**GETTING STARTED**

☛ When we mix two or more items, we get a** mixture**. A mixture can have **different quantities of items mixed together**

** ☛ Let us consider an example of alcohol and water. We have 40 litres of alcohol and 10 litres of water. When we mix both of them, the resulting solution is a mixture of alcohol and water, and we have 50 l of the mixture.**

☛ Suppose that we take 40 litres out of the mixture. Definitely, it will not be only alcohol. Water would also be present in the solution that is drawn out.

**☛ Quantity of Alcohol : Quantity of Water = 4:1 **

**(Since, 40 L of alcohol is mixed with 10 L of water)**

This would be the ratio throughout the mixture.

This ratio would be maintained even after the 40 L of mixture is drawn out.

Hence, the **quantity of alcohol in the mixture is 4/5 x 40 = 32 L**

And **water = 1/5 x 40 = 8 L**

☛ Similarly, in the 10 L mixture that was left behind, ratio of alcohol to water is same as 4:1

Hence, the **left over mixture has 8 L alcohol and 2 L water**.

**DETERMINING PRICE OF MIXTURE**

Let us consider an example of alcohol and water. We have 40 litres of alcohol and 10 litres of water. When we mix both of them, the resulting solution is a mixture of alcohol and water, and we have 50 l of the mixture.

Assuming that the cost of alcohol as Rs. 100/litre

and the cost of water to be Rs. 0.

**Total cost of the mixture**

** = ( 40 x 100 + 10 x 0)**

**= Rs.4000**

**Cost of the mixture per litre** = 4000/50 = **Rs. 80/litre**

☛ Price of the mixture is known as **‘mean price’**. If we see this example, the **price per litre of the mixture is in between the price per litre of the items mixed ie. between Rs.100/litre (cost of alcohol) and Rs. 0/litre ( cost of water)**

**☛ The price of a mixture is Rs. 80 which is closer to the price of alcohol – Rs. 100 than the price of water – Rs.0**

If we observe carefully,

Suppose in another experiment, **40 litres of water is added to 10 litres of alcohol**,** Total cost of mixture **

= 400 x 0 + 10 x 100

**= Rs.1000**

**Cost of mixture** = 1000/50

**= Rs.20/litre**

☛ In this case, **cost of mixture per litre is closer to the price of water (Rs.0) than that of alcohol (Rs.100)**

Generalizing,

**Replacing One liquid with Other**

**Another important concept in Mixture / Alligation is replacing one liquid with another.**

**☛ Let us say, a container contains 30 L milk and 10 L was drawn out of it and replaced with water. **

Quantity after 1st operation is 30-10

= 30(1-10/30)

= 30(1-1/3)

= 30 x 2/3

= 20 L

After **1st** **operation, the quantity of milk in container is 20 L and water is 10 L**.

**☛ Once again the same operation was performed**. 10 L of mixture was drawn and replaced with 10 L water.

The 10 L which is drawn out of mixture will contain 20/3 L of milk and 10/3 L of water and is replaced by 10 L water.

Thus effectively, milk quantity is reduced by 20/3 L and water quantity is increased by 20/3 L.

Quantity of milk after 2nd operation

= 20-20/3

= 20(1-1/3)

= **40/3 L**.

Quantity of milk after 2nd operation can also be written as,

**30(1-1/3) ^{2}**

Extending it further, quantity of milk after

3 rd operation = 30(1-1/3)^{3}

Generalizing, whenever a liquid is replaced by another liquid,

**Quantity of 1st liquid **

**= Q[1- Qty of 2nd liquid/Q] ^{n}**

Q – Total quantity of 1st liquid initially present

n – No. of times operation is performed