Was bored today and searched for a puzzle. Came across this one....

[haynewp]

So I am missing something in this one. The answr is 8. Can someone explain?
In a department store, with only one girl working on the gift-wrapping service, four gifts per hour were wrapped in the first 15 hours. With help from a colleague for the next 45 hours, the amount of gifts wrapped rose to seven per hour. How many gifts were wrapped each hour in the last 45 hours only?

 

[phinds]

what calculations have you done? It's trivial so at the very least you should show some effort

 

[hutchphd]

I think we know the error and a nudge is required.

With help from a colleague for the next 45 hours, the amount of gifts wrapped rose to seven per hour.

This is the new aggregate rate for the full 60 hrs. Got it now?

 

[haynewp]

The answer is 8 though.

 

[DaveC426913]

:ahem:

what calculations have you done?

The answer is 8 though.

Yup. I got 8.

Here's a couple of hints to get you started:
1] How many total hours were presents being wrapped for? (Show your work.)
2] How many total presents were wrapped? (Show your work.)

 

[hutchphd]

Yes what answer do you get?

 

[phinds]

Yes what answer do you get?

and more to the point HOW did you get whatever answer you got? SHOW YOUR WORK !

 

[symbolipoint]

with only one girl working on the gift-wrapping service, four gifts per hour were wrapped in the first 15 hours. With help from a colleague for the next 45 hours, the amount of gifts wrapped rose to seven per hour. How many gifts were wrapped each hour in the last 45 hours only?

This feels like a Constant Rates problem. The effective rate of the two people working together is the sum of the two separate rates. You have something like (rate)*(time)=countofgiftswrapped.

The one-girl-alone wraps at rate of 4/15 gifts per hour.
The combined girl&coleague rate became 7 gifts per hour.
What is really unknown is HOW MANY gifts per hour for the colleague alone?
And then you could answer the question asked in your exercise.

Can you determine what to do with this summary of information?

 

[DaveC426913]

What is really unknown is HOW MANY gifts per hour for the colleague alone?

And it can remain unknown. The colleague is a red herring. It is nothing more than a story-telling explanation to rationalize why the rate might have gone from 4/h to 7/h - but it does not factor into the math. (The narrative could just as easily have had the first girl taking a double shot of Redbull in her coffee for the next 45 hours.)

I arrived at the answer of 8 without needing to consider the present-wrapping rate of 'colleague' - only the aggregate wrapping rate.

 

[symbolipoint]

And it can remain unknown. The colleague is a red herring. It is nothing more than a story-telling explanation to rationalize why the rate might have gone from 4/h to 7/h - but it does not factor into the math. (The narrative could just as easily have had the first girl taking a double shot of Redbull in her coffee for the next 45 hours.)

I arrived at the answer of 8 without needing to consider the present-wrapping rate of 'colleague' - only the aggregate wrapping rate.

Now this makes me wonder if I am stuck in a certain way of thinking. I sometimes spot this when I actually try to analyze and then solve-through an exercise.

 

[symbolipoint]

In a department store, with only one girl working on the gift-wrapping service, four gifts per hour were wrapped in the first 15 hours. With help from a colleague for the next 45 hours, the amount of gifts wrapped rose to seven per hour. How many gifts were wrapped each hour in the last 45 hours only?

NOW I SEE. The first part, about the one girl, is not needed in order to answer the question. The question is ONLY about the last 45 hours while both personnel worked together.

 

[symbolipoint]

And it can remain unknown. The colleague is a red herring. It is nothing more than a story-telling explanation to rationalize why the rate might have gone from 4/h to 7/h - but it does not factor into the math. (The narrative could just as easily have had the first girl taking a double shot of Redbull in her coffee for the next 45 hours.)

I arrived at the answer of 8 without needing to consider the present-wrapping rate of 'colleague' - only the aggregate wrapping rate.

Yes. My thinking was stuck. This is an exercise with unnecessary information so the student must decide which information is needed and which is not needed.

 

[phinds]

Interesting. It never occurred to me to worry about how many people were doing what. If you just focus on the math, the problem is trivial, as I said in post #2, and as has now become clear to everyone, except perhaps the OP.

 

[DaveC426913]

Yes. My thinking was stuck. This is an exercise with unnecessary information so the student must decide which information is needed and which is not needed.

?? There is no unnecessary information - at least no unnecessary numbers.

Sure, there's some storytelling, but every problem has to have some of that to provide context for the math.

