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On Friday I posted this puzzle….

A carpenter is paid 2 pounds for each day that he works and forfeits 3 pounds for each day that he doesn’t work. After 30 days he has paid out exactly as much as he has received. How many days did he work?

If you have not tried to solve it, have a go now.  For everyone else the answer is after the break.

The carpenter worked on 18 days and didn’t work on 12 days

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. khaled says:

he worked 15 days……..

2. mgm75 says:

I worked it out like this:
2*3 = 6

30/6 = 5

Therefore just work out the ratio from that
I tried 3:2 first which is the right answer

6*3 = 18 (his wage) *2 = 36
6*2 = 12 (his deductions) *3 = 36

So he worked 18 days and took 12 off. Lazy sod 😛

• Alex says:

Mgm75,
I see you tried 3:2 ratio. It’s a pure guess. This kind of math problems is solved by translating literal problem description to equations. See Michael Scharf solution.
Guess doesn’t guarantee the solution but correct algebra application always solves this type of problems.
Good luck next time.

• mgm75 says:

It wasn’t a guess.

If 3:2 wouldn’t have worked, I’d have tried 2:3 and 4:1. Either way, I got the right answer and the working was going to find the right answer based on the numbers supplied.

3. Anne Elk says:

I guessed 30 days as working days at random and then worked backwards, using a Post-It note. When I read one of the spoilers on Friday I almost slapped my forehead. It was dead obvious to do it as a ratio (even though that twat got the numbers round the wrong way!). Took a couple of minutes to get 18 working days.

4. repton says:

On day 0 he realizes what a crummy job he’s got, and quits.

5. LazyT says:

I’d negotiate a better deal.

6. Dave's cousin says:

Less than a minute

7. w is the number of days he works
n is the number of days he does not work

w*2 – n*3 = 0

or

2w – 3n = 0

The total number of days is 30

w + n = 30

=> n = 30 – w

2w – 3(30-w) =0

2w – 90 + 3w =0

5w = 90

w = 18
n = 12

• Alex says:

Michael,
You transfer conditions described in this problem to math equations the right way. Good job.

8. Eddie says:

Could you explain how you get from ‘w + n = 30’ to ‘n = 30 – w’?

• Anonymous says:

Minus ‘w’ from each side to make ‘n’ the subject.

9. SpiderChops says:

I got (w=work, r=rest):

2w = 3r (since they negate each other)
w+r = 30 (since work and rest total 30 days) –> w=30-r

2(30-r) = 3r
60-2r = 3r
60 = 5r –> r = 12 –> w = 30 – 12 = 18

10. SpiderChops says:

Another way without algebra is that for the conditions stated to be correct he must have rested for 2 days (cost 6 pounds) to every 3 days worked (gain 6 pounds) within 30 days. Therefore each cost-neutral block of time is 5 days and there are 6 of those in 30 days. So he spent 6 lots of 2 days resting (12 days) and 6 lots of 3 days (18 days) working.

11. Barry Goddard says:

All solutions so far assume he has a seven day a week contract to play or pay.

If the contract does not apply at weekends then we need to know when the first Monday is so we can do the right arithmetic calculations.

This is made worse if he is not allowed to work Sundays but must pay for the privilidge.

• Lord Manley Fan Club says:

Barry,

The word “week” does not appear in the question, so I would challenge the relevance of making assumptions about weeks, seven days or otherwise.

There is an old Sufi saying that you might find helpful, which is:-

• Barry Goddard says:

@Lord Manley Fan Club

You are right that week is implied in the question and not explicitly meantioned.

If we limit ourself to things in the question then we cannot have an answer of 12 and 18 as they are not meantioned in the question either.

You cannot have it both ways, Lord Manley Fan Club

• Lord Manley Fan Club says:

Barry

Apologies for the use of intemperate shale related language.

I am not “right that week is implied in the question…” The concept of a week is no more implied in the question than the concept of temperature or cheese is.

As for your comment that – “If we limit ourself to things in the question then we cannot have an answer of 12 and 18 as they are not meantioned in the question either.” – do you mean that the answer to a puzzle has to be explicitly stated in the puzzle to be accurate? Surely not.

It may not seem so, Barry, but I have again carefully considered your comment about a seven day week and just don’t understand its relevance to the puzzle.

If I am being stupid or obtuse please forgive me – being a member of the Lord Manley Fan Club it is axiomatic that I am not playing with a full deck.

• Anonymussel says:

I hate to disappoint your sense of fair play, Barry, but it’s perfectly normal for a carpenter to work 30 days straight without a break at a weekend. I know several of them. And builders as well — such as those who were working on my house on Sunday. No reference to a weekend is required, therefore, to solve this puzzle.

12. Bill T. says:

number of days worked = w. number of days didn’t work = 30 – w.
total pay for work days = 2w.
total penalty for non-work days = -3 (30 – w) = -90 + 3w
the sume is zero:
2w -90 + 3w = 0
solve for w:
w = 18

• Bill T. says:

as with the sume, the sum is also zero.

13. One Eyed Jack says:

When did middle school algebra become a puzzler?

• Eddie says:

When it’s simultaneous. Good point though.

14. Michele Goodman says:

Once I told my 12 year-old that pounds is like a dollar in the US, he got 18 in about a minute.

15. Jerry says:

If negotiating a better compensation package were as easy as the algebra the carpenter would be paid for a non-work day, but paid more for a work day. It doesn’t pay to take a holiday or become ill. The carpenter needs to unionize; or if he knows any chemistry, un-ionize.

16. Lisa Bella says:

Richard it Tuesday. Write post please. Thank you 🙂 (No rushing, just remind you)

17. h3rne777 says:

@Alex – I disagree that “guessing” is not a mathematical approach. “Trial and improvement” is taught in schools as a legitimate way of solving problems. In fact, the set of problems that are solvable with this method is very much larger (it’s a different order of infinity) than those we currently have analytical methods for. It forms the basis of arguably the most important field in engineering – finite element analysis – which relies on computers or more usually, super-computers. (I note that the most recent noble prize in chemistry was for this kind computational simulation of chemical reactions). Even at the business and finance end of the computing spectrum, simple algebraic curve-fitting is done this way _by_the_computer.

@Jerry – unionise = develop more layers? 🙂

• Jerry says:

@h3rne777

No – unionise is the UK spelling of unionize ( http://en.wiktionary.org/wiki/unionise ).
Yes – “Guessing” is a very legitimate way to solve complex problems, such as multi-particle Hamiltonians in quantum mechanics. You keep guessing until self-consistency is reached.

18. a says:

acha

19. dayuntoday says:

Wow. Really easy. Find the least common multiple and you can see that he breaks even every 5 days with 3 on, 2 off…then it’s simple multiplication. I can’t see why people used all those elaborate methods.