Container $A$ contains a milk and water solution in the ratio $4 : 1$, while Container $B$ contains milk and water in the ratio $2 : 3$. To obtain a mixture containing $60\%$ milk, the volumes from Container $A$ and Container $B$ must be mixed in the ratio $x : 1$. The value of $x$ is ________ (round off to 1 decimal place).
Answer Key:
1.0
Step-by-Step Solution & Explanation
1. Concentration of milk in Container A $= \frac{4}{4+1} = \frac{4}{5} = 0.80$ ($80\%$).
2. Concentration of milk in Container B $= \frac{2}{2+3} = \frac{2}{5} = 0.40$ ($40\%$).
3. Desired concentration $= 60\% = 0.60$.
4. Applying alligation rule:
$\frac{V_A}{V_B} = \frac{0.60 - 0.40}{0.80 - 0.60} = \frac{0.20}{0.20} = 1$
Therefore, the volume ratio is $1 : 1$, giving $x = 1.0$.