Missing Pound

Three people enjoy a meal at a Thai restaurant. The waiter brings the bill for £30 so each person pays £10.

Later the chef realises that the bill should have only been £25 so he sends the waiter back to the table with £5. The waiter was not very good at Maths and could not figure out how to divide the £5 so he gave each person a £1 and kept £2 for himself.

So....the three people have paid £9 each for the meal.
3 x £9 = £27

The waiter kept £2
£27 + £2 = £29

What happened to the other pound? Does this make sense?

 This activity is suitable for students of mathematics all around the world. Use the button below to change the currency symbol used to make it more relevant to your students. You may wish to choose an unfamiliar currency to extend your students' experience.

• Chris, Exeter
•
• I think they are very good - some of the 'numeracy' starters were a little dry but you seem to have improved them recently. Thanks for taking the time to put these together and making them free. You've earnt your gold star!
• Anne, Blackburn, Lancashire
•
• Good Grief! That had me going (despite being a 30 year old maths graduate!) The waiter's £2 came out of the £27 paid by the three diners. They did pay £27 (3 x £9) and £25 went to the kitchen and £2 to an unscrupulous waiter. Thanks for making me think though!
• 9yma3, Northcliffe School
•
• The £27 they paid is the £25 for the meal and the £2 the waiter kept.
• Nikki Jordan, Braunton School, Devon
•
• Excellent. Thank you very much for a fabulous set of starters. I use the 'weekenders' if the daily ones are not quite what I want. Brilliant and much appreciated.
• Class 8A (Miss Yuen), St Philomena's School, Carshalton
•
• £30 shared equally between three is £10 each.

£25 shared equally between three is £8.33333......(recurring).

(£8.3333...) X 3 + £5 = £10 X 3

So the pound is still there!
• Stuart Don, Aberdeenshire
•
• Right

The £30 is spead out like this

Restaurant:- £25

Waitor:- £2

Customers:- £3

so 25 + 2 + 3 = £30 so the pound is still there

That Waitor is "NAE NICE"
• Joss Dutton, Class 9T3, National Comprehensive School
•
• Man thats hard that'll keep me thinking but if they all payed 10 came back with 5, £1 for every one = £23 + £2 = 25 help me!
• Faizah, Newcastle
•
• The £30 is still there!
• Katie n Shaz n Cyan n Jess n Jess !, Menzies High School
•
• Here's the real sum;

£25 for Bill !
£3 for People !
£2 for Waiter ! SOOOOOOOO......

£25 + £3 + £2 = £30 so the pound hasn't gone anywhere.
• Baran Ulutas, london
•
• They have not paid £27 for meal the bill was £25. yes they paid £27 but inculiding £2 which waiter kept for hiself.
so £25+£2+£3=£30
• Miss Andrew and Miss Dengure, Nether Hall Learning Campus
•
• We found this starter extremly good and the children really enjoyed it and quoted " This was the best starter.
• Elder Class, Petersfield
•
• The Pound is still there because:
£30 - £1x3 = £27.00
£27.00 - £2.00 = £25.00

So the restaurant gets
£25 + £2.00 (waiter)
Guests get £3.00
Totals £30.00

We thought this was a good starter because it got us thinking!!!
•
• The pound has not gone any where because if the people paid 10 pounds each,the total was 30 pounds.
But the waiter came back with 5 pounds meaning they only paid 9 pounds.
this is the sum.
Waiter=2 pounds,
People=3 pounds,
Bill= 25 pounds.
• MWS student, Home
•
• 10x3=30 right and the waiter gives back 1 pound to each person so thats 9x3 which is 27 plus the 2 pounds the waiter took which is £29. but it's really 30-3=27 and 27-2 so the pound hasn't gone anywhere it's to do with the times tables i think.
• Eddy, Huntcliff school
•
• So what are we looking for? the 30 pounds spent or should we look for the 25 given for the meal:
each pay £9 (9x3=27) and the waiter keeps £2 so the sum should realy be £27 - £2 = 25 which is what the restaurant got!!!
real good staters though
Thanks
• Farnborough hill Hampshire , year 8 set 1
•
• they added the 2 twice in both 9x3 and the 27+2. They should have written 27+3 because they paid for the waiters *tip*
Original cost=£30
Returned £3
Waiter kept £2
therefore they paid £27, instead of £25-the evil waiter stole £2.
THEIF!!!!!
woooooooooooooohoooooooooooooo
• Mrs O'Hagan's 2nd yr class, Holyrood RC Sec , Glasgow
•
• £25 + £3 + £2 =£30