NOW I SEE. The first part, about the one girl, is not needed in order to answer the question. The question is ONLY about the last 45 hours while both personnel worked together.

I disagree. This can't be solved without including the events of the first 15 hours.

Though the answer is only about the last 45 hours.

hutchphd's post #3 holds the key.

 

[phinds]

?? There is no unnecessary information - at least no unnecessary numbers - sure, there's some storytelling, but every problem has to have some of that (otherwise you'd never know that train 1 left Chicago at 3:15 and train 2 left New York at 2:30).I disagree. This can't be solved without including the events of the first 15 hours.

Though the answer is only about the last 45 hours.

hutchphd's post #3 holds the key.

I think we're all actually saying the same thing. The storytelling is in some ways the glue that holds the math together but in a very practical way it's irrelevant.

 

[symbolipoint]

After #14 & #15
Do I really need to try to solve this all the way through myself? It looks like a very simple constant rates exercise with distracting information included.

 

[symbolipoint]

With help from a colleague for the next 45 hours, the amount of gifts wrapped rose to seven per hour. How many gifts were wrapped each hour in the last 45 hours only?

In bold, that is a rate.
(other emphasis using italicization )

Those are TWO periods of 45 hours each.
The two-partner work rate WAS "seven per hour".
The question "how many gifts wrapped each hour in the last 45 hours only"? Already answered in the description! 7; not 8.

 

[phinds]

In bold, that is a rate.
(other emphasis using italicization )

Those are TWO periods of 45 hours each.
The two-partner work rate WAS "seven per hour".
The question "how many gifts wrapped each hour in the last 45 hours only"? Already answered in the description! 7; not 8.

I think you are misreading the problem. It doesn't say that the 7 is only in the last 45 hrs, it says that the 7 is OVERALL for the whole time. See post #3.

Also, there are not two 45 hr periods. There's a 15 hr followed by a 45 hr

 

[symbolipoint]

With help from a colleague for the next 45 hours, the amount of gifts wrapped rose to seven per hour. How many gifts were wrapped each hour in the last 45 hours only?

This is ambiguous. Most clear is the rate became 7 gifts per hour. This new rate is plainly enough stated . Unclear is if this is only one period of 45 hours or did another period of 45 hours come next. No more analysis to do. Either 45 hour period with both personnel together, number of gifts wrapped already given, as a rate - not as a count in fortyfive hours, but as 7 PER HOUR.

 

[hutchphd]

This is ambiguous. Most clear is the rate became 7 gifts per hour.

I agree the restatement by the OP is ambiguous (who knows what the verbatim original statement was). But from the answer they supplied the authors surely meant the aggregate rate for all 60 hours became 7 per hour. The alternative may be clear to you but apparently not the creators of the puzzle!
Why are we (you) still discussing this?

 

[DaveC426913]

Note that the OP provided us with the expected answer of 8. We can surmise then that the interpretation (see hutchphs's post #3) that results in 8 is surely the intended one.

I did the calculation based on that interpretation and got 8 (exactly). I'd say that's either confirmation or an incredible coincidence.

 

[phinds]

Note that the OP provided us with the expected answer of 8. We can surmise then that the interpretation (see hutchphs's post #3) that results in 8 is surely the intended one.

I did the calculation based on that interpretation and got 8 (exactly). I'd say that's either confirmation or an incredible coincidence.

PLUS the fact that the question writer would have to be either a moron or just stupidly devious to INTEND the 7/hr to apply only to the last 45 hrs and then ask for the rate for the last 45 hours.

The problem is an extremely common type and very straightforward. All this second guessing the question writer (not the OP) to make him look like an idiot is just a waste of time.

 

[Borg]

This is ambiguous. Most clear is the rate became 7 gifts per hour. This new rate is plainly enough stated . Unclear is if this is only one period of 45 hours or did another period of 45 hours come next. No more analysis to do. Either 45 hour period with both personnel together, number of gifts wrapped already given, as a rate - not as a count in fortyfive hours, but as 7 PER HOUR.

Agree 100%. The statement "for the next 45 hours, the amount of gifts wrapped rose to seven per hour" does not state that seven per hour refers to an average for the entire time that gifts were being wrapped. Reverse the sentence and tell me what you think - "the amount of gifts wrapped rose to seven per hour for the next 45 hours". The problem is at best, very poorly worded and isn't clear in its references.

 

[Jarvis323]

These types of ambiguous problems
are common. There are at least three cases why the problem could be like this:

1) Author of question just did a poor job.

