Simple Maths Problem
Discussion
We start with 2 whole cakes:
1 + 1 = 2
(Cake A + Cake B)
Person A first takes his take of cake A
(1-5/6) + 1 = 1/6 + 1 = 7/6
Person B now takes his take of cake B
7/6 - 7/8 = 7/24 or to lay it out easier (1-5/6) + (1-7/8) = (24/24 - 20/24) + (24/24 - 21/24) = (24 - 20 + 24 - 21)/24 = 7/24
7/48 is the incorrect answer as say (as someone suggested) each cake was made of 48 slices, starting with whole cakes again:
48/48 + 48/48 = 96/48 (1 + 1 = 2)
Person A first takes his take of cake A. 5/6ths of 48 slices is 40 so:
(48/48 - 40/48) + 48/48 = 8/48 + 48/48 = 56/48
Person B now takes his take of cake B. 7/8ths of 48 slices is 42 so:
8/48 + (48/48 - 42/48) = 8/48 + 6/48 = 14/48 which is the same as 7/24
Those deducing 7/48 are forgetting that they have started with two whole cakes. Yes there are 7/48 slices left on the table, but thats 7/48ths of 2 cakes which makes 7/24ths remaining of all the cake on the table.
1 + 1 = 2
(Cake A + Cake B)
Person A first takes his take of cake A
(1-5/6) + 1 = 1/6 + 1 = 7/6
Person B now takes his take of cake B
7/6 - 7/8 = 7/24 or to lay it out easier (1-5/6) + (1-7/8) = (24/24 - 20/24) + (24/24 - 21/24) = (24 - 20 + 24 - 21)/24 = 7/24
7/48 is the incorrect answer as say (as someone suggested) each cake was made of 48 slices, starting with whole cakes again:
48/48 + 48/48 = 96/48 (1 + 1 = 2)
Person A first takes his take of cake A. 5/6ths of 48 slices is 40 so:
(48/48 - 40/48) + 48/48 = 8/48 + 48/48 = 56/48
Person B now takes his take of cake B. 7/8ths of 48 slices is 42 so:
8/48 + (48/48 - 42/48) = 8/48 + 6/48 = 14/48 which is the same as 7/24
Those deducing 7/48 are forgetting that they have started with two whole cakes. Yes there are 7/48 slices left on the table, but thats 7/48ths of 2 cakes which makes 7/24ths remaining of all the cake on the table.
4x4Tyke said:
It is Maths problem not an English grammar question so it should be worded as 'What fraction of the cakes are left?'
Don't think so. It's "is" because the subject of the question is "fraction", and fraction is singular. "Of the cakes" qualifies "fraction". Shortening it by removing the qualification leaves the sense intact: "what fraction is left?".
This
Alias218 said:
When you put it like that it's plain to see what all the kerfuffle is about!
However, it is also mathematically wrong.
Lets turn the fractions into decimals, for ease of visualisation. So:
3/24 = 0.125
4/24 = 0.167
0.125 + 0.167 = 0.291, we can all agree on that.
Incidentally, 7/24 = 0.291.
What you are doing is taking a whole (24) and turning it into two wholes (48). Makes sense, there are two 24 piece cakes therefore 48 pieces.
However, if we take 0.125 of 1 cake and 0.167 of 1 cake we see that when we add them using the two cake logic:
0.125/1 + 0.167/1 = 0.291/2 (as 3/24 + 4/24 = 7/48)
But then when we strip that right back: 0.125/1 = 0.125 and 0.167/1 = 0.167 and 0.291/2 = 0.145
The two cake logic states that 0.125 + 0.167 = 0.145, which it most certainly does not. It equals 0.291 or 7/24.
Incidentally, 0.145 = 7/48.
I had to scratch my head a little to get past the two cake image, which does lead you to conclude that 3/24 and 4/24 = 7/48. But mathematically, it doesn't. It can't. by adding the denominators, the amount of cake is being doubled, which then halves the amount of cake, as a fraction, that is left!
It's a very misleading question that can lead you down the garden path, but if you stick to mathematic principles you won't go wrong.
