1. Alligation And Mixture

2. Mixture: Mixing of two or more than two type of quantities gives us a mixture.
Example:
Quantities of these elements can be expressed as percentage or ratio.(20% of sugar in water)
Fraction ( A solution of sugar and water such that sugar : water = 1:4)

3. Alligation :  Alligation is a rule which is used to solve the problems related to mixture and its ingredient.
It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of desired price.

4. Alligation Rule
When two elements are mixed to make a mixture and one of the elements is cheaper and other one is costlier then,

5. Diagram Representation

6. Explanation:
Suppose x kg of Cheaper quality is mixed with y kg of Superior quality.
Price of Cheaper Ingredient = Rs. cx
Price of Superior Ingredient = Rs. dy
Price of Mixture = Rs. (cx + dy)
and Quantity of mixture = (x + y) kg
Price of mixture(m)/kg = Rs. (cx + dy) / (x + y)
So, cx + dy = m
x + y

7. cx + dy = mx + my dy – my = mx – cx y(d-m) = x(m-c) x/y = d – m / m – c

8. Examples
Q1. How many kg of tea worth Rs. 25/kg must be blended with 30 kg of tea worth Rs. 30/kg so that by selling the blended variety at Rs. 30/kg there should be a gain of 10%?
36 kg
40 kg
32 kg
None

9. Q2. How much water be added to 14 litres of milk worth Rs. 5
Q2. How much water be added to 14 litres of milk worth Rs a litre so that the value of the mixture may be Rs a litre ?
7 litres
6 litres
5 litres
4 litres

10. Q3. In what ratio should a coffee blend containing coffee and chicory in the ration 3 : 7 and another blend with coffee to chicory ration as 2 : 3 be mixed so that the resultant blend will have to coffee to chicory ration of 7 : 13?
2 : 1
1 : 1
3 : 5
Cannot be Determined

11. Q4. In what ratio must water be added to spirit to gain 10% by selling it at the cost price?
1 : 11
1 : 5
1 : 10
1 : 9

12. Q5. Sea water contains 5% salt by weight
Q5. Sea water contains 5% salt by weight. How many kilograms of fresh water must be added to 40kg of sea water for the salt content of the solution to be 2%? 50 60 65 70

13. Q6. A man buys two cows for Rs
Q6. A man buys two cows for Rs and sells one for loss of 6% and the other for gain of 7.5% and on the whole he neither gains nor loses. What does each cow cost?
Rs. 850, Rs.500
Rs. 650, Rs. 700
Rs. 750, Rs. 600
Rs. 550, Rs. 800

14. Q7. In a mixture of 60 L, the ratio of milk and water is 2:1, If the rate of milk and water is to be 1:2, then the amount of water to be further added must be ?
12 L
20 L
40 L
60 L

15. Q8. An alcohol water mixture of 729 L is in the ratio 7:2, how much more water is to be added to get a new alcohol and water ratio of 7:3?
81 L
72 L
80 L
78 L

16. Q9. A person has Hydrogen Peroxide of Rs. 50 per liter
Q9. A person has Hydrogen Peroxide of Rs. 50 per liter. Water should be mixed in what ratio with Hydrogen Peroxide, so that after selling the mixture at 40 per liter, he gets a gain of 50%?
8:7
9:8
6:8
7:9

17. Q10. A mixture of certain quantity of juice with 32 liters of water is worth 1.50 per liter. If pure juice is to be worth 4.50 per liter, how much juice is there in the mixture?
18 L
14 L
16 L
20 L

18. Q11. Reeta has 1000 mL of a solution that contains ethanol and water in the ratio 3:1. She adds 250 mL of 3:2 solutions of ethanol and water to it and then uses 250 mL of combined mixture. How much of pure ethanol is she left with?
1000 mL mL
750 mL
720 mL

19. Q12. A mixture of 45 L of spirit and water contains 20% of water in it
Q12. A mixture of 45 L of spirit and water contains 20% of water in it. How much water must be added to it to make the water 25% in the new mixture?
3 L
4 L
5 L
6 L

