Math IC Algebra Strategies

　　There are several ways to answer most algebra problems. You could try to solve a problem by using standard algebra and setting up and solving an equation. Alternatively, you could try to avoid algebra and simply plug the answer choices back into the question until one of them works out. Or you can pick numbers to substitute into the various expressions given as answer choices. 　　None of these methods is necessarily better than the others. Remain flexible in your approach to each question and choose the method that best suits the problem. For a problem you know how to solve, using algebra is probably the quickest method. In contrast, a tough problem that you are unsure how to solve might become easy if you try to plug in some answers. When you study your practice tests and look over the algebra questions you got wrong, you should think about the method you employed. Did you plug in answers when you should have used algebra? Did you use algebra when you should have plugged in answers?　　We’ll lay out for you the different problem-solving approaches and tell you all you need to know about them. Then you can decide for yourself which method to choose.　　Let’s use a sample algebra problem to illustrate these separate approaches:

　　A baseball player travels from his home city, Jasonville, to Giambia City for a baseball game. He drives at 50 miles an hour. After the game, he travels back home and takes a flight that travels at 500 miles an hour. If the distance from Jasonville to Giambia City is 250 miles, and it took him j hours longer to drive than to fly, what is j?

　　(A)1

　　(B)3.5

　　(C)4

　　(D)4.5

　　(D)12

　　Using Algebra　　This question is a simple rate problem that can be solved with a few basic equations. Since traveling time = distance speed, it took him:　　to drive to Giambia City. To find the duration of his flight, we use the same rate formula:　　It took the player:　　longer to drive. D is the correct answer.　　Plugging In Answers　　Sometimes you might not be sure how to approach a problem or don’t have the time to think out the proper equations. In such instances, plugging in might be the best method for you, especially as you come across the more difficult questions at the end of the test. All you have to do is substitute the answer choices back into the problem, and see whether the given information holds true. 　　The process of plugging in is simple. First, you should make full use of the fact that the answer choices on Math IC are always presented in ascending value. So start by plugging in answer choice C, since if it doesn’t turn out to be the answer, you can usually tell whether to try a smaller or larger answer choice. Now, to solve the question: it takes the baseball player 25050 = 5 hours to drive to Giambia City. So, if it takes him C 4 hours more to drive, than it takes him 5 – 4 = 1 hour to fly back to Jasonville. But the question tells us that in 1 hour, he could fly 500 miles. Therefore, it must take him longer than 4 hours more to drive than to fly. Next, we try D 4.5. It takes him 5 – 4.5 = .5 hours to fly, which means that he travels 500　.5 = 250 miles on his flight. D is the answer. 　　Picking Numbers　　Picking numbers is a variation of plugging in and should only be used when the answer choices contain variables. A modified version of our original sample question shows what kind of problems might lend themselves to picking numbers.

　　A baseball player travels from his home city, Jasonville, to Giambia City for a baseball game. He drives at m miles an hour. After the game, he travels back home, and takes a flight instead at p miles an hour. If the distance from Jasonville to Giambia City is v miles, and it took him j longer to drive than to fly, what is j?

　　This question asks you to figure out which set of variables in the answer choices is the right one. But thinking in terms of variables can be confusing to some people. Picking numbers allows you to transform variables into concrete numbers. 　　To use the picking numbers method, you need to select numbers and plug them into the answer choices. You’re essentially testing the relationships between the variables in each given answer and ensuring they remain true. It doesn’t matter what specific numbers you plug into a problem. The same answer choice will always surface as long as you plug in consistently and follow all guidelines given by the problem.　　For example, in the baseball player problem, let m = 5, v = 100, and p = 10. Clearly, these numbers aren’t realistic (who flies at 10 miles an hour?), but your goal is to pick easy-to-manipulate numbers. Using our numbers, it takes the baseball player 1005 = 20 hours to drive and 10010 = 10 hours to fly. So, it takes him 20 – 10 = 10 hours longer to drive. After plugging m, v, and p into all the answer choices, we find that only D produces an answer of 10.　　Very rarely, more than one answer choice will result in the correct answer for the first set of numbers you picked. When this occurs, simply plug in a different set of numbers. You will almost never have to plug in more than two sets of numbers.　　When picking numbers, you must check through all the answer solutions with your chosen numbers. Obviously, this will slow you down, but that’s the price you pay for using this method. Picking numbers gives you a mechanical method of solving tricky problems, and it also allows you to check your math for careless calculations, but it is time-consuming.　　Finally, when you are picking numbers, avoid 0, 1, or any numbers that appear in the answer choices. Picking these numbers can overly simplify the expressions you are dealing with and cause you to pick the wrong answer.　　The Bottom Line　　As you can see, there is no “right” method to solving all algebra problems. Some methods work best some times, and others work best at other times. Part of your practice for the Math IC test will be to get comfortable with algebra questions so that you can choose which method you want to use for every question. 　　Now, we’ll review the algebra topics covered in the Math IC Subject Test.　　Equation-Solving　　There are a number of algebraic terms you should know in order to be able to talk and think about algebra: Variable. An unknown quantity, written as a letter. The letters x and y are the most commonly used letters for variables, but a variable can be represented by any letter in the English alphabet. Greek letters are also used quite often. Variables will sometimes represent specified quantities, like apples or dollars, for example. Other times, a specific meaning won’t be attached to them. You’ll need to manipulate variables just to show that you understand certain algebraic principles. Constant. A quantity that does not change. In other words, a number. Term. The product of a constant and a variable. Another way to define a term is as any quantity that is separated from other quantities by addition or subtraction. For example, in the equation below, the left side contains four terms {x3, 2x2, –7x, 4} and the right side contains two terms {x, –1}. The constants, 4 and –1, are considered terms because they are considered coefficients of variables raised to the zero power. For constant 4, 4 = 4x0. So every term, including constants, is the product of a constant and a variable raised to some power. Expression. Any combination of terms. An expression can be as simple as a single constant term, like 5. Or an expression can be as complicated as the sum or difference of many terms, each of which is a combination of constants and variables, such as {(x2 + 2)3 – 6x} 7x5. Expressions don’t include an equal sign, which is what differentiates expressions from equations. Expressions therefore cannot be solved; they can only be simplified. Equation. Two expressions linked by an equal sign. A lot of the algebra that you’ll have to perform on the SAT II Math tests will consist of solving an equation with one variable. Most of this chapter, in fact, deals with different techniques for simplifying expressions and solving different types of equations. First, we’ll review how to write an equation.