>The problem does not state that the numbers have to be integers. a and b happen to be 9 +- 2 sqrt(11)
but the problem does state that you should be able to do it in your head. who exactly should be able to formulate and reduce simultaneous equations in xy then apply the quadratic formula (with some spicy +/- to keep track of) to get an answer with an irrational number, all in their head? usually, when a problem like this is given there is a shortcut that leads to a simple, not only rational but integer, answer.
the statement "you can do it in your head" generally does not entail this much complexity, as the person who said "you can do it in your head" comes out and says after previously spending a fair amount of time working on it.
words matter, people, that's why I didn't throw in the adjective integral even though I could have.
Well, I had to write it down, but I have to write down everything these days. But from the way the problem was phrased, it was obvious you don;t have to actually find to numbers.
None of this is required for solving the problem in your head. All that is required is the ability to add 1-digit unit fractions in your head, as the problem requests.
> the statement "you can do it in your head" generally does not entail this much complexity
It's funny that you jump to accusing OP of falsely claiming you can do it in your head, without apparently considering the alternative: that the intended solution is a simpler one than you outlined.
Trust me, you can do this in your head if you know basic high school level math, and you don't need to solve quadratic equations or keep a ton of numbers in your head at the same time.
If I ask you if 123456789 is a prime number, do you complain that it's not fair to make you perform division on such a long number?
The difference between the two is that it’s clear that 123456789 can’t be prime since the sum of the digits is a multiple of 3, which doesn’t even require finding the sum since we know 1+8, 2+7 up to 4+5 are multiples of 3. I can even tell you that 43717421 isn’t prime without having to do a divisibility test on it by looking at the digits, although it is a bit more tedious than the 123456789 field.
the difference between the two is that I removed the factors of 3 from 123456789 to get 13717421. so much for your secret knowledge of a hyperspecific case.