Posted by joshua on Tuesday, May 8, 2012 at 9:45am.
Let the number be ab
a+b=15
10b+a = 10a+b - 27
a + b = 15
-9a + 9b = -27
a + b = 15
a - b = 3
2a = 18
a=9
b=6
check: 69 = 96 - 27
Answering by logic and mental calculations:
Since the order of digits is not important (they will be reversed anyway), the only two-digit combinations that add up to 15 are (8,7) and (6,9).
(8,7) does not work because 87-78=9. Now try 96-69=27, which is obviously the answer.
When we have a two digit number, it can be expressed as :
10 a + b
In this case :
10 a + b = orginal number
The sum of the 2 digits is 15
a + b = 15 Subtract a to both sides
a - a + b = 15 - a
b = 15 - a
If you reverse the digits of 10 a + b, it will be :
10 b + a
10 b + a = orginal number - 27 Add 27 to both sides
10 b + a + 27 = orginal number - 27 + 27
10 b + a + 27 = orginal number
10( 15 - a ) + a + 27 = orginal number
150 - 10 a + a + 27 = orginal number
177 - 9 a = orginal number
Also
10 a + b = orginal number
10 a + 15 - a = orginal number
9 a + 15 = orginal number
177 - 9 a = 9 a + 15 Subtract 15 to both sides
177 - 9 a - 15 = 9 a + 15 - 15
162 - 9 a = 9 a Add 9 a to both sides
162 - 9 a + 9 a = 9 a + 9 a
162 = 18 a Divide both sides by 18
162 / 18 = a
9 = a
a = 9
b = 15 - a
b = 15 - 9 = 6
orginal number = 10 a + b = 10 * 9 + 6 = 90 + 6 = 96