SOLUTION: A two digit number is 54 more than the number formed by interchanging the digits the difference between the product and the sum of the digits is 15. Find the number

Question 946177: A two digit number is 54 more than the number formed by interchanging the digits the difference between the product and the sum of the digits is 15. Find the number
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
let a = the 10's digit
let b = the units
then
10a + b = the number
and
10b + a = the number with digits reversed
:
A two digit number is 54 more than the number formed by interchanging the digits
10a + b = 10b + a + 54
10a - a = 10b - b + 54
9a = 9b + 54
simplify, divide by 9
a = b + 6
:
the difference between the product and the sum of the digits is 15.
ab - (a+b) = 15
replace a with (b+6)
b(b+6) - ((b+6)+b) = 15
b^2 + 6b - (2b + 6) = 15
combine like terms
b^2 + 6b - 2b - 6 - 15 = 0
b^2 + 4b - 21 = 0
Factors to
(b + 7)(b - 3) = 0
The positive solution is what we want here
b = 3
Find a
a = 3 + 6
a = 9
:
93 is the number.
:
:
Check this in the statement:
"A two digit number is 54 more than the number formed by interchanging the digits "
93 = 39 + 54
:
You can check it in the other statement
" the difference between the product and the sum of the digits is 15.
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