Alligation - Tips and Tricks for SSC and IBPS

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 a desired price.
Alligation is nothing but weighted average, used when a number of smaller groups are mixed together to form one larger group.

Situation of an Alligation:

Let two groups of elements are mixed together to form another group containing the elements of both the groups.
Let the first group contains n₁ number of elements and the average of first group of element is A₁. While second group contains n₂ number of elements and posses an average A₂.
Now, we need to find average of the new group formed. We can use either the weighted average equation or the Alligation equation.
That is,

Weighed average
$$ = \frac{{{\text{Sum}}\,{\text{total}}\,{\text{of}}\,{\text{all}}\,{\text{groups}}}}{{{\text{Total}}\,{\text{num}}{\text{.}}\,{\text{of}}\,{\text{elements}}\,{\text{of}}\,{\text{the}}\,{\text{groups}}}}$$
$$\eqalign{ & {{\text{A}}_{\text{w}}} = \frac{{{{\text{n}}_1}{{\text{A}}_1} + {{\text{n}}_2}{{\text{A}}_2}}}{{{{\text{n}}_2} + {{\text{n}}_2}}} \cr & {\text{Or,}}\,\frac{{{{\text{n}}_1}}}{{{{\text{n}}_2}}} = \frac{{{{\text{A}}_2} - {{\text{A}}_{\text{w}}}}}{{{{\text{A}}_{\text{w}}} - {{\text{A}}_1}}} \cr} $$
Where, A_w = equal to weighted average of third group formed.
This situation can be represented graphically:

Here, A₁ = average of elements of 1^st group;
A₂ = average of elements of 2^nd group;
A_w = average of elements of 3^rd group;
which is formed by combining 1^st and 2^nd groups of elements. n₁ and n₂ = elements of 1^st and 2^nd group.

Solved Examples

Example 1:
Average marks of two classes of students are 25 and 40 respectively. On mixing these two classes, the average becomes 30 marks. Find
a) The ratio of students in the classes
b) The number of students in the first class if second class had 30 students.

Solution:
Let n₁ is number of 1^st class and n₂ is number of students in second class; then

a) Ratio of students, $$\frac{{{{\text{n}}_1}}}{{{{\text{n}}_2}}} = \frac{{10}}{5} = \frac{2}{1}$$
b) Ratio of student in classes is 2 : 1 and 30 students are in second class then 60 students will be in 1^st class.

Example 2:
A tea merchant blends two types of tea costing Rs. 15 per kg and Rs. 20 per kg each respectively. In what ratio should these two types of tea be mixed so that the resulting mixture may cost Rs. 16.50 per kg?

Solution:

Hence, required ratio
$$\eqalign{ & = \frac{{3.50}}{{1.50}} \cr & = 7:3 \cr} $$

Example 3:
There are some pigeons and rabbits in a zoo. If the number of heads is 200 and the number of heads is 200 and the number of legs is 580, then how many pigeons are there?

Solution:
Rabbit has 4 legs and Pigeon has 2 legs.
Average legs per heads = $$\frac{{580}}{{200}} = \frac{{29}}{{10}};$$

Hence, Rabbits : Pigeons = $$\frac{{\frac{9}{{10}}}}{{\frac{{11}}{{10}}}} = 9:11$$
Thus, no. of pigeons = $$200 \times \frac{{11}}{{20}} = 110$$

Example 4:
The ratio of milk and water in a vessel is 5 : 2. In another vessel the ratio of milk and water is 8 : 5. In what ratio should the mixtures from both containers be taken so that the resulting ratio should be 9 : 4?

Solution:
Milk in 1^st vessel = $$\frac{5}{7}$$;
Milk in 2^nd vessel = $$\frac{8}{{13}}$$;
Milk in mixture = $$\frac{9}{{13}}$$;

Hence, Required ratio
$$\eqalign{ & = \frac{1}{{13}}:\frac{2}{{91}} \cr & = 7:2 \cr} $$

Example 5:
A vessel of 60 litres is filled with milk and water. 70% of milk and 30% of water is taken out of the vessel. It is found that the vessel is vacated by 40%. Find the initial quantity of milk and water.

Solution:

Hence, ratio of milk and water = 1 : 3
Quantity of milk = $$1 \times \frac{{60}}{4}$$  = 15 litres;
Quantity of water = 60 - 45 = 45 litres.

A mixture of spirit and water is kept in x equal glasses. Ratio of spirit and water in each glass is:
a1 : b1, a2 : b2, a3 : b3 . . . . . a_x : b_x.
All these glasses are emptied into a container then the ratio of spirit and water in container is given by-
$$\left[ {\frac{{{a_1}}}{{{a_1} + {b_1}}} + \frac{{{a_2}}}{{{a_2} + {b_2}}} + \frac{{{a_3}}}{{{a_3} + {b_3}}} + \,.\,.\,.\,.} \right]:$$       $$\left[ {\frac{{{b_1}}}{{{a_1} + {b_1}}} + \frac{{{b_2}}}{{{a_2} + {b_2}}} + \frac{{{b_3}}}{{{a_3} + {b_3}}} + \,.\,.\,.\,.} \right]$$

Example:
Three bowls each of 10 litres contain mixture of milk and water in the ratio of 2 : 1, 3 : 1 and 3 : 2 respectively. All these three bowls are emptied into a big container. What is the ratio of milk and water in the container?

Solution:
Reqd. ratio
 $$ = \left[ {\frac{2}{{2 + 1}} + \frac{3}{{3 + 1}} + \frac{3}{{3 + 2}}} \right]:$$     $$\left[ {\frac{1}{{2 + 1}} + \frac{1}{{3 + 1}} + \frac{2}{{3 + 2}}} \right]$$
$$\eqalign{ & = \left[ {\frac{2}{3} + \frac{3}{4} + \frac{3}{5}} \right]:\left[ {\frac{1}{3} + \frac{1}{4} + \frac{2}{5}} \right] \cr & = \frac{{121}}{{60}}:\frac{{59}}{{60}} \cr & = 121:59 \cr} $$

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