A businessman bought two types of rice at the rate of Rs. $120/\text{kg}$ and Rs. $220/\text{kg}$. How much rice (in kg) of the first type should be mixed with $72\text{kg}$ of rice of the second type so that the resulting mixture can be sold at Rs. $280/\text{kg}$ to earn a profit of $40\%$?

Let the cost price of the mixture be $C_m$.Given that by selling the mixture at Rs. $280/\text{kg}$, the profit is $40\%$:
$$\text{Selling Price (SP)}=C_m\times \left(1+\frac{40}{100}\right)$$
$$280=C_m\times 1.4$$
$$C_m=\frac{280}{1.4}=200\text{Rs./kg}$$
Now, apply the rule of alligation:Cost price of Type 1 rice ($C_1$) = Rs. $120/\text{kg}$ Cost price of Type 2 rice ($C_2$) = Rs. $220/\text{kg}$ Mean Cost Price ($C_m$) = Rs. $200/\text{kg}$ $
$\text{Ratio of Type 1 to Type 2}=\frac{C_2-C_m}{C_m-C_1}=\frac{220-200}{200-120}=\frac{20}{80}=\frac{1}{4}$$LetthequantityofType1ricerequiredbe$ x\,\text{kg}$.ThequantityofType2riceisgivenas$ 72\,\text{kg}$.$$\frac{x}{72}=\frac{1}{4}$ $
$$x=\frac{72}{4}=18\text{kg}$$
Hence, $18\text{kg}$ of the first type of rice should be mixed.