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December 6th, 2018, 04:45 AM  #1 
Newbie Joined: Dec 2018 From: UK Posts: 2 Thanks: 0  Help with deriving formula
Can you help to derive a formula that can calculate b2, b3, b4 where T = a1(b1) + a2(b2) + a3(b3) + a4(b4) a1=a2=a3=a4=0.25 b1+b2+b3+b4=1 b1<b2<b3<b4 b1 = x1 T = y For example, one solution is b1=16%, b2=26%, b3=46% for Y = 2 and x1=12% 2 = 0.25(12%) + 0.25(16%) + 0.25(26%) + 0.25(46%) How to calculate b2, b3, b4 be when Y = 2 and x1 =14% 2 = 0.25(14%) + 0.25(?) + 0.25(?) + 0.25(?) 
December 6th, 2018, 05:53 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,301 Thanks: 1971 
y = T = a1(b1) + a2(b2) + a3(b3) + a4(b4) = 0.25(b1 + b2 + b3 + b4) = 0.25, not 2.

December 6th, 2018, 07:28 AM  #3  
Senior Member Joined: May 2016 From: USA Posts: 1,306 Thanks: 549  Quote:
Second, your notation is far more complex than it needs to be. Third, if you are asking for a solution of the following system $d + p + q + r = 1, \text { and } d < p < q < r.$ Once d is given, you have three unknowns, but only one equation. No unique solution is possible. If, however, you stipulate that $0 \le d < 0.25 = \dfrac{1}{4}$, you can bound p, q, and r. EDIT: Perhaps you are trying to work with this system. $0 < y,\ 0 \le 4x < y, \ p = \dfrac{x}{y},\ p + q + r + s = 1, \text { and } p < q < r < s.$ If x and y are given, p is determined, q, r, and s are bounded, and $y = py + qy + ry + sy.$ Last edited by JeffM1; December 6th, 2018 at 08:06 AM.  

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