Alligation and Mixture - Concept with Examples

 The word Alligation literally means linking .

Alligation method is applied for percentage, value, ratio, rate, prices, speed etc. and not for absolute values.That is whenever per cent ,per hour, per kg, per km etc. are being compared , we can use Alligation.

Rules of allegation:

If the gradients are mixed in a ratio, then

Then, (Cheaper Quantity) : ( Dearer Quantity)  = (d-m) : (m-c).

Let us understand it better by doing some examples:

Question - 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 a kg?

Solution -

Required ratio = (d-m)/(m-c)   = 35/15  =7:3  

Please note that when the decimal are same in number , remove decimals or multiply by 10 or 100 0r 1000 depending upon the number of decimals .

Question - How many kg. of salt at 42 P per kg. must a man mix with 25 kg of salt at 24 P per kg. so that he may, on selling the mixture at 40 P per kg, gain 25% on the outlay?

Solution - 

Cost price of the mixture  = S.P. x 100/(100+gain %)
                                          = 40 x 100/(100+25) 
                                          = 40 x 100/125
                                          = 32 P per kg

Require Ratio: 8/10 = 4:5

Thus for every 5 kg of salt at 24P, 4 kg of salt at 42 P is used.

Required number of kgs  =25 x 4/5 = 20

Milk and mixture

Question - A mixture of a certain quantity of milk With 16 litres of water is worth 90 P per litre. If pure milk be worth Rs. 1.08 per litre. How much milk is there in the mixture

Solution -

The mean value of is 90 P  and the price of water is 0 

Ratio of milk and water = 90/18 =5:1
= Therefore quantity of milk in the mixture =5 x16 = 80 litres.

Question - In what proportion must water be mixed with spirit to gain 16 2/3% by selling it at cost price?

Solution -

Let the CP of spirit be Rs. 1 per litre.

= Then S.P. of 1 litre of mixture = Rs. 1 gain = 162/3%.

= C.P of 1 litre of mixture = Rs.[(100 x 3 x 1 )/350 ] =    Rs.  [6/7]

(Quantity of water)/(quantity of spirit)=(d-m)/(m-c)

                                                             =  (1/7)/(6/7)=1/6

Or ratio of water and spirit = 1:6

Mixture from two vessels

Milk and water are mixed in a vessel A in the proportion 5:2, and in vessel B in the proportion 8:5. In what proportion should quantities be taken from the two vessels so as to form a mixture in which milk and water will be in the proportion of 9:4?

Solution 

In vessel A, milk  = 5/7  of the weight of mixture. 

(Total parts = 5+2 =7)

In vessel B, milk = 8/13 of the weight of mixture 

(Total parts = 8+5=13). 

Now we want to form a mixture in which milk will be 9/13 ( Total parts 9+4 = 13) of the weight of the mixture

By Alligation rule:

required proportion is  (d-m)/(m-c)

= (9/13- 8/13 )/(9/13- 5/7 )

=  1/13 : 2/91     

= 7:2 

Question - A butler stores wine from a butt of sherry which contained 30% of spirit and he replaced what he had stolen by wine containing only 12% of spirit. The butt was then 18% strong only. How much of the butt did he steal?

Solution - 

By the Alligation rule we find that wine containing 30% of spirit and wine containing 12% of spirit should be mixed in the ration 1:2 to produce a mixture containing 18% of spirit.

Ratio =  6:12 = 1:2
This means that  1/3 of the butt of sherry was left, i.e. to say, the butler drew out 2/3  of the butt.
Therefore, 2/3  of the butt was stolen.