AaronPond wrote:
Let's talk strategy here. Many explanations of Quantitative questions focus blindly on the math, but remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. As a result, I might write out some steps that I would normally just do in my head on the GMAT, but I want to make sure everyone sees the complete approach. Ready? Here is the full "GMAT Jujitsu" for this question:
First, the problem obviously gives us different rates and times. Since \(Amount = Rate * Time\), the initial situation can be expressed as \(336 = R*T\). The adjusted situation (using \(R\) and \(T\) to represent the initial amounts) would be \(336 = (R-2)(T+4)\). Since this is a system of two equations, we can substitute chunks of one to eliminate what we don’t want to see in the other. There is a natural temptation to substitute the \(R*T\) in the first equation for the \(336\) in the other. However, this doesn’t get rid of either variable – in fact, it gets rid of a number. Substituting in this way would only get us down to a smaller equation, but still with two variables. Our “target” for this equation is \(T\), so let’s eliminate \(R\).
If \(336 = R*T\), then \(R = \frac{336}{T}\) and \(336=(\frac{336}{T}-2)(T+4)\)
Multiplying both sides by \(T\) gives us: \(336T=(336-2T)(T+4)\)
Foiling this out gives us: \(336T=336T+336(4)-2T^2-8T\)
This simplifies to: \(T^2+4T=2(336)\)
At this stage, there is another temptation to bring everything over to one side so we can have it in simplified quadratic form. However, factoring this quadratic might be a little messy. I like to call math like this “Mathugliness” in my classes. Whenever it looks like the math is getting ugly, try looking for other, more conceptual ways to approach the problems. After all, the GMAT is critical-thinking test, not a “let’s-see-if-you-can-solve-obnoxious-math-the-long-way-around” test. Incidentally, notice that I didn’t multiply large numbers out when I simplified the equations above. Multiplying just makes bigger numbers, so I avoid that until I have to. This not only saves time, it also allows me to more easily see factors.
Now, one underutilized tool for many test takers is what I call “Look Out Below!” Use your answer choices as assets you can leverage to think about the question. With this problem, we have really pretty, integer answers that need to be plugged into otherwise messy math. And one of them is the right answer. You can use this to your advantage.
Since our equation above simplifies down to \(T(T+4)=2(336)\), we can plug in values into our equation, and see which ones work.
In this case, it is really easy. Answer choice “C” is way too small, since \((16)(16+4) = 16*20=320\). “D” and “E” resolve to even small numbers. All three answers can be quickly eliminated. Answer choice "A" also fails, but for a different reason: The units digit if \(T=28\) doesn’t even work out in our formula \((T)(T+4) = 2(336)\). (The units digit of 28*32 is "6", but the units digit of 2*336 is "2". You don't need to do any more math than that!)
The only answer remaining is "B", which is the right answer. On the GMAT, I wouldn't even look any further. However, it might be useful here to prove why "B" works, from a critical-thinking perspective. If \(T=24\), then \(24(28)\) should be equal to \(2(336)\). Instead of multiplying this out – which just makes bigger numbers and takes unnecessary time – it is easier to factor the numbers, trying to turn \(2(336)\) into \(24(28)\). Our goal, therefore, is to factor out a \(12\) out of \(336\). This is a piece of cake. After all, \(336=6(50+6)=6(2)(25+3) = 12*28\). And \(24(28) = 2(12*28)\). We are done without doing any messy multiplication. It matches perfectly. “B” is definitely our answer.
Now, let’s look back at this problem from the perspective of strategy. This question can teach us several patterns seen throughout the GMAT. First, the GMAT tries to bait you into doing math the long way around. But if you use the answer choices as part of the analysis of the problem, look for common factors, and intelligently use math in a strategic way, you can avoid a lot of "Mathugliness!" That is how you think like the GMAT.
Can you explain to solve this problem conceptually? Not even without the math or equations. I've been a fan of
Veritas prep books where solutions to problems are only conceptual sometimes, not even with equations. That's so amazing. If there is any conceptual related solution, please add it here. Thanks in advance!