# Finding numbers such that $a+b+c+d=abcd=£7.11$ [duplicate]

Possible Duplicate:
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

It would be great if you please specify what are $a,b,c,d$ here. – Tapu Nov 8 '11 at 1:03