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Math beginner here. Say I have the following items

Marsbar: sugar 100/lb, salt 5/lb, price $4/lb, inventory: 40lb

Twix: sugar 100/lb, salt 9/lb, price $7/lb, inventory: 9lb

Bounty: sugar 105/lb, salt 4/lb, price $3/lb, inventory: 10lb

and I need to make a 3 lb blend of these items to the lowest price, where the minimum criteria for the blend is:

sugar: 101/lb, salt: 6/lb

Does anyone know how to solve this or can maybe give me a hint, and potentially with more properties than sugar and salt?

acrmuui
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  • @G.Sassatelli I believe it is $3$lb, not $31$, so $x+y+z=3$. Also, the last two equations are inequalities as $101$ and $6$ per pound are minimum values. – Dave Aug 01 '17 at 20:54
  • @G.Sassatelli Sorry if that is unclear, it is supposed to say 3 lb. – acrmuui Aug 01 '17 at 21:00

1 Answers1

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Words to equations: $$\begin{cases}100m+100t+105b\geq 101\times 3\\5m+9t+4b\geq 6\times 3\\m+t+b=3\end{cases}$$ where $m$ is the pounds of Marsbars, $t$ for Twix, and $b$ for Bounty. We want to minimize $P=4m+7t+3b$, where $P$ is the total price. The above system yields $b\geq\frac{3}{5}$ and $t\geq\frac{9}{10}$ and $m\geq\frac{3}{2}$ by simple rearranging and substitution. Hence, the minimum solution occurs at $(m,t,b)=\left(\frac{3}{2},\frac{9}{10},\frac{3}{5}\right)$ with a price of $\$14.10$.

Dave
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  • Thanks a lot for that! Now that I see your equation I think I know how to do the rearranging and substitutions, but how do I minimize 4m + 7t + 3b in that process to get your result? – acrmuui Aug 01 '17 at 21:40
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    The constraints come from the system, as if you increase any one of the quantities you add more cost. – Dave Aug 01 '17 at 21:46
  • Ah yes. But what I dont think I understand is, when you have a combination of the candybars' properties and the minimum criteria for the blend, where many different blends would meet the minimum criteria (which to my understandig solving the first three line equation in your answer will show?), then how do i go from there to knowing which of the many solutions is the cheapest? – acrmuui Aug 01 '17 at 22:04
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    Effectively, we are trying to minimize $P$ with the constraints $$\begin{cases}5b\geq 3\4t-b\geq 3\end{cases}$$by using $m=3-t-b$. Notice that the only corner point of the above system in the $bt$-plane is at $(b,t)=\left(\frac{3}{5},\frac{9}{10}\right)$. So this is where the minimum of $P$ occurs. – Dave Aug 01 '17 at 22:29