1. Alligation:
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.
2. Mean Price:
The cost of a unit quantity of the mixture is called the mean price.
3. Rule of Alligation:
If two ingredients are mixed, then
Quantity of cheaper/Quantity of dearer = C.P. of dearer  Mean Price/QMean price  C.P. of cheaper
∴ (Cheaper quantity) : (Dearer quantity) = (d  m) : (m  c).
4. Suppose 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}]units.
1. 

By the rule of alligation, we have:
Required ratio = 80 : 70 = 8 : 7.
2. 

C.P. of 1 litre of milk = Rs. (20 x 2/3) = Rs. 40/3
Ratio of water and milk = 8/3 : 32/3 = 8 : 32 = 1 : 4
Quantity of water to be added to 60 litres of milk = [1/4 x 60 ]litres =15 lt.
3. 

Let C.P. of milk be Re. 1 per litre.
Then, S.P. of 1 litre of mixture = Re. 1.
Gain obtained = 20%.
C.P. of 1 litre of mixture = Rs.[(100/120) x 1]=Rs.5/6
By the rule of alligation, we have:
Ratio of water and milk = 1/6 : 5/6 = 1 : 5
4. 

Let the C.P. of milk be Re. 1 per litre
Milk in 1 litre mixture of A = 4/7 litre
Milk in 1 litre mixture of B = 2/5 litre
Milk in 1 litre mixture of C = ½ litre
C.P. of 1 litre mixture in A = Re .4/7
C.P. of 1 litre mixture in B = Re.2/5
Mean price = Re.1/2
By the rule of alligation, we have:
Required ratio = 1/10 : 1/14 = 7 : 5
5. 

By the rule of alligation, we have:
By the allegation rule:
(Quantity of cheaper rice) / (Quantity of dearer rice) = 35/15 = 7/3
They must be mixed in the ratio 7 : 3.
6. 

S.p of one Kg of mixture = Rs. 4.80, Gain = 20%.
C.P of one kg of mixture = Rs. [(100/120) x 4.80] = Rs. 4.
(Quantity of cheaper sugar)/(Quantity of dearer sugar) = 210/115.
(Quantity of cheaper sugar)/(Quantity of dearer sugar) = 42/23.
If cheaper sugar is 42 kg, dearer one = 23 kg
If cheaper sugar is 126 kg,
Dearer one = [(23/42) x 126]kg= 69 kg.
7. 

By allegation rule:
(Wine with 40% spirit) / (Wine with 16% spirit) = 8/16 = 1/2.
i.e., they must be mixed in the ratio (1 : 2).
Thus 1/3 of the butt of sherry was left and hence the butler drew out 2/3 of the butt.
8. 

Step 1.
Mix wheats of first and third kind to get a mixture worth Rs. 1.41 per kg.
By allegation rule :
(Quantity of 1st kind of wheat)/(Quantity of 3rd kind of wheat) = 33/21
(Quantity of 1st kind of wheat)/(Quantity of 3rd kind of wheat) = 11/7
i.e., they must be mixed in the ratio 11 : 7.
Step 2.
Mix wheats of first and second kind to get a mixture worth Rs. 1.41 per kg.
By alligation rule :
(Quantity of 1st kind of wheat) / (Quantity of 3rd kind of wheat) = 3/21
(Quantity of 1st kind of wheat) / (Quantity of 3rd kind of wheat) = 1/7
i.e., they must be mixed in the ratio 1 : 7.
Thus, (Quantity of 2nd kind of wheat)/(Quantity of 1st kind of wheat) = [(Quantity of 2nd kind of wheat) x (Quantity of 1st kind of wheat)]/[(Quantity of 1st kind of wheat) x (Quantity of 3rd kind of wheat)] = [(7/1) x (11/7)] = 11/1.
Quantities of wheat of(1st kind : 2nd kind : 3rd kind) = (1 : 7 : 7/11).
Quantities of wheat of(1st kind : 2nd kind : 3rd kind) = (11 : 77 : 7).
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