Problem Solving - Platinum Techniques
Although we believe the only sure way to master the GMAT is to master the content tested on the GMAT, we recognize that knowing certain techniques can improve your score in a rather short amount of time. Consequently, we have assembled a list of techniques useful for GMAT problem solving questions.
- Simplifying Questions
- Picking Numbers
- 2x2 Matrix
- Avoid Oversolving
- Avoid Lengthy Computations
- Watch for Tricky Questions Written to Deceive
- Estimate (When Appropriate)
- Eliminate Wrong Answers and Guess
The GMAT test-writers often complicate an easy problem by obfuscating the question language. The key to this technique, which is quite possibly the most important technique, is to simplify the question. By translating the question into a simplified algebraic equation, seemingly complicated problems become easy. Consider the following example:
- 25x = (52)x = 52x. Consequently, the simplified question is:
- What is the smallest integer x for which 52x > 510? (This question can be simplified again):
- Since the bases are equal (i.e., are both 5), the only thing that will determine the relative size of the numbers are their exponents.
What is the smallest integer x for which 2x > 10? (This question can be simplified again):
- What is the smallest integer x for which x > 5?
The essence of picking numbers is that you pick numbers that meet the stipulations of the question stem and perform certain operations on these numbers. You then compare the outcome with the answer choices.
In an overwhelming majority of cases, it is quicker to solve the problem using appropriate theory and algebra. However, there are instances where it is actually faster to solve a problem by picking numbers than by algebraic manipulations (even if you thoroughly understand the algebra).
Picking numbers is an especially effective strategy in a large number of problems involving percents, mixtures, or ratios where algebraic expressions are in the answer choices. However, there are some problems for which it is essentially required that you can solve the question using algebra.
Also note that, unlike backsolving where you can stop immediately if you find the correct answer, when you pick numbers you cannot stop until you find the only answer that fits the numbers you picked.
The following two examples should help clarify this incredibly valuable technique:
- Although this can be solved algebraically, it is much easier and quicker to pick 100 and perform the appropriate calculations (in other words, assume the original cost of the coat was $100 and adjust the cost according to the problem\'s stipulations--see next steps). Since 100 is such a strategic number for percents (percent literally means per 100), the problem can be done very quickly using this number.
- A 20 percent increase (from $100) brings the cost of the coat to $120.
- A 25 percent decrease (from $120) brings the cost of the coat to $90.
- At a final price of $90, the net change in the price of the coat is a decrease of 10 percent [=($90-$100)/$100].
- Pick "four consecutive positive integers" and find their sum. 1 + 2 + 3 + 4 = 10. We now know that z = 10.
- "The sum of the next four consecutive integers" (5 + 6 + 7 + 8) is 26. 26, in terms of z (which is 10), is z + 16. This is true since 10 + 16 = 26.
The essence of backsolving is that it seeks to solve a problem backward--by starting with the answer choices and putting them into the given equations, seeing which answer choice works.
Before seeing an example, three points should be made:
- (1) When the answer choices have numerical values listed in ascending order, it is important to start with the middle answer choice and move up or down based upon the result of your first attempt to backsolve (i.e., if the first answer choice you select produces an answer that is too small, choose a larger answer choice).
- (2) Presuming the answer choices do not contain variables, once you find an answer that works, you can stop immediately (by comparison, the picking numbers strategy requires you to try all answers to ensure that the numbers you picked do not work on multiple equations).
- (3) Some individuals present this strategy as a panacea, which it is not. The picking numbers strategy is considerably more valuable than backsolving, which should be used only as a last resort. Do not depend on backsolving. Rather, invest the time to master the math content that the GMAT tests.
Consider the following example:
- The perimeter is 2(length) + 2(width). See which answer choice makes the perimeter equal to 20. Remember, however, that the answer choices must be in a ratio of 2:1 since the length is twice as long as the width.
- Begin by checking answer C and adjusting accordingly. 2[(10/2)] + 2([5/2]) = 15; Wrong Answer. Since this is too small, proceed to an answer choice with larger numbers.
- Try checking answer B. 2[(20/2)] + 2[(10/2)] = 30; Wrong Answer. Since this is too large, proceed to an answer choice with smaller numbers.
- Try checking answer D. 2[(20/3)] + 2[(10/3)] = (40/3) + (20/3) = 60/3 = 20; Correct Answer. Since we found the correct answer, there is no need to try an additional answer choice.
- Note: You can immediately rule out (A) because the numbers do not allow the length to be twice the width (i.e., 16/2 is not twice 7/2). Likewise, you can immediately rule out E because the length is not twice the width (i.e., six is not twice the value of four).
Understanding how to use a 2x2 Matrix is a valuable technique for solving a specific type of complicated word problems--namely, those with two sets of data, each with two binary subtypes. The best way to understand this technique is simply to see two examples.
- Set up a 2x2 Matrix, inputting the given information.
Team A Team B Total Scored > 10 Points 5 11 Scored <= 10 Points 4 10
- Fill in other information you can logically deduce. You know that if 10 players are on team A and 5 scored more than 10 points, 5 did not score more than 5 points. Moreover, if a total of 11 players scored more than 10 points and 5 were on team A, then the other 6 must have been on team B.
Team A Team B Total Scored > 10 Points 5 6 11 Scored <= 10 Points 5 4 10
- Fill in other information you can logically deduce. If you know that 4 players on team A did not score more than 10 points while 5 players on team B did not score more than 10 points, you know that a total of 9 players did not score more than 10 points.
Team A Team B Total Scored > 10 Points 5 6 11 Scored <= 10 Points 5 4 9 10
- Fill in all remaing information.
