<p>You’re the best Dr.Steve :)</p>
<p>@DrSteve ohh I see, didn’t read carefully. Thanks for pointing that out. I scored an 800 on math but I still screw up sometimes…lol.</p>
<p>The thing about guessing is, it works on some SAT problems but I doubt it would work on this one (too time consuming). It’s easy to show that the # of rooms is a multiple of 15, but that still leaves several numbers to try. I guess it’s partly a matter of personal preference and algebra skill. Here’s how I would do it algebraically (presuming I read the question correctly 
)</p>
<p>Let x be the number of rooms</p>
<p>x/3 rooms have 3 coats of paint
2/5 of the remaining rooms is (2/5)(2x/3) = 4x/15 rooms (2 coats of paint)
24 rooms = x - (x/3) - (4x/15) = 6x/15 → x = 60.</p>
<p>Then just do arithmetic and the answer is 116 coats.</p>
<p>I also like xiggi’s method. Can be easily visualized by drawing pictures.</p>
<p>@rspence</p>
<p>My impression is that you were probably always a relatively strong math student. This makes you naturally resistant to the method of guessing. And as a mathematician I agree completely - guessing and checking is terrible mathematics. In fact, it’s not mathematics at all. However, the SAT tutor in me has seen firsthand how effective it is for the majority of students. </p>
<p>In this particular problem, guessing is the best method for me personally. Why is that? Because my first guess would automatically be 60. How would I come to 60 right away? Well I know immediately that the number of rooms has to be a multiple of 15 greater than 24. Now, looking at the fractions in the problem I see 1/3 and 2/5 of the remaining 2/3 which leads me to multiiply 24 by something between 2 and 3. The only number satisfying all these conditions is 60. </p>
<p>Now, someone with a bit less number sense should still guess either 45, 60, or 75 - at worst, possibly 90. But even a wrong guess should point you in the direction to a better guess.</p>
<p>You may ask, well what if a student can’t come to these reasonable guesses. Well in that case we have 2 possibilities</p>
<p>(1) This student has a very weak number sense. Therefore their current SAT math score is at a place where they shouldn’t be getting to the end of the test anyway, so it is ok for them to spend the extra time with the guessing strategy until they zero in on the number even if it is a bit time consuming.</p>
<p>(2) The student simply hasn’t had enough practice with the guessing strategy. Practicing guessing leads to a better intuitive understanding of the number line and arithmetic in general.</p>
<p>In any case, I think that for most students it’s nice to have the guessing strategy in their back pocket to pull out only when needed.</p>
<p>True, there are some problems where guessing could be helpful. However there are lots of math problems (SAT and non-SAT) which are non-guessable, such as, find the largest three-digit integer n such that the sum of its divisors is odd. All in all, guessing can help for the SAT, but shouldn’t be overused. </p>
<p>Btw, #14 on the 2010 AIME I was basically a guessing problem.</p>