2) Author intended it to be ambiguous and the real problem is meant to be about understanding/guessing intended meaning from inaccurate and imprecise language based on context.

3) The author intended to make it ambiguous so that people will disagree and it will generate lots of comments and views and ad money.

Either 1 or 2 are very common in word problems and I've always wondered which one.

Case 3 is extremely popular now, as it has become an established formula for profit. Any content that can get a lot of people arguing with each other is worth money.

 

[symbolipoint]

Why are we (you) still discussing this?

Because some people will not make that "aggregate" interpretation. For example, I cannot find it literally in the problem description; and neither do I see it by inference; and have never found any such-like exercise in the algebra textbooks from which I studied and reviewed from.

 

[symbolipoint]

The problem is an extremely common type and very straightforward. All this second guessing the question writer (not the OP) to make him look like an idiot is just a waste of time.

No. I have studied from a few elementary, intermediate, and college algebra textooks and for review purpose too. I never found such a exercise requiring that way of "aggregate" interpretation. NEVER.

 

[hutchphd]

NEVER.

Until now.

 

[DaveC426913]

1.The OP did not ask for an answer; The OP stated the answer and asked for an explanation. That is how we know the best interpretation.

2.This aggregate thing is a red herring - brought up in this thread. The 'best' explanation requires no reference to it. To take a page from Occam: one should not multiply entities needlessly.

 

[DaveC426913]

Here is my solution.

Note that it results in the given answer exactly - requiring neither rounding of decimals nor fractions - integers only (which, of it weren't contrived to be so, would be astronomically unlikely).

Spoiler: No peeking!

• In session 1, (4x15=) 60 gifts were wrapped in 15 hours.
• In the final tally, a total rate of 7 gifts per hour was achieved over a total of 60 hours.
• That means a total of (7x60=) 420 gifts were wrapped.
• But 60 of those occurred in session 1 (15h).
• Which means the all the rest (420-60=) 360 were wrapped in session 2 (45h).
• Which makes for a session 2 rate of (360/45=) 8.

 

[phinds]

Exactly.

 

[symbolipoint]

Here is my solution.

Note that it results in the given answer exactly - requiring neither rounding of decimals nor fractions - integers only (which, of it weren't contrived to be so, would be astronomically unlikely).

Spoiler: No peeking!

• In session 1, (4x15=) 60 gifts were wrapped in 15 hours.
• In the final tally, a total rate of 7 gifts per hour was achieved over a total of 60 hours.
• That means a total of (7x60=) 420 gifts were wrapped.
• But 60 of those occurred in session 1 (15h).
• Which means the all the rest (420-60=) 360 were wrapped in session 2 (45h).
• Which makes for a session 2 rate of (360/45=) 8.

Only very, very, very, very slowly, I am beginning to understand. The original problem is badly composed. It may even be said to be ambiguous.

 

[mfb]

With help from a colleague for the next 45 hours, the amount of gifts wrapped rose to seven per hour. How many gifts were wrapped each hour in the last 45 hours only?

Asking for the last 45 hours "only" makes no sense if the 7 per hour are referring to the same time frame. And who would ask a homework problem that answers itself? Have you ever seen this in homework?

Agree 100%. The statement "for the next 45 hours, the amount of gifts wrapped rose to seven per hour" does not state that seven per hour refers to an average for the entire time that gifts were being wrapped. Reverse the sentence and tell me what you think - "the amount of gifts wrapped rose to seven per hour for the next 45 hours". The problem is at best, very poorly worded and isn't clear in its references.

You changed the sentence. The correct inversion would be "the amount of gifts wrapped rose to seven per hour because a colleague helped for the next 45 hours".

 

[Jarvis323]

Asking for the last 45 hours "only" makes no sense if the 7 per hour are referring to the same time frame. And who would ask a homework problem that answers itself? Have you ever seen this in homework?

It's not a homework question though. If you google it, you'll find it was posted by someone on a puzzle site, with the tag "brain teaser". And the poster posts other questions with tags, "tricky" and "riddle", etc. The answer, 8, is one user's answer, and isn't necessarily correct. It is the top ranked answer out of 1. So who is to say really if the answer is meant to be 7 or 8, or whether it has a correct answer at all? It might just have a subjectively best answer, and it is left to the audience to interpret and vote/debate what the correct answer should be?

 

[Borg]

You changed the sentence. The correct inversion would be "the amount of gifts wrapped rose to seven per hour because a colleague helped for the next 45 hours".

