ALLIGATION & MIXTURE
Alligation and Mixture is a mathematical concept used to solve problems related to mixing two or more ingredients with different prices, concentrations, or values to obtain a mixture with a desired price, concentration, or value. This topic is often covered in quantitative aptitude for various competitive exams.
Key Concepts:
- Alligation Rule:
The alligation rule helps in finding the ratio in which two or more ingredients at given prices must be mixed to produce a mixture at a given price.
- Mean Price: The price of the mixture obtained by mixing two or more ingredients in a certain ratio.
- Cheaper Price: The price of the cheaper ingredient.
- Dearer Price: The price of the dearer ingredient.
The formula used is:
Ratio of Mixing =
- Types of Problems:
- Mixing Two Quantities: Find the ratio in which two quantities of different values must be mixed to get a mixture of a given value.
- Replacement Problems: If a part of the mixture is replaced by another quantity with a different concentration or value, find the final concentration or value of the mixture.
- Finding the Price of the Mixture: Given the quantities and prices of the ingredients, find the price of the resulting mixture.
- Problem Solving Techniques:
- Alligation Method:
- Identify the cheaper and dearer components.
- Apply the alligation rule to find the ratio of quantities to be mixed.
- Use the obtained ratio to find the required quantity or value.
- Weighted Average Method:
- This method involves calculating the average price, concentration, or value of the mixture by taking into account the weights (quantities) of the components.
Weighted Average=
- Step-by-Step Calculation:
- Step 1: Determine the values for the cheaper, dearer, and mean prices.
- Step 2: Use the alligation rule to find the ratio.
- Step 3: Apply the ratio to find the required quantities.
2.Types of Alligation Problems
a. Simple Alligation:
This involves mixing two components.
EXAMPLE: A shopkeeper has two types of sugar, one costing ₹20/kg and the other ₹30/kg. In what ratio should they be mixed to get a mixture costing ₹25/kg?
Solution:
Ratio= = = 1:1
This means the two types of sugar should be mixed in equal quantities.
b. Alligation with More Than Two Ingredients:
When more than two ingredients are mixed, the concept is extended by applying the rule iteratively.
EXAMPLE : Three varieties of tea costing ₹20/kg, ₹30/kg, and ₹40/kg are mixed to get a mixture worth ₹35/kg. If equal quantities of the first two are used, find the ratio of the third variety.
Solution:
- First, find the mean price of the first two varieties:
Mean Price of first two= = 25/kg
- Now, apply the alligation between ₹25/kg (mean price) and ₹40/kg (third variety) to achieve ₹35/kg.
Ratio = = = 1:2
So, the ratio of the first two types of tea to the third is 2:1.
c. Replacement Problems:
In these problems, a part of the mixture is removed and replaced with another substance.
EXAMPLE : A container contains 50 liters of milk. 5 liters of milk are removed and replaced with water. This operation is repeated two more times. What will be the final quantity of milk in the container?
Solution:
- After the first operation:
Milk left = 50 × = 45 Liters
After the second operation:
Milk left = 40 × = 40.5 Liters
After the third operation:
Milk left = 40.5 × = 36.45 Liters
The final quantity of milk left in the container is 36.45 liters.
3. Advanced Concepts in Alligation
a. Weighted Average and Alligation:
The weighted average can also be derived using the alligation rule.
For two ingredients mixed in ratio r1 : r2 the mean price M is given by:
M = C1 ×
Where:
- C1 and C2 are the prices of the two ingredients.
- r1 and r2 are the quantities of the two ingredients.
b. Successive Replacement:
If a mixture is repeatedly replaced, the quantity left after each operation follows a geometric progression.
Let V be the initial volume, R be the volume replaced each time, and nnn be the number of operations.
The remaining quantity after nnn operations:
Remaining Quantity = V ( )n
c. Mixing Multiple Solutions:
When multiple solutions are mixed, we can use the concept of alligation with weighted averages to determine the final concentration or price.
4. Application of Alligation in Different Fields
- Chemistry: To determine the concentration of solutions after mixing.
- Economics: To calculate the average price or cost in a market basket.
- Agriculture: To mix fertilizers or pesticides in a desired proportion.
- Retail: For mixing products of different qualities and prices.
5. Practice Problems with Solutions
Problem: A merchant has 100 kg of rice, part of which he sells at ₹5/kg and the rest at ₹8/kg. He gains 20% on the whole. What is the quantity of rice sold at ₹8/kg?
Solution:
- Let the quantity sold at ₹5/kg be x kg and at ₹8/kg be 100−x100 – x100−x kg.
- The cost price (C.P.) of 100 kg rice = 100 × ₹5 = ₹500.
- Selling price (S.P.) = 1.2 × ₹500 = ₹600.
- Therefore, 5x + 8(100 – x) = ₹600.
- Solving, x=40 kg.
- Hence, 60 kg is sold at ₹8/kg.
Problem: A mixture of 40 liters of milk and water contains 10% water. How much water must be added to make water 20% of the mixture?
Solution:
- Initially, water = 10%×40=4 liters.
- Let x liters of water be added.
- The new mixture = 40+ x liters.
- Water = 4 +x liters.
- Now, = 20% =
- Solving, x=2 liters.
6. Formulas Recap
- Ratio of Mixing:
Ratio =
- Weighted Average:
M = C1 ×
- Successive Replacement:
Remaining Quantity = V ( )n
