A gardener has supply of fertilizer A which consists of 10\% nitrogen and 6\% phosphoric acid and fertilizer B which consists of 5\% nitrogen and 10\% phosphoric acid. After testing the soil conditions, he finds that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for his crop. If fertilizer A costs ₹ 10.60 per kg and fertilizer B costs ₹ 8.40 per kg, what is the minimum cost at which the farmer can meet the nutrient requirement by using a combination of both types of fertilizers?
(a)₹ 1488
(b)₹ 1576
(c)₹ 1648
(d)₹ 1732
Answer
Answer (as printed): D
Explanation
Let the quantity of fertilizer A required be $x \mathrm{kg}$ and that of fertilizer B be $y \mathrm{kg}$. Then, $$\begin{aligned} & 10 \% \text { of } x+5 \% \text { of } y=14 \Rightarrow 0.1 x+0.05 y=14 \\ & 6 \% \text { of } x+10 \% \text { of } y=14 \Rightarrow 0.06 x+0.1 y=14 \end{aligned}$$ Subtracting (ii) from (i), we get: $0.04 x-0.05 y=0$ $$\Rightarrow \quad x=\frac{5}{4} y .$$ Substituting $x=\frac{5}{4} y$ in (i), we get: $0.5 y+0.2 y=56$ $$\Rightarrow \quad 0.7 y=56 \Rightarrow y=80 .$$ Putting $y=80$ in (ii), we get: $x=100$. $$\therefore \quad \text { Required cost }=₹(100 \times 10.60+80 \times 8.40)=₹ 1732 .$$