The meaning of a sentence like that one shouldn't change if you reverse it. If you have to modify the reversal in order for it to make sense, then the original sentence wasn't clear. When your modified phrase is reversed, it makes more sense as to its intent.

because a colleague helped for the next 45 hours, the amount of gifts wrapped rose to seven per hour

 

[DaveC426913]

The answer, 8, is one user's answer, and isn't necessarily correct. It is the top ranked answer out of 1. So who is to say really if the answer is meant to be 7 or 8, or whether it has a correct answer at all?

Because, again, the answer works out to 8 exactly, no rounding, no fractions. It would be a fabulous coincidence if that happened by accident due to a misinterpretation. The riddle author has chosen the numbers carefully to arrive at the answer without any math more complicated than 3-digit arithmetic.

 

[phinds]

Because, again, the answer works out to 8 exactly, no rounding, no fractions. It would be a fabulous coincidence if that happened by accident due to a misinterpretation. The riddle author has chosen the numbers carefully to arrive at the answer without any math more complicated than 3-digit arithmetic.

It appears that you are stating this, correctly I believe, to deaf ears. For reasons I don't understand, some people just don't seem to want to believe it.

 

[DaveC426913]

It appears that you are stating this, correctly I believe, to deaf ears. For reasons I don't understand, some people just don't seem to want to believe it.

In their defense, it's not a black and white issue. In our opinions "it's dark grey enough that I think it was meant to be black".

And others are of the opinion that "Dark grey is just a mix of black and white, and opinions are cheap." which is not, technically, wrong.

 

[phinds]

True. I DO understand the logic behind their point of view but I firmly believe that you have it right. As I posted in post #2 I consider the problem to be trivial. I immediately rejected the, in my view, incorrect interpretation and solved the problem in my head almost immediately, so it did not occur to me to take the alternate interpretation seriously and I still can't take it seriously.

Part of the reason for all that is that, as you have pointed out, the math is trivial using the, to me, obvious interpretation.

 

[mfb]

The meaning of a sentence like that one shouldn't change if you reverse it. If you have to modify the reversal in order for it to make sense, then the original sentence wasn't clear. When your modified phrase is reversed, it makes more sense as to its intent.

because a colleague helped for the next 45 hours, the amount of gifts wrapped rose to seven per hour

My reversal was keeping the sentence the same, yours wasn't. The ability to change a sentence by changing it isn't telling us anything.

 

[Borg]

My reversal was keeping the sentence the same, yours wasn't. The ability to change a sentence by changing it isn't telling us anything.

OK, let's try it another way.

In a department store, with only one girl working on the gift-wrapping service, four gifts per hour were wrapped in the first 15 hours. With help from a colleague for the next 45 hours, the amount of gifts wrapped rose to seven per hour. How many gifts were wrapped each hour in the last 45 hours only?

I still find the problem to be highly misleading given the stated answer. As I read through the paragraph, the first sentence states a gift wrapping rate that is an hourly rate and also happens to be the total average hourly rate. The second sentence refers to a time period of 45 hours and a rate of seven per hour. However, it does not specify whether that rate was an hourly rate for the 45 hour period or was a total average hourly rate for the entire (or per @phinds an OVERALL) 60 hour period. With the way that it's written, "the next 45 hours", can easily be construed as being the object of the stated seven per hour rate and would therefore be an average for that time period.

To me, this looks like a trick question that's asking for an answer that they give in the second sentence. If I change the question being asked in the third sentence, the meaning of the information given in the first two sentences should not change (I hope everyone agrees on that at least). How would everyone solve this question given no change in the first two sentences?

What was the average number of gifts wrapped per hour for the entire 60 hours?

Would people assume that this one was the trick question because the 7/hour rate given in the second sentence was an overall rate as many are stating about the original question? Or would the assumption now be that the reader should perform a different calculation because the question is different?

 

[collinsmark]

I totally did not expect to contribute to this thread, but here we go.

Long story short, the ambiguity is whether we interpret the given rates ("four gifts per hour were wrapped in the first 15 hours," and "the amount of gifts wrapped rose to seven per hour") as instantaneous rates, or average rates. Do we have enough information to tell? I think we do. Let's break it down.

Assumptions:
We should assume that the actual instantaneous rates (given or not) are uniform. Sure it could happen that the girl or girls slowly and gradually improved their gift wrapping skills over the course of the 60 hours. But let's rule that out; there's not enough information in the puzzle statement to justify continuously variable, instantaneous rates. Instead, let's assume that for the first 15 hours, there is a constant, uniform instantaneous rate, and for the next 45 hours there is a separate, but constant, uniform, instantaneous rate.

