The following question (number 15 of this test) has become infamous as a poor "Common Core" question. What is the correct answer?
Juanita wants to give bags of stickers to her friends. She wants to give the same number of stickers to each friend. She's not sure if she needs 4 bags or 6 bags of stickers. How many stickers could she buy so there are no stickers left over?
On
This "infamous" question is so poorly worded, but any mathematical answer that can be given would reinforce the perception that "the question is OK, see, it has a valid answer!"
The question is not OK.
Word problems should not be about mulling over "what did the author want to say?". Even the "trick" questions are normally about spotting some well-defined linguistic (or mathematical) trick, rather than based on complete lack of clarity. Plus, note it was not meant to be a trick question, but some standard question meant to assess how well students of certain age comprehend divisibility and common multipliers.
I can personally think about a dozen different ways to write a word problem which would boil down to the same mathematical problem and will be better worded.
As for this question, I hope if it was on some exam, that it did not affect anyone's passing or failure. I can relate to a poor methodical soul who got stuck for two hours on that question, trying to get their head around it and failing to comprehend it (and also wasting precious time for the following questions) not realising that it is not their fault.
On
Short Version
The correct answer is "any multiple of 12", but they really probably just want "12".
Long Version
I agree that the question is poorly-worded, but after some reflection upon existing commentary and answers, I also agree it is possible to work out the proper mathematical problem and its possible solutions without requiring many leaps of logic or unfounded assumptions. However, I also believe that the actual answer to the question is not the answer that test reviewers would consider correct.
First, about how to solve the problem:
The first thing a student needs to do is eliminate what they don't need from the problem.
This is a common assessment objective of word problems - finding whether the student can determine what is or is not relevant to the question.
The scenario talks a lot about bags, stickers, and friends. We know that we need to fill bags with stickers, and the bags are being given to friends. We don't know how many friends Juanita has and, at first glance, this would seem to be part of the problem. However, the number of friends is irrelevant because Juanita has already narrowed the quantity of bags to two possibilities - 4 or 6.
Now the student can determine the mathematical problem, and its possible solutions.
We have either 4 or 6 bags to fill, and we want to buy an amount of stickers that divides evenly among the bags no matter which number is actually true. The easiest way to do this is to multiply the two numbers, for which you get 24.
It is important to note here that 24 is actually a valid answer to the question that has been presented to the student.
Now the student could conceivably extrapolate the ideal result, and determine that solution.
Considering that Juanita is buying these stickers, she probably doesn't want to pay more than necessary. And, if they understand the concept of the Least Common Multiple, the student could realize that there might be lower quantities of stickers that could satisfy Juanita's needs.
At this point, they would then do the appropriate work and arrive at the answer of 12 - which is probably the answer the instructors want to see on the standardized test.
The question as written leaves open the possibility of an infinite number of correct answers being considered "wrong".
That last part is where this question, in my opinion, fails to serve its purpose horribly. There's an important difference between this (from the original question):
How many stickers could she buy so there are no stickers left over?
And this (change in italics):
What's the fewest number of stickers she could buy so there are no stickers left over?
The former leaves all multiples of 12 as possible valid answers. In fact, the most correct answer would be for the student to write (as it's a free-form answer field anyway) "Any multiple of 12".
However, the latter form narrows the field of possibilities down to only the answer that costs the least amount of money. For that question, the only correct answer (assuming the stickers are individually priced, and there's no buy-one-get-one-free sales on) is 12.
A Better Question
Personally, I'd probably suggest a rewrite similar to this:
Juanita is going to the store to buy some stickers for her friends. To distribute them equally, she's going to put the stickers into bags. However, she left her bags at home and can't remember whether she has 4 or 6 bags to fill. She wants to be able to give the same number of stickers to each friend. How many stickers should she buy, to avoid spending any more money than she needs to while still equally dividing the stickers?
That puts the question in simpler terms, and links it to a practical real-life need (saving money), while keeping all of the elements from the original question.
On
Juanita buys zero bags. Then she gives zero to each friend. After that, there are zero stickers left over.
(Note that, consistent with other answers, zero is a multiple of twelve.)
Beyond that, I keep wondering how many bags of stickers Juanita intends to keep for herself after supplying her friends with some. It may well be that there are no circumstances under which there would be any "left over".
On
The question is terribly worded. Here is another solution which is consistent with the wording:
The only way Juanita can be unsure whether she needs 4 or 6 bags is if she has exactly 1 or 2 friends; any other number of friends does not divide evenly into both 4 and 6 bags and she would know in those cases that at least one choice was inappropriate. The question clearly states that Juanita has more than one friend, so she must have two. To ensure that no stickers are left over, she should buy the smaller of the two allowed choices, i.e. four bags.
The difficulty arises from the confusing wording and strange premise. Juanita is buying "stickers", not "bags of stickers". Each friend is to receive a single bag of stickers, with the number of stickers in each bag the same. For reasons unexplained, Juanita does not know exactly how many friends she is giving stickers to, but she does know that it is either $4$ or $6$. Thus, the question is how many stickers to buy so that she will be able to divide them evenly among the friends, whether there turn out to be $4$ or $6$ of them. The answer, then, is any multiple of both $4$ and $6$. Equivalently, the number of stickers can be any multiple of $12$, which is the least common multiple of $4$ and $6$.