Question 198448: The sum of the digits of a three-digit number is 13. If the tens and hundreds digits are interchanged, the new number is 90 less than the original, and if the units and hundreds digit are interchanged, the resulting number is 99 less than the original. Find the original number.
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! The sum of the digits of a three-digit number is 13.
If the tens and hundreds digits are interchanged, the new number is 90 less than the original,
and if the units and hundreds digit are interchanged, the resulting number is 99 less than the original.
Find the original number.
:
The three digits, x, y, z
The original number: 100x + 10y + z
:
Write an equation for each statement:
:
"The sum of the digits of a three-digit number is 13."
x + y + z = 13
:
"If the tens and hundreds digits are interchanged, the new number is 90 less than the original,"
100y + 10x + z = 100x + 10y + z - 90
:
100y - 10y + z - z = 100x - 10x - 90
:
90y = 90x - 90
Simplify,divide by 90
y = x - 1
:
:
"if the units and hundreds digit are interchanged, the resulting number is 99 less than the original."
100z + 10y + x = 100x + 10y + z - 99
:
100z - z + 10y - 10y = 100x - x - 99
:
99z = 99x - 99
Simplify, divide by 99
z = x - 1
:
:
Using the digit sum equation, substitute (x-1) for y and (x-1) for z
x + y + z = 13
x + (x-1) + (x-1) = 13
3y - 2 = 13
3x = 13 + 2
x = 
x = 5
then
y = 4 and z = 4
:
The original number: 544
:
:
Check solution in the statement:
"If the tens and hundreds digits are interchanged, the new number is 90 less than the original,"
454 = 544 - 90
and
"if the units and hundreds digit are interchanged, the resulting number is 99 less than the original."
445 = 544 - 99
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