Since the shaded region is divided into two squares, we know that the distance from E or F to the base of the figure must be 5. So the shaded area is 50, because we multiply 10 (the length) by 5 (the height) to find the area. So (C) is correct.
There are other ways to get this answer, as well. We could also realize that AC must be 10√2(√2), or 20 (again because of the 1:1:√2 ratio for sides of a 45o-45o-90o triangle). Then we could realize that the height of triangle ABC must be 10 (because BD is one of the legs for which BC is a hypotenuse, and this is another 45o-45o-90o triangle for which the 1:1:√2 ratio applies). That means the area of the entire triangle ABC must be 100 units. The shaded region represents 1/2 of the area of ABC (we know this either by eyeballing the scaled diagram, or by realizing that the four unshaded small triangles are the same total area as the shaded rectangle).
Now let’s turn to the answer choices. An awareness of the patterns that frequently appear in answer choices on the SAT Math section would help us to realize that 50 looks like a very likely option to be the correct answer. It fits the halves-and-doubles pattern and it’s also the middle number in a series (the series is 25, 50, 100). It also doesn’t have √2 in it, which is probably good because 3 out of the 5 choices don’t have √2.
Page 423, Question 11
Like most SAT Math questions, this one rewards us for reading very carefully and thinking about the definitions and properties of basic terms.
We’re told that 2 of the faces are black and the rest are white; this means there must be 4 white faces, since the total number of faces on any cube is 6.
If the total area of the white faces of the cube is 64 square inches, and if there are 4 white faces, then the total area per face is 64/4, or 16.
That means each face is 16 square inches. And since the dimensions of a cube are all identical, that means each face is a 4 x 4 square, which means the cube is 4 inches in each dimension.
The volume of the cube, then, is 4 x 4 x 4, or 64. So the correct answer is (A).
Notice that some of the answer choices differ from the correct answer by a factor of 2 or 4. This is a strong, strong reason to go back over your work and check it for small mistakes. If we misread the setup, or if we accidentally mis-multiplied or mis-divided, we can easily be off from the right answer in a way that will be accounted for in the answer choices. Also notice that (B) is 5 x 5 x 5, and (D) is 6 x 6 x 6. This reinforces our belief that we should find the correct answer by cubing something, but it also means we have to make sure we were right to cube 4 instead of 5 or 6.
Page 424, Question 14
As will often be the case on the SAT, there are multiple valid approaches to this question. If we wanted to use a simple permutation solution, we could realize that there are 5 options for one color, and 4 options for the other color (there are only 4 options for the second color because we’re not allowed to repeat the selected color—otherwise there would be 5 options for both). 5 x 4 = 20, so there would be 20 possible arrangements.
But the SAT doesn’t make us do things like this in the formal way—the SAT doesn’t care how we get the answer, as long as it’s right. So, if we want, we can just list out the different possible arrangements and then count them up. (To be clear, this list-and-count approach will take a good bit longer than we would normally like to spend on a question, but it can be a very concrete way to arrive at the answer if you don’t feel comfortable with permutations. Remember, too, that the reason we try to go through most questions as quickly and efficiently as possible is so that we can have more time if we need it on questions like this.)
So if we call the colors a, b, c, d, and e, and then list things out, here are the different arrangements:
zone 1 / zone 2
a/b
a/c
a/d
a/e
b/a
b/c
b/d
b/e
c/a
c/b
c/d
c/e
d/a
d/b
d/c
d/e
e/a
e/b
e/c
e/d
That means there are 20 different arrangements, so (B) is correct.
Page 457, Question 20
Most people try to approach this using some variation on the slope formula (which is
(y2 – y1)/(x2 – x1), not that I recommend it here).
But we have to remember that this is the SAT, and the College Board often likes to leave short cuts hidden in the questions for people who think to look for them.
Instead of going through the hassle of working out the slope of the original line, the slope of the perpendicular line, and the missing y-value that would make the whole thing work out correctly, let’s take a second and actually notice the answer choices for a minute.
Remember that we always want to check out the answer choices before we commit to a course of action. They’ll help us understand a lot of stuff in a lot of questions if we just pay attention to them a little.