__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

** = **

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.

__Example:__

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 ?

__Answer:__

=

=

Quantity of cheaper rice / Quantity of dearer rice = 30 / 25 = 6 / 5

**So, They must be mixed in the ratio 6: 5 .**

__Example:__

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?

__Answer:__

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.**

__Example:__

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?

__Answer:__

We know that: =

= =

= =

**So , They must be mixed in the ratio 1 : 2 .**

__Example:__

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

__Answer:__

The already existing solution has 30% rice .

So the other solution has 100% rice ,

We know that: =

= = .

The two mixtures added in ratio 5 : 2 .

required rice is = 400 x = 160 gm .

__Shortcut tricks:__

solution

**400**** = 160 gm .**

__Example:__

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.**