I've been working with the Advanced Quant guide by ManhattanGMAT for some time now and really benefited a lot from their excellent strategy guide on DS questions. I decided to summarize the basic steps of their DS section with an emphasis on what I think to be the most important points and combined this with my own experiences. I thought this might be beneficial to other users as well. I do not intend to violate any copyrights, I will merely focus on the basic principles without using any of the examples. The Advanced Quant Strategy guide is excellent and I think you take the most from it by really going through all the "Try-It" examples and all the questions in the back of the book. The book has a lot more to offer than the DS section and the other sections will definitely make you a smarter and better test taker, so buy the book if you find this helpful.

DS Strategy:

• Make sure to write out all the important information in the question stem, whether it be certain (integer, pos/neg etc.) constraints, equations or the actual question. I noticed that this definitely helps you to prevent a lot of silly mistakes, such as forgetting about a constraint that is crucial to answering the question, but it also takes a lot of time. In my opinion it's best to try to pick up the constraints and then keep them in the back of your mind. When you analyze each statement, quickly get back to the question stem and look for the constraints, you save a couple of seconds this way and avoid silly mistakes due to forgetting about important information as good as possible.
• Always rephrase any equations, inequalities, fractions etc. Most of the simplifications that the GMAT uses are of the same pattern. For example try to write expressions with exponents in such a way that all of the bases are the same, or whenever you see a sum with root contained in it, try to multiply by its conjugate. The more problems you do the more you become accustomed to the GMAT style of simplifying expressions.
• Separate the statements on your paper cleary. I think this is really important, because statement carryover is a lot more likely, if you don't do it. Also you should always actively think about the danger of statement carryover, that way you are less likely to do it.
• Be a contrarian. You always have to look at all the possible scenarios when analyzing the statements, one trick that sometimes help you to detect that a statement is insufficient is to look at the other statement and deliberately try to violate that statement. I found this tip to be really helpful, because a lot of the times the opposite of the other statement is exactly the scenario that proves insufficiency.
• Assume nothing. On most questions where there aren't any constraints given, people sometimes still assume that there is e.g an integer constraint and thus only test integer cases. By really paying attention to what is given and whats NOT given, you can find the correct answer on a lot of questions.
• Compute the completion. When people see quadratic expressions, they sometimes assume that there are two solutions and that this proves insufficiency, since you can't find a single solution. This is a big mistake, on some occasions the value you are looking for is a product of two variables and plugging the two solutions in the product you actually get the same result. I found myself commiting this mistake quite often. You should always finish your calculation, unless you're 100% sure that you're correct. You don't want to spend a lot of time on a question, only to find yourself getting the question wrong, because you skipped the last calculation step, which would have cost you maybe 10 seconds.
• Beware of inequalities. Inequalities are really important in DS questions. You should especially look out for inequalities that combine with integer constraints, this will often give you the solution. If you forget about the constraint, the inequality in the statement might be of little use. Also look out for variables in the denominator in an inequality, never multiply the inequality by a variable, unless you know the sign of the variable. The GMAT uses this rule a lot in inequality DS questions.
• Use a scenario chart. In some problems there are only a handful of different options, this is especially true in positive/negative and even/odd type of questions. Create a table with each variable and each statement in it's own column and different scenarios in each row. Then pick numbers that satsify the given statements and see if there's only one solution to the question or multiple solutions. As soon as you find two possible solutions the statement is insufficient and you can go on to the next statement. If both statements are insufficient, use only those values for the variables that satsify both statements.
• Test numbers systematically. I think this is the most important skill in DS questions. Obviously you need to know all the basics to perform well on this test and it's always good to be able to prove sufficiency or insufficiency elagantly in an algebraic way. But sometimes a statement is just too difficult to handle algebraically or it takes too much time to solve algebraically. No matter how difficult a statement is, you can always pick numbers to prove sufficiency or insufficiency. But don't just test numbers randomly, you should have a systematic approach to it. MGMAT suggests to always have a standard set of numbers, which can be adjusted to the constraints given in the question and statement. The numbers are -2,-3/2,-1,-1/2,0,1/2,1,3/2 and 2. On some occasions it's also crucial to pick boundary values, e.g if a question ask whether x is less than or equal to 1 and a statement gives you x² less than or equal to 1.3, you can e.g pick x=1.1 to prove insufficiency.

Those are in my opinion the most important stategic points to remember for all DS questions. There a couple of more points in the guide, but I didn't think they were that helpful.

Thank You very mu ch.That was very helpful.