Question

How many kilograms of tea worth $300$ rupees per $\mathrm{kg}$ should be mixed with $10\mathrm{kg}$ of tea worth $240$ per $\mathrm{kg}$ to produce a mixture to cost $260$ per $\mathrm{kg}$?

Answer 1

Use the rule of allegation

Given, cost of expensive tea, ${\text{CP}}_{\mathrm{E}}=\mathrm{Rs}.300$ per $\mathrm{kg}$
Cost of cheaper tea, ${\text{CP}}_{\mathrm{C}}=\mathrm{Rs}.240$ per $\mathrm{kg}$
Weight of cheaper tea in the mixture $m_{\mathrm{C}}=10\mathrm{kg}$
Price of the mixture, ${\text{CP}}_{\mathrm{M}}=\mathrm{Rs}.260$ per $\mathrm{kg}$
Let $x$ be the quantity of the expensive tea needed

By the rule of allegation,
$\frac{\text{Quantity of cheaper variety}}{\text{Quantity of costlier variety}}=\frac{\text{Price of costlier variety}-\text{Mean price}}{\text{Mean price}-\text{Price of cheaper variety}}$

Thus,
$\begin{array}{cc}& \frac{m_C}{x}=\frac{{\text{CP}}_{\mathrm{E}}-{\text{CP}}_{\mathrm{M}}}{{\text{CP}}_{\mathrm{M}}-{\text{CP}}_{\mathrm{C}}}\\ \Rightarrow & \frac{10}{x}=\frac{300-260}{260-240}\\ \Rightarrow & \frac{10}{x}=\frac{40}{20}\\ \Rightarrow & x=5\mathrm{kg}\end{array}$

Hence, $5\mathrm{kg}$ of tea worth $300$ rupees per $\mathrm{kg}$ should be mixed with $10\mathrm{kg}$ of tea worth $240$ per $\mathrm{kg}$ to produce a mixture to cost $260$ per $\mathrm{kg}$