Picking Numbers The strategy of “picking numbers” works on a wide range of different math problems on standardized tests (such as the SAT, ACT and GRE) in all topics and difficulty levels. It can often be used to make a difficult problem much easier to understand, and if you are careful in its use, you will usually get the answer without too much trouble. The idea is simple – replace the unknowns in the problem with specific values. Here are some guidelines when picking numbers. Pick a number that is simple but not too simple. In general you might want to avoid picking 0 or 1 (but 2 is usually a good choice). Try to avoid picking numbers that appear in the problem. When picking two or more numbers try to make them all different. Most of the time picking numbers only allows you to eliminate answer choices. So do not just choose the first answer choice that comes out to the correct answer. If multiple answers come out correct you need to pick a new number and start again. But you only have to check the answer choices that have not yet been eliminated. If there are fractions in the question a good choice might be the least common denominator (lcd) or a multiple of the lcd. In percent problems choose the number 100. Do not pick a negative number as a possible answer to a grid-in question. This is a waste of time since you cannot grid a negative number. If your first attempt does not eliminate all choices except one, try to choose a number that is of a different “type.” Here are some examples of types: A positive integer greater than 1. A positive fraction (or decimal) between 0 and 1. A negative integer less than -1. A negative fraction (or decimal) between -1 and 0. If you are picking pairs of numbers try different combinations from (8). For example you can try two positive integers greater than 1, two negative integers less than -1, or one positive and one negative integer, etc. Remember that these are just guidelines and there may be rare occasions where you might break these rules. For example sometimes it is so quick and easy to plug in 0 and/or 1 that you might do this even though only some of the answer choices get eliminated. Examples Okay, so let’s try solving a math practice problem by picking numbers. The following problem could appear on the ACT or GRE. Level 4 Number Theory For nonzero numbers a, b, and c, if c is three times b and b is 1/5 of a, what is the ratio of a² to c² ? A) 9 to 25 B) 25 to 9 C) 5 to 9 D) 5 to 3 E) 3 to 5 Looks scary? This ACT problem is actually pretty easy if you pick numbers. Let’s choose a value for a, say a = 5. Then b = 1, c = 3, and therefore the ratio of a² to c² is 25 to 9, choice B. Simple, right? Maybe a bit too simple? You can of course solve this ACT math problem algebraically as well, but an algebraic solution is more likely to lead to careless errors, and in this problem will actually take more time than picking numbers. Here is one more math problem for more practice: Level 4 Algebra If a = 3b and b = c + 4 , what is a/27 in terms of c ? A) c + 1 B) c C) 3c+1 D) 3c + 1 E) 3c + 2 Using the same method as above, let’s pick a number for c, say c = 2. It follows that b = 2 + 4 = 6. So using our calculator, a = 36 = 729. We then have a/27 = 729/27 = 27. Put a nice big, dark circle around this number so that you can find it easily later. We now substitute a 2 for c into each answer choice and use our calculator. A) 2 + 1 = 3 B) 32 = 9 C) 33 = 27 D) 32 + 1 = 9 + 1 = 10 E) 32 + 2 = 9 + 2 = 11 We now compare each of these numbers to the number that we put a nice big, dark circle around. Since (A), (B), (D) and (E) are incorrect we can eliminate them. Therefore the answer is choice C. Remember by picking a number, we can only eliminate answer choices: C is not the correct answer simply because it is equal to 27. It is correct because all four of the other choices are not 27. You absolutely must check all choices! You can see how powerful picking numbers can be to solve math problems on standardized tests. Follow the guidelines I have provided above and you should be solving more difficult ACT, SAT and GRE math problems with ease. If you would like to see how to apply the strategy of picking numbers to problems involving percents, click the following link: Picking A Number To Solve Percent Problems More Math Practice Problems More information on this extremely useful strategy, as well as many more problems to practice with, can be found in the Get 800 collection of test prep books. Click on the picture below for more information about these books. Until next time… Comments comments