# A population (P) of worms in a field varies directly as the amount of water (W) and soil nutrients (S) and inversely to other populations (I) and toxic materials (T).  What will be the population when W= 500, S =30, I = 100, and T = 3 if we know that P= 100 when W=2000, S=50, I=200, and T=500? To be directly proportional, the relationship must follow

` ` to be inversely proportional the relationship must follow

` ` combining these relationships we get

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We can use the initial conditions to determine the value for k:

`k = (PIT)/(WS) = (100*200*500)/(2000*50) = 100` this gives us the general equation:

`P = 100(WS)/(IT)` This can now be used to answer the question:

`P = 100(500*30)/(100*3) = 5000`

Alternate solution:

Label given quantities with subscript 2 and new quantities with subscript 1.

`k = (P_1I_1T_1)/(W_1S_1)` and  `k = (P_2I_2T_2)/(W_2S_2)` by the transitive property of equality

`(P_1I_1T_1)/(W_1S_1) = (P_2I_2T_2)/(W_2S_2)` solving for `P_1` gives

`P_1 = (P_2I_2T_2W_1S_1)/(W_2S_2I_1T_1) = (100*200*500*500*30)/(2000*50*100*3) = 5000`

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