Here’s another post in our GMAT Case Study series, where we dig into the key concepts behind GMAT practice questions. This week: word problems.
After you practice long enough for the GMAT, you may find yourself answering certain common types of problems on autopilot. But always read carefully –sometimes a problem looks like one you’ve seen a million times before, and yet it’s actually about something else altogether.
Let’s try out this sample problem:
Eunice sold several cakes. If each cake sold for either exactly 17 or exactly 19 dollars, how many 19-dollar cakes did Eunice sell?
- Eunice sold a total of 8 cakes.
- Eunice made 140 dollars in total revenue from her cakes.
(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
(D) EACH statement ALONE is sufficient to answer the question asked.
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Try it out for yourself, then read on for a full explanation!
If you chose option C for this problem, you wouldn’t be alone. You’ve probably had it drilled into your head (especially if you’re a Knewton student) that if you have two variables and only one equation, you can’t solve for both variables. In this problem, you’ve got two unknown quantities (the number of 17-dollar cakes and the number of 19-dollar cakes that Eunice sold), and you automatically expect to need two separate equations to find those numbers.
But sometimes, particularly in tricky word problems like this one, the GMAT actually gives you more information than you realize. A problem like this, that seems to be testing the simple concept of an insufficient system of equations, for example, is actually be testing you on your ability to understand number properties.
The difference in this problem is that the word “several” in the question stem refers to integers. Eunice could have sold three cakes, or three hundred, but even so, there’s a big difference between, for example, “Eunice sold several cakes” and “Eunice used a certain amount of flour to make her cake.” She can only sell her cakes in whole numbers.
So there’s an infinite set of numbers that can’t be the answer to this question: 3.5, Ï€, âˆš7. (Of course, there’s still pretty much an infinite set that could; even if it’s unlikely that Eunice sold, say, a million cakes all by herself, we can’t rule the possibility out.) Still, you’ve eliminated a whole lot of answers before you even get to the statements. As you’ll see, this thought process is key to arriving at the correct answer.
Now, let’s evaluate the statements.
Statement 1 says that Eunice sold a total of 8 cakes. Â But this tells you nothing about what each cake cost. Half of them could have cost 17 dollars, or maybe all but one. So statement 1 is definitely insufficient, even though there are only a few possible answers (1, 2, 3, 4, up to 8).
Statement 2 says that Eunice made 140 dollars total. If this were a regular algebra problem, and not a word problem, you might write something like this: 17x + 19y = 140, where x and y were the numbers of 17- and 19-dollar cakes. And in that case, you would be able to graph all the solutions to this equation on a line, which means that there would be infinite solutions.
But you know that x and y are integers, and that they’re positive (otherwise Eunice would be buying cakes from her own customers, which isn’t a great business model!). So instead of using straight algebra, use this information to figure out how many cakes Eunice sold.
First, 17 goes into 140 about 8 times. If Eunice sold 8 cakes for 17 dollars each, she’d make 8 * 17 = 136. But if she sells any more than that, say 9, her least possible revenue is 17 * 9 = 153. So she couldn’t have sold more than 8 cakes, because she would have made at least 153 dollars. Meanwhile, 19 goes into 140 about 7 times. If Eunice sells only 7 cakes, she makes, at most, 7 * 19 = 133. So she couldn’t have sold 7 or fewer cakes.
You’ve narrowed it down: Eunice must have sold 8 cakes. Now you can figure out how many of each type she sold using a simple system of equations: x + y = 8 and 17x + 19y = 140.
In fact, you could stop here. 2 equations with 2 variables is clearly sufficient information, so answer choice B is correct. But here’s a quick explanation of how to solve the system (just remember not to worry about solving on test day — instead, move on as soon as possible after establishing sufficiency!).
x + y = 8 becomes x = 8 — y (remember you’re looking for y, so substitute x). Next, plug that into the other equation: 17(8 – y) + 19y = 140, or 136 + 2y = 140. Then you have y = 2, so Eunice sold 2 19-dollar cakes.
“Wait,” you might be thinking. “What if I happened to find this same combination intuitively? I wouldn’t have had to go through all these steps.” Â That’s the difference between problem solving, where guess-and-check often works fine, and Data Sufficiency, where you have to know for sure that there’s only one possible answer. So you would still have to make sure that it was impossible for any integer number of cakes to have been sold other than 8.
The takeaway? Counting words, like “several,” and any discrete objects that appear in a word problem are your tip-off that you’re dealing with integers. It may seem strange since a majority of GMAT problems have integer solutions regardless, but knowing that the solution is an integer is a key piece of information to solve many problems!