Brainteaser/mathmatical question
 

its an old 1.... some1 will know the answer and my brain aint up to the job today so hoping scoobynet will work its magic for me...

its the 1 where 3 people chip in £10 each for a room = £30.... guy at the counter realises hes over charged the room was only £25 so send the porter up with £5 who pockets £2 and gives the guys a pound each back. theyve all paid £9 each 9x3=27 and the porter has 2 quid wheres the missing pound?

Ive owrked this out no end of times but not today I really cant be bothered using the head

thanks :)

The porter's two quid takes it down to £25, the price of the room, not up to £29.

There isn't one..

By your theory if they all pay £9 (3x9=27) and the room costs £25 then there would be £2 left over which the porter kept. Or am I just being dim?

Carter, not sure if I understand what your saying although think Ive just worked it out again, if theyve all only paid £9 each then they've only paid £27 in total and £2 of that is in the porters pocket, so the answer is there is no missing pound its just twisted the way its worded with the maths involved, correct?

The missing pound has been used to purchase a picture of a 4wd jap car in a pond

Yes. You're using the two different totals with the different amounts being paid.

£25 room charge, plus £3 they get back = £28 (not £27). Divided by 3 = £9.33 each

The muddle creeps in here because you end up counting the same money more than once, and not counting other money!

Consider the following transaction:

• Bobby gives Sally ten pounds.
• Sally gives Bobby back eight pounds.

In effect, Bobby has paid two pounds, since he gave Sally ten and got back eight. Sally has kept two pounds.

If you were to make the same mistake as in the original problem above, you would say "Bobby has paid two pounds, and Sally has kept two pounds, making a total of four pounds, so where did the other six pounds from the original ten pounds go?"

What if Bobby originally gave Sally twenty pounds, and she again kept only two and gave back eighteen? Would we then be missing sixteen pounds from the original twenty?!

The problem here is that you're counting the same money twice - the two pounds that Bobby paid to Sally is the same money as the two pounds that Sally has. Adding these two values together doesn't tell you anything meaningful. If you add up the actual moneys held at the end of the transaction - the two pounds that Sally keeps and the eight pounds that Bobby gets back - you get ten pounds, as expected.

Now, in your problem there were thirty pounds in cash originally received by the hotel. At the end of the transaction, the total amount of cash in the "system" (25 + 2 + 1 + 1 + 1) still adds up to thirty, so there is no missing pound. In other words, all the different chunks of money add up to the original total of thirty pounds.

The fallacious missing pound creeps in when you start adding up the same money more than once.

In the story, it says that "each employee paid 9 pounds each for their room, costing them 27 pounds in total. Together with the 2 pounds that their porter has kept that makes a total of 29 pounds."

However, the two pounds that the porter has kept is part of the 27 pounds the employees have paid - it is not different money, so it can't be added to the employees' 27 pounds. You're counting it twice, just like with Bobby and Sally.

The fact that this meaningless value adds up to one less than thirty, which causes the "missing pound", is pure coincidence. If the employees had overpaid by fifteen pounds rather than five, and their boss had still kept two pounds, then by this argument you would now have eleven missing pounds:

• Employees pay 40 pounds
• Shopkeeper gives boss 15 pounds back
• Boss keeps 2 pounds, gives employees 13
• Net result: Employees have paid 27 pounds, boss keeps 2, sum 29.
• Where's the other 11?

In other words, by choosing the numbers properly, you can make arbitrary amounts of money "go missing", because the underlying line of reasoning is completely wrong. Sadly, this isn't going to help you on your tax return!

They had £30
They have paid £27

The b&b has £25 and the porter has £2

Its not £9 x 3 = £27 + £2 = £29

the £2 is already used in the 3 x £9.

It should be the £27 + the £3 ie £25 room, £2 porter and £3 change

Dave.

It's all in the wording

If the room cost £25, then this would have cost £8.33 each, so the porter should have given them £1.66 each, as it is he only gave them £1 so he stole 66/7p of each of them. This equates to his £2.

Mike

This is the phrase in which the error creeps in. There is no 'missing' pound.

They've paid £9 each, total £27, which equates to £25 for the room plus the £2 stolen by the porter.

Or, to look at it another way, the original £30 goes as £25 for the room, £2 for the porter and £3 refunded.

The problem is just in the assertion that for some reason the £2 should be added to the £27. This is nonsense - the £2 has already been included in the £27.

PMSL :)

Alternatively
 

3 people chip in £10 each for a room = £30.... guy at the counter realises hes over charged the room was only £25 so send the porter up with £5 who pockets £3 and gives the guys 66p each back. theyve all paid £9.33 each 9.33x3=28 and the porter has 3 quid, generating a pound from thin air:eek2: Keep doing this and you'll soon be a rich man ;)