It’s a real waste to miss a question like this! Make sure you pay proper attention to every single question on the entire test. Even the so-called ‘easy’ ones can trip you up if you rush.
Page 789, Question 18
This is one of those questions in which knowing the unwritten rules of the SAT Math section comes in especially handy.
Most people who try to answer this question will either use some version of a summation formula or they’ll actually try to add up every integer starting from -22 until they hit a running total of 72.
Either approach could conceivably work, but neither is very fast, nor very easy.
Instead, we want to remember that there must be some way to do this question in under 30 seconds relying only on some combination of basic arithmetic, algebra, and/or geometry.
One thing that would jump out at me right away is that we’re looking for a positive sum, but we’re starting by adding up some negative numbers in the beginning of our series. That means we’ll have to add quite a few numbers before our running total becomes positive, because at the beginning of our series (-22, -21, -20, . . .), the running total will just become increasingly negative.
So wait a second . . . when does the running total start to creep towards a positive balance?
Well, that won’t happen until we start adding in some positive numbers, of course. So our series will have to go past zero, for sure.
And when it passes zero, for a while, each positive number added to the running total will only serve to erase one of the negative numbers already added to it. Positive 1 undoes -1, for instance, and positive 2 undoes -2, and so on.
If we think about it, then, we’ll see that every number from -22 to positive 22, added together, gives a sum of zero, because every positive number in the series is outweighed by a negative number and vice-versa.
So all the numbers from -22 to positive 22 add up to zero. Then what happens?
We add in positive 23, positive 24, and positive 25, and we’ve reached our grand total of 72.
That means that all the numbers from -22 to positive 25 add up to positive 72. So (B) is the answer.
Notice some of the wrong answers here. (C) is the sum of the two numbers in the original question, while (E) is the difference. (A) is the first positive number that produces a positive value when added to the running total, so some test-takers will probably accidentally choose it because they’ve realized it marks the beginning of the positive running total without remembering what the question was actually asking.
Page 799, Question 14
People often get frustrated by this question because they can’t figure out the values of b and c. In school math classes, we’re often programmed to find the values for every variable in a question, but on the SAT Math section we often have to let go of that. Sometimes there’s not enough information in a question to figure out the value of every variable, but the question can still be answered anyway. One of the clearest signs of that situation is when the question involves multiple variables but doesn’t actually ask you for each one of their values.
For questions about function graphs, I often like to look at the intercepts. In this case, we know that c is positive, so the y-intercept must also be positive, because the y-intercept is what we get when x = 0.
Only (E) has a graph with a positive y-intercept, so only (E) works.
Page 800, Question 15
I often resist the idea of identifying ‘types’ of math questions on the SAT Math section, because I want you to realize that you’ll basically have to figure out stuff you’ve never seen before when you take the SAT. But if there were going to be a type of question that I might specifically train somebody for, it would be the kind of thing described in this question. (By the way, this comes up less than once per test in the College Board’s Blue Book. The odds that it will ever come up for you on a real test are less than 50%. And, even if it does come up, it’ll probably only count for one question. These are the kinds of reasons why I generally avoid thinking in terms of question types.)
Anyway, for these questions that ask you about the distance from one point of a 3-dimensional figure to another point, you basically have to apply the Pythagorean theorem twice.
Let's label the corner underneath point B as point Q, and let’s label the corner to the right of point A as point T.
So then triangle ATQ has legs of 1 and 2 units, which means that AQ has a distance of √5.
So now we can use AQ, which was the hypotenuse of ATQ, as a leg in the triangle ABQ. Using the Pythagorean theorem again, we get this:
AB2 = √52+ 12
AB2 = 5+ 1
AB2 = 6
AB = √6
So the answer is (D).
Note that one of the wrong answer choices is √5, for people who forget to finish the question.