IMO suffers from not reading the question, which is "What fraction of the cakes is left?". It's dreadfully worded, but were one to start with 10 cakes and eat one, the "fraction of the cakes" left is obvious: 9/10 (it's even more obvious if one asks "what percentage of the cakes is left"). However, it is also mathematically wrong.
Lets turn the fractions into decimals, for ease of visualisation. So:
3/24 = 0.125
4/24 = 0.167
0.125 + 0.167 = 0.291, we can all agree on that.
Incidentally, 7/24 = 0.291.
What you are doing is taking a whole (24) and turning it into two wholes (48). Makes sense, there are two 24 piece cakes therefore 48 pieces.
However, if we take 0.125 of 1 cake and 0.167 of 1 cake we see that when we add them using the two cake logic:
0.125/1 + 0.167/1 = 0.291/2 (as 3/24 + 4/24 = 7/48)
But then when we strip that right back: 0.125/1 = 0.125 and 0.167/1 = 0.167 and 0.291/2 = 0.145
The two cake logic states that 0.125 + 0.167 = 0.145, which it most certainly does not. It equals 0.291 or 7/24.
Incidentally, 0.145 = 7/48.
I had to scratch my head a little to get past the two cake image, which does lead you to conclude that 3/24 and 4/24 = 7/48. But mathematically, it doesn't. It can't. by adding the denominators, the amount of cake is being doubled, which then halves the amount of cake, as a fraction, that is left!
It's a very misleading question that can lead you down the garden path, but if you stick to mathematic principles you won't go wrong.
If one were to eat another half a cake, the fraction left would be 8.5/10, or 17/20 (or 85%).
Once you see that the the wording of the question - intentionally or otherwise - requires you to compare what you are left with with the total number of cakes that you start with, there's only one answer, with is 7/48.
The answer 7/24 is one to a different question, namely "What fraction of a cake is left?". Forex: you are left with 12.5% of one cake and 16.7% of one cake. Squish them together and you have 29.2% of one cake, or 7/24. But the question isn't that; it's what percentage do you have of the two cakes that you started with? You have 29.2% of 200%, ie 14.6% of 100%.
So, as always, RTQ.
That said, were I the chief examiner I would direct full marks be given for each answer because the question is so poorly put, and I'd sack the person who wrote the question.
Edited by anonymous-user on Thursday 11th January 18:55
Greg66 said:
That said, were I the chief examiner I would direct full marks be given for each answer because the question is so poorly put, and I'd sack the person who wrote the question.
I'd offer bonus marks if both answers were given with an explanation as to why each might be considered valid. 
Alias218 said:
Again, this is making the mistake of doubling the quantity.
If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
Once again, you haven't read the question. It's not "what fraction of a cake is left?". It's "what fraction of the cakes is left?". The line of yours that starts "The two cake logic" is the line where you depart from the question and choose to answer a different question - the former one above. 7/24 is precisely the fraction of a single cake that you'd be left with. Try it. If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
ETA: in fact, try this (but give answers to both questions).
You've got two pizzas. Each one is divided into four equal slices.
You eat one slice.
(a) what fraction (or percentage) of a single pizza do you have left?
(b) what fraction (or percentage) of the two pizzas do you have left?
Edited by anonymous-user on Thursday 11th January 19:05
Greg66 said:
Alias218 said:
Again, this is making the mistake of doubling the quantity.
If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
Once again, you haven't read the question. It's not "what fraction of a cake is left?". It's "what fraction of the cakes is left?". The line of yours that starts "The two cake logic" is the line where you depart from the question and choose to answer a different question - the former one above. 7/24 is precisely the fraction of a single cake that you'd be left with. Try it. If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
ETA: in fact, try this (but give answers to both questions).
You've got two pizzas. Each one is divided into four equal slices.
You eat one slice.
(a) what fraction (or percentage) of a single pizza do you have left?
(b) what fraction (or percentage) of the two pizzas do you have left?
Edited by Greg66 on Thursday 11th January 19:05
(b) 3/4 + 4/4 = 7/4 or 1 3/4 (not 3/4 + 4/4 = 7/8 which is exactly what a lot of people, including you, have done on the cake question).