Team A Team B Total Scored > 10 Points 5 6 11 Scored <= 10 Points 5 4 9 10 10 20
- You are now able to easily answer the question ("how many players did not score more than 10 points?"). Clearly, the answer is 9.
- Set up a 2x2 Matrix, inputting the given information. (Note that a ratio of men to women of 2/5 means that men comprise 2/(5+2) = 2/7 or about 29% of the student population while women comprise 5/7 or about 71% of the student population.) Also, note that the 50% is out of the number of men, not the total number of students. Likewise, the 25% is out ot the total number of women, not the total number of students. Also, let x = total number of students.
Men Women Total Above 3.5 GPA 50% of 29% of Total Population = 14.5% of Total Population Not Above 3.5 GPA 25% of 71% of Total Population = 18% of Total Population (2/7) = 29% of Total Population (5/7) = 71% of Total Population x
- Simplify the information from above.
Men Women Total Above 3.5 GPA 14.5%x Not Above 3.5 GPA 18%x 29%x 71%x x
- Fill in other information.
Men Women Total Above 3.5 GPA 14.5%x 53%x 67.5%x Not Above 3.5 GPA 14.5%x 18%x 32.5%x 29%x 71%x x
- We are not ready to answer the original question ("what percent of the student body is comprised of women with GPAs not above 3.5?") The answer will be (the percent of women with GPAs not above 3.5)/(total student body) = 18%x/x = 18%
Avoid Oversolving Problems
One common trap the GMAT test writers lay for intermediate-level test-takers is writing problems designed to ensnare you in unnecessary calculations. This can cause test-takers to waste precious time and run circles around the correct answer. Do not solve for more than is necessary out of habits from previous mathematics instruction. This is best illustrated by an example:
- Since 10k = 20, by division, k = 20/10 = 2.
- Since k = 2 and n2 = 4k4, n2 = 4(2)4 = 64.
- At this stage, most individuals out of habit immediately solve n = 8, even though the problem never asks for n, but instead asks for n2. Although this is not too costly in this problem, on more difficult questions there is no other way to solve except for recognizing the principle of substitution at work in this problem.
- The key to solving this problem quickly is to substitute n2 = 64 into n2 + k
This yields: 64 + 2 = 66
Do not do 82 + 2.
Avoid Lengthy Computations
Virtually every GMAT Problem Solving question does not require lengthy computation. Since time is precious, it is absolutely essential that you not spend time on lengthy computations when a shorter method almost certainly exists. Consequently, if you find yourself beginning lengthy computations, re-think your strategy as there is almost certainly a quicker method. For example, check the answers to see if approximation is possible (i.e., the answer choices are not too close together so you can perform rounded calculations that are significantly faster).
Watch for Tricky Questions Written to Deceive
Some GMAT questions are written in order to prey upon common high-school math rules. Consequently, you must be careful about making assumptions that you are not allowed to make.
Common traps include:
- Confusing a percent increase with an absolute percent. For example, 100 percent of 50 is not the same as a 100 percent increase from a base of 50.
- Confusing the percent of a whole versus the percent of a part. For example, the following two statements are not equal: (1) 50 percent of all the lights in the factory were red. (2) 50 percent of the small lights in the factory were red.
- Confusing the distance traveled with the distance remaining.
- Confusing the volume left in a can with the volume extracted.
- Confusing the units of measurement.
- Confusing the area inside a figure with the area outside a figure.
The bottom line is that you need to be very careful when reading GMAT problems. Moreover, you must be precise and thoughtful when labeling your variables.
Estimate (When Appropriate)
Some GMAT problems appear as though they will require long and tedious calculations. However, as mentioned above, the GMAT rarely requires long tedious calculations. Consequently, on these problems you can often approximate. This is especially true if the answers are not close in value. For example, instead of calculating 53% of 980, simply round and calculate that the answer is approximately half (or 50%) of 1000, which is 500.
- Clearly, the test does not expect you to find the exact square root of 85. So, approximate it as a little more than the square root of 81, which is 9. Let the square root of 85 be 9.2.
- The question now reads: what is 30% of 9.2. You could carry out the multplication, which would not be cumbersome. Or, since the answers are not close together, you could approximate 30% as 1/3, allowing you to divide by 3. The question now becomes, what is 9.2/3, which you know is a little more than 3. Since you know that this number will be a little larger than the true answer (you are dividing by 3 instead of taking 1/3), look for an answer a tad bit smaller.
- Since the answer is "a little less than 9.2/3," the correct answer choice is clearly 2.7.
Moreover, although the GMAT test-writers do not have to construct figures to scale, if you are stumped on a problem involving geometry, your best shot is to take a guess based upon appearances, ruling out clearly wrong answers. For example, if a question asks for the measure of angle x and you can see that it is probably 20-60 degrees (based upon a labeled angle), and the answers are: (A) 30 (B) 40 (C) 70 (D) 80 (E) 90, you should eliminate C, D, and E, and take a guess between A and B. Instead of guessing with odds of 1/5, your odds stand at 1/2.
Eliminate Wrong Answers, and Guess
On most questions, there is at least one answer that can be quickly ruled out. For example, there are problems for which you know that the answer must be positive yet a negative answer exists as an answer choice. If you are short on time or stumped about how to find the correct answer, eliminate the obviously wrong answers and guess. Often, by simply determining the sign of the answer (e.g., whether it is positive or negative) and the rough value of the answer (100, 1000, or 1000000), you can rule out an answer or two and improve your odds of guessing the correct answer.