THE WAITER SHOULD BE FIRED!!
• year 8 top set, Brockington College
•
• Charlotte says no it doesn't make sense, but we liked it anyway. More like this please.
• Simran Kaur Kainth, Holyrood RC Secondary School
•
• The one pound is still there because the waiter =£2
people =£3
bill =£25

so they all add up to £30.there we have it there was no one pound missing after all!
• Ghaya Rashid, Northbury Junior School
•
• I think he kept the pound for him self!
• Godwin Amarikwa-Obi, St. Mary's COE School, Hendon, London
•
• I really enjoy using your starter of the day with my students. The missing pound one was a very hard nut to crack.
• Lewis, School
•
• The money was added up incorrectly. They payed £27. The waiter kept £2 pounds but these 2 pounds are in £27 already we should add the £3 they were given back!
• Tirzah Cooper,
•
• £30 - £3=£27 so the waiter kept £2.The 3 people have £1. £27 - the £2 that the waiter has = £25. There is no missing coin. The 3 people have £1 each so £25 + £3 = £28. The waiter has £2 so £28 + £2 = £30.
• Cassie Heart Of The National Forest &+ Jack, Castlerock High School!
•
• Right
whilst one was calculating the sum, one come across a discovery that there was no missing pound.
2+3=5
£25 SPENT
25+5=30.
So therefore there was no missing pound sterling.
• Mr P, Dorset
•
• The bill totalled £25 so the customers would each have paid £8.33r NOT £9. So each would get £1.67 change
It is not the waiter's right to keep any money left over but the customers really should have given a tip. However, if the waiter was that dishonest I wouldn't tip him at all!
No missing pound, just customers out of pocket and a cheeky waiter not declaring his tips.
•

• Mr C, Bexhill HIgh
•
• Initially they paid £30, the waiter kept £2 so in fact they paid £28 - £3 returned = £25. There is no missing £1.
• Yr 6/7 Class, Stirling East PS
•
• We really enjoyed the question. It provoked a lot of discussion.The waiter's 2 pounds should be taken not added and 27 pounds subtract 2 pounds is 25 pounds which was the original bill.
• Farhana Khatun, Azhar Academy 9B
•
• The sum which says 27+2 is wrong it is suppossed to be 27-2 which equals 25 so inturn the sum is wrong.
• Mr Barker's Year 7 (Best Set?), William Farr Welton
•
• The £30 was the Red Herring. The bill was actually £25 plus a £2 tip totalling £27 so they paid the correct amount. Usually the tip should be 10% so if I was the witer I would be looking for an extra 50p!
• Becky Taylor, Crookfur Primary
•
• If they were ment to pay £25 and then they got £3 back that would be £28
then add the £2 the waitor got is £30.
• Gwindley, St Peter In Thanet Ce Junior School
•
• £8.33 per person for meal ( £25/3)
£0.67 per person went to waiter = £2.00 (3 x £0.67)

