An Easy Way to Solve Theoretical Math Problems on the SAT

SAT Tip of the Week - FullOne of the biggest tricks the SAT uses is to confuse students is putting a question in theoretical terms instead of in practical terms. This simply means the questions on the SAT will sometimes reference a general term, for example an even integer, rather than giving a concrete number that fits that description, such as two or four.

The good news is that it is easy to correct this by simply plugging in concrete numbers when the question gives general terms.


Here is an easy example:

An even and an odd integer are multiplied together. Which of the following could not be the square of their product?

(A) 36

(B) 100

(C) 144

(D) 225

(E) 400

One way to approach this problem is to start with an even and an odd integer and plug them in to the parameters set by the problem. If we begin with two and three, we see that the product is six and the square of the product is thirty six.

(2)(3) = 6  6² = 36

Similarly we can see that two and five, three and four, and four and five all give us possible answer choices.

(2)(5) = 10  10² = 100

(4)(3) = 12  12² = 144

(4)(5) = 20  20² = 400

Answer choice (C) is also a perfect square, but if we take the square root of it, we see that the result is fifteen, which is not divisible by an even and an odd number. Thus the only answer that could not be the squared product of an even and odd integer is answer choice (C).

Here is a slightly more difficult question.

A right triangular fence is y inches on its smallest side and z inches on its largest side. If y and z are positive integers, what represents the formula for the area of the fenced in region in square feet?

(A) ?(z² – y² ) (y)

(B) 24 (z² – y² )

(C) ?(z² – y² )  (y/12)

(D) ?(z² – y² )  (y/24)

(E) (½) ?(z² – y³)/ 12

At first glance, this problem may seem complex, but we can simply plug in real numbers into this problem and solve by seeing which answer choice gives the same response as the answer we derive. This is a right triangle, so if z is five and y is 3, then the third side, which is also the height, would be four. The total area in inches would then be one half base times height. To convert inches to feet we would have to divide the area by twelve.

y = 3

z = 5

H = 4

1/2 (3)(4) = 6 in²

6/12 = 1/2 ft²

Only answer choice (D) gives the correct answer of one half when the numbers we chose are plugged into the equation. We can also see that, if multiplied by 12 to account for the change to feet, answer choice (D) is essentially the formula for the area of a triangle with ?(z² – y² ) as the height.

It is easy to get frustrated when given a theoretical problem, but when real numbers are inserted for the theoretical ones, the problem becomes surprisingly simple. So throw some real numbers into the mix and see what happens. The only thing to be wary of is that in certain contexts, it may be necessary to plug in different combinations of numbers that fit the given parameters to make sure that the general equation works with different sets of specific numbers. Even with this caveat, with a little practice, this technique can make even very confusing problems seems quite simple. Happy studying!

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David Greenslade is a Veritas Prep SAT instructor based in New York. His passion for education began while tutoring students in underrepresented areas during his time at the University of North Carolina. After receiving a degree in Biology, he studied language in China and then moved to New York where he teaches SAT prep and participates in improv comedy. Read more of his articles here, including How I Scored in the 99th Percentile and How to Effectively Study for the SAT.