Math: 2b15d65f

 ID: 2b15d65f

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

Comment: As always, whenever dealing with a word problem, it is crucial to tease out any pertinent figures, relative to what the question is asking, and to form a plan of attack as you encounter the information in the passage.

Method 1: Because we are told that this is a linear function, we can use ratios to determine the answer by jotting down nothing more than a few numbers. This method, called allegation, can help with mixtures as well, and I did not learn it until I started teaching graduate-level exams such as the GRE® and GMAT™. Here, since we have three prices, we can track the difference between them in the following manner:

$40---$55-$60

I have used a dash, -, to represent $5 increments. We can see that there are 3 dashes between $40 and $55 and that there is a single dash between $55 and $60. We could also use numbers to track this distance, 55 - 40 = 15 and 60 - 55 = 5, and 15/5 reduces to 3/1. However, I like to represent the ratio graphically in simplest terms to avoid making careless errors.

Now, look at the difference in units from the setup: 20,000 - 15,000 = 5,000 units. To get the answer, we apply the 3:1 ratio to these 5,000 units as a part-to-whole ratio. That is, the total number of parts must equal the whole value, and once we figure out what each part is worth, we can determine how many units we need to count up or down from either 20,000 or 15,000.

3 + 1 = 4 parts

5,000 units/4 parts = 1,250 units/part

For the final step, since we are dealing with an inverse relationship—demand decreases as the price increases—and since $55 is closer to $60 than it is to $40, we know the ratio applies in one of two ways:

- 3 parts less than the number of units demanded at a price of $40

- 1 part greater than the number of units demanded at a price of $60

Using the second, less labor-intensive point of entry, we can simply add the 1 part to the number of units demanded at $60:

15,000 + 1,250 = 16,250

The answer must be (A).

Of course, there is nothing wrong with going the other way, but generally speaking, people find it easier to add than to subtract (I certainly do), and adding a single unit is simpler than subtracting three times a unit. In any case,

20,000 - 3(1,250)

20,000 - 3,750 = 16,250 

The answer must be (A).

Allegation is an advanced technique that may take some time and practice to get used to, but it can really help with certain types of questions, linear modeling among them. This question, for instance, might take 30-45 seconds using the method I have outlined above.  

Method 2: You could use Desmos to model the linear relationship using the points (40, 20,000) and (60, 15,000), but, as we have explored in at least one other question (2937ef4f) there is nothing wrong with using smaller values to keep the axes more manageable. Here, we can use (4, 20) and (6, 15), chopping off a few 0's that might otherwise lead to resizing the graph several times. Now, work out some of the basic algebra.

y = mx + b

y = [(20 - 15)/(4 - 6)]x + b

y = -(5/2)x + b

Substitute one of the two points.

(20) = -(5/2)(4) + b

20 = -10 + b

30 = b

Thus, y = -(5/2)x + 30 is our equation.

To answer the question, add a slider, a second equation: x = 5.5. The graph will show you the answer. 

Remember, 5.5 is the decimal equivalent of 55 as the actual input value, and 16.25 corresponds to 16,250. It should be clear that the answer is (A).

Alternatively, you can go to Edit List (the gear icon, which I will highlight in bright green) and Create Table to input the value 5.5 for x to see the answer.

Again, it should be clear that the answer is (A).

Method 3: There is nothing wrong with using the above method, minus the graph. A calculator could quickly churn out the correct answer using the right input values. You could even keep all the numbers the same as those used in the passage if you were uncomfortable altering them in any way. The equation from earlier would look a little different, even if the result would end up being the same.

y = mx + b

y = [(20,000 - 15,000)/(40 - 60)]x + b

y = -(5,000/20)x + b

y = -250x + b

Substitute one of the two points.

15,000 = -250(60) + b

15,000 = -15,000 + b

30,000 = b

Thus, y = -250x + 30,000, and now you can solve with an input of 55.

y = -250(55) + 30,000

y = -13,750 + 30,000

y = 16,250

Bigger numbers are unwieldy for most people to work with, but in any case, the answer must be (A).