of the digits is 11", so,
x+y+z = 11
Let's replace z with y+3 and simplify
x+y+z = 11
x+y+y+3 = 11
x+2y+3 = 11
x+2y = 11-3
x+2y = 8
Then we can solve for x
x+2y = 8
x = 8-2y
which will be useful later in a substitution step.


Let A = 100x+10y+z be the original three-digit number, and B = 100z+10y+x be the reversed version of the original number (eg: A = 123 and B = 321)
As you can see, I swapped the hundreds digit and units digit to reverse the number.


We're told that "if the units and hundreds digits are interchanged the number increased by 99."
meaning,
new = old+99
B = A+99
100z+10y+x = 100x+10y+z+99


From here, we will plug in these items
x = 8-2y
z = y+3
so that everything is now in terms of y. This will allow us to solve for y.


100z+10y+x = 100x+10y+z+99
100(y+3)+10y+8-2y = 100(8-2y)+10y+y+3+99
100y+300+10y+8-2y = 800-200y+10y+y+3+99
108y+308 = 902-189y
108y+189y = 902-308
297y = 594
y = 594/297
y = 2


Then use this to find x and z
x = 8-2y = 8-2*2 = 4
z = y+3 = 2+3 = 5


The original number is:
A = 100x+10y+z
A = 100*4+10*2+5
A = 400+20+5
A = <font color=red>425</font>
The reversed number is
B = 100z+10y+x
B = 100*5+10*2+4
B = 500+20+4
B = 524


Notice how B-A = 524-425 = 99
which helps show that B = A+99 is correct.
Or you could verify like so:
B = A+99
524 = 425+99
524 = 524


Also, 
x+y+z = 4+2+5 = 6+5 = 11
confirms the digits add up to 11


Lastly, the units digit (5) is exactly three more than the tens digit (2).


Therefore, we have fully confirmed the <font color=red>answer of 425</font>
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