£1.00 per person returned =£3.00
total £25 +£3 + £2= £30
Meal costs £8.33 per person not £9 as stated in question.
• Jonathan, St Aidans C O E Tech College
•
• The restaurant gets £25
the people altogether get £3
and the waiter gets £2
add them together and you get £30.
•
• Easy you minus the £2 from 27
why on earth would you add it to make £29
30 - 3 + 2 will always leave you a pound short
100 - 3 + 2 = 99 (1 pound short)
50 - 3 + 2 = 49 (1 pound short)
5 - 3 + 2 = 4 (1 pound short)
the answer is 30 - 3 - 2 = 25.
• Cherie Glennon, Brooksbank School
•
• If you approach the problem from the other direction or backwards i.e.
The 3 men have paid £25 in total for the meal. They then recieve £1 each
making £28 and the 'dumb waiter' gets £2 = £30.
• Shannon Lohan, Stevenagw
•
• Initially they paid £30, the waiter kept £2 so in fact they paid £28 - £3 returned = £25. There is no missing £1.
• Jack McNeill,
•
• In fact, you shouldn't have added 2, you should have subtracted 2, to get to 25, the cost for the meal!
• Why On Earth,
•
• Easy you minus the £2 from 27
why on earth would you add it to make £29
30 - 3 + 2 will always leave you a pound short
100 - 3 + 2 = 99 (1 pound short)
50 - 3 + 2 = 49 (1 pound short)
5 - 3 + 2 = 4 (1 pound short)
the answer is 30 - 3 - 2 = 25.
• Sebastian Calvert, East Marden PS, SA
•
• This had me thinking for a while! I finally thought, " Well, the bill is \$25, and the customers have three and the waitor has two! Why didn't I think of that before?"
Anyway, It's nice to have a strategical starter once in a while.
• Gabie, Dorset
•
• £5 is returned so each of the costumers get £1 each back and there is 3 costumers so thats £3 back and the waiter kept £2. so £25+£3+£2=£30.
• Will,
•
• The waiter gave back £3 pounds so that is £3 of the £5 gone so 30-3=27
and he keeps £2 so 27+2=29 but you gave the men £3 so the pound that is not counted the man has.
• Alan Courtenay, Leytonstone
•
• For a bill of £25 each person should have paid £8.33, but they paid £9 each. This means that each person paid an extra 67p (just under) and this times 3 gives the £2 that the waiter kept. What's the problem?
• Miriam, Lewisham
•
• If people keep adding up the total from another angle, it will always add up to £30, but that's not the question. From this angle it doesn't make sense because there is an error in the calculations. You're adding an extra £2, to the total. The £9 that each of the people paid already includes the £2 that the waiter took. 9x3=27 and that 27 is made up of th2 £25 that the hotel has and the £2 that the waiter has. Now all you have to do is add the amount left, the £3 given to each of the men by the waiter, which makes £30.
• JOHN BARNES, JPBARNESJR@MSN.COM
•
• Suppose the three gentlemen who initially put down \$10.00 each take one of the 5 one dollar bills and leave the restaurant.
Have they not parted with a total of 9.00 each?
Isn't a total of \$2.00 left on the table?
Don't we still have a dollar missing?
• Class Haw, Woodfarm Junior School
•
• This had us going for an hour or so. The waiter had been given 5 pounds to give to the people and he gave them 1 each so he has 2 and he has accidently given 1 person 2 pounds.
• Lizzie Griffiths, Kenley
•
• You're all adding back on... you should be taking away.
30 takeaway 3 (for each person) equals 27, then takeaway the final £2 that the waiter keeps makes £25 which is the total of the meal! job done!!! (stop adding and start taking away!!!) .
• Alma Reasoner, Abc, El Salvador ,Central America
•
• There is no missing pound! She took 2 pounds for herself and gave 1 to each person=1+1+1=3 2+3=5 And 5 is the pounds she took to the people.
• Catherine, Chester
•
• Where is the pound £30/3 = 10
£30-5=25
5-2=3
10(each) -1 (returned) =9
9x3=27 +2 =29
Where is the pound????? you can work it one way but not the other!!!!!!!!!!
• Sambhaw Jain, Delhi
•
• Breaks the pound 25 .
1) 24+1
2) 24/3= 8
3) 8 pound is given by 3 men.
4) return 1 pound then they gien by 9 pound
so 27 +2 =29 so 1 pound is included in 25 pound.
• Pupil From Raglan Schools, Ella A
•
• Each person [A B C] pay £10 pounds to pay the bill of £30.
Later the chef realises that the bill only should have been £25
So he sends the waiter with £5 to give them back.
The waiter taks £2 and gives the people [A B C] £1 each.
So the money the waiter took, plus the money he gave back to the people= £2+£1+£1+£1=£5
There is no pound missing.
• Tia Brown, St Petrocs School, Bodmin
•
• That was really difficult! !!!!!!!
You have to think about it in a different way. You have to add instead of times.
30-5=25
People=£1x3 £3
Waiter=£2
2+3 =5
25+5=30.
There is no missing pound!
•

