Pistachios, Cashews, Hazelnuts, and Almonds

Algebra Level 5

You survey the cost of bulk nuts from a wholesale outlet. You find that Pistachios cost you $19 / kg; Cashews are at $26 / kg; while Hazelnuts cost $34 / kg, and Almonds cost $15 / kg. You want to make a mixture of these nuts in certain proportion and sell the mixture at competitive pricing. To make the best selling price, you want to minimize the cost of the mixture per unit weight, but there are constraints. The minimum content of any kind is 15%, and the maximum content is 30%. Under these conditions, what is the minimum possible cost of a mixture per kg in dollars ?

The answer is 21.8.

2 solutions

Zee Ell
May 16, 2018

This problem can be solved very easily, without knowing anything about the simplex method (e.g. by a good Grade 8 student), just with a little common sense and a simple calculator (or even without a calculator, let alone any online application).

The mix has to include at least 15% of each ingredient, that is 60% altogether. Now it is easy to see, that we will get the cheapest mix, when the remaining 40% is the cheapest, so we take as much as we can from the cheapest ingredients: another 15% (30% altogether, the maximum) from the cheapest, an additional 15% from the 2nd cheapest as well and the remaining 10% from the 3rd cheapest.

Hence our formula:

$0.15*(19+15+26+34)+0.15*15+0.15*19+0.1*26 = \boxed { 21.80 } \text { dollars per kg}$

Mark Hennings
May 16, 2018

We need to apply the Simplex method to solve the system of needing to minimize $X = 19p + 26c + 34h + 15a$ subject to the constraints $p+c+h+a = 100$ and $15 \le p,c,h,a \le 30$ . Happily, there are online implementations of the Simplex method, so we do not have to do any work!

The minimum value of $X$ is $2180$ , achieved when $p=30$ , $c=25$ , $h=15$ and $a=30$ . Thus the minimum cost is $\boxed{21.80}$ dollars per kg.

Pistachios, Cashews, Hazelnuts, and Almonds

The answer is 21.8.

2 solutions

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