# Allegations and Mixtures

**Allegations and Mixtures**

In the context of quantitative ability, "allegations and mixtures" is a problem-solving concept related to finding the ratio in which two or more components of different strengths or qualities are mixed to form a mixture of a specified strength or quality. This concept is commonly encountered in various quantitative aptitude tests and competitive exams.

Here's a brief explanation of allegations and mixtures:

**Allegation:**

Allegation is a rule that helps find the ratio in which two or more components should be mixed to obtain a mixture of the desired strength. It assumes that the strength of the components is inversely proportional to the mixing ratio.

Components:

In allegations and mixture problems, there are multiple components, each having a specific strength or quantity. For example, these components could be different liquids, powders, or solutions with varying concentrations.

**Mixture:**

The mixture is the final combination obtained by mixing the components together. The objective is to determine the quantity or ratio of each component needed to achieve a particular desired strength or concentration.

Rule:

The rule of allegation states that if two components are mixed to form a mixture, and their quantities and strengths are represented by (A₁, A₂) and (B₁, B₂), respectively, then the ratio in which they should be mixed to obtain a mixture of a desired strength (M) is given by:

(A₁ - M) / (M - A₂) = (B₂ - M) / (M - B₁)

Solving an allegation problem involves finding the unknown value, which is either the quantity or the ratio of the components. Once the unknown value is determined, it can be used to calculate the required quantities to form the desired mixture.

Let's go through an example to illustrate how an allegation and mixture problem is solved:

**Examples:**

Example 1:

Suppose you have two containers of different types of fruit juices - orange juice and apple juice. The first container contains 10 litres of orange juice with a 20% concentration, and the second container contains 15 litres of apple juice with a 30% concentration. You want to mix these two juices to obtain a mixture with a 25% concentration. In what ratio should you mix the orange and apple juices?

Solution:

Let the ratio in which orange and apple juices should be mixed be x:y.

According to the rule of allegation:

(10x - 25y) / (25 - x - y) = (30y - 25x) / (x + y - 25) = 5y / 5x

=> x / y = 1 / 1

=> x = y

So, the orange and apple juices should be mixed in a 1:1 ratio to obtain a mixture with a 25% concentration. Therefore, 12.5 litres of each juice should be mixed to achieve the desired strength.

Example 2:

A chemist has two solutions of different concentrations: Solution A, which contains 40% acid, and Solution B, which contains 25% acid. How many litres of each solution should be mixed to get 20 litres of a new solution containing 30% acid?

Let the quantity of Solution A be x litres and the quantity of Solution B be (20 - x) litres.

According to the rule of allegation:

(40x - 30(20 - x)) / (20 - x - x) = (30(20 - x) - 25x) / x

Solving for x:

(40x - 600 + 30x) / (20 - 2x) = (600 - 5x) / x

40x + 30x - 600 = 600 - 5x

75x = 1200

x = 16 litres (approx.)

So, 16 litres of Solution A (40% acid) and 4 litres of Solution B (25% acid) should be mixed to get 20 litres of a 30% acid solution.

Example 3:

A coffee shop owner has two types of coffee beans: Type X, which costs $12 per kg, and Type Y, which costs $8 per kg. How many kilogrammes of each type should be blended to create a 20 kg mixture worth $10 per kg?

Let the quantity of Type X coffee beans be x kg and the quantity of Type Y coffee beans be (20 - x) kg.

According to the rule of allegation:

(12x + 8(20 - x)) / (x + (20 - x)) = 10

Solving for x:

(12x + 160 - 8x) / 20 = 10

4x + 160 = 200

4x = 40

x = 10 kg

So, 10 kg of Type X coffee beans and 10 kg of Type Y coffee beans should be blended to get a 20 kg mixture worth $10 per kg.

Example 4:

A farmer wants to create a mixture of two types of fertilisers: Fertiliser A, which contains 10% nitrogen, and Fertiliser B, which contains 20% nitrogen. The farmer wants to make 100 kg of a mixture containing 15% nitrogen. How much of each fertiliser should be used?

Let the quantity of Fertiliser A be x kg and the quantity of Fertiliser B be (100 - x) kg.

According to the rule of allegation:

(10x + 20(100 - x)) / (x + (100 - x)) = 15

Solving for x:

(10x + 2000 - 20x) / 100 = 15

-10x + 2000 = 1500

-10x = -500

x = 50 kg

So, 50 kg of Fertiliser A (10% nitrogen) and 50 kg of Fertiliser B (20% nitrogen) should be mixed to get 100 kg of a 15% nitrogen mixture.

These examples demonstrate how to use the rule of allegation to solve mixture problems involving different types of components with varying strengths or concentrations.

Keep in mind that the rule of allegation assumes an inverse relationship between the quantity and strength of the components. So, as one component's quantity increases, its strength decreases, and vice versa.