Guest Post by Chris Seymour

How long have you known what a variable is? If you're like most students, you've known about variables since middle school algebra, or maybe even since late elementary school pre-algebra. That's a pretty long time.

But now let me ask you: How long have you known what a number is?

This time, if you're like most students, you can't even remember a time when you didn't know what a number was. It seems ridiculous to even bother answering such a question.

No matter how comfortable we are working with variables, we will never be more comfortable with variables than we are with numbers. We've been working with numbers our entire lives, and even today, we use concrete numbers far more often than we use abstract variables to solve everyday problems.

Suppose you came across the following statement in a math problem: "John weighs x pounds." When we read that statement, it's impossible to get a mental picture of what John might look like. It's too abstract.

However, suppose you were to say, "Okay, let's just make x equal 300 for now." Suddenly, you have a clear image of John as a hulking linebacker or sumo wrestler. This image allows you to think about the problem in concrete, real-life terms. It also makes it easier to figure out other pieces of information that you might need to solve the problem.

Perhaps you'd like to figure out how much John weighs compared to other people. Perhaps you want to know how much weight he'd have to lose to get to a certain goal. Solving for these pieces of information no longer requires any complicated algebra if you imagine the variable as a number; all it takes is simple arithmetic.

This technique has many different names, but we'll just call it 'Replacing Variables with Numbers'. You might be surprised how often you can take advantage of this trick.

Here's an example:

Yuck! This doesn't seem like a very easy question. Most people would have to reread it several times just to wrap their heads around the given information. But take heart. The problem is much simpler than it seems, and the only things that make it seem more complicated are those pesky variables. Let's try replacing those variables with some concrete numbers:

Now let's plug those numbers into the original question:

Unfortunately, we have a problem with the numbers we chose. The person will be 2 years old in 3 years—that means the person hasn't been born yet! But even though our numbers don't work, we've already learned something valuable about the problem: x must be greater than y. Now let's revise the numbers we've chosen:

Again, let's plug those numbers into the original question:

Look at that! Suddenly, the problem is much simpler. The person will be 10 years old in 3 years, so the person must be 7 years old right now. Then, in 4 years, the person will be 11 years old. So our answer is 11.

But oops, we don't have 11 in the answer choices! Don't worry, all we have to do is replace the variables in the answer choices with the same numbers we used in the problem:

Choice B is the only one that comes out to 11, so B is the correct answer.

Let's try a tougher one now:

Take a look at the answer choices. When the answer choices contain variables, replacing the variables with numbers is almost always a good option. Since the total area is 5,000, the math will work out more easily if we choose something that goes evenly into 5,000. Let's say x = 100. Area of a rectangle is base multiplied by height. So that would mean that y = 50 (since 100 x 50 = 5,000). Let's label the diagram with our values for x and y:

Now, just add up the lengths of the fences. We have two fences of length 100, and three fences of length 50, so the total length of the fencing will be 350 feet. Once again, all we have to do now is replace the variables in the answer choices with the number we've chosen and find out which one gives us 350.

D'oh! We have 350 for both C and D. Although this is not an ideal situation, it's nothing to worry about. It just means that we will have to replace our variable with a different number to figure out which of these two answers is correct. Let's say x = 200 this time. Then you know y = 25 to give us an area of 5,000 (200 x 25 = 5,000). This time, if we add up the lengths of the fences, we get 475. Now we can go back to the answer choices. Remember, as a time saver, you only have to check choices C and D—the other choices are already out of the running.

Therefore, C is the correct answer.

So just remember these steps:

- Choose numbers to replace variables.
- Use these numbers to solve for the answer.
- Replace variables in the answer-choices you're given with the numbers you chose to find which ones match your answer from Step 2.
- If get more than one answer-choice that matches your Step 2 answer, repeat Steps 1-3 using new numbers.

I have to say that replacing variables with numbers is probably the single most valuable trick you can learn for the math section of the SAT. Much like magic, you're making all of the variables in the problem simply disappear! So, the next time you do a math section, look for opportunities to replace variables with numbers. You just might find that the problems are much easier than you thought.

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