The point of fact is that we are talking about two cakes. This very fact changes the way the question is answered.
Show me how 3/4 + 4/4 = 7/8 is any different from 3/24 + 4/24 = 7/48 in the context of question (b).
Edited by Alias218 on Thursday 11th January 19:31
Greg66 said:
4x4Tyke said:
It is Maths problem not an English grammar question so it should be worded as 'What fraction of the cakes are left?'
Don't think so. It's "is" because the subject of the question is "fraction", and fraction is singular. "Of the cakes" qualifies "fraction". Shortening it by removing the qualification leaves the sense intact: "what fraction is left?".
Fraction is the subject of the question, but not the subject of the sentence.
AIUI 'plural and singular subject is' is American usage,
while plural 'subjects are' and singular 'subject is' British usage.
Alias218 said:
Greg66 said:
Alias218 said:
Again, this is making the mistake of doubling the quantity.
If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
Once again, you haven't read the question. It's not "what fraction of a cake is left?". It's "what fraction of the cakes is left?". The line of yours that starts "The two cake logic" is the line where you depart from the question and choose to answer a different question - the former one above. 7/24 is precisely the fraction of a single cake that you'd be left with. Try it. If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
ETA: in fact, try this (but give answers to both questions).
You've got two pizzas. Each one is divided into four equal slices.
You eat one slice.
(a) what fraction (or percentage) of a single pizza do you have left?
(b) what fraction (or percentage) of the two pizzas do you have left?
Edited by Greg66 on Thursday 11th January 19:05
(b) 3/4 + 4/4 = 7/4 or 1 3/4 (not 3/4 + 4/4 = 7/8 which is exactly what a lot of people, including you, have done on the cake question).
The point of fact is that we are talking about two cakes. This very fact changes the way the question is answered.
Show me how 3/4 + 4/4 = 7/8 is any different from 3/24 + 4/24 = 7/48 in the context of question (b).
Edited by Alias218 on Thursday 11th January 19:31
1 or 1/2?
V8LM said:
Alias218 said:
Greg66 said:
Alias218 said:
Again, this is making the mistake of doubling the quantity.
If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
Once again, you haven't read the question. It's not "what fraction of a cake is left?". It's "what fraction of the cakes is left?". The line of yours that starts "The two cake logic" is the line where you depart from the question and choose to answer a different question - the former one above. 7/24 is precisely the fraction of a single cake that you'd be left with. Try it. If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
ETA: in fact, try this (but give answers to both questions).
You've got two pizzas. Each one is divided into four equal slices.
You eat one slice.
(a) what fraction (or percentage) of a single pizza do you have left?
(b) what fraction (or percentage) of the two pizzas do you have left?
Edited by Greg66 on Thursday 11th January 19:05
(b) 3/4 + 4/4 = 7/4 or 1 3/4 (not 3/4 + 4/4 = 7/8 which is exactly what a lot of people, including you, have done on the cake question).
The point of fact is that we are talking about two cakes. This very fact changes the way the question is answered.
Show me how 3/4 + 4/4 = 7/8 is any different from 3/24 + 4/24 = 7/48 in the context of question (b).
Edited by Alias218 on Thursday 11th January 19:31
1 or 1/2?
If taken as two cakes as per the OP's question, then 1.
ETA: As I have said previously, having two cakes changes the way you answer the question, as shown in the above example.
Edited by Alias218 on Thursday 11th January 20:22
Alias218 said:
V8LM said:
Alias218 said:
Greg66 said:
Alias218 said:
Again, this is making the mistake of doubling the quantity.
If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
Once again, you haven't read the question. It's not "what fraction of a cake is left?". It's "what fraction of the cakes is left?". The line of yours that starts "The two cake logic" is the line where you depart from the question and choose to answer a different question - the former one above. 7/24 is precisely the fraction of a single cake that you'd be left with. Try it. If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
ETA: in fact, try this (but give answers to both questions).
You've got two pizzas. Each one is divided into four equal slices.
You eat one slice.
(a) what fraction (or percentage) of a single pizza do you have left?
