Page 800, Question 16
This is a question with a made-up function. For these kinds of things, it’s always very important to read and follow directions very carefully.
In this case, people often fail to realize that the right-hand side of the equation should be
(a-2)2 - (a -2). We have to apply the oval function to the whole expression inside the oval, and the whole expression inside the second oval is (a - 2).
Once we do that correctly, all that remains is to solve:
a2 - a = (a-2)2 - (a -2) (substitute the starting values from the question)
a2 - a = (a2 - 4a + 4) - a +2 (FOIL the expression on the right)
a2 - a = a2 - 5a + 6 (combine like terms on the right)
- a = - 5a + 6 (subtract a2 from both sides)
0 = - 4a + 6 (combine a terms)
4a = 6 (separate a term from constant)
a = 6/4 (isolate a)
a = 3/2 (simplify)
So (C) is correct.
As we might expect, one of the wrong answers is twice as much as the right answer. The other wrong answers seem to reflect what we might have ended up with if we had handled the algebra incorrectly.
By the way, another approach to this question, which would probably take longer but would avoid so much algebra, would be to take each answer choice and plug it in to the original expression until you find one choice that results in a true mathematical statement. But even if you do the question in this way, you still need to make sure you handle the substitution properly by reading the question very carefully.
Page 835, Question 18
To answer this one we have to think about what we're given and what we're asked for. We're given the averages of the two groups and asked for the ratio of their sizes. Note that it's impossible to know the actual number of people in each group--one major clue to this is the fact that the test asks for the ratio.
Many test-takers try to approach this question by figuring out individual values for p and n, but it literally can’t be done. It is possible, however, to figure out the value of p/n.
We'll figure out the ratio by using the definition of the term "average," which tells us that if p students have an average of 70, then the total number of points scored by those p students is 70p.
Similarly, the total number of points scored by the n students is 92n.
Using the definition of average, again, we find that the average of all the n students’ scores plus the p students’ scores must be (70p + 92n)/(n + p) = 86. Then we solve:
70p + 92n = 86p + 86n (multiply both sides by n + p)
6n = 16p (combine like terms)
6n/16 = p (isolate p)
6/16 = p/n (isolate p/n since the question asks for it)
3/8 = p/n (simplify)
Page 848, Question 6
Many test-takers miss this question because they mistakenly think that a square yard is the same thing as 3 square feet. But a square yard is 9 square feet, because a square yard is 1 yard x 1 yard, or 3 feet x 3 feet, or 9 feet2.
Once we know that, it becomes easier to realize the correct answer is 12 * 18 / 9, or 24.
Another way to approach the situation is to convert the dimensions of the floor to yards right from the beginning, so that we calculate 4 yards by 6 yards, again getting 24 yards2.
So (C) is correct. Note that (A) is there for us in case we divide by 3 one time too many, and (E) is there in case we accidentally find the square footage instead of the square yardage. Things like this are why it’s always so important to read carefully on the SAT Math section.
Page 851, Question 15
This question often frustrates people, but it’s actually not as hard as people often think. The question asks for the shortest distance between the center of the cube and the base of the cube. If we visualize the cube, we'll see that the distance from its center to its base is the same as the distance from the center of one of its sides to the base—or, in other words, if you looked at the cube straight on it would look like this:
where C is the center of the square/cube and the bottom line of the square/cube is the base.
If the volume of the cube is 8, then the sides are all length 2, which means the distance from the center to the base is 1 (because it's half the distance from the base to the top side of the cube). So the answer is (A).
Page 852, Question 18
This question provides one more terrific example of the difficulty that the College Board can manage to infuse into an SAT Math question without actually using any math.
This question throws a lot of people. But, as always, all we really need to do here is follow what the text says, alternately switching the left or the right wire with the center wire. So the pattern goes like this:
step 1: BAC
step 2: BCA
step 3: CBA
step 4: CAB
step 5: ACB
step 6: ABC
So the answer is (D). Notice that it’s possible to end up with the wrong answer of 7 if we accidentally count the “Start” as a step. One more moment when critical reading skills really come in handy on the math part of the SAT.