When to Plug In Numbers!

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Consider the following problem:

The sum of two numbers that differ by 1 is $x$. In terms of $x$, what is the value of the smaller of the two numbers?
[ A) frac{x}{2} ] [ B) frac{x}{2}+1 ] [ C) frac{2x-1}{2} ] [ D) frac{x-1}{2} ] [ E) frac{x+1}{2} ]
Most people are intimidated by this kind of SAT problem. It certainly looks formidable. Yet this is one of the easiest kinds of math problems on the test. How can you solve a question like this? By using a concept called “plugging in.”

“Plugging in” involves substituting a number for a variable. The reason that this problem seems difficult is that we’re not given what x equals. But if the answer choices contain variables, you can “plug in” (in other words, substitute) a number and come up with a specific answer. Then, you can go to the answer choices and substitute the same number for that variable, every time. The correct answer choice will yield the answer you initially got. Here’s how it works for the abovementioned problem.

What is x? According to the problem, it’s the sum you get when you add two numbers that differ by 1. So let’s select two numbers that differ by 1. Let’s choose 4 and 5. That means, in our case, x = 9. Write this down and circle it—you’ll have to refer to it.

Now, reread the question. Don’t be confused about the phrase “in terms of x”—that merely means “you’re going to see the letter ‘x’ in your answer.” What do you need to find out? You have to find the smaller of the two numbers. Well, that’s easy. We had chosen 4 and 5 as the numbers, so which of those is the smaller? Four. That’s our answer. Write your answer down and circle it—you’ll have to refer to it.

Now, go to your answer choices, one by one, and “plug in” 9 every time you see an x. Do the math each time. The correct choice will give you the answer you had gotten, which, in this case, is 4. Let’s use a calculator and try each choice:

[ A) frac{x}{2}] Plug in 9 for x and do the math. This gives us 4.5, not 4. Wrong answer.

[ B) frac{x}{2}+1] Plug in 9 for x and do the math. This gives us 5.5, not 4. Wrong answer.

[ C) frac{2x-1}{2} ] Plug in 9 for x and do the math. This gives us 8.5, not 4. Wrong answer.

[ D) frac{x-1}{2} ] Plug in 9 for x and do the math. This gives us 4. This answer choice could be the right one. But try the remaining answer choice to insure that it’s incorrect.

[ E) frac{x+1}{2} ] Plug in 9 for x and do the math. This gives us 5. Wrong answer.

The correct answer is D.

See? Easy.

A few more words of advice are in order, though:

Follow the directions. Sometimes the question gives you certain plug-in restrictions: the number may have to be positive, greater than 100, even, or consist of four digits. Just read carefully and choose a number that fits the given situation.

Choose easy numbers, if the requirements allow them. And most of the time, they will. Don’t get cute by choosing difficult numbers. Fractions, decimals, negative numbers, and particularly large numbers make computation trickier than it has to be. On occasion, the directions to the problem may force you to use one of these types of numbers, but most of the time you can choose small, easily manageable numbers.

But avoid using “1” and “2.” These numbers will often yield duplicate correct answers.

Try all answer choices, even after you get the correct answer. Occasionally, more than one answer choice will give you the answer you had gotten.

So what happens if you get duplicate correct answers? Let’s use the abovementioned problem as an example. Let’s pretend both answer choices B and D had given you the correct answer, which was 4. What do you do? Eliminate the other choices, choose different numbers, plug them in, and do the problem again. You’ll come up with another answer. Then plug in the numbers you used into only answer choices B and D. See which one gives you the correct answer.

Be careful when plugging in more than one variable. If you’re plugging in for two different variables, such as in the equation $x=y^4$, plug in only one number. Choose the variable that makes up the more difficult part of the problem (in this case, $y^4$), then use it to calculate the remaining variable. If you plug in “3” for y, you’ll get x = 81. So, x automatically becomes that number.

Always write down your chosen numbers and your answer—and circle them. This was already mentioned, but it bears repeating. If you lose track of what number you’ve chosen for x, you’re going to get lost while doing the problem. If you don’t write down the answer to the actual problem, you’ll forget what to do when you approach the answer choices. Stay focused by writing down the important information.

Use your calculator, even if you’re a math whiz. The time you save by using your calculator is enormous—and you should have as much time as possible.