A 4th Grade standardized test practice question from my 4th grade daughter

[hughee99]

At the Village Gift Shop, balloons are sold in two different sizes. The picture below shows the cost of each size of balloon.

<There's a picture of a small balloon for $1.25 and a large balloon for $2.99>

Then there's several easy questions before the final question, which is this...

Sam has $10 to buy balloons at the Village Gift Shop. He will follow all of the rules listed below to choose his balloons.

He will buy at least one of each size of balloon
He will buy as many balloons as he can
He will spend as close to $10 as he can without going over.

Using his rules, how many small balloons and large balloons can Sam buy for $10? Show your work or explain your answer.

My daughter came up with two different answers following the rules above, and found, what I think, a flaw in the question.

She asked me. "I'm in 4th grade and I found the problem with the question in a few minutes. Some adult has a full time job coming up with these questions. How could they not see this?"

For those of you more familiar with these sort of tests (my oldest just started taking them), are poorly worded or ambiguous questions unusual?

[jeff47]

1. It's not unusual, or new.

I took the SAT twice a long time ago. Both time there were multiple questions where more than one answer could be correct. You had to try and figure out which correct answer the test author meant to really be correct.

[Warren Stupidity]

2. I'll bite. I think there is only one correct answer.

Please provide the two answers.

[hughee99]

4. The issue is that there are 3 rules, so it depends on which one you give precidence to.

<There's a picture of a small balloon for $1.25 and a large balloon for $2.99>

Sam has $10 to buy balloons at the Village Gift Shop. He will follow all of the rules listed below to choose his balloons.

He will buy at least one of each size of balloon
He will buy as many balloons as he can
He will spend as close to $10 as he can without going over.

The first rule is simple, you spend $4.24 on 1 large balloon and 1 small balloon. Now you have 5.76 cents left.

You could buy 4 more small ballons for $5.00, and you'd now have 6 total balloons and will have spent $9.24.
You could buy 1 large balloon for $2.99 and 2 small balloons for $2.50 and will have spent $9.73

If the primary goal was to buy the most balloons, answer 1 gives you 6 balloons (compared to 5).
If the primary goal was to come as close to $10 without going over, answer 2 gives you $9.73 (compared to $9.24).

The question says follow ALL of the rules listed below, but two of the rules give you different answers and there is no single answer that follows ALL the rules.

[Lithos]

3. Just a stupid test

The problem is how to maximize the number of balloons given $5.76 cents

Or rather, $10 minus $1.25 and $2.99 (= $5.76) While getting close to $5.76

Assume that the number of balloons is more important than getting closer to $5.76

The closest would be 4 balloons ($5), but the permutations are:

4 small balloons - cost $5.00 w/ remainder of $0.76
3 small balloons + 1 large balloon - cost = 3.75 + 2.99 = 6.74 (or over - not viable)
2 small balloons + 1 large balloon - cost = 2.50 + 2.99 = 4.99 < 4 small balloons @ $5.00

[hughee99]

5. That was my initial reaction as well, but when my daughter explained her two answers,

I didn't see the flaw in her logic. It only makes sense if we assume things that aren't actually in the question.

[CBGLuthier]

6. Rule 3 disallows two different answers.

As displayed in another post there is only one answer that satisfies all of the well written rules.

[hughee99]

7. Only if you assume the rule of getting the most balloons is more important than

the rule of coming as close as you can to $10 without going over.

[CBGLuthier]

8. I assume following all three rules is what is required.

It would have been better worded as Buy as many balloons as you can while coming as close to spending all ten dollars. I would have to see the original wording to know just how badly it was phrased to properly evaluate the quality of the question. Were there three distinct clause in three separate sentences?

[hughee99]

9. It was written on the question exactly as I wrote it in the post.

There were 3 rules listed on three separate lines, they weren't numbered, and you were told to follow all 3 rules.

I'm sure they wanted you to buy as many balloons as you can, but it was poorly worded enough that I think there can be more than one interpretation. If the last rule was simply, "Don't spend more than $10", there would have been only one answer.

[Perogie]

10. The rules are simple

There is only one correct answer. One Large and five small for $9.24. It's the only answer that follows all the rules.

Your other answer of 1 large balloon for $2.99 and 2 small balloons for $2.50 and will have spent $9.73 does not follow all the rules. It only follows rule 1 and 3.


He will buy at least one of each size of balloon
He will buy as many balloons as he can
He will spend as close to $10 as he can without going over.

[hughee99]

11. It depends on how you read it.

There are TWO objectives...
1. buy as many balloons as you can
2. spend as close to $10 as you can without going over.

You are reading it, as it was probably intended, that the MOST important objective was to buy as many balloons as possible, and the secondary objective was to spend as much of the $10 as possible, but that's doesn't necessarily follow from the wording. I listed them as the first and second objective, but the wording simply says follow all three rules and doesn't specify an order of precedence. That is inferred by the reader, but is not explicitly stated. If the last rule was "Don't spend more than $10" then it's simply a constraint, not an objective, and there's only one answer.

