Mixture And Alligation

Alligation  

The Alligation is a rule that helps us calculate the price of a mixture if that mixture is a mix of two elements with different prices. The Alligation is used to find the mean value of a mix when the ratio and amount of the ingredients mixed are different and also to see the proportion in which the elements are mixed. 

Alligation Formula 

The basic formula which is used to find the ratio in which the ingredients are mixed is 
 

This is called the rule of the Alligation, and it is also represented as below -

Points to consider 

• Mean price is the price of 1 unit of the mixture. Also, everything has to be calculated by considering one unit only.
• Another point to note is, the mean price would always be between a lower price and a dearer price.
• In questions of replacement where we initially have some quantity of pure element, and we keep on replacing a fixed portion of this pure liquid every time with some other element, thus making it a mixture, we apply the following formula for finding the quantity of pure element after ‘N’ replacements :

P * [1 – (R / P)] ^ N

P: initial quantity of pure element

R: quantity replaced every time

N: number of replacements

Sample Problems

Let’s understand these concepts with the below examples -

Question 1 - From a vessel of 25 liters of pure milk, 1 liter is taken out and replaced with water to keep the volume constant to 25 liters. This process is repeated 5 times. Find the percentage of pure milk left in the vessel after 5 replacements.

Solution - we can apply the formula discussed above -

P * [1 – (R / P)] ^ N

P = Initial quantity of pure element = 25 liters

R = Quantity replaced every time = 1 liter

N = Number of replacements = 5

Quantity of pure milk after 5 replacements = 25 * [1 – (1 / 25)] ^ 5

Quantity of pure milk after 5 replacements = 25 * (0.96) ^ 5 = 25 * 0.81537 = 20.38425 liters

Hence, percentage of pure milk left in the vessel after 5 replacements = (20.38425 / 25) * 100 = 81.53 %.

Question 2 - A shopkeeper mixed cheaper quality rice, priced at Rs. 15 / KG with good quality rice, priced at Rs. 30 / KG, and sold the mixture at Rs. 20 / KG. Find the ratio in which he mixes the two qualities of rice.

Solution - We can use the Alligation rule explained above to solve this problem.

Hence, from the above diagram, we can find out the ratio of quantities of cheaper and good quality rice, Alligations, i.e., 10:5 = 2:1

Question 3 - Find out the ratio at which the shopkeeper mixes sugar at Rs 30 / kg and Rs 32.5 / kg so that by selling the mixture at Rs 34.1 / kg, he may gain 10%? 

Solution - Let’s try to solve it using the Alligation method. The Alligation method for this question looks like below -

Selling Price of 1 kg of mixture = Rs 34.1 

Profit = 10% = 10 / 100 = 1 / 10 

Sugar Price = 1 + 10 = 11 unit 

11 units = 34.1 

1 unit = 3.1 

10 unit = 31 

That means CP of the mixture = Rs 31. 

Hence, the ratio in which he mixed is 3 : 2 

Frequently asked questions

1) What is Mixture?

Ans) A Mixture, as its name suggests, is a combination of 2 or more elements to create a 3rd element. There are 2 types of Mixtures - Simple Mixture and Compound Mixture.

2) What is Alligation?

Ans) The Alligation is a rule that helps us to calculate the price of a mixture if that mixture is a mix of two elements having different prices. The Alligation is used to find the mean value of a mixture when the ratio and amount of the mixed ingredients are different and to see the proportion of the elements mixed. 

3) What is the Alligation Formula?

Ans) Below is the formula used to solve questions using the Alligation method -
 

Key Takeaways

This article discussed Mixture and Alligation and later discussed various questions to understand this concept more clearly.

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