20. Q13. One mixture contains 25% milk and other contains 30% milk and the rest water. A jar is filled with 6 parts of first mixture and 4 parts of second mixture. The percentage of milk in the mixture is
23%
67%
27%
16%

21. Q14. How many Kg of wheat which costs Rs
Q14. How many Kg of wheat which costs Rs. 42 per kg should a vendor mix with Rs. 25 per kg and thus he gains 25% on selling the mixture at Rs. 40 per kg?
22.5 kg
11.5 kg
2 kg
20 kg

22. Q15. In what ratio must a grocer mix two varieties of pulses costing Rs. 15 and Rs. 20 per kg respectively so as to get a mixture worth Rs kg?
7:3
4:5
6:4
None

23. Q16. A dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is:
20%
10%
11%
None

24. Q17. In what ratio must a grocer mix two varieties of tea worth Rs
Q17. In what ratio must a grocer mix two varieties of tea worth Rs. 60 a kg and Rs. 65 a kg so that by selling the mixture at Rs a kg he may gain 10%?
3:2
3:5
3:6
None

25. Mixture of more than two elements.
Q18. In what ratio must a person mix three kinds of tea costing Rs.60/kg, Rs.75/kg and Rs.100 /kg so that the resultant mixture when sold at Rs.96/kg yields a profit of 20%?
1 : 2 : 4
3 : 7 : 6
1 : 4 : 2
None of these

26. Q19. Find out the ratio of new mixture so that it will cost Rs 1
Q19. Find out the ratio of new mixture so that it will cost Rs 1.40 per kg from the given three kinds of rice costing Rs 1.20, Rs 1.45 and Rs 1.74.
39 : 20 : 20
30 : 20 : 30
30 : 29 : 29
None of these

27. Q20. A merchant mixes three varieties of rice costing Rs. 20/kg, Rs
Q20. A merchant mixes three varieties of rice costing Rs.20/kg, Rs.24/kg and Rs.30/kg and sells the mixture at a profit of 20% at Rs.30/kg. How many kg of the second variety will be in the mixture if 2 kg of the third variety is present in the mixture?
1 kg
5 kg
3 kg
6 kg

28. Removal and Replacement
If a vessel contains “x” liters of liquid A and if “y” liters be withdrawn and replaced by liquid B, then if “y” liters of the mixture be withdrawn and replaced by liquid B, and the operation  is repeated ‘n’ times in all, then :
F.C = I.C(1-y/x)n
FC= Final concentration
IC= Initial concentration
y = no. of liters replaced
x = Total concentration
n = total number of iterations

29. Q21. A vessel contains 125 litres of wine
Q21. A vessel contains 125 litres of wine. 25 litres of wine was taken out of the vessel and replaced by water. Then, 25 litres of mixture was withdrawn and again replaced by water. The operation was repeated for third time. How much wine is now left in the vessel?
49 L
64 L
72 L
56 L

30. Q22. A container contains 40 liters of milk
Q22. A container contains 40 liters of milk. From this container 4 liters of milk was taken out and replaced with water. This process was repeated further three times. How much milk is now contained by the container?
27.36 L
26.24 L
29.36 L
24.36 L

31. Q23. From a vessel containing only milk, 5 litres are drawn and replaced with water. This action is repeated once more. The ratio of milk to water now is 36 : 13. How many litres of solution does the vessel hold? 35 30 25 40

32. Q24. From a cylindrical can containing milk of 30 liters, 5 liters are drawn and replaced with water. If the same process is repeated for 3 more times, what is the amount of milk left in the cylindrical can?
14.46 L
15.56 L
17 L
20 L

33. Q25. Product M is produced by mixing chemical X and chemical Y in the ratio of 5 : 4. Chemical X is prepared by mixing two raw materials, A and B, in the ratio of 1 : 3. Chemical Y is prepared by mixing raw materials, B and C, in the ratio of 2 : 1. Then the final mixture is prepared by mixing 864 units of product M with water. If the concentration of the raw material B in the final mixture is 50%, how much water had been added to product M?
328 units
368 units
392 units
616 units