Table of Contents
Understanding the concept of alligation medial is essential for solving problems related to mixture and concentration. This article provides a variety of practice problems with different types and detailed solutions to help students grasp the concept effectively.
Introduction to Alligation Medial
Alligation medial is a method used to find the average value or concentration when mixing two or more quantities with different values. It is particularly useful in problems involving mixtures of liquids, solutions, or solids with varying concentrations.
Basic Concept and Formula
The alligation medial rule involves comparing the quantities and their concentrations to determine the average concentration of the mixture. The key formula is:
Average concentration = (Sum of individual quantities’ contributions) / Total quantity
Practice Problems with Solutions
Problem 1: Mixing of Two Solutions
A chemist mixes 10 liters of a 30% acid solution with 20 liters of a 50% acid solution. Find the concentration of the resulting mixture.
Solution:
- Quantity of first solution (Q₁) = 10 liters, concentration (C₁) = 30%
- Quantity of second solution (Q₂) = 20 liters, concentration (C₂) = 50%
Total quantity = Q₁ + Q₂ = 10 + 20 = 30 liters
Contribution of each solution to the acid content:
- First solution: 10 × 30% = 3 liters of pure acid
- Second solution: 20 × 50% = 10 liters of pure acid
Total pure acid = 3 + 10 = 13 liters
Concentration of the mixture:
= (Total pure acid / Total quantity) × 100 = (13 / 30) × 100 ≈ 43.33%
Problem 2: Alligation of Three Solutions
A farmer has three types of water: one with 10% salt, another with 20% salt, and the third with 30% salt. He wants to prepare 100 liters of a mixture with 20% salt. How much of each type should he use?
Solution:
- Let x liters of 10% salt solution
- Let y liters of 20% salt solution
- Let z liters of 30% salt solution
Total volume: x + y + z = 100 liters
Desired salt content: 20% of 100 liters = 20 liters
Equation for salt contribution:
- 0.10x + 0.20y + 0.30z = 20
- x + y + z = 100
Express z in terms of x and y:
z = 100 – x – y
Substitute into the salt equation:
0.10x + 0.20y + 0.30(100 – x – y) = 20
0.10x + 0.20y + 30 – 0.30x – 0.30y = 20
Combine like terms:
(0.10x – 0.30x) + (0.20y – 0.30y) = 20 – 30
-0.20x – 0.10y = -10
Multiply through by -1:
0.20x + 0.10y = 10
Multiply entire equation by 10 to clear decimals:
2x + y = 100
Express y in terms of x:
y = 100 – 2x
Calculate z:
z = 100 – x – y = 100 – x – (100 – 2x) = 100 – x – 100 + 2x = x
Choose x such that y and z are positive:
For y ≥ 0: 100 – 2x ≥ 0 ⇒ x ≤ 50
For z = x ≥ 0: x ≥ 0
Thus, x can vary between 0 and 50 liters. For example, if x = 20 liters:
y = 100 – 2(20) = 60 liters
z = x = 20 liters
Conclusion
Alligation medial is a powerful tool for solving mixture problems efficiently. Practice with various types of problems to strengthen your understanding and improve problem-solving speed.