GMAT Math Strategies — Estimation, Rounding and other Shortcuts
This tutorial explores the uses and limitations of GMAT math strategies such as estimation, rounding and other shortcuts. You'll learn when and when not to resort to these sorts of time-saving techniques.
Q: Does the GMAT reward test takers who know certain shortcuts for arithmetical calculations and for manipulating numbers?
A: It depends on the type of shortcut. On the one hand, a GMAT test taker who can perform complex arithmetical calculations quickly would hold absolutely no advantage. Number-crunching is simply not what the GMAT is about. For example, you won't need to:
Using columnar multiplication or long division to combine multiple-digit numbers
Deal with unwieldy numbers to determine root and exponential values
Carry decimal points beyond one or two places
On the other hand, if by "shortcuts" you mean the combining of multiple computational steps, then the GMAT does indeed reward test takers who know how to use shortcuts.
Q: Can you provide a few examples of the sorts of shortcuts GMAT test takers can use to their advantage?
A: A GMAT question might call for you to remove radical signs from a fraction's denominator (or rationalize the fraction). You can do this simply by copying the radical term to the numerator and removing the radical sign from the denominator—omitting the intermediary steps of multiplying both numerator and denominator by the root value, and then canceling. Here's an example:
Here's another useful shortcut: If two fractions are equal, you can "factor out" terms across numerators or denominators, and set the fractions' "cross-products" equal to each other. Consider, for example, the following equation:
To solve for x, first factor out 5 from the two numerators, and then set the cross-products equal to each other:
To isolate x from here, there's really no shortcut. Multiply both sides by √2, and then divide by 2:
Be sure that the steps in solving equations like this one are second nature to you by the time you sit for the GMAT.
Q: In preparing for the GMAT, should test takers memorize certain formulas or computational tables?
A: Yes, but the learning curve is neither steep nor long. In gearing up for the GMAT you need to learn only a handful of formulas. Most of these formulas involve geometry, but you should also know certain others — formulas that help you set up and solve algebra problems cast in a real-world setting (so called "story" problems):
- Areas and perimeters of certain triangles: right, isosceles, equilateral
- Area and circumference of a circle
- Area and perimeter of a square, rectangle, rhombus, and trapezoid
- Angle measures of any polygon
- Area and volume of a rectangular solid
- Area and volume of a right cylinder
Algebraic formulas for "story" problems:
- Rate of work
- Rate of motion (speed)
- Weighted average
Understanding the Pythagorean Theorem (for determining the area of a right triangle and the relationship among its three sides) will be especially helpful on the GMAT. Any good GMAT-prep book will provide the geometry formulas listed above and will explain how they come into play in GMAT Quantitative questions.
In gearing up for the GMAT you should also memorize tables for determining:
- fraction-percent-decimal equivalents
- certain square roots and cube roots that are integral values (no decimals)
- squares of integers up to 15, along with 25
- divisibility (for factoring numbers)
Again, any good GMAT-prep book will provide these tables.
Q: For the GMAT, would you suggest memorizing conversion tables for units of measurement — such as weight, length, and monetary units?
A: For the GMAT, there's no need to memorize conversion tables. If a GMAT question requires you to convert across systems of measurement (e.g., from kilometers to miles) or within a system (e.g., from ounces to pounds), the question itself will provide the conversion information you need — for example, "1 pound = 16 ounces."
But what every test taker should be concerned about is making sure their calculated solution is expressed in terms of the specific unit of measurement called for in the question. A GMAT question might express units in pounds, then ask for a solution in terms of ounces. If you neglect to convert — by either multiplying or dividing a key figure by 16 at some point in your calculations — you'll come up with the wrong solution, of course. And if the question is in the Problem Solving format, chances are that your wrong solution will appear among the four incorrect answer choices!
Q: Do the test designers frequently resort to this ploy — determining common errors and listing wrong-answer choices that reflect those errors? If so, how can test takers avoid falling victim to this ploy?
A: Yes, the test makers incorporate this ploy into nearly every GMAT Problem Solving question. To increase the difficulty level of a question, they load a question with three or four of these sucker-bait choices; to decrease the difficulty level, they reduce the number to one or two.
The best way to avoid falling prey to this ploy is to predetermine, if possible, the sort of answer choice you're looking — in other words, determine what meets the criteria for a viable correct response. If the question asks for a numerical solution — without variables — ask yourself how large or small a number would make sense as the correct answer in the context of the problem:
- A single-digit number?
- A very small fractional number?
- A large percentage?
In so-called "story" problems — questions in a real-world setting — you can often define parameters for a viable answer choice based on common sense, then eliminate at least one answer choice based on those parameters. This technique also helps if you're in a time crunch during the Quantitative section. If you can eliminate one or two answer choices without doing any pencil-work, simply because they are unrealistic in size, this will help increase your odds.
