MIXTURE & ALLIGATION

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GETTING STARTED

☛ When we mix two or more items, we get a mixtureA 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

PRACTICE PROBLEMS

0

MIXTURE & ALLIGATION

1 / 25

A dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 40%. The percentage of water in the mixture is:

2 / 25

A milk vendor has two cans of milk. The first contains 30% water and the rest milk. The second contains 60% water. How much milk should he mix from each of the containers so as to get 18 litres of milk such that the ratio of milk to water is 3 : 2?

3 / 25

A can contains a mixture of two liquids X and Y in the ratio 4 : 5. When 9 litres of mixture are drawn off and the can is filled with Y, the ratio of X and Y becomes 4 : 9. How many litres of liquid X was contained by the can initially?

4 / 25

A vessel is filled with liquid, 4 parts of which are water and 6 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

5 / 25

Tea worth Rs. 140 per kg and Rs. 170 per kg are mixed with a third variety in the ratio 1 : 2 : 1. If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be:

6 / 25

In what ratio must a grocer mix two varieties of flour costing Rs. 25 and Rs.30 per kg respectively so as to get a mixture worth Rs.27 per kg?

7 / 25

In what ratio should water and an 70% alcohol solution be mixed to obtain a 50% alcohol solution

8 / 25

Vessel A has 20 litres of a mixture of milk and water having 75% milk. Vessel B has x litres of a mixture of milk and water having 50% milk. The contents of the vessels are mixed to form a mixture having 60% milk. Find the volume of mixture in vessel B.

9 / 25

A container with capacity 25 litre is 60% filled with a solution of milk and water. Milk is 80% of the solution. If 10 litre water is added to the solution, what amount of the new mixture will be milk?

10 / 25

Suresh purchases five kilograms of sugar costing Rs.15 per kg and ten kilograms of sugar costing Rs.12 per kg. He mixes both the varieties of sugar. The mixture is sold at 20% loss. Find the selling price of the mixture (in Rs./kg).

11 / 25

Ramesh is a retailer who purchases two varieties of coffee. In what ratio must he mix the two kinds of coffee which cost Rs.100 per kg and Rs.120 per kg such that the resultant mixture costs Rs.108 per kg?

12 / 25

In a class of 100 students, there are 30 boys and 70 girls. In a test, the average score of boys was 85 and that of girls was 95. What was the average mark of the entire class?

13 / 25

In what ratio must two varieties of Tur dal costing Rs.35 per kg and Rs.45 per kg be mixed,so that by selling the mixture at Rs.45 per kg the shopkeeper would make  25% profit?

14 / 25

A vessel has 200 litres of pure milk. 40 litres of milk is removed from the vessel and replaced by water. 40 litres of the mixture thus formed is replaced by water. This procedure is repeated once again. Find the percentage of milk in the resultant solution.

15 / 25

A vessel contains 70% milk in a 100 litre mixture of milk and water, . How much water should be added to the solution to dilute the milk content to 60%.

16 / 25

Raju has 2 vessels containing a mixture of milk and water. The ratios of milk and water are 3 : 10 and 4 : 5 in the first and second vessels respectively. In what ratio should he mix them so that the resulting mixture contains milk and water in the ratio 79 : 155?

17 / 25

How many kilograms of rice costing Rs.19 per kg should be added with 9 kg of rice costing Rs.21 per kg, so that the shopkeeper incurs a cost of Rs.20.2 on the mixture?

18 / 25

A bag contains a total of 60 coins in the denominations of 50 p and Rs.1. Find the number of 50 p coins in the bag if the total value of the coins is 50.

19 / 25

Consider that you mix 5 litres of 70% alcohol with 10 litres of 40% alcohol. Find the concentration of the resulting solution.

20 / 25

The cost of Type A wheat is Rs. 45 per kg and Type B wheat is Rs. 70 per kg. If both Type A and Type B are mixed in the ratio of  3 : 2, then the price per kg of the mixed variety of wheat is:

21 / 25

A milkman has 25 litres of pure milk. He adds water to the milk and sells the resulting mixture at cost price, still managing to make a profit of 60%.Find the percentage of water in the resulting solution.

22 / 25

A shopkeeper mixes three varieties of rice priced at Rs.100 per kg, Rs.120 per kg and Rs.180 per kg. He sells the mixture at Rs.140 per kg. What would be the proportion of the three varieties?

23 / 25

There are 2 vessels containing copper and zinc metals in the ratio 4 : 5 and 7 : 11 respectively. If equal weights of the two are melted together to form a third alloy, find the ratio of the weights of copper and zinc in the third alloy named C.

24 / 25

Two solutions of hydrochloric acid are mixed in the ratio 2 : 3. The concentrations of hydrochloric acid in the first and second solutions are 30% and 40% respectively. What would be the concentration of hydrochloric acid in the final solution?

25 / 25

A container has 40 litres milk solution with 60% milk content in it. How much water should be added to the solution in order to reverse the content of milk and water?

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