Math: 95cad55f

ID: 95cad55f

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

Comment: Watch out for assumptions. I have seen more students miss this question because of a certain assumption than I have seen miss other more difficult questions.

Method 1: Read the question first, then lean on the answer choices. Since all the inequalities start with one that adds detergent and softener to get a sum ≤ 300, think about what that models. We are told that deliveries will consist of no more than 300 pounds, and also the respective weights of a container of detergent and softener. Conceptually, this means

weight of each detergent (d) + weight of each softener (s) ≤ 300 pounds

All we have to do is insert the correct numbers for the two weights.

(7.35)d + (6.2)s ≤ 300

Please note that answer choices (C) and (D) incorrectly double the weight of each container of detergent, and the passage only tells us that an order will consist of at least twice as many containers of detergent as of softener. Ordering more containers does not affect the weight of a given container.

Now, between (A) and (B), think through just what the two (second) inequalities are saying. Otherwise, it is easy to jump into the trap answer.

(A) d ≥ 2s

Implication: For every 1 softener, there must be at least 2 detergents. This correctly models the situation outlined above.

(B) 2d ≥ s

Implication: For every 1 softener, there must be at least 1/2 a detergent, or, reading left to right, for every 1 detergent, there must be no more than 2 softeners. This is not what the passage describes, but I have seen plenty of students blunder into this answer because 2d = twice as many containers of detergent can feel right. Take the time to consider what you are modeling before jumping to conclusions.

The answer must be (A).

Method 2: You could create two separate graphs for the two inequalities, even though doing so will not immediately reveal the answer. (You still have to think about what the graphs mean.) How about using x for d and y for s?

An easy way to check which one is correct is to consider the x-intercept, since the y-intercept is the same for both inequalities. Could the service purchase close to 20 or close to 40 containers of softener, if it were allowed to buy 0 containers of detergent? There is no need to carry out the math: 6.2 * 20 is going to be between 100 and 200, while 6.2 * 40 will be between 200 and 300, so the red inequality is the one we want. Get rid of (C) and (D).

For graph 2, use the same substitution as before: x = d and y = s.

To my eye, the blue inequality stands out more, so how about starting there? Pick an easy point such as (2, 4). Again, if x = d, then this graph is telling you that for every 2 detergents, the service must purchase at least 4 softeners, and that is backwards. If (B) cannot be correct, then (A) is the correct answer. You could check the red inequality in the same way if you needed reassurance.

Method 3: You could guess and check using the answer choices, but again, you would still need to ensure that the inequalities accurately described the information in the setup. The second inequality might be easier to work through, since you could keep the numbers small. For instance, look at (A) and (C).

d ≥ 2s

Try something simple such as 1, but to avoid dealing with fractions, it makes more sense to start with s than with d.

d ≥ 2(1)

d ≥ 2 (if 1 softener is purchased)

This correctly describes the information from the passage. The other inequality from (B) and (D) does not.

2d ≥ s

2d ≥ (1)

d ≥ 1/2 (if 1 softener is purchased)

This would mean that the service could purchase 1 container of detergent without violating the given condition that there must be at least twice as many containers of detergent as of softener, and that is simply false: the two cannot be equal.

Between (A) and (C), it should be clear that the latter is an imposter for reasons explained above, in method 1. The correct answer must be (A).