Mixture and Alligation: Notes, Methods and Questions_00.1

Mixture and Alligation: Notes, Methods and Questions

Quantitative Aptitude is a very important section for government recruitment exams like SSC, railway and banks. In this article, we are going to discuss the rule of mixtures and allegations. Generally, there are questions asked related to basic concepts and formulas of the Mixture and Alligation. The rule of alligation enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of the desired price. We have covered important notes and questions focusing govt. exams that aspirants can practice and prepare for the topic.

Mixture : Aggregate of two or more than two type of quantities gives us a mixture.

Alligation: It is a method of solving arithmetic problems related to mixtures of ingredients. This rule 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.

Alligation Method 1:
  • It is a modified form of finding the weighted average. If 2 ingredients are mixed in a ratio and the cost price of the unit quantity of the mixture, called the Mean Price is given then,Mixture and Alligation: Notes, Methods and Questions_60.1

The above formula can be represented with the help of diagram which is easier to understand.Here ‘d’ is the cost of dearer ingredient, ‘m’ is mean price and ‘c’ is the cost of cheaper ingredient.

Mixture and Alligation: Notes, Methods and Questions_70.1

Mixture and Alligation: Notes, Methods and Questions_80.1Thus, (Cheaper quantity) : (Dearer quantity) = (d – m) : (m – c).

Alligation Method 2: Repeated Dilution
  • This is used to calculate pure quantity left after ‘n’ number of processes of repeated replacement is done on the pure quantity. Suppose, a container contains ‘x’ units of a liquid from which ‘y’ units are taken out and replaced by water. After ‘n’ operations quantity of pureMixture and Alligation: Notes, Methods and Questions_90.1

1.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 litres
B. 29.16 litres
C. 28 litres
D. 28.2 litres
2. 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. 1/3
B. 1/4
C. 1/5
D. 1/7
3. Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety of tea in the ratio 1 : 1 : 2. If the mixture is worth Rs. 153 per kg, what is the price of the third variety per kg ?
A. Rs.182.50
B. Rs.170.5
C. Rs.175.50
D. Rs.180
 4.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
5. Two vessels A and B contain spirit and water in the ratio 5 : 2 and 7 : 6 respectively. Find the ratio in which these mixture be mixed to obtain a new mixture in vessel C containing spirit and water in the ration 8 : 5 ?
A. 3: 4
B. 4 : 3
C. 9 : 7
D. 7 : 9
6. 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
7.8 litres are drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of the water is 16 : 65. How much wine did the cask originally hold?
A. 30 litres
B. 26 litres
C. 24 litres
D. 32 litres
8. 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%
9. A jar full of whiskey contains 40% alcohol. A part of this whisky is replaced by another containing 19% alcohols and now the percentage of alcohol was found to be 26%. The quantity of whisky replaced is
A. 4/3
B. 3/4
C. 3/2
D. 2/4

Answers & Solutions

1( B)
Explanation
Assume that a container contains x of liquid from which y units are taken out and replaced by water. After n operations, the quantity of pure liquid
=x(1-y/x)^n
Hence milk now contained by the container = 40(1-4/40)^3
=40(1-1/10)^3
=40×9/10×9/10×9/10 =(4×9×9×9)/100 =29.16
2(C)
Explanation
Suppose the vessel initially contains 8 litres of liquid.
Let x litres of this liquid be replaced with water.
Quantity of water in new mixture = (3 – 3x/8 + x) litre
Quantity of syrup in new mixture = (5 – 5x/8) litres
So  (3 – 3x/8 + x)  = (5 – 5x/8) litres
=> 5x + 24 = 40 – 5x
=>10x = 16
=> x = 8/5 .
So, part of the mixture replaced = (8/5 x 1/8) = 1/5
3(C)
Explanation :
Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed in the ratio 1 : 1
So their average price = (126+135)/2=130.5
Hence let’s consider that the mixture is formed by mixing two varieties of tea.
one at Rs. 130.50 per kg and the other at Rs. x per kg in the ratio 2 : 2, i.e., 1 : 1. Now let’s find out x.
By the rule of alligation, we can write asCost of 1 kg of 1st kind of tea              Cost of 1 kg of 2nd kind of tea
(130.50)                                                     (x)Mean Price
(153)(x – 153)                                                                     (22.50)=>(x – 153) : 22.5 = 1 : 1
=>x – 153 = 22.50
=> x = 153 + 22.5 = 175.54(C)Explanation:
Suppose the can initially contains 7x and 5x of mixtures A and B respectively.
Quantity of A in mixture left = (7x – 7/12 x 9)litres
= (7x – 21/4) litres.
Quantity of B in mixture left = (5x – 5/12  x 9) litres
= (5x – 15/4)  litres.
So (7x – 21/4)/((5x – 15/4) +9) = 7/9
=> (28x – 21)/(20x + 21) = 7/9
=> 252x – 189 = 140x + 147
=> 112x = 336
=> x = 3.
So, the can contained 21 litres of A.

