Home » Banking » Quantitative Aptitude » Alligation Or Mixture
Important Formulae

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.

Practice Questions

1.

 In what ratio must rice at Rs. 9.30 per kg be mixed with rice at Rs. 10.80 per kg so that the mixture be worth Rs. 10 per kg ? A.7 : 8 B.8 : 7 C.9 : 7 D.7 : 9

By the rule of alligation, we have:

Required ratio = 80 : 70 = 8 : 7.

2.

 How much water must be added to 60 litres of milk at 1( ½) litres for Rs. 20 So as to have a mixture worth Rs.10(2/3) a litre ? A.12 litre B.13 litre C.14 litre D.15 litre

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.

 In what ratio must water be mixed with milk to gain 20% by selling the mixture at cost price? A.1 : 5 B.5 : 1 C.3 : 5 D.5 : 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.

 The milk and water in two vessels A and B are in the ratio 4 : 3 and 2: 3 respectively. In what ratio, the liquids in both the vessels be mixed to obtain a new mixture in vessel C containing half milk and half water? A.5 : 7 B.7 : 5 C.6 : 7 D.7 : 6

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.

 In what proportion must rice at Rs. 3.10 per kg be mixed with rice at Rs. 3.60 per kg so that the mixture be worth Rs. 3.25 per kg ? A.7 : 3 B.7 : 5 C.7 : 6 D.5 : 7

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.

 How many kilograms of sugar costing Rs. 6.10 per kg must be mixed with 126 kg of sugar costing Rs. 2.85 per kg so that 20% may be gained by selling the mixture at Rs. 4.80 per kg? A.67 kg B.68 kg C.66 kg D.69 kg

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.

 A butler stole wine from a butt of sherry which contained 40% of spirit and he replaced what he had stolen by wine containing only 16% spirit. The butt was then of 24% strength only. How much of the butt did he steal? A.2/3 B.4/5 C.6/5 D.3/2

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.

 In what ratio must a person mix three kinds of wheat costing him Rs. 1.20,Rs. 1.44 and Rs. 1.74 per kg., so that the mixture may be worth Rs. 1.41 per kg? A.7 : 8 : 9 B.11 : 77 : 7 C.12 : 88 : 9 D.11 : 88 : 7

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).