First, it’s important to realize that the question tells us that a and b are equal, so f(a) and f(b) must also be equal.
We also want to remember that questions with roman numerals for answer choices are often based on abstract properties.
Finally, we want to remember that we don’t need to know what the actual function is. All we need to know is that it has the property that f(x + y) = f(x) + f(y).
So now let’s try to reframe each Roman numeral in terms of that property, and see if we can do it:
For I, 2f(a) = f(a) + f(a) = f(a + a) = f(a + b). We can substitute b for a at the end to arrive at f(a + b).
For II, f(a) * f(a) has nothing to do with what the question told us about; the question only told us about an additive property of the functions, not a multiplicative one. So we can’t substitute or manipulate anything further here.
For III, f(2a) = f(a + a) = f(a) + f(a) = f(b) + f(b). So this one works too.
So we can see that (C) works because roman numerals I and III are valid statements given that
f(x + y) = f(x) + f(y).
This is a question that won’t allow us to use a concrete example (unless you get extremely lucky in making up a function for f). The only practical way to approach it is to make substitutions and follow the rules of algebra to see which roman numerals contain valid equations.
For most test-takers, this will be one of the hardest questions. So this is a good time for me to remind you that your primary goal probably shouldn’t be trying to improve your performance on the occasional tough question like this; it’s much more important to make sure you lock down all the questions that seem easier first. Once you get to a point where you never make any ‘careless’ errors, you should feel free to start worrying about questions like these. But if you try to tackle these kinds of questions without first making sure that you’re executing correctly on the questions you can handle more comfortably, you’ll just be wasting your time, and your score won’t improve.
I’m not saying that a question like this can’t be answered, or that if follows different rules. This question can be solved with basic math and careful thinking, just like every other SAT Math question. I’m just saying that it’s important to focus on eliminating mistakes before you focus on figuring out questions that seem more challenging to you.
Page 519, Question 20
This question really helps drive home the importance of considering the answer choices along with the rest of the question.
When we look at the answer choices, we can see that they all involve y, with no x. That means we need a way to express x in terms of y. The only way to do that is to realize that x times y is 4000, since that's the area of the rectangle. So x is 4000/y.
Furthermore, the total length of rope needed is y + 4x, because there are 4 vertical line segments with length x in the diagram.
We can re-write y + 4x as y + 4(4000/y), which is the same as y + 16000/y. So (B) is correct.
When I look back over the other answer choices, I would definitely want to notice that more of the answer choices include 3y than just y in the denominator, and that would worry me for a second, because I know that elements of the correct answers tend to appear in wrong answers in questions like this. So I would double-check my work again.
Page 527, Question 8
Despite my general dislike of algebra for the purposes of SAT Math solutions, sometimes it can’t be avoided, and I think this question represents one of those times for most test-takers. This question is also one of those questions that can be expected to take most test-takers more time than usual. It’s because of questions like this that we have to try to work through other questions as quickly as possible without sacrificing accuracy.
For most people, the obvious first step will be to multiply out the expression on the left (using FOIL), which gives us this:
(x-8)(x - k) = x2 - kx - 8x + 8k
In terms of what we would normally get from FOIL-ing out two binomials, this expression is a little odd, because it has two terms with x instead of one. (This is because the original binomials involved more than one unknown value—they had a k in addition to the x-es.) So let’s try to fix that by combining 8x and kx:
(x-8)(x - k) = x2 - kx - 8x + 8k
(x-8)(x - k) = x2 - (8 + k)x + 8k
Now it looks a bit more normal—we’re still stuck with that weird k, but at least now we have one term with x2, one term with just x, and one term with no x at all, which is our normal arrangement after FOIL-ing out two binomials.
So now our entire equation looks like this:
x2 - (8 + k)x + 8k = x2 - 5kx + m
Now we realize that (8 + k)x must correspond with 5kx, and that 8k corresponds with m. In other words, the two x terms on both sides must correspond, and the two terms on both sides with no x at all must also correspond. So we can solve for k, and then use k to solve for m: