To Algebra or Not to Algebra?
To Algebra or Not to Algebra?
When faced with an algebra question on the SAT, you could, as you might expect, try to solve it by using standard algebra—setting up and working out the equation. But there are often alternative ways to attack. You might be able to plug the answer choices back into the question until one of them works out. You could pick numbers to substitute into the various expressions given as answer choices.
For problems you know how to solve, using algebra is probably the quickest method. In contrast, a problem that’s giving you trouble may suddenly become much easier if you start plugging in numbers.
We’re not telling you to pick just one method and always use it. Far from it. Flexibility is the key. Some methods work for some problems, and others work better with others. When you study your practice tests and look over the algebra questions you got wrong, think about more than just what the right answer was. Ask yourself if you approached the question correctly. Did you plug in answers when you should have used algebra? Did you use algebra when plugging in answers would have simplified the problem?
Here’s an example of an algebra question. We solve it using each of the different problem-solving methods, explaining what you need to know about each one in the process.
A man flipped a coin 162 times. The coin landed with heads side up 62 more times than it landed with tails up. How many times did the coin land on heads?
(A) 100
(B) 104
(C) 108
(D) 112
(E) 116
Solving by Algebra
To answer this problem with algebra, you first have to translate it into algebra. You have to set up an equation. If you assign the variable x to stand for the number of times the coin landed on heads, then tails are represented by x – 62, since the coin landed on heads 62 times more times than it landed on tails. And since the coin was thrown 162 total times,
As you can see, setting up the question takes a little bit of time and knowledge, but once you’ve set it up, the math is quick and easy.
Using algebra will only take you longer than plugging in if you have trouble coming up with the equation x + (x – 62) = 162. So here’s a quick rule of thumb to help you decide whether to use algebra or to plug in: If you can quickly come up with the equation, then use algebra to solve algebra problems. If you have the sense that it will take you a while to figure out the equation, then plug in.
Solving by Plugging In
There are two ways to plug in: intelligently and maniacally. Don’t be a maniac. How can you avoid this? Simple. The answer choices on the SAT that contain numbers (rather than variables) always appear in either ascending or descending order. The first answer choice will be the lowest and the last will be the largest, or vice versa.
Let’s say the answer choices are in ascending order. If you start by plugging in the middle number, the answer choice for answer C, then even if that choice doesn’t work, you can use the outcome to determine whether you need to plug in a smaller or larger number. If you need a smaller number, move to answer choice B. If you need a larger number, try D. If you follow this method, instead of having to check all five answer choices, you shouldn’t ever have to check more than three. That’ll save you time. (5 – 3) / 5 100 = 40% of your time, to be exact.
To answer the coin-flip problem by plugging in, pick C, 108, as the first number to try. So, if the coin came up heads 108 times, then how many times did it land on tails? It landed on tails 162 – 108 = 54. Are 108 heads 62 more than 54 tails? No: 108 – 54 = 54. In order for the problem to work out you need more heads. You can eliminate A and B as possibilities. Choose D, 112, as your next plug-in number: 162 – 112 = 50. Does 112 – 50 = 62? Yes.
Picking Numbers
Picking numbers is a variation of plugging in. It should only be used when the answer choices contain variables. A modified version of our original sample question shows what kind of problem lends itself to picking numbers.
A man flipped a coin z times. The coin landed on heads y more times than it landed on tails. If the number of times the coin landed heads is h, then, in terms of h and y, how many times was the coin flipped?
(A) z = h + y
(B) z = hy
(D) z = 2hy
Instead of testing your ability to set up and solve an equation, this question asks you only to set up an equation based on a word problem. While using algebra to set up the equation would be the faster tactic, for some people, thinking in terms of variables can be confusing. Picking numbers allows you to transform variables into concrete numbers.
To use the picking numbers method, select numbers and plug them into the answer choices. It doesn’t matter what specific numbers you pick for each variable as long as you always plug the same number in for each variable and follow all guidelines given by the problem.
In the coin-flip problem, you are given three variables, z, y, and h. The question asks you to find z in terms of h and y. We’ll pick some numbers. Let’s say the coin landed on heads (h) 5 times, and that it landed on heads on 2 more flips (y) than it landed on tails. That would mean that the coin landed on tails 3 times, since 5 – 2 = 3. Since the coin landed on heads on 5 flips, and on tails on 3 flips, the coin must have been flipped a total of 5 + 3 = 8 times. Now plug 5 for h and 2 for y into all the equations and see which one comes out to 8: only D, which is the right answer.
In addition to giving you a method for solving tricky problems, picking numbers is also a good way to check your math for careless calculations.
Solving by Being an Amazing Genius
It’s quite possible that you just looked at this problem and said to yourself, “Other than the 62 more heads, all the other flips were equally heads and tails. So: If I take the 62 out of the total of 162, then I know that the other 100 flips were 50 heads and 50 tails. Now I can just add 62 + 50 = 112. Man, I am an amazing genius!”
Yes, you are. No one knows how to teach other people how to be an amazing genius, though, and you can rest assured that almost no one taking the test will be an amazing genius on every question.
The moral of the story: Know that amazing-genius shortcuts exist, and keep a lookout for them, but don’t stress over them. Only a fool would waste time looking for shortcuts. And you’re no fool.
Algebra: The Bottom Line
There isn’t any “right way” to answer an SAT algebra question. Some methods work best for some types of questions, and others for others. The best way to learn which methods work best for you is to take and study practice tests.
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