Most people here have zeroed in on the underdefined statement of "90% good". I would try to be a good Bayesian here and start with a few of my personal observations about the real world as priors.
1a) people who have formal training in assessing the quality of prediction models actively avoid using unspecific terms like "90% good".
1b) most managers, especially those bearing a title like "partner", have no formal training in metrics for prediction quality.
From these two, we can conclude that
he must be a layperson who doesn't know the basics of prediction quality metrics.
Which we can combine with another set of priors:
2a) most laypeople faced with the results of a prediction algorithm intuitively come up with the metric of precision and think it is a perfect metric for prediction quality.
2b) a hiring manager has no data available for calculating true and false negatives, since he typically loses connection to rejected job candidates and never learns whether they would have been good hires. On the other hand, the information of which hire turned out to be a bad call (false positive) is highly salient for him, for business reasons. Therefore, if he is able to calculate any metric for his prediction quality, it must be based on true and false positives only.
2c) A question whose intended answer is "you didn't provide enough data for me to give you the answer" is a very poor choice for one-upmanship. He must believe that his problem description is sufficient for an exact answer (which he knows, but hapless candidates might not find).
The above observations point in one direction: the metric he uses must be
precision, defined as: true positive cases divided by predicted-positive cases
This assumption about the metric leads to the simplest possible solution:
The question is, what percentage of people who got hired shouldn't been. This is the false positive rate of the hirer, which is 10%.
The reason why this is likely considered a brain teaser is
The problem is stated in a way inviting the answerer to use both numbers provided. But the question asks for an answer as a quotient of the people hired, so the first number (how many applied) is not relevant.
A few further thoughts on the underspecified metric
Under a realistic scenario, there are people who reject a job offer - making not just for missing data, but for data which is certainly not missing at random, because it is the best hires who can afford it. Also, it is very rare that absolutely all of the hired people were approved by a single hiring manager. So, if he has observed that 90% of people at the firm are good hires, his calculation that he is "90% good" (has a 90% precision) at hiring is also wrong. He not only picked a bad metric, but miscalculated it.
Morale of the story: Metrology is not for hobbyists.
Also, this guy is a self-important jerk, and I wouldn't want to work for him anyway.