GMAT question 31-33

If 12 ounces of a strong vinegar solution are diluted with 50 ounces of water to form a three-percent vinegar solution, what was the concentration of the original solution?  19.3%  17%  16.67%  15.5%  12.5%

Solution:
We know that the new solution is 3% vinegar and has a total of 62 ounces of liquid.  Using the percent equation:  % = part/whole   3/100 = ounces vinegar/62  (3/100)62 = ounces of vinegar  Now, before we actually calculate, let's think about what we're doing next. We want to know the % vinegar in the original solution. So, once again:   % = part/whole * 100%  % = (3/100) (62)/12 * 100%  % = (186)*(1/12) %  Doing some cancellation:  % = (93/6) %  % = 15.5%... choose D

A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale?  A. 20  B. 36  C. 48  D. 60  E. 84

Solution:
The R scale measurements went from 6 to 24; the S scale measurements went from 30 to 60.  So, for every increase of 18 in R, we have an increase of 30 in S.  change in R/change in S = 18/30 = 3/5   In our last jump, S goes from 60 to 100 (change of 40). Plugging that into our ratio we get:  change in R/40 = 3/5  change in R = (3/5)40 = 24   R started at 24; 24+24 = 48, choose C.

In the xy-plane, does the line in question y=3x+2 contain the point (r,s)?  (1) (3r+2-s)(4r+9-s)=0  (2) (4r-6-s)(3r+2-s)=0

Solution:
When two terms multiply to 0, at least one of them must be 0. Let's look at the statements in those terms.
(1) either (3r+2-s)= 0 or (4r+9-s)= 0  Since r is our x coordinate and s is our y coordinate, we can rewrite in "y=mx+b" form:  either:  s = 3r + 2; or s = 4r + 9  In the first case, if s = 3r + 2, does s = 3r + 2? Definitely YES.  In the second case, if s = 4r + 9, could s = 3r + 2? Sure, we really have no clue, it depends on the values of s and r.  Therefore, (1) is insufficient.   (2) Jumping ahead to our final calculations, we get either:  s = 3r + 2; or s = 4r - 6  We have the same first case as for (1), so we know we can get a "YES".  In the second case, if s = 4r - 6, do we know if s = 3r + 2? Nope, no clue. Insufficient.  Eliminate A, B and D.   Now we need to combine the statements. We now know that:  either:  s = 3r + 2; or s = 4r + 9  AND  s = 3r + 2; or s = 4r - 6   Now, could both second equations be true? That is, could it be true that:  s = 4r + 9 AND s = 4r - 6?  Well, if we subtract the second equation from the first, we get:  0 = 15  which is patently absurd. Therefore, it's impossible for both of those equations to be true.   Since both of them cannot be true, we know for sure that, to make the original statements correct, it must be true that: s = 3r + 2, giving us a definite "YES" answer to the original question: choose (C).