Math: 1035faea

ID: 1035faea 

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Algebra)

Comment: If the inequalities look confusing, do not make assumptions. Break down the information line by line so that you can be sure your answer makes sense.

Method 1: Sometimes it is not a bad idea when encountering a lengthy word problem to glance ahead at the question so that you know how to frame the information in the passage. What will help you solve that question? In this case, we see a conditional, an if statement, that tells us that more than 20% of all participants selected a certain picture. This conditional is followed by a question that asks us to model possible values of p. The line before the question tells us that p refers to the number of people who chose this particular picture from the remaining 150 participants. What about the first group of participants? The line in the middle of the paragraph tells us that 36 of the first 150 chose this same picture. Okay, we now have enough information to put everything together.

36 of 150 participants + p from among 150 participants > 20% of 300

36 + p > 0.2(300)

We can stop right here, since this looks almost exactly like answer choice (D). Notice that the latter inequality in each answer choice, p ≤ 150, is the same, so we can ignore that part altogether, knowing it must be correct. In short, the correct answer must be (D). All we had to do was transcribe a few pieces of information from the passage.

Method 2: Plug in the inequalities to Desmos and graph. You can use y for p.

You can hover over the y-intercept for each graph to get a crucial point to make sense of.

The red line shows a forbidden intercept at 52.8. (Remember, we are dealing with an inequality, hence the dashed lines.) Since there cannot be fractions of participants, we can round up to 53. 53 participants + 36 participants (from the first group) = 89 participants who chose the first picture, and that is more than what you need: 20 percent of 300 is 60, however you calculate it. Because these numbers do not align, you know that this inequality has a lower limit that is too high—there could have been fewer people in the second group who preferred the first picture. Furthermore, anything with a lower limit that is higher than this one in the graph must also be incorrect, since you cannot get down to 60 by bumping the inequality up. A quick hover reveals that the blue line shows an intercept at 67.2 and that the green line shows an intercept at 96. The purple line shows the proper intercept at 24—36 + 24 = 60. Again, the correct answer must be (D).

Method 3: You could work backwards from the inequalities if the algebra does not intimidate you, even if it does create more work than is necessary.

A. p > 0.20(300 - 36) --> p > 0.20(264) --> p > 52.8

B. p > 0.20(300 + 36) --> p > 0.20(336) --> p > 67.2

C. p - 36 > 0.20(300) --> p - 36 > 60 --> p > 96

D. p + 36 > 0.20(300) --> p + 36 > 60 --> p > 24

As explained above, in method 2, since 36 participants (from group 1) + 24 participants (from group 2) = 60 total participants from the combined groups who, at a bare minimum, must have selected the picture in question, we know that the correct answer is (D).