On the standardized test, your choices are limited to only five, which is more manageable than the plethora of choices you encounter every day. However, even five answer choices can cause a lot of frustration for people who have difficulty differentiating among them.
The good news is, the exam is mandated to have four to fivw different answer choices on every question, but some of these answer choices are redundant. While you won’t actually see the same answer choice twice on the test (unless you’re seeing double), many answer choices don’t differ from another answer choice in a meaningful way.
As an example, if you’re looking for the product of two even integers, such as 4 and 6, you know the product can never be odd. So while one answer choice may be 25 and another may be 33, they can both be eliminated for the same reason, greatly streamlining your task if you’re eliminating possible answer choices based on sound reasoning. Sometimes, a question may have two or three answer choices you can eliminate without having to do any math, as long as you can sort multiple answers into the same bucket (think Gryffindor).
Let’s look at such a question and how we can consider eliminating answer choices without actually calculating them longhand:
If x^4 > x^5 > x^3, which one could be the value of x?
This question seems complicated because it is very abstract. We’re dealing with some unknown variable x raised to various uncomfortable powers. A great strategy here would be to try and make it easier to understand by using actual numbers. This will allow us to better visualize what is actually happening in the problem.
Let’s begin with the base case. Say we set x to be a simple positive integer, such as 2. If we square 2, we get 4. If we multiply by 2 again, we get 8. This is 2^3. We can continue by multiplying by 2 again and getting 16 for 2^4, and one final time to get 32 for 2^5. It should come as no surprise that the variable gets bigger as the powers increase.
However, this situation does not satisfy our original premise of x^4 > x^5 > x^3 because x^5 is the biggest value. Beyond eliminating the number 2 from contention, we can eliminate 3, 4, and every other positive integer bigger than 1. This is because all positive integers greater than one will increase in amplitude as the powers increase. Knowing this, we can eliminate answer choice E, which follows the same mould.
The remaining answer choices seem to either be negative, fractional or both. We might also recognize that numbers smaller than 1 will follow a different pattern, because successive increases in power will make the number smaller and smaller. Furthermore, negative numbers can break the pattern as well, as they will oscillate between positive results for even powers and negative results for odd powers. In fact, these two axes will be the only determining factors in identifying the correct result. The answer will be only one of the following structures: positive and less than 1, negative and less than 1, positive and more than 1, or negative and more than 1. Our job is to sort these out (like the sorting hat at Hogwarts).
We have already observed that positive and greater than 1 doesn’t satisfy the given inequality, so let’s look at positive and less than 1. We can take ½ as an example and extrapolate that to any result 0 > x > 1. If we square ½, we get ¼. If we continue to multiply by ½, we get 1/8, 1/16 and 1/32 respectively. Unsurprisingly, these are the reciprocals of the values found for x = 2. This batch doesn’t satisfy the inequality either, as x^3 is actually the biggest number here. This eliminates answer choice D. If it’s not obvious, the relative sizes of the exponents are easier to see if we use the number line:
0 1/32 1/16 1/8 1
x^5 x^4 x^3
Now that we’ve eliminated two possibilities, let’s look at the remaining choices: -3, -2 and -2/3. At this point, it should make sense that all negative numbers with absolute value greater than 1 will behave the exact same way in this inequality. This means that the answer cannot be either -3 or -2, as they are indistinguishable inputs on this question . Thus, if -2 worked, so would -3, and vice versa. Since only one answer choice can be correct, neither of these will be correct, and the answer must be -2/3. Let’s go through the calculation to confirm, but we already know it must be correct.
When we square a negative number, we are multiplying a negative by a negative and yielding a positive. When we multiply that number by a negative again, we revert to negative numbers. Thus, every odd numbered power will be negative and every even numbered power will be positive. Knowing this, we can easily calculate that x = -2/3, then x^2 = 2^2/3^2. Multiplying by -2/3 again, we get -2^3/3^3 for x^3. The next values will be 2^4/3^4 for x^4 and -2^5/3^5 for x^5. If it’s easier to see, you can calculate each of these values and get:
x^2 = 4/9
x^3 = -8/27
x^4 = 16/81
x^5 = -32/243
Using the number line again as a visual aid (roughly to scale):
-1 -8/27 -32/243 0 16/81 1
x^3 x^5 x^4
This confirms that x^4 is the biggest (most to the right) value while x^3 is the smallest and x^5 is the middle value. This also highlights the issue that -2 and -3 would have, as the amplitude increases, x^5 would be much smaller than x^3. Of the choices given, the only value that works is answer choice C: -2/3.
On the CAT, one of the four answer choices must always be correct, but the other three can give you insight into what you should consider to solve the question. Oftentimes, you can figure out what the key issues are by perusing the choices provided. And more often than not, you can eliminate swaths of answer choices based on a logical understanding of the question. On test day, you don’t want to waste time considering answer choices that are obviously incorrect. If you can sort through the various answer choices quickly, you’ll end up in the house of your choice.