• Y6, Teacher
•
• 30-10-10-10=00 but there was a mistake so the waiter gives each person 1 pound and he keeps the 2 pounds so the calculation should be
£1=1 person + £1= 2nd person + £1= 3rd person + £2= the waiter= £5.
• EmilyHannah, Kent
•
• Where it says £27.00, that's only 2 people when there is meant to be 3. So then the waiter keeps the £2.00 and that makes £5.00 which is the change.
• Kira, FMS
•
• The three people pay £30 (£10 each).
The waiter takes back £5
Each of the three people are given £1 (Which is a total of £3 out of the £5)
The waiter keeps two pounds (Which is £2 out of the remaining £2)
There was £30
Minus £3 that was given back to the visitors
£27
Minus £2 the waiter kept for himself
£25.
• Lock, Lock
•
• \$25+3 from the waiter to the people and the waiter keeps \$2.
\$25+\$3=\$28
\$28+\$2=\$30.
• 8L1, North Yorkshire
•
• Pippa thinks they need to get a new waiter.
We all agree that they should have taken the £2 away rather than adding it.
• Heenu, Batleey
•
• Firstly the addition and subtraction were done at the wrong time. The pound was gone in the money that the people used to pay for the meal.
• Les, UK
•
• This is an old (have seen versions quoted in shillings) but fascinating conundrum because of its apparent simplicity but underlying complexity. There is NO problem with the commonsense real-world 'solution', which is what most posters are addressing: The cashier has £25 in the till after giving £5 of the £30 back, the waiter has £2 in his pocket and each of the diners have £1 each, so obviously £25 + £2 + £1 + £1 + £1 = £30. However, working it the other way, from the diners' subjective point of view, it IS true they paid £10 each, got £1 back, therefore paid £9 each, added to the waiter's ill-gotten gains of £2, (3 x £9 + £2) which comes only to £29. I was just wondering if anyone can give a proper mathematical explanation.
I'm afraid I am the opposite of a mathematical genius - a digital dunce? - but I think the true explanation lies in some fascinating interplay between the real world of discrete numbers, where a pound is a single, complete pound and the abstract world of mathematics and notional fractions, not translatable into cold hard cash. It may come down to the fact they did not pay £9 each - their actual transaction with the restaurant (minus the unofficial £2 'tip') - comes to £28 so divided by 3, they each paid, in effect, £9.333...{repeating}, plus £0.666...{repeating} towards the tip. Those fractions don't resolve in mathematics but do in the real world and so seem to disappear or are conflated in the figures.
So there is a genuine counter-intuitive conundrum still to be explained. I was just wondering if someone with a mathematical background could find this page and demonstrate an explanation mathematically or at least show a commonsensical way of looking at it that would have the penny - or the pound - dropping for me.
• Les, UK
•
• Sorry, my last post was a bit off - the actual transaction with the restaurant is £25 divided by 3 (that is, less their £3 refund), making £8.333...{repeating}. However, my point is that there is some sleight of hand involved in it all because it does logically add up and yet can't be right and I just wonder where that sleight of hand resides. This Wikipedia explanation:
http://en.wikipedia.org/wiki/Missing_dollar_riddle
again is a re-statement of the commonsense explanation but shows the figures as \$8-33 x 3 which obviously adds up to £24-99 and 3 x \$0.66 which obviously adds up to \$1-98. I just wonder does the sleight of hand reside in the unresolvable fraction that has meaning in the mathematical world but does not apply to the real world of discrete entities.
• A Person, A Place
•
• The pound is still there. The £5 is made by the 2 pounds (kept by the waiter) and 3 pounds (which was because the waiter gave each of them 1). I found this problem in a Sherlock Holmes Book and it says not to let other people word a question for you.
• Nick Steblay, USA
•
• This is a deceptive problem. The issue is in the final assumption that adding the resulting payment to what the waiter kept should be equal to the original 30 dollars. That is a false assumption. The resulting dollars spent are the only dollars in play and that should be compared to what the restaurant and waiter kept. If you want to compare to 30 you need to add 3 to each side of the equation: ((10 - 1) * 3) + 3 = 2 + 25 + 3. This is counter intuitive. Instead intuition says (10-1) * 3 + 2 = 30 which is always false. Interesting how the brain works and the cognitive dissonance this problem produces.
• Steve Rackley, Norwich, UK
•
• If you take the waiter out of the equation (excthe pun) - just focus on the customers and their bill to start with. £30-£25= £5 change = £1.67 (per person).
Now think of the customers giving the waiter a tip, they decide to pocket a pound each, and give him the rest.
So they retain 3x£1, the remainder (3x£0.67) goes to the waiter.
3x£0.67 is £2 :)
£25 bill, £3 change, £2 tip
The weird thing is that £1-£0.67=£0.33, and £0.33x3 =£1 - That's the maths behind the missing pound.
So, it's been there all along.
All rounded to 2dp.
•
• This created a lot of discussion in my S3 class the other day. Good wee brain teaser courtesy of @Transum
• Jamie, New Zealand
•
• If I borrow £50 off my mum, £50 off my dad, that's £100.
I buy a shirt for £97, and have £3 change.
I give £1 to my mum, £1 to my dad and keep £1.
I owe my mum £49, and my dad £49, together that's £98, plus the £1 I kept is £99.
Where is the other £1?
• Sarah V, Myddelton College
•
• This was great - I was looking for a starter for today's lesson on ' Identifying Miseading Graphs' , so this misleading information just the right tone!
• Willow 2, Horsell Junior
•
• We worked out that the real bill was £30 minus a £2 tip. Therefore each person payed £9.33. The missing pound is in the 0.33 x 3.