(b) what fraction (or percentage) of the two pizzas do you have left?
Edited by Greg66 on Thursday 11th January 19:05
(b) 3/4 + 4/4 = 7/4 or 1 3/4 (not 3/4 + 4/4 = 7/8 which is exactly what a lot of people, including you, have done on the cake question).
The point of fact is that we are talking about two cakes. This very fact changes the way the question is answered.
Show me how 3/4 + 4/4 = 7/8 is any different from 3/24 + 4/24 = 7/48 in the context of question (b).
Edited by Alias218 on Thursday 11th January 19:31
1 or 1/2?
If taken as two cakes as per the OP's question, then 1.
ETA: As I have said previously, having two cakes changes the way you answer the question, as shown in the above example.
Edited by Alias218 on Thursday 11th January 20:22
!00% of the cakes is [are] left? Really?
ETA: It doesn't ask what fraction of a cake is left? Or how much of a cake is left? Or what fractions of the cakes are left? The question refers to a single fraction of the total cakes. It ask what fraction [a single number] of the cakes [together, not individually] is left.
Appallingly worded question.
Edited by V8LM on Thursday 11th January 20:41
Alias218 said:
Greg66 said:
Alias218 said:
Again, this is making the mistake of doubling the quantity.
If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
Once again, you haven't read the question. It's not "what fraction of a cake is left?". It's "what fraction of the cakes is left?". The line of yours that starts "The two cake logic" is the line where you depart from the question and choose to answer a different question - the former one above. 7/24 is precisely the fraction of a single cake that you'd be left with. Try it. If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
ETA: in fact, try this (but give answers to both questions).
You've got two pizzas. Each one is divided into four equal slices.
You eat one slice.
(a) what fraction (or percentage) of a single pizza do you have left?
(b) what fraction (or percentage) of the two pizzas do you have left?
Edited by anonymous-user on Thursday 11th January 19:05
(b) 3/4 + 4/4 = 7/4 or 1 3/4 (not 3/4 + 4/4 = 7/8 which is exactly what a lot of people, including you, have done on the cake question).
The point of fact is that we are talking about two cakes. This very fact changes the way the question is answered.
Show me how 3/4 + 4/4 = 7/8 is any different from 3/24 + 4/24 = 7/48 in the context of question (b).
Edited by anonymous-user on Thursday 11th January 19:31
You've got two pizzas. Each one is divided into four equal slices.
You eat one slice.
(a) what fraction (or percentage) of a single pizza do you have left?
Answer: you have seven quarters left. A single pizza consists of 4 quarters. The fraction of a single pizza that you have is 7/4.
(b) what fraction (or percentage) of the two pizzas do you have left?
Answer: you have seven quarters left. Two pizzas is 8 quarters. The fraction of the two pizzas that you have is is 7/8.
So you got the right answers. Just the wrong way round.
Here's the problem with your answer (a): the question is what fraction of a single pizza you have. One of the pizzas is untouched. And there's some more pizza next to it. You cannot conceivably have <1 of a single pizza; the evidence of your eyes tells you that you have >1 pizza.
Here's the problem with your answer (b): the question is what fraction of both pizzas you are left with after you've eaten a quarter of one. Your answer is 7/4 - ie >1. Your answer means you end up with more than you started with, despite having eaten a quarter of a pizza.
If you can make that work in real life, I urge you to open a pizza restaurant. You will end up very fat, but ver, very rich.
Trying to distinguish two cakes vs two pizzas shows, I think, that you're grasping, and don't really have a grip on the principles at work.
The original question
That is the question verbatim, and I agree about the answer.The example was
It's (3+4)/8 = 7/8, or (3+4)/48 = 7/48. You've derived the wrong calculation.
dr_gn said:
Greg66 said:
dr_gn said:
There are two identical cakes. Person A takes 5/6 of one cake, person B takes 7/8 of the other cake. What fraction of the cakes is left?
If that's the question verbatim, the answer is 7/48. Greg66 said:
(b) what fraction (or percentage) of the two pizzas do you have left?
