Mixture and allegation

Alligation: This 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 a desired price. Mean Price: The cost Price of a unit quantity of the mixture is called the mean price.
 Rule of Alligation:

If Two ingredients are mixed in a ratio , then
Quantity of cheaper/Quantity of dealer =C.P dealer-Mean price/Mean price-C.P Cheaper    

We present it under as
C.P.M.1 = Cost Price of First Materiel in a Mixture.
C.P.M.2 = Cost Price of Second Materiel in a Mixture.
C.P.M.3 = Cost Price of Mixture.
So,
C.P.M.1 : C.P.M.2 = ( C.P.M.1 – C.P.M.3 ) : ( C.P.M.2 – C.P.M.3 ).
OR,
(Cheaper quantity ) : ( Dearer quantity ) = ( d – m ) : ( m – c ).
m = mean price.
d = C.P of dearer.
c = C.P of cheaper.
 Here is another important formula which is,
Consider a container contain x units, and from which we are taken out y units and replace it by water.
After that n operation,
the quantity of pure liquid = [ x ( 1 – y / x )n] units.
Here is some sub link of Examples about this topic which help you better understanding.

In what proportion must rice at 3.25 per kg be mixed with rice at Rs 3.80 per kg , So that the mixture be worth Rs 3.50 a kg ?

 = Quantity of cheaper/Quantity of dealer =C.P dealer-Mean price/Mean price-C.P Cheaper    

= 380-350/350-325
Quantity of cheaper rice / Quantity of dearer rice = 30 / 25 = 6 / 5
So, They must be mixed in the ratio 6: 5 .

In a What ratio Wheat at Rs. 12.30 per kg be mixed with wheat 18.30 per kg so that the mixture be worth Rs.15 per kg?

We know that: (Cheaper quantity): (Dearer quantity)=(d – m): (m – c).
C.P.M.1=(Cost price of First material) = 12.30 = 1230
C.P.M.2=(Cost price of Second material) = 18.30 = 1830
M.P = (Mean price of both Material) = 1500
So,
(C.P.M.1 – M.P):(C.P.M.2 – M.P)= (1500 – 1230):(1500 – 1830) = 270 : 330 = 9 : 11.
Required Ratio of both materials is 9: 11.


A goldsmith has two qualities of gold one of 14 carats and another of 17 carats purity . In what proportion should he mix both to make an ornament of 16 carats purity?

= 17-16/16-14 = 1/2
So , They must be mixed in the ratio 1 : 2 .

400 gm of rice solution has 30% rice in it  . How much rice should be added to make it 50% in solution?

The already existing solution has 30% rice .
So the other solution has 100% rice ,
100-50/50-30
= 50/2=5/2
The two mixtures added in ratio 5 : 2 .
required rice is = 400 x 2/5= 160 gm .

In What ratio stone be mixed with rice so as to gain 40% by selling at C.P?
Answer: Let be
C.P of pure rice is = 1
and S.P of mixture is = 1
than % profit = 40%
C.P in a mix is = 1 / 140 X 100 = 5 / 7.

here we applied alligation formula
C.P.1 = 1(rice).
C.P.2 = 0(stone).
M.P(mix) = 5 / 7.
= ( C.P.1 – M.P ):( C.P.2 – M.P ) = 2 / 7 : 5 / 7 = 2 : 5.
So Ratio between Rice and Stone to gain 40% Profit = 2 : 5.