MIXTURE AND ALLEGATIONS

Mixture and alligation form a significant concept in the field of mathematics, particularly in problem-solving and quantitative aptitude. This concept deals with the mixing of different elements or substances to achieve a desired mixture with a specified concentration or ratio.

In simpler terms, alligation is a method used to calculate the proportions in which two or more ingredients at different prices or concentrations must be mixed to obtain a mixture at a given price or concentration. This concept finds practical applications in various real-life scenarios, such as mixing different grades of materials to achieve a blend with specific properties.

To comprehend the concept of mixture and alligation, it is essential to understand the basic principles:

• Basic Components:

  • Ingredients: These are the individual components or substances that are mixed together.
  • Mixture: The resulting combination of the ingredients after mixing.

• Alligation Rule:

  • The alligation rule involves placing the prices or concentrations of the ingredients along with the mean or average value of the mixture in a line or column.
  • The distances of the ingredients from the mean represent their respective quantities in the mixture.
  • This rule helps in determining the ratio in which the ingredients must be mixed to achieve the desired characteristics of the mixture.

• Use in Quantitative Aptitude:

  • Mixture and alligation problems are commonly encountered in the quantitative aptitude sections of various competitive exams.
  • These problems often involve finding the quantities of two or more components that need to be mixed to obtain a mixture with specified properties.

• Practice and Application:

  • Mastery of mixture and alligation requires regular practice in solving problems of varying complexity.
  • Candidates preparing for competitive exams, especially those with a quantitative aptitude component, should dedicate time to understand and practice problems related to this concept.

Important Formulas for Mixture and Alligation

1.Ratio of Mixture Components:

• Two components: Ratio of component A to component B = (Difference in B's property and mean value) : (Difference in mean value and A's property).
• Multiple components: Use the logic above for each pair of components in the alligation table, considering the "mean value" as the point where a new pair's connection lines intersect.

2. Mean Concentration or Cost:

• If x liters of component A with concentration a and y liters of component B with concentration b are mixed, the mean concentration c of the mixture is: c = (ax + by)/(x + y).
• Similarly, for cost calculations, replace concentration with cost per unit quantity.

3. Dilution or Enrichment:

• To find the final volume V of a solution after replacing x liters of a solution with concentration c1 with a solution of concentration c2: V = Original volume * (Original concentration - desired concentration) / (desired concentration - c2).

4. Replacement Operations:

• To find the remaining quantity P of a substance after n replacements of x units with a weaker/stronger substance, assuming equal replacement volume each time: P = Original quantity * (1 - replacement ratio)^n.

5. Profit and Loss:

• To find the selling price S of a mixture when buying price per unit of component A is a1, price per unit of component B is a2, and desired profit percentage is p: S = (a1x + a2y + total cost) * (1 + p/100), where x and y are the quantities of components A and B used.

 

 

Practice Mixture and Alligation Questions   Question 1: A chemist has two solutions. The first solution contains 20% acid, and the second solution contains 50% acid. In what ratio should the chemist mix the two solutions to obtain 30% acid in the final mixture? Solution: Let's assume the chemist mixes the solutions in the ratio $x:y$. The equation using the alligation rule is: (50−30​)/(30−20)=x/y​ Solving for $x:y$, we get: $x:y=2:1$   Question 2: A container has 40 liters of milk. 10 liters of water are added to it. What is the percentage of water in the mixture? Solution: The ratio of milk to water is $40:10$, which simplifies to 4:1. The percentage of water is: (Total Quantity)/(Quantity of Water)​×100=10/50​×100=20% Therefore, the percentage of water in the mixture is 20%   Question 3: A solution of alcohol and water contains 25% alcohol. If 5 liters of water is added to 15 liters of the solution, find the new percentage of alcohol. Solution: The initial quantity of alcohol in the solution is $25\%\times 15=3.75$ liters. After adding 5 liters of water, the new total quantity becomes $15+5=20$ liters. The new percentage of alcohol is: (Total Quantity)/(Quantity of Alcohol​)×100=(3.75/20​)×100   Question 4: A mixture contains alcohol and water in the ratio 2:3. If the total mixture is 50 liters and 10 liters of water is added, find the new ratio of alcohol to water. Solution: Initially, the quantity of alcohol is (2/5)×50 and the quantity of water is (3/5)×50. After adding 10 liters of water, the new quantity of water becomes 3/5×(50+10)5.