Quantitative Aptitude is avery 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 a desired price. We have covered important notes and questions focusing govt. exams that aspirants can practice and prepare 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,

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.

Thus, **(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 pure

**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?**

**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?**

**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 ?**

**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?**

**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 ?**

**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?**

**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?**

**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:**

**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**

**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 = 175.5

**4(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

@Mosaic

_/_ 🙂

Thank u sir,,very useful

1.B

..

Gladi maths k post pe aa jaya kro discuss krne

solve krne mei baht problem h abhi bhi 🙁

I hate dis chapter,bt will try

..

Aap yaha bhi:))

2.B ?

jaha mam bulaenge waha ham jaenge 😛

Hihi :)) mega kuch tips de so sbi or Po k liye is baar clear ho jaye

1=b

3=c

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 😀

Gladi kaha ho. Its me Sam

Hanji jaroor 🙂

1-b ,2-c ,3-c

:O mixture n alligation _/_

lunch k baad try krta hoon..:P

Mt kro lunch

😛 ammmma bhuk lag gyi

Shayar yar ye chapter samajh me aata kabhi bhi plz help.after Lunch

ok.. lunch k baad aata hoon..

tab tak try krte rhiyo

1-B, 2-C, 3-C, 4-C, 5-D6-C, 7-C, 8-C, 9-D, 10-B

Hmm..

B.B.C.C.?.C.C.C.D.B

explain ???

????..

explain..?

which one

4-10 pls explain??

B,?,C,C,D,C,C,C,D,B

Let 100 ml was there and it contained 40 alcohol

No 40(1-x)+19*x= 26

X= 2/3

saare..

wait answers explanations ke sth ajaenge

any problem.. i hav problem dis chapter

Without explanation :((

Explain to mam krengi naaa

Han…

Aajkal quiz ne hoti kya yaha @mosaic ki??

dikha nhi..mujhe kal se vo……

Han mai bhi usey hi search kar rahi hu kal se.. 🙁

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

1-10

BCCCD

CCCDB

D most easiest one

D most easiest one

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

ur from which region?

Uff…..sam…

Pehchan hi nahi saka…

Hi sam …hw r u??

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

Why in questions no. 2 it is multipled with 1/8. Please explain

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

Same doubt…what happened in que no 7

Plz explain 2 & 4 questions. …

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..

Thanks a lot priyanshu ji for clear my doubt

Yaar ye Slov kase kaiya h samja hi nhi aa rha

Post office. Ka exam h