I used to teach GRE and SAT test prep for a large, well known company and also for a small boutique firm - both places taught logic strategies heavily.
Yes, you do need math skills to get questions right. But you'd be surprised how often you can logic your way to the right answer quickly without doing any math once you are very proficient at the strategies. They are very awkward and cumbersome at first, but once you are good at them they will make you a much faster and better test taker.
Reasoning strategies are particularly helpful for quickly eliminating 2-3 wrong answers which increases your chances of guessing correctly significantly if you dont know how to solve it or if you do something wrong in your calculations.
Many wrong answers in the math section are wrong because you made an easy math mistake. For example, maybe you forgot a negative sign in your calculation. Logic would tell you before you started that you will need a negative answer. As a result, you can eliminate any positive answer choices right away. Then, if you calculate out a positive you know you are wrong and need to try again. If you are running short on time or aren't sure how to proceed you can take a guess and you've upped your odds from a 20% chance to a 50% chance of guessing correctly. If you have to guess on 10 questions you just theoretically upped your score on those questions from 2 out of 10 correct to 5 out of 10 correct, which will impact your end score.
If you are relying solely on your ability to mathematically solve problems you'll never up your odds from that 20% to 50% and you'll have less time for harder questions. Also, a lot of questions aren't difficult mathematically but they are written in really jerk-tactic ways that are designed to trick you into picking a wrong answers that are tempting for some reason or another. You could be reeeeeeallly good at the fundamental mathematics needed and still get bamboozled.
When I was tutoring my high school students used to get mad because if we got new SAT prep material into the testing center I would do practice tests with them and finish in about 1/2 the allotted time with 95%+ scores while they struggled through. That speed and accuracy was as much due to reasoning as it was to actual math skills.
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