# Mixture and Alligation

- Mixture is a combination of two or more elements to form a third element. The questions from mixture are usually based on Ratio and Proportion and do not require any special formula.
- Alligation is a rule that enables us to quickly calculate the price of a mixture, given that it is a mix of two elements having different prices.
- Alligation Rule :

Here, mean price is the price of 1 unit of the mixture. Please note that in the above formula, everything has to be calculated by considering one unit only.

Another point to be noted is that mean price would always be between cheaper price and dearer price. - In questions of replacement where we initially have some quantity of pure element (like petrol), and we keep on replacing a fixed portion of this pure liquid every time with some other element (like water) thus making it a mixture, we apply the following formula for finding the quantity of pure element after ‘n’ replacements :

P x [1 – (R / P)]^{n}, where

P is the initial quantity of pure element

R is the quantity replaced every time

n is the number of replacements

### Sample Problems

**Question 1 : **From a vessel of 20 liters pure milk, 1 liter is taken out and replaced with water, so as to keep the volume constant to 20 liters. This process is repeated 5 times. Find the percentage of pure milk left in the vessel after 5 replacements.

**Solution : **Here, we need to apply the formula P x [1 – (R / P)]^{n}

P = Initial quantity of pure element = 20 liters

R = Quantity replaced every time = 1 liter

n = Number of replacements = 5

So, quantity of pure milk after 5 replacements = 20 x [1 – (1 / 20)]^{5}

Quantity of pure milk after 5 replacements = 20 x (0.95)^{5} = 20 x 0.773780938 = 15.4756 liters

Therefore, percentage of pure milk left in the vessel after 5 replacements = (15.4756 / 20) x 100 = 77.378 %

**Question 2 : **A dishonest shopkeeper mixed cheaper quality of rice, priced at Rs. 10 / KG with good quality rice, priced at Rs. 25 / KG and sells the mixture at Rs. 15 / KG. Find the ratio in which he mixes the two qualities of rice.

**Solution : **

Thus, ratio of quantities of cheaper and good quality rice = 10 : 5 = 2 : 1

### Problems on Mixture and Alligation | Set 2

**Programs on Mixture and Alligation**

This article has been contributed by **Nishant Arora**

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