16k = 10k2
Solving, we get
16 = 10k (divide through by k)
1.6 = k (isolate k)
And that’s all. Notice that this question only required us to know the definitions of the word “perimeter” and “area,” and basic algebra. There was no formula involved, apart from the formula for the area of a square, which the test provides for us at the beginning of the section. There was also no real need for a calculator, since the question only involves dividing by 10, which we can accomplish by moving the decimal point.
Again, these attributes are pretty typical of the ‘hardest’ SAT Math questions.
Page 483, Question 8
This question stumps a lot of people. In my experience, almost everybody who misses it does so because they don’t read it carefully, or they don’t notice the answer choices.
If the probability of choosing red is 3 times that of choosing blue, that means that there are 3 times as many red beads as there are blue beads. Since there are 12 red beads, then, there must be 4 blue beads. Further, the number of glass beads altogether is 4 times the number of wooden ones, so if there are 16 total glass beads (12 red and 4 blue), then there are 4 wooden ones. Adding that all up, we get that there are 20 beads.
That’s the more mathematical way to approach this.
But an easier way to think about this might be to realize that there are more red beads than anything else, and there are only 12 of those, so 45 is already way too big of a number, and anything bigger than 45 is obviously also way too big. That means the only answer choice that can possibly work is (A).
Remember to pay attention to details and answer choices!
Page 485, Question 12
This is another SAT Math question that a lot of people struggle with, even though it only involves one of the simplest ideas in all of geometry: the idea that there is an infinite number of points in a circle (or in any geometric figure).
I think the easiest way to approach this is to say that every single point on the circumference of the circle could be a point that served as the corner of a rectangle like the rectangles in the original diagram. So there is an infinite number of rectangles with perimeter 12 that can be inscribed in the circle. Since infinity is bigger than 4, we know that the answer is (E).
Page 486, Question 15
For this question, once more, I would just read the question carefully and think about what it's describing.
20% of Tom's money was his spend on the hotel. He spent $240 overall, so he spent $240 * 0.2 = $48 on the hotel.
If he only paid for 1/4 of the hotel, then the hotel cost $48 * 4, or $192.
The fact that the last three answer choices all differ from one another by $48 should alert us to the fact that we need to be really careful here, because there are ways to misread or miscalculate and end up on either wrong answer. The far most common mistake is to misread the thing about sharing with 3 other people, and treat it like it just says the room was split among 3 people.
This is one more situation in which paying attention to the relationships among the answer choices can alert us to mistakes that the College Board wants us to make, and can reassure us that (D) is the correct answer.
Page 486, Question 16
Like many other SAT Math questions, this is one that manages to be fairly challenging even though it only involves basic arithmetic. It’s also a question that will require very careful reading, and a question for which there is no ready-made formula. In other words, it’s a typical SAT Math question.
So let’s just think about what the question is describing, and how we might figure out what it’s asking us.
One approach would be to try to make a square board that would have a number of border tiles somewhat near each answer choice, and see if only one answer choice can be made to work like that. This will be a little tedious, and probably extremely time-consuming, but it will work if we do it right:
(A) doesn't work--if there are 10 on the boundary, it couldn't be 3 x 3 or 4 x 4.
(B) doesn't work either. 7 x 7 would give you 24.
(C) doesn't work because 9 x 9 would give you 32.
(D) doesn't work because 11 x 11 would give you 40.
(E) works because if the board is 14 x 14 there are 52 tiles on the border.
Another approach, possibly slightly faster, is to try drawing out a few small game boards to see if we can figure out some kind of pattern that would help us eliminate all the wrong answer choices.
If, for instance, n = 2, then the square would look like this:
In this case, the number of things on the boundary would be 4.
If n is 3, then the board looks like this:
In that case, the k number is 8.
And so on. Now we need to try and understand what's going on here. Basically, k must always end up being a multiple of 4. So we need an answer choice that's a multiple of 4. Only (E) is.