supposed you've seen a nice outfit in a mall worth $97.  You borrowed 50 from your mom and another 50 from you dad so you have $100 now. You bought the outfit.  You still got $3 dollar change.  You returned a dollar to your mom so you still owe her $49.  You gave another dollar to your dad so you still owe him $49.  The other dollar left is in you.  Here's the problem.  49 + 49 = 98  +  dollar that's in you = $99.  Why is it $99 only?  It should have been $100 right?  Where's the dollar?
208325
the other dollar is in your pocket.... you spent $97 dollars and had 3 dollars change... you payed back $2. of the $100.  you spent $97..... no matter how you look at it you had $3. and if you payed back 2 of the $3. the other Dollar is in your pocket.  the equation is 100. - 97 = 3 then 3 - 2 = 1 the dollar is in your pocket
71367
the trick to this question was in the wording and how the math was worked out in error\nthe  $3 they got back is what needs to be added to $72... NOT the $2 the waiter kept \nsince they got that back that is what you add... since the waiter kept $2, it is already included in what they paid
209079
It is a matter of wording.\n\nThe bill was $75.  The cashier gave them $5 discount, so now the bill is just $70.  So the waiter gives back $1 each, so everybody paid $24, giving us $72.  $70 of which went to the cashier, and $2 into the waiters pocket!
135128
This is not really a Math question. It's more of a logic question. Think of it this way: The three soldiers had given out 15 dollars. The manager kept 10 dollars, the bell boy kept 2 dollars, and the soldiers got 3 dollars back. Nothing is missing.\n\nThe way in which the question counts the total number of dollars is wrong. Yes, each soldiers only had to pay $4. That's a total of $12. Think about it. These $12 included the $2 that's secretly kept by the bell boy already (Again, $10 to the manager/hotel, $2 to the bell boy). Why count that again? The question is trying to fool you.\n\nThe correct way to count is: 15 dollars were circulating in this whole story. The soldiers paid out $12 and got $3 back. See? There's no missing dollar!\n\nIf you're still a little confused, draw a diagram. It helps.
130659
it doesnt matter how much a b and c "spent". they all gave d ten dollars. that equals 30 dollars. person d spends 25 dollars. then he gives 1 dollar back to a b and c. a b and c all have 1 dollar, which equalls 3 total $. person d has $25, plus the 2 he has left=$27, plus the three owned by a b and c=$30.
659701
The two dollars that was a tip to the bellhop is not added to the $27 to give $29 as the last sentence states.  It is subtracted from the $27 paid by the three men ($9 each), giving $25 (which is how much the room cost).\n\nSo basically, each man paid $8.33 for the room, and $0.67 each to the bellhop.  This gives the $25 room charge, $2 tip and the $3 change from the original $30.\n\nThere is no missing dollar.
577006
A short mathematical explanation:\n\nAssume you have two negative numbers u,v.  Then there exists two numbers positive number a and b such that u=-a and v=-b. Therefore v+b=0, and u(v+b)=0.\n\nBy the distributive law:\nu(v+b)=uv+ub=uv+(-a)b=uv-ab=0.  Thus uv=ab, and the product of two negative numbers is the product of two positive numbers.\n\nWhy does this make sense in the real world?\n\nAssume that you owe 5 people 5 dollars.  We can consider owing to be a negative number of having, so you have -5 dollars for 5 people or -25 dollars total.\n\nBut if it was only 4 people, you would only have -20 dollars, for 3 it would be -15, for 2 it would be -10, for 1 it would be -5 and for 0 it would be 0.  So, each time you remove a person, you add 5 dollars.  When you have -1 people that you have -5 dollars for (you can also say you have -1 people that you owe 5 dollars to, or there is 1 person that owes you five dollars) you continue to add 5 dollars.  Now you have -1*-5=5, so two negatives multiplied together to make a positive.
578632
100/2=50\n 50    50\n-1      -1\n49      49\n49+49= 98\nyou gave them each 1 dollar back so 100-2=98.\nYou aren't missing a dollar.
624737
each man paid 23.5 but they both got the 1.5 back, so to the $47 you andd not the money kept by the keeper but the money got back by the men who paid for it. so it would be 47 + 3 = 50 it's all there. the money kept by the keeper is in the $47.00 ( 45 for the gift and 2 for him self)
165753
theres not an extra dollar.see the bellboy kept $2 right.he gave back 1 dollar to each person.theres 3 people.so...\n\n25+3=28.\n30-28=2\nthe bellboy kept the 2 so really theres no etra money.
