Don’t be Intimidated by Word Problems
So many phrases, so little time!
Word problems are a major component of the SAT MATH exam, and many students are not immediately comfortable with the task of translating mathematically that which has been described. While the SAT test-makers do have a variety of ways of expressing what ultimately will equate to simple mathematical operators, this shouldn’t be cause for undue concern. Generally, just some basic common sense will allow you to quickly write one or more valid algebraic expressions and/or equations that proceed directly from the word problem at hand.
For instance, if it’s stated that Ginny sold two more than half of Brenda’s cookie sales, then just translate this piece by piece. So first what represents half of Brenda’s sales? That would be—
0.5B
Next add “two more”—
0.5B + 2
And then set this expression equal to Ginny’s total—
G = 0.5B + 2
And, of course, if it’s later given that Brenda sold 26 boxes of cookies, then we can substitute 26 for B, and easily solve for Ginny’s sales total—
G = 0.5(26) + 2 = 13 + 2 = 15
So Ginny sold 15 boxes of cookies, whereas Brenda sold 26. If it’s additionally stated that one box of cookies costs five dollars and you’re asked how much more revenue did Brenda generate than did Ginny, then you could just reason that Brenda sold 11 more boxes (26 versus 15), so the difference in revenue must be 11 X 5 dollars or 55 dollars.
Somewhat more difficult than the above would be a scenario in which you’re not explicitly given as much information, and asked to derive the appropriate equation versus calculate a specific value. So what if you’re asked to identify the equation that calculates the extra revenue that Brenda generates, and you’re not given Brenda’s actual sales total? Well, it stands to reason that the extra revenue (R) must be the difference in cookie sales multiplied by the price. So how do we mathematically express the difference in sales? It must be—
B – (0.5B + 2)
Makes sense, right?? This expresses Brenda’s total minus Ginny’s total. And after subtracting both parenthetical terms, we derive the following expression—
0.5B – 2
So 0.5B – 2 expresses the difference between Brenda’s and Ginny’s sales. To now express the difference of revenue generated (R), we just multiply by the five-dollar price that we were given—
R = 5(0.5B – 2)
And now distribute the terms—
R = 2.5B – 10
So on the exam, you would select the above equation as the correct answer choice. Now as a final check, let’s utilize our derived equation, but plug in the sales total for Brenda that we were given in our original discussion (i.e.,. 26 boxes)—
R = 2.5(26) – 10 = 65 – 10 = $55
And yes, this generates the same $55 extra revenue that we determined in our first workup of this problem. So both approaches (the more literal and the more generic) are in agreement.
Eat, Drink and Text–An American Tradition
As you may have noticed, the SAT test-makers have a thing for food and beverage questions! They also seem to have more recently embraced text-messaging teens! In this typical workup, the hourly rates of messaging will be given as variables (let’s say x and y for Tony and Tanya respectively) and then later in the question, you’ll be given the actual number of hours of messaging for each teen. So the hourly rate (i.e., texts per hour) TIMES hours equals the total amount of texts for each student, correct? And if asked to express the total texts of the two teens, then the only real issue is to pair the corresponding rates and hours. And so in this type of word problem, if Tony texts for five hours and Tanya for four hours, then you would select—
T = 5x + 4y
—as the correct equation, AND you would take care NOT to select T = 4x + 5y (in which the two coefficients (i.e., hours) have NOT been correctly matched to their corresponding rates (i.e.,. x and y)). In other words, don’t multiply Tony’s hours by Tanya’s rate, right??!! I mean cmon, Tony’s rate has got to go with Tony’s hours!
Inequalities as SAT Word Problems
Another potentially nettlesome area of word Problems on the SAT is the appearance of inequalities. But again, just use some common sense. How could “no more than” be expressed as a mathematical operator? It’s “≤”, right? If Bill can work “no more than” 20 hours per week, then we could write—
B ≤ 20
—where B represents Bill’s hours. In fact, we could graph this simple inequality on a number line as follows—
And consider these variations– “Bill can work up to 20 hours per week,” or “Bill can work as many as 20 hours in a given week.” Right?? Both variations still describe the same simple inequality— “B ≤ 20” !!
