SOLUTION: The sum of three digits is 12. The tens digit exceeds the hundreds digit by the same amount that the units digit exceeds the tens digit. If the digits are reversed, the new numbe
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Question 172625This question is from textbook Algebra Structure and Method Book 1
: The sum of three digits is 12. The tens digit exceeds the hundreds digit by the same amount that the units digit exceeds the tens digit. If the digits are reversed, the new number exceeds the original number of 198. Find the original number. This question is from textbook Algebra Structure and Method Book 1
You can put this solution on YOUR website! The sum of three digits is 12. The tens digit exceeds the hundreds digit by the same amount that the units digit exceeds the tens digit. If the digits are reversed, the new number exceeds the original number of 198. Find the original number.
:
x = 100's digit
y = 10's digit
z = units
:
The number 100x + 10y + z
;
Write an equation for each statement:
:
"The sum of three digits is 12. "
x + y + z = 12
:
"The tens digit exceeds the hundreds digit by the same amount that the units digit exceeds the tens digit."
y - x = z - y
y + y - x - z = 0
2y - x - z = 0
:
We can use these first two equations to find y using elimination
x + y + z = 12
-x +2y - z = 0
------------------addition eliminates x and z
0 + 3y + 0 = 12
y =
y = 4
:
"If the digits are reversed, the new number exceeds the original number of 198.
100z + 10y + x = 100x + 10y + z + 198
Simplify
100z - z + 10y - 10y + x - 100x = 198
99z - 99x = 198
Simplify divide equation by 99, results:
z - x = 2
z = x + 2
:
Find the original number.
:
In the 1st equation, substitute 4 for y, and (x+2) for z; find x
:
x + 4 + (x+2) = 12
2x + 6 = 12
2x = 12 - 6
x =
x = 3
:
An easy step to find z:
z = 3 + 2
z = 5
:
The number 345
:
:
Check the solution using the statement:
"If the digits are reversed, the new number exceeds the original number of 198.
543 = 345 + 198
