Length of a wrapping paper and a box of chocolate

[Jignesh77]

Good evening,

I saw the following problem in the past kS3 ( year 7,8 and 9) maths paper while helping my son who is in KS3 ( year 7). I have attached the question and the correct answer which is 45. I have asked this question on another forum as well but still don't understand the logic behind the correct answer as mentioned in the attached marking scheme. I am asking for my own understanding as I love the subject. I work in healthcare but love maths and helping my son as and when required. Please note that it's not a homework question. it's a genuine doubt.

Suitable length of a wrapping paper

Morning everyone, My son is in year 7. Yesterday he solved questions relevant to his curriculum from past KS3 question paper. There was a question about wrapping paper aroun a box of chocolate. I have attached the question. He tried but neither he nor I understood the logic. The correct answer...

mathforums.com

I appreciate your help. Thank you in advance.

 

[The Highlander]

Good evening,

I saw the following problem in the past kS3 ( year 7,8 and 9) maths paper while helping my son who is in KS3 ( year 7). I have attached the question and the correct answer which is 45. I have asked this question on another forum as well but still don't understand the logic behind the correct answer as mentioned in the attached marking scheme. I am asking for my own understanding as I love the subject. I work in healthcare but love maths and helping my son as and when required. Please note that it's not a homework question. it's a genuine doubt.

Suitable length of a wrapping paper

Morning everyone, My son is in year 7. Yesterday he solved questions relevant to his curriculum from past KS3 question paper. There was a question about wrapping paper aroun a box of chocolate. I have attached the question. He tried but neither he nor I understood the logic. The correct answer...

mathforums.com

I appreciate your help. Thank you in advance.

Hi @Jignesh77

I am not entirely sure what your difficulty is with this problem; I feel I would have been better placed to help you if you had provided a little bit more information about exactly what it is you don't understand about the answer and how it is arrived at.

However, I will make my best guess as to what you need explained and offer this....


If you look at the net of the chocolate box, you (should) see that you need a length of at least 60 cm of paper to wrap around the box (in the '60 cm direction') but the roll is 70 cm wide so the width of the paper is more than long enough to achieve that. So you want to find the minimum length to cut that will cover the box (in the '50cm direction').

So I can only presume that you don't understand how cutting just 45 cm will cover that 50 cm length, yes?

You have to imagine the paper having been wrapped around the box with just 2.5 cm sticking out at each end. That adds up to 2.5 + 40 + 2.5 = 45 but when you fold down the paper on the top surface (2.5 cm) and fold up the 2.5 cm from the bottom surface they will meet in the meddle at the sides of the box, thus just covering it and no more (ie: that is the minimum you can get away with).

Of course, in real life you would want these edges to overlap (not just meet precisely) so it would be better to use a length that is slightly greater than 45 cm (but well under 60).

Hope that helps.

 

[Jignesh77]

Hi @Jignesh77

I am not entirely sure what your difficulty is with this problem; I feel I would have been better placed to help you if you had provided a little bit more information about exactly what it is you don't understand about the answer and how it is arrived at.

However, I will make my best guess as to what you need explained and offer this....

View attachment 39405
If you look at the net of the chocolate box, you (should) see that you need a length of at least 60 cm of paper to wrap around the box (in the '60 cm direction') but the roll is 70 cm wide so the width of the paper is more than long enough to achieve that. So you want to find the minimum length to cut that will cover the box (in the '50cm direction').

So I can only presume that you don't understand how cutting just 45 cm will cover that 50 cm length, yes?

You have to imagine the paper having been wrapped around the box with just 2.5 cm sticking out at each end. That adds up to 2.5 + 40 + 2.5 = 45 but when you fold down the paper on the top surface (2.5 cm) and fold up the 2.5 cm from the bottom surface they will meet in the meddle at the sides of the box, thus just covering it and no more (ie: that is the minimum you can get away with).

Of course, in real life you would want these edges to overlap (not just meet precisely) so it would be better to use a length that is slightly greater than 45 cm (but well under 60).

Hope that helps.

Thank you so much. I got confused by the phrase "length" which is 400 cm as per the diagram. I don't understand why the correct answer is not given with respect to length which is 400 cm in this case. I am so sorry for asking such a basic question. I appreciate your help. Your answer has certainly helped me so thank you for your help and time.

 

[Dr.Peterson]

Thank you so much. I got confused by the phrase "length" which is 400 cm as per the diagram. I don't understand why the correct answer is not given with respect to length which is 400 cm in this case. I am so sorry for asking such a basic question. I appreciate your help. Your answer has certainly helped me so thank you for your help and time.

"Length" in the question refers to how much of the 400 cm you need to take off the roll in order to wrap the box. 400 cm is the maximum possible length.

