Mixture problems with best and easy solutions all together

Mixture: Mixing of two or more than two type of quantities gives us a mixure.
Quantities of these elements can be expressed as percentage or ratio.
i.e. Percentage (20% of sugar in water)
Fraction ( A solution of sugar and water such that sugar : water = 1:4)

Alligation: Alligation is a rule which is used to solve the problems related to mixture and its ingredient.
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.

Alligation Rule
When two elements are mixed to make a mixture and one of the elements is cheaper and other one is costlier then,

Quantity of cheaper/Cost of Costlier = [CP of costlier (D) - Mean Price (M)]/[Mean Price (M)- CP of Cheaper (C)]


Example:
In what proportion must sugar at Rs 40 per kg be mixed with sugar at Rs 60 per kg so that the mixture be Rs 55 per kg?
Sol:

Here, CP of Cheaper(C) = 40,
CP of Costlier(D) = 60 and
Mean Price(M) = 55
So from the rule of alligation we can say that
Cheaper sugar (C)/Costlier sugar (D)= [60-55]/[55-40] = 5/15 = 1/3