The $9.24 answer doesn't follow rule 3. He could have spent closer to $10 on balloons without going over.
The $9.73 answer doesn't follow rule 2. He could have bought more balloons.

[Warren Stupidity]

12. What part of you have to meet all three rules is unclear?

1. He will buy at least one of each size of balloon
2. He will buy as many balloons as he can
3. He will spend as close to $10 as he can without going over.

All three. Not pick two or one. There is one answer that meets all three rules. The fact that there are other answers that don't meet all three rules is relevant only in that this is part of the problem - you have to choose the answer that meets all three.

By your logic there is a third answer: buy one large balloon and one small balloon. That meets rule 1. Another answer, according your interpretation is to buy infinity balloons of both sizes, meeting rules one and two.

The real world is full of conflicting requirements.

[hughee99]

15. Yes, the real world is full of conflicting requirements.

Should a child's standardized test math question have them though? Is the goal to determine the child's proficiency in math, or their ability to infer the "right" things into a question?

NEITHER answer meets all three rules unless you decide that one rule is more important than the other.

You have 2 constraints, both of which must be followed:
buy at least one of each balloon type
Don't spend more than $10

You have 2 goals:
Buy as many balloons as you can.
Come as close to $10 as you can, without going over.

Any answer that doesn't fall within the constraints is clearly wrong. These two answers meet all the constraints, but each sets a precedence on which of the two goals is more important.

If the primary goal was to buy the most balloons, answer 1 gives you 6 balloons (compared to 5).
If the primary goal was to come as close to $10 without going over, answer 2 gives you $9.73 (compared to $9.24).


As I said below, without changing the 3 rules at all, let me impose an order on them

The most important goal is that he will spend as close to $10 as he can without going over.
He will buy at least one of each size of balloon
He will buy as many balloons as he can

I've changed none of the rules, but I have specified what the most important goal is. Does this change your answer?

[Perogie]

13. I got what you meant the first time you posted it

You don't need to talk down to me and show me your work.

You don't seem to understand you have to meet all three rules and no your $9.73 option doesn't meet rule 2.

$9.24 option meets all three. NO, he couldn't spend more because he would have to break rule two do spend more.

Again it's a simple problem with the correct instructions, you're just reading more into it than what is there.

[hughee99]

14. Didn't mean to offend you, but I believe you're mistaken when you say

I'm reading MORE into than what is there. I didn't read ANYTHING into it, which is why I think it could be confusing. You are reading something into it to impose an order that's not specified into the question.

Without changing the 3 rules at all, let me impose an order on them

The most important goal is that he will spend as close to $10 as he can without going over.
He will buy at least one of each size of balloon
He will buy as many balloons as he can

I've changed none of the rules, but I have specified what the most important goal is. Does this change your answer?

[Mass]

16. It is well worded. But math is not only number facts. This question tests number sense

and priorities, two notions she should have learned to be able to solve this question.

Priority tells us that rule 1 is more important than rule 2 and rule 2 more important than rule 3.

Number sense tells us that to get the highest number of balloons, you only buy the smaller ones after having bought one of each.

And the third rule fixes the upper limit and the operation (divide what remains after having bought one balloon of each type by the price of the smallest balloon.

This said, what is clear is that either those notions have not been taught to your daughter OR she did not master them. It is too bad because those are important notions for everyday maths.

[hughee99]

17. So in other words, she should have made assumptions not explicitly listed in the question,

instead of reading the question carefully and following the explicit instructions, which is something that has been stressed to the class in the past.

My daughter recognized both possibilities, provided both answers, and then explained why she felt the question was ambiguous. In my estimation, and based on the fact that following instructions has been stressed, I think my daughter's math and reading comprehension skills are just fine. It does get confusing for a child when you tell them to follow exactly the instructions that were given, and then complain that they didn't make the right assumptions to get the answer you really wanted.

And the third rule does more than just fix the upper limit. "He will not spend more than $10" fixes an upper limit, "He will spend as close to $10 as he can without going over" both sets an upper limit and provides another objective.

[Dr. Strange]

18. This is an example of an "open response" problem.

In fact, here's a slide show with that problem and some others:
https://docs.google.com/presentation/d/1GVVgk-kMxTJvjT-ZbqqwvK-6MWW2aJ9p7hE5dC4ONrE/embed?hl=en&size=m#slide=id.i0

The idea here (I think) is not having the students get "the" answer, but have them justify what they're doing. (See also "common core", "critical thinking", and other similar buzz words.) A good teacher isn't going to look at the final answer and say right or wrong: they should be looking at the process and seeing if the student recognizes that two different solutions exist. If I were the teacher, I wouldn't even care how they picked their final answer--just that they recognized these two solutions.