One of the most versatile and fruitful approaches to solving ACT Math problems is also one of the most basic tactics: using the answers provided in the multiple choice options to solve backwards. This often takes the form of simply “plugging in,” or taking each choice and feeding it into an equation or expression until one of the choices gives you the answer you’re looking for. There are more and less efficient ways of pulling this off, but in general, this method is reliable. This, however, doesn’t mean the technique is perfect, or that it can be applied in all scenarios that first appear to be conducive to plugging in.
First, let’s see a problem for which this works very nicely.
This can be solved algebraically or even graphically, but it’s just not worth the while: for the average student, an algebraic approach would be quite difficult, error-prone, and time-consuming. The easiest route is clearly to plug in each number listed in the multiple choice answers to see which ones make the left side of the equation equal to zero. If you do a thorough job (i.e., you don’t go with choice F, which works but is incomplete), you’ll pick choice G, the correct answer. Try it for yourself!
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The Pitfalls of Plug ’n Chug
Sometimes, a problem can seem like a great candidate for the working-backwards strategy, but in reality is only masquerading as such. Here’s a great example.
Your initial instinct may be to simply plug in each choice for x until one of them makes the equation true. We’ll see why this is the wrong approach from a conceptual standpoint, but first, let’s see what happens when we proceed.
A. (–30)2 + (–30) – 30 ==> 900 – 30 – 30 = 840 NO
B. (–6)2 + (–6) – 30 ==> 36 – 6 – 30 = 0 YES
C. (–1)2 + (–1) – 30 ==> 1 – 1 – 30 = –30 NO
D. (0)2 + 0 – 30 ==> –30 NO
E. (5)2 + 5 – 30 ==> 25 + 5 – 30 = 0 YES
Two answer choices seem to work! What happened? By plugging in the options, we assumed that each represented a potential solution to the problem. Upon more careful reading, it becomes clear that the question isn’t asking for the solutions of the equation (i.e., what it would need to ask for in order for our method to have been a valid approach), but the sum of the solutions of the equation. That means that plugging in choices will not get us the answer directly, because what we’re solving for and what we need to be solving for aren’t the same thing.
It is true, however, that –6 and 5 are the two solutions to the equation—they do, after all, make the equation work. Because their sum—that is, the thing we’re supposed to be solving for—is –1, the correct answer is choice C. That the individual solutions to the quadratic equation were included as answer choices is a deliberate attempt by the ACT to penalize students who uncritically plug ’n chug.
Just Watch Out
As a math nerd who enjoys thinking about the “real” way to understand and solve math problems, I sometimes find backward solving by trial-and-error cringeworthy. But I’m not kidding myself: your goal as an ACT test taker is to answer as many questions correctly as possible in the time allotted, sophistication be darned. For some students, this approach makes the difference between an 18 and a 23 on the ACT Math—a very significant jump. In summary: use the answer choices to the greatest advantage possible, but don’t ever assume that plugging them in directly will solve the problem straight away. You may need to manipulate the options before or after plugging in to answer the question being asked.