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.

Previous Day | This starter is for 19 June | Next Day

Danny Baker on his wonderful BBC radio 5 live programme suggests a unique solution to the missing pound puzzle.

Danny Baker's Missing Pound Solution

Danny Baker Show Transum Podcast Transum Podcast RSS

The final paragraph of the story should read:

The waiter kept £2
£27 − £2 = £25, the correct cost of the meal.

Missing Baht

Here is a similar puzzle from Thailand: "You borrow money from your Dad (500 baht) and your Mom (500 baht) to buy a phone that costs 970 baht. You then you have 30 baht change from the shop so you return 10 baht to Dad and 10 baht to Mom and you keep 10 baht yourself. But 490 + 490 = 980 and the 10 baht that you keep totals 990 baht. Where is the missing 10 baht?"

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Here is the URL which will take them to a related student activity.

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Extension

A man had an apple stall and he sold his larger apples at 3 for a pound and his smaller apples at 5 for a pound.

When he had just 30 apples of each size left to sell, he asked his son to look after the stall while he had lunch. When he came back from lunch the apples were all gone and the son gave his father £15.

The father questioned his son. "You should have received £10 for the large apples and £6 for the 30 small apples, making £16 in all."

The son looked surprised. "I sold them all at the average price of 2 small and 2 large for £1. Four into 60 goes 15 times so I am sure £15 is correct.

Where is the missing pound?

This extension is adapted from a puzzle in Amazing Brain Teasers by Erwin Brecher

The average cost of the large apples is £1 ÷ 3 = 33⅓p.

The average cost of the small apples is £1 ÷ 5 = 20p.

So the 2 small and 2 large apples should have been sold for
33⅓p + 33⅓p + 20p + 20p = £1.06⅔
to earn the £16

For Students:

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