I agree with Vlad, brainteasers are not very common (referring to my experience, when I applied, I got 1 out of 20+ interviews). Still, answering to your question, why do they ask them?
The main reason is to understand how you structure complex problems. Brainteasers usually require a more out-to-the-box approach that is more difficult to prepare than market sizing or cases. As a consequence they let the interviewer understand some peculiar elements of the logic of the candidate even when he/she is extensively prepared on cases.
The negative side of brainteasers is that they are usually a one-dimension type of question - that is, you need one key inside only to solve it. Business cases and market sizing include instead more key elements to be analyzed and thus offer more insides on the candidate.
Mental math is a different topic. You have to know how to do mental math in your head, as you could have to do it in front of the client. You cannot have a math problem and tell the client "Please give me one second, I will calculate 67% of 67 on my phone". Part of the skills you have to show to clients is credibility, and good mental math is one of the elements that will support that.
To improve on brainteasers, the best thing would be to download a list of the most common ones and simply practice on them - you will recognize after a while there are some commonalities between them. A good idea would be to do it gradually, allocating them proportionally in the days before the interview to keep learning constant. As mentioned, they are pretty unlikely nowadays though.
To improve on the math, you can consider the following tips:
- Divide complex math in smaller logical steps: if you have to compute 96*39, you can divide it in 96*40 - 96*1 = 100*40 - 4*40 - 96*1 = 4000 – 160 – 100 + 4 = 3744
- Use shortcuts: the most useful to know are:
- Fractions, at least 1/6, 1/7, 1/8, 1/9
- Cubes, ideally until 19^2
- Practice: there are several online services you can find to improve your speed; I would recommend to keep track of the performance in an excel sheet to assess whether you are improving and if not, analyze the reasons for that