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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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    Ops, I forgot I couldn't convert to pence. Thanks for pointing out2011-11-08

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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.

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    That link shows the answer (and other nice stuff) but not the reasoning. http://answers.yahoo.com/question/index?qid=20070913141854AAH3Rqp may be a better place to go. Also http://www.casact.org/pubs/actrev/feb99/puzzle.htm2011-11-08