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Two sisters, Saraha and Jane, go to a book store to buy an algebra book. Saraha is 10 dollars short for the book, and Jane is 1 dollar short. However, if the sisters combine their money, they still don't have enough for the book!

If neither of the sisters have any change in their pocket (Any coin or note lower than a dollar), what is the price of the book?

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closed as off-topic by Deusovi, mmking, AJL, Shkeil, Ken Y-N Apr 21 '16 at 1:26

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  • $\begingroup$ Because 8 of the 11 countries that use a currency named "Dollar" have a $1 coin instead of a bank note, the answer to this puzzle is ambiguous. $\endgroup$ – Ian MacDonald Apr 20 '16 at 17:53
  • $\begingroup$ @IanMacDonald It's only ambiguous if that dollar has a fractional note. From Wikipedia, there are a handful of dollars for which the smallest banknotes are 2 and 5, in which case the expected answer still holds. For countries whose bank notes start at 5 or higher, then the reasonable answer is: "This puzzle doesn't occur in that country, but in another one." And, after all that, the Bahamian dollar does offer half-dollar notes (though Wikipedia says they are rarely used), in which case the book could cost BSD9.50 or BSD10. $\endgroup$ – Matt Apr 20 '16 at 18:35
  • $\begingroup$ @Matt, not true. Since the currency is not precisely defined, and the vast majority of "dollar" currencies have no $1 note, the majority answer is "there is no solution". It is only in the minority of cases that there could potentially be a solution. This is what makes answering ambiguous: there is more than one possible answer given the information in the puzzle. $\endgroup$ – Ian MacDonald Apr 20 '16 at 18:44
  • $\begingroup$ Unless you are willing to assume that the question or the asker is unreliable (lol there is no book!) and that this is the situation that our intrepid protagonists Saraha and Jane do indeed find themselves in, then it must be the case that the specific variant of the dollar they are using makes this possible, whether or not the question specifies it. If the answer is that there is no answer, then there is no question. I agree with you that the question is ambiguous, but I can supply you with a dollar that has no \$1 note yet still satisfies the conditions with a unique solution. $\endgroup$ – Matt Apr 20 '16 at 19:10
  • $\begingroup$ Or we just get over the discussion, work out a generalised solution, and then within the assumptions discussed, find a specific 'no change' solution. O wait, we did. $\endgroup$ – Tim Couwelier Apr 22 '16 at 6:23
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Answer:

Sarah has anywhere from 0 to 99 cents, Jane has 9 dollars more, and the book costs at least 9.01 dollars + twice what Sarah has.

Reasoning:

If the book has a cost of 'x', Sarah has 's' dollars and Jane has 'j' dollars, the following equations true: x = s + 10 (1)
x = j + 1 (2)

There's also an inequity we know of: s + j < x (3)

From (1) and (2) we find that j = s + 9
Substitute in (3) to find:
2s +9 < x (4)

Then from (2) we know x = s+10, so substitute in (4):
2s + 9 < s+10 (5)

We know s cannot be negative, so we are certain boh sides are positive. If we deduct (s+9) from both sides of (5) we get the following inequity:
s < 1

EDIT:
As pointed out in comments, 'no change' is deemed to imply 'no coins, only bills'. Given there is no bills smaller then 1 dollar, and the monetary amount cannot be negative, by definition s has to be 0. Therefor Sarah has 0 dollars, Jane has 9 dollars and the book costs 10 dollars.

Details on the non-integer solution:
0.99 dollars, 9.99 dollars and a 10.99 dollar book works (and any smaller amount for s). If Sarah has ONE dollar, and Jane has 10 dollars, they'd both be able to combine their money for an 11 dollar book.

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  • $\begingroup$ Read the last line of the question. I made the same mistake. (Unless we assume these girls were carrying purses.) $\endgroup$ – SendersReagent Apr 20 '16 at 17:30
  • $\begingroup$ Meh, poor phrasing then. I'm used to euro's, there's no notes below 5... I'll edit conclusions to reflect .(also, yay for generalised solutions) $\endgroup$ – Tim Couwelier Apr 20 '16 at 17:32
  • $\begingroup$ This is clearly the best answer so far. $\endgroup$ – cbmanica Apr 20 '16 at 21:43
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The book costs:

\$10. Saraha has \$0 and Jane has \$9, so combined they have just $9.

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  • $\begingroup$ can you show why no other price will meet the criteria? $\endgroup$ – Kate Gregory Apr 20 '16 at 17:11
  • $\begingroup$ @SendersReagent yea classic mistake on these is to only consider integer values. I have a full answer written up by now. $\endgroup$ – Tim Couwelier Apr 20 '16 at 17:29
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    $\begingroup$ @TimCouwelier Right, I was interpreting the question's "neither of the sisters have any change in their pocket" as an instruction to limit to integers. A classic mistake is not reading the whole question. $\endgroup$ – Dan Russell Apr 20 '16 at 17:33
  • $\begingroup$ You got me there. Thing is, 'change' doesn't mean the same everywhere. Still, you have an answer (which works) but doesn't explain how you got it, or why it's unique. $\endgroup$ – Tim Couwelier Apr 20 '16 at 17:36
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    $\begingroup$ Good point, Tim. As an American, I confess to assuming that the whole world operates in the same way as my personal experience. (We're practically taught this in school!) As for other cases, it seems more intuitive to simply just go one dollar down (and realize that you're in negative territory) or go one up (and realize you'd be able to pay for the book with the combined total). $\endgroup$ – Dan Russell Apr 20 '16 at 18:09
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Assuming "neither have change in their pocket" means that they have a natural (0,1,2,...) amount of dollars:

Jane is \$1 short, but Sarahas money is not enough to cover for this single dollar. Therefore, Saraha has 0 dollars. Since Saraha has \$0 and the book costs \$10 more, the book costs \$10.

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