When and WHY are we allowed to multiply\add\subtract statements on DS?

Question

Hi everyone,

This one is one of the greatest mysteries of GMAT for me.

I'll give you one example, by Bunuel.

What is the value of  \frac{x}{yz}  ?

(1) x= \frac{y}{2}  and z=  \frac{2x}{5} .

If y=10, then x=5, z=2 and in this case  \frac{x}{yz}  =  \frac{5}{20}  = \frac{1}{4}  BUT if y=20, then x=10, z=4 and in this case  \frac{x}{yz}  =  \frac{10}{80}  =  \frac{1}{8} Not sufficient.

(2) \frac{x}{2}  =  \frac{5}{2} and \frac{1}{y} = \frac{1}{10}

Multiply these two equations :  \frac{x}{yz} = \frac{5}{20} . Sufficient.

Answer: B.

Question : Why and when can we do the part in red? We are we allowed to multiply here?.. How did Bunuel decide that the multiplication is allowed and it can lead us to a conclusion?

In this case it happened on the second statement but usually this happens when we decide that the two statements are insufficient separately and we need to test them together to decide C or E.

How can we detect that a DS question involves the addition\subtraction,multiplication of the two statements when we get to the stage of testing both together? How can we detect that we can multiply\add\subtract expression within the same statement??..

For me this is always an unexpected "turn", every time I see the " and now lets add the statements and we get ... " or " now multiply the expressions and get..." I am like "why did you just do it??? Who said that this is allowed?"

This always seems a very spontaneous and impossible to detect move for me!

I hope I made myself clear on what I am trying to ask.

I'd appreciate your help!

mikemcgarry · Answer

Dear iliavko , I'm happy to respond. :-)

First of all, I believe there might be a misprint in your Statement #2. It should be: (2) \frac{x}{z} = \frac{5}{2} and \frac{1}{y} = \frac{1}{10} The first fraction is x/z , not x/2 . Here's what I will say. One of the most profoundly underestimated symbols in all of mathematics is the equal sign. The equal sign is not a decoration. It is not simply a bit of mathematical etiquette. By contrast, it is one of the most profound philosophical statements imaginable. When we say A = B, think about that. We are saying that whatever the value of A is, the value of B must be 100% identical. In any equation in which A appears we could substitute B, and vice verse. These are are wholly mathematically interchangeable, to the extent that God Himself (or any Supreme Being of your choice) would not be able to distinguish them. This is not simply limited to GMAT DS. This is universally true. If A = B and C = D, then we have two different pairs of numbers with this same profound relationship, and we can perform any mathematical operation with them: A + C = B + D A - C = B - D A*C = B*D A/C = B/D (as long as C = D is not equal to zero) C/A = D/B (as long as A = B is not equal to zero) A^C = B^D C^A = D^B etc. etc. If we were to include logarithms and trigonometric functions, we could add to this list considerably, but you get the idea. When we say that x/z = 5/2, this means these are two equal and identical things, so if we multiply one side of any equation on the planet by 5/2, of course we are allowed to multiply the other half by x/z, because, once again, no one on earth and no Supreme Being could tell the difference between these things!! Also, I will say, part of math is rule-following (left-brain) but a big part of GMAT math is pattern-recognition (right-brain). Bunuel the super-genius is very good at both. It's a bit of right-brain insight to see that if we multiply (x/z) by (1/y), we will get x/yz, which is precisely the value that the prompt question seeks. If this leap is something you find difficult, I would recommend this blog: How to do GMAT Math Faster Does all this make sense? Mike :-)

iliavko · Answer

Hi, mikemcgarry ! Wow.. thank you so much for your explanation, yes now it does make sense! I never thought of this "connection" or equivalence between two statements before! It's funny that you replied because now I am following the Magoosh course and it has lots of interesting stuff

So, just to confirm, in case we are using both statements together to decide C or E, if have like 1) x=a and 2) y=b we are allowed to do x+y=a+b because of the logic you explained and because both statements are talking about the same thing, that "thing" being the question stem . I used to get confused with the add\subtract\multiply technique because it seemed too much of a freestyle thing to do, but now I am thinking, the +\-\* is also allowed because both statements convey information about the same topic. Am I correct? Example: if we have an equation of physics that says x=y and a pie recipe that says a=b can we add them to get x+a=y+b? it wouldn't make any sense right? It would be correct mathematically, but wrong from the logical stand. This is to say that I used to look at the statements as two independent and disconnected pieces of information. Because this is what you are supposed to do when checking for A,B,D. But when you go for the C,E part you have to assume that both statements are a single piece of information and this fact allows us to logically use the +\-\* operation based on the logic that you explained. Am I correct? Maybe this sounds obvious and nooby, but I just never looked at the statements in this way! Ones again, thank you so much Mike!