When using this technique, keep in mind that numerical answer choices are always listed in ascending order by value (except for questions that ask which of the five choices is largest/smallest in value). In other words, the smallest value among the five choices will be listed first among the five, while the largest value will be listed last. So if you determine parameters up front, and only the first two listed choices fall within them, chances are that the last two listed choices are both wrong. Thus defining parameters can help speed up the elimination process a bit.
Q: The process-of-elimination technique you just mentioned applies only to the Problem Solving format. What about the Data Sufficiency format? Is there any such technique that might be useful in handling questions in this format?
A: Yes; the Data Sufficiency format does suggest a particular process of elimination. Let's first look at the five answer choices for every Data Sufficiency question:
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient
EACH statement ALONE is sufficient to answer the question asked
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
These five answer choices suggest the following process of elimination:
Consider statement (1) by itself. If you can answer the question based on (1) alone, eliminate (B), (C) and (E) as viable choices.
Consider statement (2) by itself. If you can answer the question based on (2) alone, eliminate (A), (C) and (E) as viable choices.
If you were able to answer the question based on (1) alone and based on (2) alone, the correct response is (D).
If you were not able to answer the question based on either statement alone, the correct response must be either (C) or (E).
As you can see, built into the Data Sufficiency format is the opportunity to make reasoned guesses when you're in a time crunch or have trouble analyzing one of the two numbers statements.
Q: Are there any visual shortcuts to answering GMAT geometry questions that are accompanied by pictures of geometric figures? In other words, can the test taker analyze these questions by estimating lengths and sizes visually?
A: First of all, keep in mind two basic ground rules that you'll see as part of the directions for the Quantitative section:
In Problem Solving questions, assume that figures are drawn to scale unless a figure indicates that it is not drawn to scale.
In Data Sufficiency questions, figures are not necessarily drawn to scale, unless otherwise indicated.
So with respect to Data Sufficiency questions, the answer to your question is clearly "no."
But the answer is also "no" for Problem Solving questions. Why? The test makers draft geometry questions so as to eliminate any advantage of visual measurement. For example, you're unlikely to encounter a question that asks you to compare one linear length in a figure with another? And if you do, the test makers will intentionally distort the figure's proportions and indicate that the figure is not drawn to scale. The bottom line is: Don't rely on your eye to answer Quantitative questions, regardless of whether its format is Problem Solving or Data Sufficiency.
There is one important exception, however, to this "bottom-line" advice. Handling a Data Interpretation question in the Problem Solving format might necessarily require certain visual measurements — for instance, determining the height of a certain bar on a bar graph, or the vertical position of a point on a line chart.
Q: When analyzing a bar graph or a line chart for a data-interpretation question, how precisely do you need to determine the quantity indicated by a point, a line, or a bar in the figure?
A: Some degree of precision is needed. But keep in mind that the test makers don't require you to split hairs. They design the bar graphs and line charts — along with the answer choices — so that a rough estimate suffices to zero-in on the correct response. For instance, if a bar appears to extend midway between the quantity 10 and 15 (as indicated on the vertical axis of a bar graph), rest assured that you can use either 12 or 13 in your calculations and zero-in on the correct answer among the five choices. In other words, you won't see two answer choices so close in value that the correct answer depends on which number — 12 or 13 — you used in your calculations.
The most common mistake test takers make when estimating the length of a bar or the position of a point is to round off more than once, in a direction that distorts their solution to the problem. Let's assume the quantity represented by a bar appears to be 11 to 11.5 units, but you round down to 10 for the purpose of facilitating your calculations. Let's also assume that another bar extends up to about 18.5 or 19 units, and that you decide to round that number up to 20 — again just to make your calculations easier and quicker. If the question asks you to compare the two numbers in percentage terms, by rounding the two numbers in opposite directions you end up with a distorted ratio (or fraction or percentage), and you might very well select the wrong answer. So the lesson here is: Be sure to round your numbers in the same direction — either up or down — before comparing them!
Q: What other blunders do test takers frequently commit in responding to data-interpretation questions?
A: There's nothing conceptually difficult about Data Interpretation questions. The math is simple, and there are no abstract concepts to master. Where most test takers go wrong is carelessness — either misreading the question or referring to the wrong chart, bar, or line for the data they need to answer the question. How can you avoid these blunders? Here are two suggestions:
Read the question carefully, paying close attention to the sort of number the question is asking you to calculate. Ask yourself whether the solution is a quantity or a percentage based on a quantity. If the solution is a comparative number — a ratio, fraction, or percentage — ask yourself what two numbers from the chart or graph you're being asked to compare.
Be very careful to glean your data from the correct bar, line, or chart in the figure. With your finger, point to the correct bar, line, or chart, and keep your finger there until you've confirmed that you're referring to the appropriate portion of the figure.