Divide And Conquer

The SBC GMAT Files

Quant: Divide And Conquer Part 1: A Top Down Approach

Any teacher who has prepared students for the GMAT, SAT, or any other standardized test, will recognize the following typical student exclamation: “It’s looks so easy when you solve it!”

Why do students so often feel this way? Now, I’m not referring to cases where the problem has a unique “twist” setting it apart from common GMAT problems; I’m talking about typical problems, employing concepts the student already comprehends and solvable with techniques the student has already practiced. More often than not, all the teacher does is show his or her student how a certain problem is solved without actually teaching anything new. So what could possibly be the difference? What explains this sudden, magical clarity?

The main difference between solving a problem on your own and having it explained to you can be summed up as follows: A teacher divides the problem into several distinct, well outlined sub-questions. Viewed at one glance, a GMAT problem may seem complex and unyielding. Divided into sub-questions, the same problem is revealed to consist of several simple ”˜mini-problems’, each easily conquered on its own. A good teacher knows both how to break the problem apart correctly, and how to highlight the conclusion of each phase as the premise of the following one. The solution is reformulated as a chain, each link leading seamlessly to the next.

This point is not meant to make you into GMAT instructors. I’m emphasizing it because my experience has taught me that students often fail to solve a problem because they’re attempting to take it down with one blow, instead of breaking it into smaller, manageable sub-parts.

The Nuts and Bolts
Take a look at the following Official Guide problem (12th ed. Problem solving Q #156):

A tank contains 10,000 gallons of a solution that is 5 percent sodium chloride by volume. If 2,500 gallons of water evaporate from the tank, the remaining solution will be approximately what percent sodium chloride?

When read in its entirety, this problem looks pretty nasty. Breaking it apart, however, reveals that there is nothing threatening about it: it’s no more than a string of simple questions and calculations.

Start by asking yourself what steps will be necessary to answer the question. Finding the percent of sodium chloride in the solution demands nothing more than answering three rather simple sub-questions:

1. What is the amount of sodium chloride in the tank?
2. What is the total amount of fluid in the tank?
3. What percentage does the first constitute from the second?

Question #1: What is the amount of sodium chloride in the tank?

Since sodium chloride constitutes 5% of the original 10,000 gallons in the tank, the amount of sodium chloride is (5/100) × 10,000 = 5 × 100 = 500. Write this down – 500 gallons of sodium chloride.

Question #2: What is the total amount of fluid in the tank?

This one’s a breeze. After 2,500 gallons of water have evaporated, the remaining amount of fluid in the tank is 10,000 – 2,500 = 7,500. Write this down as well – 7,500 gallons in the tank.

Question #3: What percent is 500 out of 7,500?

This one isn’t as simple as the first two, but it isn’t that bad either. 500 out of 7,500 is 500 / 7,500 = (1 / 15). Multiply by 100 to transition to percents: (1/15) × 100 is 100 / 15, which is ”¦ Hmm… That’s odd; 100 isn’t divisible by 15. What previously seemed like a minor calculation turned out to be not-so-simple after all.

This is where many test-takers panic or lose their train of thought, but it’s precisely at these moments when the strategy of tackling the separate steps of the solution as distinct sub-questions can help us the most. We’ve encountered an unexpected obstacle in our 3-step solution; no big deal – forget about the big picture for a moment, and focus on this problem. Considered on it’s own, 100 divided by 15 isn’t that bad.

Question #4: What is 100 divided by 15?

Start with a rough ballpark. 100 isn’t divisible by 15, but 90 is. Since 90 / 15 is 6, 100 divided by 15 should be slightly more than 6. Even this rough ballpark is sufficient in order to eliminate 3 out of the 5 possible answer choices. Now we only need to decide between the two remaining answer choices, 6.25% and 6.67%.

Question #5: Is 100 divided by 15 equal to 6.67 or 6.25?

Choosing the better of two options is much simpler than actually dividing 100 by 15 to an accurate result; Instead of dividing 100 by 15, Multiply each of the answer choices by 15. Multiplying the correct answer choice by 15 will result in 100.”¨6.67 multiplied by 15 is (6×15) + ((2/3)×15) = 90 + 10 = 100. Therefore, the correct answer is 6.67%.

Take another look at the 5 questions above. Are any of them beyond your capacity? Certainly not. And yet, the complete problem is considered relatively difficult; many test-takers would probably get it wrong, or let the final division psyche them out. In this case, the whole is by far more difficult then the sum of its parts. Although dividing the problem into sub-questions might seem complicated and time consuming when written down, a skilled student can easily complete the entire process in less than a minute.

Do it Yourself

Solving a complex problem demands a structured, linear solution. The optimal line of attack could be summed up in the following 5 points:

1. Begin by planning your approach: divide your solution into a series of smaller questions, in view of the problem and the specific value it asks you to find. Although at first, envisioning the entire logical path of every problem might be challenging (and in certain extreme cases, even downright impossible), the important thing is to keep trying. The more experienced you become with a specific question type, the easier this is to accomplish. In any event, spending a few seconds on mapping out your solution is never a waste of time.

2. Define to yourself, as clearly as possible, what you are actually trying to accomplish in each step: what is the sub-question you’re trying to answer?

3. Focus only on the step you’re currently working on, while temporarily disregarding the big picture. As shown in the above example, each sub-question presents a minor challenge at most. Conquer this mini-step, and don’t let the rest of the problem confuse you.

4. Write down the result of each step clearly. This is key – completing a step is useless if you don’t write down what you got. The whole point is to keep your mind on the current step alone. Trying to keep the conclusions of earlier steps in mind is exactly why test-takers fail to solve otherwise simple sub-questions. Unburden yourself by keeping track of your work. Additionally, writing everything down makes sure you won’t need to backtrack – which is a major cause of poor solving speed.

5. Don’t let minor setbacks throw you off track. Unexpected obstacles such as complicated calculations or missing pieces of data might come up – when they do, define the new problem as a sub-question. Forget about the rest of the problem for a few seconds, concentrate on solving it, write it down, and keep going. Don’t panic! Even the ugliest calculation is still a simple problem when taken out of context.

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