mikemcgarry · Answer

Dear iliavko , I'm happy to respond. :-)

The first thing I'll say is that, yes, you are right about your point of adding equations from different contexts. It's not mathematically incorrect but just logically flawed. Just take two examples in pure math: we know for a circle, A = (pi)(r^2), and for line in the coordinate plane, y = mx + b. Both are true. If we create a monstrosity such as A + y = (pi)(r^2) + mx + b, that has no mathematical problems, but it is pure gibberish in terms of logic. Much in the same way, if I start making statements along the lines of " I am the pope and I was alive when Julius Caesar was alive "----- that statement is 100% grammatically and idiomatically correct, but the fact that it doesn't have grammar or idiom problems doesn't mean that it doesn't have serious problems!! Unlike those scenario, any mathematical problem, whether PS or DS, is a self-contained context unto itself. As a general rule, all the equations in a DS question are about the same topic; there may be a few exceptions, if the DS setup quite explicitly tells us that in one case, certain things are true, and in another cause, other things are true. Such a question would be possible, but if such warnings are not given, as is usually the case, the whole question concerns one topic. Suppose some equations are given in the DS prompt and equations are given in the two DS statements. When I consider statement #1 by itself, I am allowed to combine, by any mathematical operation, the equations in the prompt and the equations in statement #1. When I consider statement #2 by itself, I am allowed to combine, by any mathematical operation, the equations in the prompt and the equations in statement #2. If neither statement is sufficient on its own, and I consider both statements together, then at this point I can combine all the equations given in the problem. Remember that the combining can be not only +, -, *, /, but also exponents & roots as well as any combination. For example, if we are given A = B and C = D, then we know: \frac{A^3}{A-B} = \frac{C^3}{C-D} Sometimes, a DS prompt might ask for the value of some bizarre algebraic expression in terms of one variable, and it may be possible to figure out the same expression in terms of another variable. Mathematics is full of possibilities, and the more possibilities you understand, the more effective a problem-solver you will be. Does all this make sense? Mike :-)

iliavko · Answer

Hello, mikemcgarry Sorry for my late reply! Thank you so much for your explanations! It's been a small revolution for me in DS since I read your post

I get the overall logic of the statements so much better and faster now! Just one more question about the a+c=b+d concept, is it also applicable to inequalities? if I try simple examples such as 4>3 ; 2>1 then 4+2>3+1, so it appears to work. However there are rules that say you can only add inequalities with the ineq. pointing in the same direction and subtract ineq. with opposite signs. Does it have an impact on this logic? So 4>3 ; -1<5 so 5>-2 ones again appears to work. But perhaps I am missing something? Regards!

mikemcgarry · Answer

Dear iliavko , Great question! I'm happy to respond.

There are two different cases. Case I : One Inequality and One Equation Suppose we have an inequality A > B and C = D. This is much more straightforward. We certainly can add the equation to the inequality or subtract it from the inequality. A + C > B + D A - C > B - D Assuming that C & D are positive numbers, then we can also multiply and divide the inequality by the equation. (Multiplying or dividing by a negative reverses the order of the inequality.) AC > BD (A/C) > (B/D) Everything with exponents gets much trickier, because even powers of negatives are positive. There are no reliable rules for exponentiating both sides of an inequality. Case II : Two Inequalities This is much trickier. What you read is the correct rule for addition and subtraction. When the inequalities are in the same direction, we can add. When the inequalities are in opposite directions, we can subtract. (Be careful, in math speak, not to refer to the opposite direction of an inequality as an " opposite sign .") Thus, if A > B and C > D, it would be true that: A + C > B + D A - D > B - C Notice that these are not two separate pieces of mathematical information, but restatements of the same. If we start with the first, then subtract C & D from both sides, we get the second. Unlike in the cases above, these we get no new information in second version. For a more detailed discussion of what you can & can't do with inequalities, see: GMAT Quant: Arithmetic with Inequalities Mike

When and WHY are we allowed to multiply\add\subtract statements on DS?

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