# Where exactly I am going wrong here?

A can of juice was $80\%$ full. $80\%$ of the contents were emptied into a glass and $81$ ml of juice was added to the can. Then the can became full to the brim. What is the capacity of the can ?

If $x$ ml be the full capacity of the can then $$\frac 4{25}x + 81 = x$$ but then solving for $x$ from here won't give $225$ ml which is the required answer for this problem. What exactly I am missing here?

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Two people have agreed that the problem is wrong. 225 would be correct if the 20% left in the can were in fact 80%. I suspect that was the error made in the book. –  Ross Millikan Nov 21 '11 at 5:14
@Ross:Please check Joel's answer, his interpretation is exactly similar to that of my module but however no explanation is provided in either case. –  Quixotic Nov 21 '11 at 5:19
Joel has recanted, and is now in agreement with the rest of the readers here. The way it is written there should be 16%=80%*20% left after the emptying, while to get 225 you need 64%=80%*80% left. Presumably your answer was 25*81/21=96 3/7 ml, which I agree with. –  Ross Millikan Nov 21 '11 at 5:35

To obtain the given solution requires $\rm\ F (1-E) = 0.64 = (225-81)/225,\$ where $\rm F$ is the initial fraction of full, and $\rm E$ is the fraction emptied, e.g. $\rm F = 0.8,\ E = 0.2\:.\:$ So it appears that the problem should say all but $80\%$ were emptied.

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Please check Joel's answer, his interpretation is exactly similar to that of my module but I am not sure about his way to that equation. –  Quixotic Nov 21 '11 at 5:24
@MaX You need to choose F,E to satisfy the above equation. The most natural choice is as Joel and I say - which is consistent with an obvious braino while composing the problem. –  Bill Dubuque Nov 21 '11 at 5:27
Yes, after his current edit. –  Quixotic Nov 21 '11 at 5:29

Unless I made a mistake in my own algebra/logic- what you have seems to make sense to me. So, I would say that the answer is wrong. Here is my way of doing things:

1. Capacity= $x$

2. Original amount of juice: $0.8 \ x$

3. Amount discarded: 80%

4. Thus, juice left in can: $0.8 \ x \ 0.2$

5. Amount added to fill up to brim: 81 ml

6. Thus, we have:

$$0.8 \ x \times \ 0.2 + 81 = x$$

which is the same as yours.

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As was noted in the comments below, something is wrong with the question. $x$ would be 225 if the equation was $\frac{64}{100}x+81=x$. It seems that what was meant was the contents in the glass (which is 80% of 80% of the can) plus 81 mL (from somewhere else) fills the can. (Sorry for my initially incorrect response.)

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I am not sure I follow. How is: 81 ml 80% of 80% of x? –  tards Nov 21 '11 at 5:13
How exactly are you are forming this equation ? –  Quixotic Nov 21 '11 at 5:15
@tards:This is exactly what the book did in the solution, but even they have not given any explanation. –  Quixotic Nov 21 '11 at 5:17
@MaX In that case, see Ross's comment to your question. Either the answer key is wrong or the question is wrong. You would get the above equation if we assume that 20% of the juice is discarded instead of 80%. –  tards Nov 21 '11 at 5:19
@tards:Perhaps the problem solver of the module made the same mistake as Joel did before?! –  Quixotic Nov 21 '11 at 5:27

if $x$ is the capacity of a can.

stage 1: $80$% of x is in the can: $0.8x$

stage 2: $80$% of the content is emptied: $0.8x-(0.8x)0.8$

stage 3: $81$ ml is added back to the can $0.8x-(0.8x)0.8+81$

final: can is full $x$

overall

$0.8x-(0.8x)0.8+81 = x$

Solving this equation, $x$ is around $94$ml can not be $225$ml as the answer in the book.

I think there must some other interpretation of the original question.

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