SOLUTION: Find a two digit integer that is increased by 1/5 of its value if its digit are reversed

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Question 696128: Find a two digit integer that is increased by 1/5 of its value if its digit are reversed
Found 2 solutions by ptaylor, KMST:
Answer by ptaylor(2198) About Me  (Show Source):
You can put this solution on YOUR website!
10x+y=the integer
When we reverse the digits, we have:
10y+x, soooo
10y+x=10x+y+(1/5)*(10x+y)
10y+x=10x+y+2x+y/5
10y-(6/5)y=11x
(44/5)y=11x
x=((44/5)y)/11
x=(44/55)y or
x=(4/5)y
y=5; x=4 This is the only possibility for 2-digit integers
So the integer has to be 45
CK
54=45+(1/5)*45=54
Hope this helps---ptaylor

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
a = first (tens) digit
b = second (ones) digit
The number looks like ab,
but its value is 10a%2Bb.

The number formed when the digits are reversed looks like ba,
and its value is 10b%2Ba , which is larger.
The reversal of the digits increases the value by 1/5 to
1%2B1%2F5=6%2F5 of the original value,
making the reverse number equal to 6%2F5 times the original number.
So 10b%2Ba=%286%2F5%29%2810a%2Bb%29

First we simplify:
10b%2Ba=%286%2F5%29%2810a%2Bb%29 --> 5%2810b%2Ba%29=5%286%2F5%29%2810a%2Bb%29 --> 5%2810b%2Ba%29=6%2810a%2Bb%29 --> 50b%2B5a=60a%2B6b

Next we solve:
50b%2B5a=60a%2B6b --> 50b%2B5a-5a-6b=60a%2B6b-54-6b --> 44b=55a --> 44b%2F11=55a%2F11 --> highlight%284b=5a%29

Since a and b are integers,
for 4b to be the multiple of 5 indicated by 5b,
b must be a multiple of 5.
Since b is a digit and a multiple of 5, it can only be highlight%28b=5%29.
So 4%2A5=5a --> 4%2A5%2F5=5a%2F5 --> highlight%28a=4%29,
and the original two-digit number is highlight%2845%29.