# Mixture and Alligation For Competitive Exams

When you prepare for the competitive exams, which section do you think is the most challenging one? While some find the Logical Reasoning section tough, most of the others find the quantitative section a tough nut to crack. This section, which includes major topics of maths, assesses the analytical and problem-solving skills of the candidates through a plethora of topics such as LCM and HCF, Permutation and Combination, Data Interpretation, Mensuration formulas, Number Series questions, Time and Work, etc. Another such topic that often leaves the candidates scratching their heads is Mixture and Alligation. Since this topic forms an integral part of various exams, we will help you understand the concepts related to this topic by solving some questions on the same.

## Understanding Mixture and Alligation

Mixture and alligation is a prevalent topic in the quantitative section of various competitive exams like SSC CGL, IBPS PO/Clerk, SBI PO/Clerk, GMAT, GRE, etc. Counted amongst the most important modules, this topic can be scoring if practised with the right approach. Before we head on to the questions, let us first understand what these terms exactly mean.

What is a Mixture?

In quantitative aptitude, when a new component, which is formed by mixing two or more ingredients of a certain quantity or ratio to produce the desired amount is called a mixture. For example, when two varieties of rice are mixed either in the same ratio or different ratio, a new variety of rice is formed which is known as a mixture.

What is Alligation?

Alligation, in layman terms, means linking. It enables us to determine the ratio in which two or more components, at a given price, must be mixed to produce the desired combination.

For example, there are two ingredients A and B, priced at x and y respectively. The ingredients are mixed and the resulting product is priced at the mean price (M). Finally, the ratio (r) in which A and B are mixed is given by the rule of alligation.

## Calculating Mixture and Alligation

To provide conceptual clarity, let us now have a look at some of the main methods with suitable examples which can be used to solve mixture and alligation questions:

### Method 1

In this kind of a mixture and alligation method, when there are two ingredients and their prices are given, the ratio in which the mixture of a known price is obtained by blending the two ingredients is given by the formula:

The above formula can be represented through a diagram as well.

Which means,

Cheaper Quantity : Dearer Quantity = [d-m] : [m-c]

Where d  = Dearer Quantity, c = Cheaper Quantity, and m = Mean Price

Sample Question

A shopkeeper mixes two types of rice worth Rs. 50/kg and Rs. 70/kg. In what ratio must he mix the two varieties so that the average cost of the entire mixture goes to Rs. 65/kg?

Solution: Let us use the diagram for solving the above question.

[optin-monster-shortcode id=”xf2mlnjiouddzrshykdb”]

Now,

[d-m] = 70-65 = 5, and

[m-c] = 65-50 = 15

So, according to the formula,

Cheaper quantity/ Dearer quantity = [d-m]/[m-c]

= 5/15

= ⅓ or 1 : 3

Hence, the required ratio is 1 : 3.

### Method 2

Known as Repeated Dilution, this method of mIxture and alligation is used when we want to calculate the quantity left after a number (n) of repetitive replacements or other processes done to the purify the ingredient. For example, a container contains ‘x’ units of milk from which we take out ‘y’ units of it and add water instead of it. After numerous processes, the pure quantity left is:

Sample Question

A milkman has a container which is filled with 50 litres of milk. He takes out 10 litres of milk and adds water to the remaining quantity. He performs this process twice. Determine, how much milk is left in the container.

Solution: Adhering to the formula of repeated dilution, the amount of milk left in the container is calculated below.

Amount of milk left= 50 [ 1-10/50]^2

= 50 x ⅘ x ⅘

= 32 litres

Hence, the amount of milk left in the container equals 32 litres.