Mixtures & Alligations | Guide to Aptitude Problems

Mixtures and Alligations is a powerful technique used to calculate the proportions in which two or more ingredients should be mixed to achieve a desired mixture at a specific price or concentration. Whether you’re preparing for competitive exams or campus placements, mastering this concept can save you a lot of time in aptitude tests.

In this article, we’ll break down Mixtures and Alligations with easy-to-understand explanations, helpful examples, and useful tricks to solve related aptitude questions effectively.

What is Mixtures and Alligations?

Mixtures and Alligations refer to a mathematical method used to find the ratio in which different ingredients should be mixed to obtain a desired average cost, concentration, or other qualities. This concept is often used to solve problems related to blending items with varying costs or concentrations.

Key Concept:

Allegation Rule: This rule helps in determining the ratio of two ingredients that must be mixed to achieve a mixture with a specific average price or concentration.
Mean Price: The cost price of the unit quantity of the mixture, representing the weighted average of the components in the mixture.

Types of Mixtures and Alligations Problems

Mixtures and Alligations can be applied to various scenarios, including:

Finding the Average Price: Given the quantity and price of two ingredients, determine the average cost of the mixture.
Determining Mixing Ratios: If the desired price or concentration is known, calculate the ratio of the ingredients to achieve that mixture.
Concentrating or Diluting a Mixture: Determine the resulting quantity when concentrating or diluting a mixture.
Selling for Profit: Calculate the ratio of ingredients that should be mixed to sell the mixture at a profit.

Mixtures and Alligations Formula

The main formula to remember is the weighted average for two ingredients:

$\frac{(p_1q_1) + (p_2q_2)}{q_1 + q_2}$

Where:

$p_1$ and $p_2$ are the prices of the ingredients.
$q_1$ and $q_2$ are the quantities of the ingredients.
$p$ is the mean price (final price of the mixture).

Alternatively, the ratio can be found using the formula:

$q1q2=p2−pp−p1\frac{q_1}{q_2} = \frac{p_2 – p}{p – p_1}$

Examples and Tricks for Solving Mixtures and Alligations Problems

Let’s go through some examples to illustrate how to apply the mixtures and alligations rule.

Example 1: Tea Mixture Problem

Problem:
In what ratio must a grocer mix two varieties of tea, priced at Rs. 60 per kg and Rs. 65 per kg, so that he can sell the mixture at Rs. 68.20 per kg and gain 10%?

Solution:

Selling Price (S.P.) = Rs. 68.20
Gain = 10%
Cost Price (C.P.) = $100110×68.20=Rs.62\frac{100}{110} \times 68.20 = Rs. 62$

Using the allegation rule, calculate the ratio of mixing the two teas:

Example 2: Liquid Mixture Replacement

Problem:
A vessel contains 2 parts water and 6 parts syrup. How much of the mixture should be replaced with water to make it half water and half syrup?

Solution:

Let $x$ be the amount to replace.
Calculate the new quantities of water and syrup after the replacement using the allegation method.

Example 3: Mixture of Liquids A and B

Problem:
A can contains a mixture of two liquids A and B in a ratio of 8:4. When 10 litres of the mixture are removed and replaced with B, the ratio becomes 6:10. How many litres of A were in the can initially?

Solution:

Start by setting the initial quantities of A and B as 8x and 4x.
Use the allegation rule to calculate the resulting quantities of A and B after removal and replacement.

Advanced Mixtures and Alligations Problems

Once you’re comfortable with basic problems, try tackling more complex scenarios like successive replacement and percentage-based profit problems.

Example 4: Milk Replacement Process

Problem:
A container has 30 litres of milk. 3 litres of milk are removed and replaced with water, and this process is repeated two more times. How much milk remains?

Solution:

Use the formula for repeated processes: $remaining=30×(1−330)3\text{Milk remaining} = 30 \times \left(1 – \frac{3}{30}\right)^3$

Example 5: Replacing Whisky with Alcohol

Problem:
A jar of whisky contains 50% alcohol. Some whisky is replaced with another containing 21% alcohol. After replacement, the alcohol percentage is 32%. Find the amount of whisky replaced.

Solution:

Use the allegation rule to calculate the ratio of whisky and alcohol.

Conclusion

Mastering Mixtures and Alligations is essential for cracking quantitative aptitude problems. By practicing different types of problems and using the allegation rule, you can solve these questions quickly and accurately.