I have a pretty simple straightforward question.
Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$
Instinctively, I do the quickest thing I know how to do. $$\times10!\implies x = \frac{10!}{8!} + \frac{10!}{9!} = 9 \times 10 + 9 = 99$$ That seemed pretty easy and the back of my textbook says it's the right answer.
Just for fun, I do it again. Except this time, I initially multiply the equation with $9!$ instead of $10!$,
$$\times 9!\implies\frac{9!}{10!}x = \frac{9!}{8!} + \frac{9!}{9!}$$
$$\implies\frac{x}{10} = 9 + 1$$
$$\times 10\implies x = 10\times10 = 100$$
Sure, by multiplying by $10$ in the last step, I've still multiplied by $10!$.
But now, this $1\%$ variation in the answer has popped up.
So, I did what any highschooler would do in this situation;
I consulted Wolfram Alpha which told me that my $2^{\text{nd}}$ trial was correct.
Who should I believe? A computational knowledge engine or the back of my textbook?
There's a mistake in your first attempt:
$$ x = \frac{10!}{8!} + \frac{10!}{9!} = 10 \times 9 + 10 = 100 $$