Suppose four horses – $A, B, C$, and $D$ – are entered in a race and the odds on them, respectively, are $6$ to $1$, $5$ to $1$, $4$ to $1$, and $3$ to $1.$ If you bet $\$1$ on $A$, then you receive $\$6$ if $A$ wins, or you realize a net gain of $\$5$. You lose your dollar if $A$ loses. How should you bet your money to guarantee that you win $\$12$ no matter how the race comes out?
— Problem 10, page 293 Fisher and Ziebur "Integrated Algebra and Trigonometry" 1957, 1958 by Prentice-Hall, Inc., Sixth printing June, 1961
I can't figure out how to make a comment because I don't have enough reputation – so I'm editing this question, but thank you cjferes for your help. I got my answer with much matrix manipulation. Your suggestion got me on the road to success! From what I can see you need to drop $\$228$ in bets to win back a guaranteed $\$12$. Very interesting!
Best Answer
Let $a$, $b$, $c$ and $d$ the bets on each horse.
HINT 1
Suppose the scenario where horse $A$ wins. Then, the net gain is given by: $$(6-1)a-b-c-d$$
Note it's $(6-1)$ because for each dollar on horse $A$, you have a net gain of $5$. On every other horse, you lose those dollars.
HINT 2
Also, we want the net gain to be equal to $12$, so $$(6-1)a-b-c-d=12$$
HINT 3
Use the reasoning of Hint 1 and Hint 2 to find the net gain in each scenario (horse $B$ wins the race, horse $C$, and horse $D$).
HINT 4
Now you should now have 4 equations... and you can solve the equation system as you like.
Good luck!