What Equates to Equal
There are even more variations for phrases that equate to the equal sign(!), such as “is”, “has”, “was”, “earned”, “got”, and on and on. But once again, just use some common sense. At the end of the week, Becky still has 5 unsold (or remaining) boxes of chocolates. So “R = 5”. And maybe we’re given a rate; Becky sold an average of 3 boxes per day for seven days. And we’re asked how many boxes did she start with? So if B represents the original total number of boxes of chocolates (and R represents remaining unsold boxes), then—
B = (3 X 7) + R
Then multiplying and substituting 5 for R—
B = 21 + 5 = 26
So Becky started with 26 boxes of chocolate, sold 21, and therefore still has five remaining boxes.
Percentage Problems as SAT Word Problems
Percentage Problems frequently appear as word problems on the SAT. The main issue here is to be crystal clear on the difference between 27% of something versus a 27% reduction of something. So again, the SAT test-makers love working with food and beverages, so as above a 27% reduction from 300 equates to a reduction of 81 cookies, and hence 219 cookies remain in her inventory. Alternatively, you can create a multiplicative factor of 0.73 (i.e., 1.00 – 0.27) and multiply 300 by 0.73. This approach will also derive the correct inventory of 219 remaining cookies.
And of course, a 27% REDUCTION of inventory is not the same as saying, “Sheryl has only 27% of her original inventory.” Now the inventory of cookies is 81, not 219. In fact, now the reduction is 219! So be vigilant in terms of raw percentages versus percentage reductions and/or percentage increases. The SAT loves percentage reductions or discounts, especially if followed by a sales tax!
System of Equations Problems as SAT Word Problems
System of Equations problems are often presented as word problems, which can add an extra layer of difficulty, unless you’re comfortable with translating the key phrases into the two requisite equations. And I should add that the SAT test-makers seem to really love System of Equations problems that revolve around food and/or beverages!! Why is that, I wonder? So let’s try one, and bear in mind that if you can properly set up one of these, then you can do a million of them because they really are highly formulaic on the exam—
Tina runs a food truck business and sells coffee and Danish pastries. Coffees cost $2.00 and Danishes cost $3.25. At day’s end, Tina has pocketed $525 and sold 200 items. How many Danish pastries did Tina sell?
OK, so I’ve made this an unrealistically easy question because I just want to focus on the algebraic set up. What are the two equations that we must translate from the above word problem? For starters, let’s just name two variables (one to represent the number of Coffees and one to represent the number of Danish). So how about C and D? We could use x and y, but perhaps C and D will better insure that we don’t mess up pairing the right price with the right item. OK, so C and D it is. And then the next question is what would algebraically represent the amount of money that Tina earns in a given day. Clearly it’s the number of coffees times $2.00 and the number of Danishes times $3.25.
So we set up the following equation—
2.00C + 3.25D = Total revenue
Oh, and we’re given the total revenue, right?? That’s $525. So now we have—
2.00C + 3.25D = 525
Now we write the other equation we’ll need to complete the system of equations. And this is the easier one. It’s given that Tina sold a total of 200 items. So this we can express simply as—
C + D = 200
So now we have completed the required system of equations. I’m not going to review how to solve a System of Equations, as we practice that extensively in our full curriculum, and that’s not the focus of this article. Suffice it to say that Tina sold 100 coffees ($200 revenue) and 100 Danish ($325 revenue) for a total revenue of $525 and a total sales volume of 200 items. Hey, maybe Tina’s like the Soup Nazi?? If you try and just order coffee, then it’s, “No Danish For You”!! Or perhaps this is just one of those odd coincidences that famously on the SAT? Actually, on the exam the totals will almost assuredly be different, and you’ll determine them by applying the basic strategy for solving a Systems of Equations problem that we practice multiple times in our full curriculum.
Alright! And as a final aside, in the last example, the test-makers could have omitted the total sales number (i.e., 200) and instead said that the number of coffees sold was equal to the number of danishes sold. So then, instead of writing C + D = 200, you would have simply written C = D as the second equation in your System of Equations. Either way you would then solve for Danishes and arrive at 100 Danishes sold as the correct answer.
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