Useful in Solving Questions in Tests
This is one of the most basic of all the rules. Whenever you get stuck, before just taking a wild guess, first try to eliminate as many silly answers as possible. You'll find that some answers simply don't make sense. You might not be able to solve the problem, but you may have a good idea that answer is a fraction, not a whole number, or else it's a positive number, not a negative one. Before taking a guess, eliminate all the answers which seem way off base. You'll be surprised at how many silly answer are in most standardized exams if you just look carefully.
Replacing Variables
If the same variable appears in both the question and the answers, you can replace the variable with an easy-to-use number. For example, let's take the following:
If y + x + 7 = 16, then what is x?
(A) 9 + y
(B) 9 - y
(C) 23 + y
(D) 23 - y
(E) y
Now, this can be simplified by taking the variable y and making it into a 2.
If 2 + x + 7 = 16, then what is x?
(A) 11
(B) 7
(C) 25
(D) 21
(E) 2
Now x = 16 - 7 - 2, so your answer is clearly (B). You can choose any random number for y and obtain the same result.
This is a very simple example, but you'll find that this method works even better the more complex the problem. You can even assign numbers to more than one variable, as long as those variables appear in both the question and the answers.
This technique not only helps you to solve problems faster, but can reduce the number of mistakes you make. A lot of students feel more comfortable with numbers than variables, so the simple act of replacing variables with numbers often makes it easier for them to see the solution.
Rules for Replacing Variables
There are some good rules of thumb when it comes to replacing variables with numbers.
1) The best numbers to use tend to be 0, 1 and -1. These numbers make it simple to get rid of unnecessary variables that are cluttering up the equation. Take the example above where y + x + 7 = 16, and imagine if we substituted 0 for y, instead of using 2. The problem becomes even simpler to solve.
2) If a variable appears multiple times within a problem, make sure to use the same number every time. Don't replace y with 0 in one part of the equation and 2 in another part. That would make no sense.
3) On the same note, if there is more than one variable, why bother to use different numbers for each variable. Who said that each variable cannot be the same? To start with, why not plug in zero for each variable in the question and the answer and see what happens. You'll be surprised at how easy to solve the problem becomes.
4) At times, it's smart to use fractions between 0 and 1.
5) Other times, it's better to use bigger numbers, like 100 and 1000.
6) Don't change the equation or add conditions that aren't explicitly stated within the question. For instance, if you're asked to pick a number than lies between 2 and 4, don't assume that it has to be 3. It might be 2.5 or 3.7 or pi or any other number that is within range. By telling yourself that the number has to be 3, you're imposing a condition on the problem that doesn't exist. You'd be surprised at how many students make this mistake and as a result become blinded to the right answer.
Manipulating Numbers
Sometimes you can manipulate numbers in an equation and make it easier to solve. For example,
If x + 0.54% = 0.21%, then what is x?
(A) -0.33%
(B) -0.75%
(C) 0.10%
(D) 0.33%
(E) 0.75%
It's easier to see the answer if you simply remove the % symbol and multiply by 100. This is the same as multiplying all the numbers that appear in the equation by 10,000. Let's try that.
If x + 54 = 21, then was it x?
(A) -33
(B) -75
(C) 10
(D) 33
(E) 75
The answer becomes transparent: x = 21 - 54 = - 33, which is answer (A). You can apply the same technique to problems involving fractions and ratios. For questions involving fractions, it's best to use the least common denominator of all the fractions in the question and the answers.
Here's another example, which is slightly different:
If (x / 5) + (1 / 10) = (7 / 100), then was it x?
(A) 0.07
(B) 0.01
(C) 7
(D) 1
(E) 0.5
Simply multiply both sides of the equation by 100, and you get:
(20x) + 10 = 7
That's a much easier equation to solve, and we can clearly see that the answer is (C).
If you look carefully, you'll notice a key difference between our two examples. In the first example, we multiplied all the numbers (but not x) in both the equation and the answers by 10,000. In the second equation, we did not multiply the answers by 100 because we multiplied both sides of the equation (including x) by 100. If we made the mistake of multiplying the answers by 100, we would have chosen (A) instead of (C).
This is an important point. If you multiply both sides of an equation by the same number, you don't have to modify the answers. But if you change only the numbers and not the complete equation, then you also have to change the answers by the same amount. Don't get confused by this. It's best to practice this technique a bit until you feel comfortable manipulating numbers.
Working Backwards
Sometimes it pays to work backwards. This is especially true with equations involving algebra. If you're asked to solve an unknown, you can save time and effort by simply taking the possible answers and plugging them back into the equation. If an answer works, then you've solved the problem. If not, try another answer until you find the one that does work.
Keep in mind, if it's simple to solve the math problem, don't bother working backwards. You should only resort to working backwards when it appears that it will take some time to actually work through the problem.
A good strategy for working backwards is to start with answer (C). This is because on most standardize exams the answers are listed in increasing or decreasing order, and since (C) is in the middle, it will give you a solid reference point. If (C) happens to work, you're done. But if it doesn't work, you should have a good idea whether you need to test a larger number or a smaller number. This saves you a bit of time by allowing you to eliminate either (A) (B) or (D) (E).
The beauty of working backwards is that it always produces the right answer. Not only can you eliminate some answers, but if you work through a maximum of three out of five of the possible answers, you should come to the solution.
The only time you wouldn't want to start with answer (C) is if one of the other answers seems much easier to plug in to the equation. For instance, if (D) seems easy but (C) appears difficult, it's smart to begin with (D). In general, it's easier to test whole numbers than fractions, and positive numbers than negative, and so on.
Now, sometimes there are questions that ask which of the five possible choices satisfies a specific set of conditions. In this case, it's usually impossible to solve the problem directly. The only way to come up with the answer is to work backwards, as described above.
Grid Problems
Working backwards is a good strategy for approaching grid problems. You can think of many grid problems as simply multiple-choice questions where the choices were inadvertently removed. In these cases, work backwards, plugging in numbers and seeing what makes sense. This works especially well when the grid question contains variables.
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