5(D)
Explanation :
Let Cost Price(CP) of 1 litre spirit be Rs.1
Quantity of spirit in 1 litre mixture from vessel A = 5/7
Cost Price(CP) of 1 litre mixture from vessel A = Rs. 5/7
Quantity of spirit in 1 litre mixture from vessel B = 7/13
Cost Price(CP) of 1 litre mixture from vessel B = Rs. 7/13
Quantity of spirit to be obtained in 1 litre mixture from vessel C = 8/13
Cost Price(CP) of 1 litre mixture from vessel C = Rs. 8/13 = Mean Price
By the rule of alligation, we can write as

CP of 1 litre mixture     CP of 1 litre mixture
from vessel A (5/7)      from vessel B (7/13)
Mean Price
(8/13)

8/13 – 7/13 = 1/13                 5/7 – 8/13 = 9/91

=> Mixture from Vessel A : Mixture from Vessel B = 1/13 : 9/91 = 7 : 9 = Required Ratio

6(C)
Explanation:
By the rule of alligation:
Cost of 1 kg pulses of       Cost of 1 kg pulses of
1st kind Rs.  (15)               2nd kindRs. (20)

Mean Price
Rs. (16.50)

(3.50)                                    (1.50)

Required rate = 3.50 : 1.50 = 7 : 3.

7(C)
Explanation :
Let initial quantity of wine = x litre
After a total of 4 operations, quantity of wine = x(1-y/x)^n=x(1-8/x)^4
Given that after a total of 4 operations, the ratio of the quantity of wine left in cask to that of water = 16 : 65
Hence we can write as (x(1-8/x)^4)/x =16/81
(1-8/x)^4 = (2/3)^4
(1-8/x) = 2/3
(x-8/x) = 2/3
3x-24=2x
x=24

8(C)
Explanation:
Let C.P. of 1 litre milk be Re. 1
Then, S.P. of 1 litre of mixture = Re. 1, Gain = 25%.
C.P. of 1 litre mixture = Re.(100/125 x 1) = 4/5
By the rule of alligation, we have:

C.P. of 1 litre of milk       C.P. of 1 litre of water
Re. (1)                                     (0)

Mean Price
Re. 4/5

4/5                                       1/5

Ratio of milk to water = 4/5 : 1/5 = 4 : 1.
Hence, percentage of water in the mixture = (1/5 x 100)% = 20%

9(D)
Explanation :
Concentration of alcohol in 1st Jar = 40%
Concentration of alcohol in 2nd Jar = 19%
After the mixing, Concentration of alcohol in the mixture = 26%
By the rule of alligation,
Concentration of alcohol in Concentration of alcohol in
1st Jar(40%)                                 2nd Jar (19%)

Mean
(26%)

(7)                                             (14)

Hence ratio of 1st and 2nd quantities = 7 : 14 = 1 : 2

 

Mixture and Alligation: Notes, Methods and Questions_110.1

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  1. Mixture and Alligation: Notes, Methods and Questions_40.1

57 Comments

  1. ye jo gladiator ji dikh rhe haina.. ye abhi medical clear karwake aye hai.. direct inhi se tips lelo.. hm bhi inse he lete hain 😀

  2. Bhai log SSC cgl me challan banvane ke bad kya krna hoga.pic and sign phle hi upload ho gya hai

  3. Hi, after the payment step of SSC CGL application I didnt get any email or msg confirmation from SSC. Is this happening with everyone?????

  4. Hello,,gladi yr mujhe sbi po and ibps po k liye tips do,,I wanna clear exam this year abhi tk 1single paper clear ne hua hai almost 1 1/2 yr ho gaye hai

  5. Dear admin please suggest me in question no 7 the value is 16/81 or 16/65 you mention here 16/65 please explain

  6. Bcz of in qustion mention, ratio(16:65) is obtain after drawn liquid so when we equate this then we take total quantity of liqued(16/16+65)=16:81..

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