What does a 31 on the ACT Math section convert to for SAT Math (the regular SAT, not the subject test)? If you do it mathematically, a 31/36 is approximately a 688/800, but I've heard that the scores are actually equalized to something else because of percentiles - closer to a 720-840 for a 31. Is this true?
Unfortunately, there is no official answer. Why? The official ACT-to-SAT (or SAT-to-ACT) conversion charts compare SAT Reading + Math to ACT composite scores and SAT Writing scores to ACT English + Writing scores. All of these comparisons bundle in multiple sections to compare to each other, which makes it a bit tricky. According to these concordance charts, a 31 ACT composite score is equivalent to a 1360-1390 on the old SAT Reading + Math sections. Assuming you got the same (or close to the same) score on SAT Math and Reading, that would mean a 680-690 score on each of those subsections.
Comparing percentiles is probably a more accurate way to go about getting an equivalent SAT Math score for your 31 ACT Math score. If you look at the ACT percentile charts, you'll see that a 31 on ACT Math puts you in the 96th percentile. To figure out what the equivalent score for the 96th percentile on SAT Math would be, you can then take a look at the SAT percentile charts (although they're from the old SAT, the Math section hasn't changed that much, and so it's probably safe enough to compare). Using this method, you find that an SAT Math score of 720-760 puts you in the 95th-97th percentiles. Therefore, the closest approximation for a 31 on ACT Math is probably a 740 on SAT Math.
Warning: SAT Math and ACT Math do not cover exactly the same material, and some students will undoubtedly find one test easier than the other. So just because you got a 31 on ACT Math doesn't mean that you're guaranteed to score a 740 on SAT Math, should you take the SAT.