Alias218 said:
Show me how 3/4 + 4/4 = 7/8 is any different from 3/24 + 4/24 = 7/48 in the context of question (b).
Because its not 3/4 + 4/4, or 3/24 + 4/24. It's (3+4)/8 = 7/8, or (3+4)/48 = 7/48. You've derived the wrong calculation.
V8LM said:
Alias218 said:
V8LM said:
Alias218 said:
Greg66 said:
Alias218 said:
Again, this is making the mistake of doubling the quantity.
If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
Once again, you haven't read the question. It's not "what fraction of a cake is left?". It's "what fraction of the cakes is left?". The line of yours that starts "The two cake logic" is the line where you depart from the question and choose to answer a different question - the former one above. 7/24 is precisely the fraction of a single cake that you'd be left with. Try it. If you go through my numbers, you are doubling up (where I put 0.291/2 - the same as your 29.1% of 200%). This, as I prove on the line below, is incorrect.
The answer is 7/24. It's all it can be.
ETA: in fact, try this (but give answers to both questions).
You've got two pizzas. Each one is divided into four equal slices.
You eat one slice.
(a) what fraction (or percentage) of a single pizza do you have left?
(b) what fraction (or percentage) of the two pizzas do you have left?
Edited by Greg66 on Thursday 11th January 19:05
(b) 3/4 + 4/4 = 7/4 or 1 3/4 (not 3/4 + 4/4 = 7/8 which is exactly what a lot of people, including you, have done on the cake question).
The point of fact is that we are talking about two cakes. This very fact changes the way the question is answered.
Show me how 3/4 + 4/4 = 7/8 is any different from 3/24 + 4/24 = 7/48 in the context of question (b).
Edited by Alias218 on Thursday 11th January 19:31
1 or 1/2?
If taken as two cakes as per the OP's question, then 1.
ETA: As I have said previously, having two cakes changes the way you answer the question, as shown in the above example.
Edited by Alias218 on Thursday 11th January 20:22
!00% of the cakes is [are] left? Really?
ETA: It doesn't ask what fraction of a cake is left? Or how much of a cake is left? Or what fractions of the cakes are left? The question refers to a single fraction of the total cakes. It ask what fraction [a single number] of the cakes [together, not individually] is left.
Appallingly worded question.
Edited by V8LM on Thursday 11th January 20:41
Greg66 said:
The original question
That is the question verbatim, and I agree about the answer.The example was
It's (3+4)/8 = 7/8, or (3+4)/48 = 7/48. You've derived the wrong calculation.
I hate to say it, and I'm more than a little embarrassed, but I think you may be correct. dr_gn said:
Greg66 said:
dr_gn said:
There are two identical cakes. Person A takes 5/6 of one cake, person B takes 7/8 of the other cake. What fraction of the cakes is left?
If that's the question verbatim, the answer is 7/48. Greg66 said:
(b) what fraction (or percentage) of the two pizzas do you have left?
Alias218 said:
Show me how 3/4 + 4/4 = 7/8 is any different from 3/24 + 4/24 = 7/48 in the context of question (b).
Because its not 3/4 + 4/4, or 3/24 + 4/24. It's (3+4)/8 = 7/8, or (3+4)/48 = 7/48. You've derived the wrong calculation.
ETA: an extraordinarily large penny has dropped and I feel like a knob.
I'm going to go and have 7/48 of a cake now to cheer myself up.
Edited by Alias218 on Thursday 11th January 21:27
Alias218 said:
I hate to say it, and I'm more than a little embarrassed, but I think you may be correct.
S’ok.For a laugh once I said to a very clever guy I know with a science background: metals expands when you heat it, right? He agreed. I have a metal washer. I heat it. What happens to the hole in the middle?
So keen was he to press upon me how rapidly he could solve this that he blurted out: “it gets smaller!”
Pause
“No! It stays the same size!”
Then looked a bit anxious, he said “I need to go away and think about this” and literally ran from the room a la Sheldon Cooper.
He came back 20 mins later with the right answer confessing that he had rarely been so embarrassed.
I was very understanding. I pointed at him and laughed loudly for about ten minutes.
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