655233
You've given two cases:\nPricing scheme 1: \nA = 5 cents, B = 3 cents, C = 0.5 cents\n\nYou could just buy all C, but that would result in 40 candies.  This is no good.\n\nHowever, you could exchange 6 pieces of C for 1 piece of B.  This is a net change of -5 pieces.  Thus it is readily apparent that {0,4,12} is one solution.\n\nYou can't do a straight trade of A pieces for C pieces because that is a net change of -9 pieces, and 20 isn't divisible by -9...\n\nLet's try other combinations of A and B.\n1A and 1B, 2 pieces for 8 cents = 16 pieces for 0.5 cent.\nThis is a net change of -14, so that won't work.\n\n1A and 2B, 3 pieces for 11 cents = 22 pieces for 0.5 cent.\nThis is a net change of -19, so that won't work either\n\n1A and 3B, 4 pieces for 14 cents = 28 pieces for 0.5 cent.\nThis is a net change of -24, so that's no good.\n\n2A and 1B, 3 pieces for 13 cents = 26 pieces for 0.5 cent\nThis is -23 pieces, no good.\n\nSo there is no way with integer combinations to buy at least one of A, B and C.\n\nThe only solution to part 1 is:\n0 pieces of A\n4 pieces of B (12 cents)\n16 pieces of C (8 cents)\n--------------------------\n20 pieces for 20 cents\n\nPricing scheme 2: \nA = 2 cents, B = 3 cents, C = 0.5 cents\n\nAgain we can start with 40 pieces of C.  We need to exchange 20 pieces.\n\nWe already know that {0,4,16} is one solution.\n\nLet's try the other combinations:\n1A + 1B:  2 pieces for 5 cents  = 10 pieces for 0.5 cent\nNet change of -8, not good.\n\n1A + 2B:  3 pieces for 8 cents = 16 pieces for 0.5 cent\nNet change of -13, not good\n\n1A + 3B:  4 pieces for 11 cents = 22 pieces...\nNet change -18, bad\n\n1A + 4B:  5 pieces for 14 cents = 28 pieces...\nNet change - 23, bad\n\n2A + 1B:  3 pieces for 7 cents = 14 pieces...\nNet: -11 bad\n\n2A + 2B (same as 1A + 1B case) - bad\n\n2A + 3B:  5 pieces for 13 cents = 26 pieces...\nNet - 21 bad\n\n3A + 1B:  4 pieces for 9 cents = 18 pieces...\nNet -14 bad\n\n3A + 2B:  5 pieces for 12 cents = 24 pieces...\nNet -19 bad\n\n3A + 3B, same as 1A + 1B - bad\n\n4A + 1B:  5 pieces for 11 cents = 22 pieces...\nNet -17 bad\n\n4A + 2B, same as 2A + 1B - bad\n\n4A + 3B:  7 pieces for 17 cents = 34 pieces...\nNet -27 bad\n\n5A + 1B:  6 pieces for 13 cents = 26 cents\nNet -20 GOOD!\n\nFinally another solution and one that allows us to buy one of each candy:\n\nThe two solutions to part 2 are:\n0 pieces of A\n4 pieces of B (12 cents)\n16 pieces of C (8 cents)\n--------------------------\n20 pieces for 20 cents\n\n5 pieces of A (10 cents)\n1 pieces of B (3 cents)\n14 pieces of C (7 cents)\n--------------------------\n20 pieces for 20 cents
57031
Basically: There is an initial $30 charge.  It should have been $25, so $5 must be returned and accounted for.  $3 is given to the 3 friends, $2 is kept by the manager - there you have the $5.  \n\nThe trick to this riddle is that the addition and subtraction are done at the wrong times to misdirect your thinking - and quite successfully for most.  Each of the 3 friends did indeed pay $9, not $10, and as far as the friends are concerned, they paid $27 for the night.  But we know that the clerk will tell us that they were charged only $25 and when you add the $3 returned with the $2 kept by the bellhop, you come up with $30.
103765
Suppose:\nI lend you one dollar, now you owe me one dollar. \n\nIf you spend the dollar you are holding ( the one I gave you), you have no money, right?\n\nNo.  You have minus 1 dollar.  You still owe me a dollar, so, you can't say "I have 0 money", as it's not true.  You do have money, simply a negative quantity of it.
170916
Say the 10 items sell for $10. You paid $10 for 11 items.\nyou paid $0.909 per item and sold it for $1.\n\nProfit is $.091 on $1.00  or 9.0909%
279336
you're mixing apples and oranges...\n\nYou owe them each $24 so 48 total, and you \nhave 3 left, so 48-3=45 which is the amount \nyou spent.\n\nAdding what you have left to what you owe\nthem has no meaning...
555045
You treat this as an addition and subtraction problem. Anything she put into her account gets added, anything she took out of her account gets subtracted.\n\n625 - 68.74 - 29.95 - 57.65 = 468.66\n\nTherefore her account increased by $468.66
152886
why should "balance after" add up to anything like the same number?   Try this:\n\nwithdrawal..... balance after\n99..............1\n1...............0\ntotal=100......total=1