Question

The Archer family raises cattle on their farm in West Midlands. They also have a large garden in which they grow ingredients for making two types of relish - SauceA and SauceB. These they sell at local stores.

The profit per kilogram of SauceA is £4 and the profit per kilogram of SauceB is £4. The ingredients in each relish are cabbage, tomatoes, onions, and oil. One kilogram of SauceA must contain at least 65% but no more than 73% cabbage, and at least 6% onion, and at least 8% oil. One kilogram of SauceB must contain at least 60% but no more than 75% tomatoes, and at least 7% onion, and at least 7% oil. Both relishes contain no more than 11% onion and no more than 10% oil.

The family has enough time to make no more than 870 kilograms of relish. They know also that they will sell at least 40% more SauceA than SauceB. They will have this year 390 kilograms of cabbage, 370 kilograms of tomatoes, and 130 kilograms of onion. They can use any amount of oil needed.

What is the maximal profit that the family can gain by producing and selling the relish?

Answer 1

The archer family will have a total of 870 kgs of relish to sell. As we know that sauce A will have 40% more sales than sauce B so we will divide 870 by 2.4 (1.4 for sauce A and 1 for sauce B) to get the individual kgs of sauce that the family expects to sell.

We get 362.5 kgs. so using our multiple (1.4 for sauce A and 1 for sauce B): the archer family expects to sell 507.5 kgs of sauce A and 362.5 kgs of sauce B.

Now we will get the bifurcation of the ingredients that must be used in sauce A and B respectively.

Particulars	Cabbage	Tomatoes	Onions	Oil	Total
Sauce A	370.475	0	64.75	72.275	507.5
Sauce B	0	271.875	65.25	25.375	362.5
Total	370.475	271.875	130	97.65	870

The above table as made as per the conditions given in the question. As we have no maximum limit for onion we can increase its content. We also have the freedom to increase the oil content as well but as we have produce we can utilize it to the maximum.