The main idea, I think, is that there are two directions in which you might wrap. If you wrapped in the following way, you couldn't use the 70 cm width of the roll to cover the long dimension of your sheet, so you'd have to take at least 90 cm from the roll, and waste a lot of the width (unless perhaps you used two pieces taped together).



The efficient way is like this:


Here you can cut off just 45 cm or so of the length of the roll, and waste very little.

This exercise is partly a matter of developing a practical sense of how things are wrapped, and visualizing the paper required. (The math involved is mostly about perimeter: in one direction you need the entire perimeter, and in the other, you need half of the perimeter.)

 

[Jignesh77]

View attachment 39403View attachment 39404

"Length" in the question refers to how much of the 400 cm you need to take off the roll in order to wrap the box. 400 cm is the maximum possible length.

The main idea, I think, is that there are two directions in which you might wrap. If you wrapped in the following way, you couldn't use the 70 cm width of the roll to cover the long dimension of your sheet, so you'd have to take at least 90 cm from the roll, and waste a lot of the width (unless perhaps you used two pieces taped together).

View attachment 39407

The efficient way is like this:
View attachment 39406

Here you can cut off just 45 cm or so of the length of the roll, and waste very little.

This exercise is partly a matter of developing a practical sense of how things are wrapped, and visualizing the paper required. (The math involved is mostly about perimeter: in one direction you need the entire perimeter, and in the other, you need half of the perimeter.)

Thank you so much for a very good answer. Your explanation has certainly helped me to clear my doubt. I am grateful for the help I get on this and other forums where Kind people share their knowledge and make learning interesting.

 

[The Highlander]

Thank you so much. I got confused by the phrase "length" which is 400 cm as per the diagram. I don't understand why the correct answer is not given with respect to length which is 400 cm in this case. I am so sorry for asking such a basic question. I appreciate your help. Your answer has certainly helped me so thank you for your help and time.

You really do seem to have confused yourself lol. I don't see 400 cm mentioned anywhere!
(Edit: Oh, I see, you are referring to the length of the whole roll? I only just noticed that looking back.)

Having had a bit more time, I thought it might be useful to add these pictures (as they may aid the understanding of anyone who doesn't get my written explanation. Your son, perhaps?).

If you were to use blue wrapping paper, then a piece cut 45 cm long from the roll 70 cm wide could be laid out as shown below with the box (in white) sitting on top of it. As the (flat, 2 dimensional) net is folded up to form the solid (3 dimensional) box, then the (blue, 70 x 45 cm) wrapping paper can be wrapped around it (from left to right) with a 10 cm overlap...

​

Then the 2.5 cm overhangs at each side can be folded down to meet each other in the middle of the sides, thus (almost, lol) completely covering every surface of the box. As illustrated below...

​

As I already suggested, using a 45 cm length is really cutting it a bit fine and you are likely to be left with a (small) gap where the edges of the wrapping paper meet. In real life, therefore, you would almost certainly use a cut of more than 45 cm but you wouldn't need much more to get sufficient coverage which I trust explains the marking scheme for you?

Hope that helps.

 

[Jignesh77]

You really do seem to have confused yourself lol. I don't see 400 cm mentioned anywhere!
(Edit: Oh, I see, you are referring to the length of the whole roll? I only just noticed that looking back.)

Having had a bit more time, I thought it might be useful to add these pictures (as they may aid the understanding of anyone who doesn't get my written explanation. Your son, perhaps?).

If you were to use blue wrapping paper, then a piece cut 45 cm long from the roll 70 cm wide could be laid out as shown below with the box (in white) sitting on top of it. As the (flat, 2 dimensional) net is folded up to form the solid (3 dimensional) box, then the (blue, 70 x 45 cm) wrapping paper can be wrapped around it (from left to right) with a 10 cm overlap...

View attachment 39410​

Then the 2.5 cm overhangs at each side can be folded down to meet each other in the middle of the sides, thus (almost, lol) completely covering every surface of the box. As illustrated below...

View attachment 39408​

As I already suggested, using a 45 cm length is really cutting it a bit fine and you are likely to be left with a (small) gap where the edges of the wrapping paper meet. In real life, therefore, you would almost certainly use a cut of more than 45 cm but you wouldn't need much more to get sufficient coverage which I trust explains the marking scheme for you?

Hope that helps.

Thank you so much for your help. I really appreciate your time and effort to hemp me clear my doubt. All these diagrams make sense.

 

[khansaheb]

Thank you so much for your help. I really appreciate your time and effort to hemp me clear my doubt. All these diagrams make sense.

Please do not provide hemp to highlander - he will surely land high with that sword in hand and there will be carnage,,,,,