Kellogg is a new cereal formed of a mixture of bran and rice that contains at least $88$ grams of protein and at least $36$ milligrams of iron. Knowing that bran contains $80$ grams of protein and $40$ milligrams of iron per kilogram, and that rice contains $100$ grams of protein and $30$ milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs $\mathrm{Rs} 5$ per $\mathrm{kg}$ and rice costs $\mathrm{Rs} 4$ per $\mathrm{kg}$

A dietician wishes to mix together two kinds of food $X$ and $Y$ in such a way that the mixture contains at least $10$ units of vitamin $A$, $12$ units of vitamin $B$ and $8$ units of vitamin $C$. The vitamin contents of one $\mathrm{kg}$ food is given below:

One kg of food $X$ costs $₹ 16$ and one $\mathrm{kg}$ of food $Y$ costs $₹ 20$. Find the least cost of the mixture which will produce the required diet?

A fruit grower can use two types of fertilizer in his garden, brand $P$ and $Q$. The amounts (in $\mathrm{kg}$) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicates that the garden needs at least $240 \mathrm{kg}$ of phosphoric acid, at least $270 \mathrm{kg}$ of potash and at most $310 \mathrm{kg}$ of chlorine.

If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?