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4 items add up to and multiply to 7.11 what are the value of the items?

This is a question from the nrich website. I think I might have solved it, but the numbers it produce are not exact. The problem is in pounds and so I am assuming you can round the number to the nearest hundredth of a pound (i.e. pence).

My progress so far is as such: $$a+b+c+d=abcd=711$$(I just changed £7.11 to 711 pence so that it is easier to work with) $$d={a+b+c\over abc-1}$$ $$\text{Thus } a+b+c<711$$ Since $abcd=711$, a, b and c should be prime factors of a number close to 711 (lets call it $\alpha$) and should add up to close to that number. (i.e. $\alpha <711, \alpha=a+b+c\approx abc$)

Lets say that $\alpha = 710$, $\space a=708,\space b=1,\space c=1$

$$d={708+1+1\over708\cdot1\cdot1-1}={710\over707}\approx1.0042$$ $$abcd=708\cdot1\cdot1\cdot{710\over707}=a+b+c+d=708+1+1+{710\over707}\approx711$$

The general equation I came up with is (where $P_1$ is the total price): $$a=P_1-3$$ $$b=1$$ $$c=1$$ $$d={P_1+1+1\over P_1 -1}$$

I am pretty sure that this method is incorrect and I don't know if all my assumptions are correct, but I haven't though of any better way. Does anyone have any suggestions?

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marked as duplicate by Gerry Myerson, Asaf Karagila, t.b., Henning Makholm, J. M. Dec 5 '11 at 18:19

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

The question makes no sense, dimensionally. For the sum to be a number of pounds, each unknown must be a number of pounds, but then the product has units of pounds-to-the-fourth. Let's ignore that. I imagine the unknowns are to be exact numbers of pence, which makes them factors of 711, so the place to start is by factoring 711 into primes and using that to make a list of all the numbers that go into 711. – Gerry Myerson Nov 8 '11 at 0:55
It would be great if you please specify what are $a,b,c,d$ here. – Tapu Nov 8 '11 at 1:03
Changing to pence would mean a+b+c+d=711 and abc*d=711000000. – GEdgar Nov 8 '11 at 1:14
Ops, I forgot I couldn't convert to pence. Thanks for pointing out – E.O. Nov 8 '11 at 1:28

1 Answer 1

up vote 4 down vote accepted

The numbers are meant to be pounds, not pence. Changing the numbers to pence would change the answer, since the equations are dimensionally inconsistent (as Gerry notes). This problem is solved in the rec.puzzles FAQ here:

I don't understand the custom of posting answers to elementary questions as comments, so I'll post my answer as an answer.

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That link shows the answer (and other nice stuff) but not the reasoning. may be a better place to go. Also – Gerry Myerson Nov 8 '11 at 4:51

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