The mathematical concept of mixtures is based in the concept of weighted average.

Weighted average is defined as:
Weighted avg. = Sum of totals of all groups / Total number of elements in all groups together

For two groups:
Aw = (n1A1 + n2A2) / (n1 + n2)

If we solve it further, we get the equation as
n1/n2 = (A2-Aw)/(Aw-A1)
This equation is known as allegation equation, which is used to solve the problems on mixtures.

Alligations formula is rearranged below for easy solutions.
The basic formula which is used to find the ratio in which the ingredients are mixed is
(Quantity of cheaper)/(Quantity of dearer)= (C.P.of dearer -Mean price)/(Mean Price-C.P.of cheaper)
It is also called the rule of allegation and can also be represented as

Practice Questions:

Q1. A vessel is filled with liquid, 3 parts of which are water and 5 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?
a. 1/6
b. 1/5
c. 2/5
d. 3/5

Q2. Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety in the ratio 1: 1: 2. If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be:
a. Rs 175.5
b. Rs 180.0
c. Rs 159.5
d. Rs 170.0

Q3. A can contains a mixture of two liquids A and B is the ratio 7:5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7:9. How many litres of liquid A was contained by the can initially?
a. 10
b. 20
c. 21
d. 25

Q4. A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second contains 50% water. How much milk should he mix from each of the containers so as to get 12 litres of milk such that the ratio of water to milk is 3 : 5?
a. 4 litres, 8 litres
b. 6 litres, 6 litres
c. 5 litres, 7 litres
d. 7 litres, 5 litres

Q5. 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. 16.50 kg?
a. 3:7
b. 5:7
c. 7:3
d. 7:5

Q6. 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:
a. 4%
b. 6 1/4%
c. 20%
d. 25%

Q7. How many kilogram of sugar costing Rs. 9 per kg must be mixed with 27 kg of sugar costing Rs. 7 per kg so that there may be a gain of 10% by selling the mixture at Rs. 9.24 per kg?
a. 36 kg
b. 42 kg
c. 54 kg
d. 63 kg

Q8. A container contains 40 litres of milk. From this container 4 litres of milk was taken out and replaced by water. This process was repeated further two times. How much milk is now contained by the container?
a. 26.34 litres
b. 27.36 litres
c. 28 litres
d. 29.16 litres

Q9. A jar full of whisky contains 40% alcohol. A part of this whisky is replaced by another containing 19% alcohol and now the percentage of alcohol was found to be 26%. The quantity of whisky replaced is:
a. 1/3
b. 2/3
c. 2/5
d. 3/5
e. None of these

Q10. In what ratio must water be mixed with milk to gain 16% on selling the mixture at cost price?
a. 1:6
b. 4:25
c. 2:3
d. 4:3

11. A bottle contains 20 litres of liquid Q. 4 litres of liquid A is taken out of its and replace by same quantity of liquid B. Again 4 litres of the mixture is taken out and replaced by same quantity of liquid B. What is the ratio of quantity of liquid A to that of liquid B in the final mixture?
a. 4:1
b. 5:1
c. 16:9
d. 17:8

12. Two equal glasses of same type are respectively 1/3 and 1/4 full of milk. They are then filled up with water and the contents are mixed in a pot. What is the ratio of milk and water in the pot?
a. 9:21
b. 11:23
c. 7:17
d. 1:3

13. 30g of sugar was mixed in 180 ml water in a vessel A, 40 g of sugar Was mixed in 280 ml of water in vessel B and 20 g of sugar was mixed in 100 ml of water in vessel C. The solution in vessel B is (UPSC 2016)
a. sweeter than that in C
b. sweeter than that in A
c. as sweet as that in C
d. less sweet than that in C

14. There is a milk sample with 50% water in it. If 1/3 of this sample is added to equal amount pure milk, then water in the new mixture will fall down to. (UPSC 2017)
a. 25%
b. 30%
c. 35%
d. 40%

15. 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. 68.20 a kg he may gain 10%?
a. 3:2
b. 3:4
c. 3:5
d. 4:5

16. Two vessels A and B contain spirit and water mixed in the ratio 5: 2 and 7: 6 respectively. Find the ratio in which this mixture be mixed to obtain a new mixture in vessel C containing spirit and water in the ration 8: 5?
a. 4:3
b. 3:4
c. 5:6
d.7:9

17. The cost of Type I rice is Rs. 15 per kg and Type 2 rice is Rs 20 per kg. If both Type 1 and Type 2 are mixed in the ratio 2: 3, then the price per kg of the mixed variety of rice is: Rs
a. 18
b. 18.50
c. 19
d. 19.50
e. None of these

18. In a pot, there is a mixture of milk and water in the ratio 4 : 5. If it is filled with an additional 8 litres of milk, the pot would be full and ratio of milk and water would become 6 : 5. Find the capacity of the pot ?
a. 11 litres
b. 44 litres
c. 33 litres
d. 22 liters

19. Priti possessing ₹ 10,000, lent a part of it at 5% simple interest and the remaining at 20% simple interest. Her total income after 5 years was ₹7500. Find the sum lent at 20% rate.
a. ₹2000
b. ₹4000
c. ₹3000
d. None of these

20. In what ratio should water be mixed with soda costing ₹12 per litre so as to make a profit of 50% by selling the diluted liquid at ₹15 per litre?
a. 10:1
b. 5:1
c. 1:5
d. 6:1

21. How much pure alcohol must be added to 400 ml of a solution containing 16% of alcohol to change the concentration of alcohol in the mixture to 40%
a. 160 ml
b. 100 ml
c. 128 ml
d. 68 ml

22. Wheat costing Rs. 30/kg, Rs. 35/kg and a third variety of wheat are mixed in the ratio of 3 ∶ 4 ∶ 2. If the mixture costs Rs. 34/kg, then what will be the cost (in Rs./kg) of the third variety of wheat?
a. 46
b. 42
c. 32
d. 38

23. The ratio of Iron and Zinc in an alloy is 4 : 5. In another alloy, the ratio of Iron, Copper and Zinc is 3 : 2 : 7. Equal amounts of the two alloys are molten and mixed together. What will be the ratio of Iron, Copper and Zinc in the resultant alloy?
a. 6 : 25 : 41
b. 25 : 41 : 6
c. 25 : 6 : 41
d. 4 : 5 : 7

24. In what ratio should Assam Tea costing Rs.300 per kg be mixed with Darjeeling Tea costing Rs. 400 per kg, so that by selling the mixture at Rs. 408 per kg there is gain of 20%.
a. 1 : 2
b. 3 : 2
c. 2 : 5
d. 1 : 6

25. A hospital uses a mixture of salt and water at Rs. 7.62/litre. This mixture contains 5 % salt. Another mixture containing 75 % water costs Rs. 7.82/litre. How much does the patient pay if he buys 5 litres of mixture containing 18% salt?
a. Rs. 83.75
b. Rs. 73.85
c. Rs. 37.85
d. Rs. 38.75

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