How to share a cash prize among entries in position A, B and C. Each position can have multiple entries. Each entry in position A should have 50% more of the cash prize than each entry in position B. Each entry in position B should have 10% more than each entry in position C.
How can we calculate the prize share for an entry in position A, position B and position C ?
Let $a,b,c$ be the number of entries in A,B,C respectively. Then we know $ax_a + bx_b + cx_c = 1$ where $x_i$ is the prize share for an entry in position $i$. Given the other two constraints, $x_a = 1.5 x_b$ and $x_b = 1.1x_c$, we get $$cx_c + 1.1bx_c + 1.65ax_c =1,$$ $$x_c = \frac{1}{c+1.1b+1.65a}.$$ From there, clearly $$x_b = \frac{1.1}{c+1.1b+1.65a}$$ $$x_a = \frac{1.65}{c+1.1b+1.65a}.$$