The sum of a two digit number and the number formed by interchanging its digits is 110. If 20 is subtracted from the original number, the new...

Hello!

The original number may be written as `bar(ab),` where `a` and `b` are one-digit natural numbers, `0lt=blt=9,` `1lt=alt=9.` The value of the number `bar(ab)` is obviously `b + 10a.`

When we interchange its digits, the resulting number is `bar(ba) = a + 10b.` The sum is `(b + 10a) + (a + 10b) = 11a + 11b = 110` (this is given). So

`a + b = 10.`

The second condition gives that

`(b + 10a) - 20 = 4(a + b) + 4,` or `6a - 3b = 24, or 2a - b = 8.`

Thus we have a linear system for `a` and `b.` If we add up these two equations, we obtain `3a = 18,` or `a = 6.` From the first equation `b = 10 - a = 4.` The initial restrictions for `a` and `b` are also satisfied.

The answer: the original number is 64.