# 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

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:

What is the smallest integer x for which 25x > 510?
 A) 0 B) 5 C) 6 D) 15 E) 100
Consider how you can write 25 with a base of 5.
1. 25x = (52)x = 52x. Consequently, the simplified question is:
2. What is the smallest integer x for which 52x > 510? (This question can be simplified again):
3. 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):
4. What is the smallest integer x for which x > 5?

## Picking Numbers

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:

The cost of a coat increased 20 percent, only to decrease 25 percent. What was the final change in the price of the coat relative to the original price of the coat?
 A) 10% increase B) 10% decrease C) 5% increase D) 5% decrease E) No Change
Let 100 equal the original cost of the coat and perform the subsequent percent change calculations on 100.
1. 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.
2. A 20 percent increase (from \$100) brings the cost of the coat to \$120.
3. A 25 percent decrease (from \$120) brings the cost of the coat to \$90.
4. At a final price of \$90, the net change in the price of the coat is a decrease of 10 percent [=(\$90-\$100)/\$100].
The sum of four consecutive positive integers is z. What is the sum of the next four consecutive integers in terms of z?
 A) 10z B) 5z C) z + 4 D) 2z + 4 E) z + 16
Let z = the sum of 1, 2, 3, and 4; Work from these numbers.
1. Pick "four consecutive positive integers" and find their sum. 1 + 2 + 3 + 4 = 10. We now know that z = 10.
2. "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.

## Backsolve

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:

A rectangular door to a bank's safe is twice as long as it is wide. If its perimeter is 20 feet, then the dimensions of the bank door are?
 A) 16/2 by 7/2 B) 20/2 by 10/2 C) 10/2 by 5/2 D) 20/3 by 10/3 E) 6 by 4
1. 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.
2. 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.
3. 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.
4. 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.
5. 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).

## 2x2 Matrix

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.

In a recent college basketball game where team A played team B, 5 of the 10 players for team A scored more than 10 points. If 4 of the players on team B did not score greater than 10 points and a combined total of 11 players scored greater than 10 points, how many players did not score more than 10 points?
 A) 20 B) 10 C) 9 D) 5 E) 4
1. Set up a 2x2 Matrix, inputting the given information.
 Team A Team B Total Scored > 10 Points 5 11 Scored <= 10 Points 4 10
2. 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
3. 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
4. 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
5. You are now able to easily answer the question ("how many players did not score more than 10 points?"). Clearly, the answer is 9.
At a nearby college, 50% of the men have a GPA above 3.5 while 25% of the women do not have a GPA above 3.5. If the ratio of men to women is 2/5, about what percent of the student body is comprised of women with GPAs not above 3.5?
 A) 60% B) 40% C) 20% D) 18% E) 5%
1. 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
2. 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
3. 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
4. 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:

If n2 = 4k4 and 10k = 20, what is the value of n2 + k?
 A) 66 B) 64 C) 32 D) 10 E) 8
1. Since 10k = 20, by division, k = 20/10 = 2.
2. Since k = 2 and n2 = 4k4, n2 = 4(2)4 = 64.
3. 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.
4. 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.

What is 30% of the square root of 85?
 A) 62.4 B) 35.7 C) 15.6 D) 2.7 E) 0.23