So the real question is, does the "seven per hour" figure in the "for the next 45 hours, the amount of gifts wrapped rose to seven per hour" statement refer to the instantaneous rate or average rate?

Instantaneous hypothesis:
If we assume that the "seven per hour" figure refers to the instantaneous rate, then the puzzle makes no sense as a puzzle at all. The crux of the riddle -- its final question -- is clearly asking for the instantaneous rate during the last 45 hours only. So if the "seven per hour" refers to that, it's not much of a riddle at all. It just gave you the answer in the preceding sentence.

Average hypothesis:
The key wording leading to this hypothesis is "rose to." The verb is ambiguous in that the change can happen instantaneously, but there is a connotation in "rose to" that the change happens gradually; that's the connotation. Accordingly, this leads to reader to surmise that the "seven per hour" figure is the average rate, and true (i.e., true to be "seven per hour") only after considering the entire 60 hour timeframe.

Conclusion:
So (in my opinion) there is enough information in the riddle to conclude that @DaveC426913's analysis in post #29 is correct. Although there is some ambiguity in the riddle, I believe the puzzle's wording sufficiently addresses the ambiguity such that there is only one, clear interpretation. Thus the clear answer is 8 gifts per hour.

 

[Borg]

Since this is boiling down to an English grammer question, I decided to post this question on one of those forums. The question hasn't been answered yet but they are looking at this thread.

https://www.usingenglish.com/forum/threads/poorly-worded-math-story-problem.293499/

I will abide by their decision which I hope is more clear cut than this thread.

 

[Borg]

It was an informative discussion and they all clearly agreed that the wording was a mess. However, I am now changing my opinion of what the wording meant.

One post pointed out this pair of options for the placement of the comma in the second sentence:

With help from a colleague for the next 45 hours, the amount of gifts wrapped rose to seven per hour.
With help from a colleague, for the next 45 hours the amount of gifts wrapped rose to seven per hour.

The original question is written like the first sentence which places the emphasis for the rate away from the colleague's help. It's still a linguistic mess but if I had to pick two meanings for the rates in those two sentences, it's pretty clear which one goes with each sentence.

 

[DaveC426913]

Dunno, I'm kinda partial to this:

2) Author intended it to be ambiguous and the real problem is meant to be about understanding/guessing intended meaning from inaccurate and imprecise language based on context.

Remember, this was an online riddle - not necessarily strictly a math problem.

 

[gmax137]

I mentally translated it to this:

"I stopped to fill up with gas, and reset my trip odometer and mileage. At 15 miles the average mileage showed 4 miles per gallon. At 60 miles, the average read 7 miles per gallon. What was my average mileage for the last 45 miles?"

But I have recently been on a long driving trip and have been thinking of such calculations. Plus (confirmation bias?) this works out to the given answer, 8 (as mentioned several times, above).

 

[symbolipoint]

I mentally translated it to this:

"I stopped to fill up with gas, and reset my trip odometer and mileage. At 15 miles the average mileage showed 4 miles per gallon. At 60 miles, the average read 7 miles per gallon. What was my average mileage for the last 45 miles?"

But I have recently been on a long driving trip and have been thinking of such calculations. Plus (confirmation bias?) this works out to the given answer, 8 (as mentioned several times, above).

This is further less clear than the originally given giftwrappers girl & colleague example.

 

[DaveC426913]

This is further less clear than the originally given giftwrappers girl & colleague example.

Maybe because I do a lot of driving, and I am used to grappling with the concept of a running average of gas mileage.

But the way gmax phased it removes the ambiguity in the original problem:

- reset my trip odometer
- At 15 miles the average mileage showed 4 miles per gallon
- At 60 miles, the average read 7 miles per gallon.

Since the reset was explicitly mentioned at the start, we can logically conclude that it was not reset anywhere else - (such as: at the 15 mile mark).

Which means the average of 7mpg must needs apply to the entire 60 miles.

 

[haynewp]

Hmmm...appears to have turned out that the question actually wasn't trivial at over 40 replies later. This is because of the question's poor wording. It is a trivial question if it is worded correctly. Thanks for the solution that was posted which cleared up the confusion for me. And no it wasn't a homework problem.

 

[symbolipoint]

@haynewp
Where did the original question come from or where did you find it?

 

[haynewp]

https://www.mensa.org.uk/puzzles/brainteasers?page=470