SIXPHRASE-THE FINISHING SCHOOL
APTITUDE TRAINING
QUATITATIVE SECTION
PART 7 - ALLIGATIONS & MIXTURES
INSTRUCTOR-SARAVANA KUMAR
ALLIGATIONS & MIXTURES
Sub Topics
√ Components of Alligations & Mixtures
√ Is it the same as Weighted Average?
√ Alligation & Mixtures Formula Derivation
√ Alligation & Mixtures Formula Formation
√ Alligation & Mixtures Formula - Diagram
√ Replacement Problems
ALLIGATIONS & MIXTURES
Components of Alligation & Mixture
In any Alligation & Mixture problem you will always have,
Example
Ingredient 1
+
Ingredient 2
+
Mixture
55kg
of Type 1 Rice costs Rs 70/kg
45kg
of Type 2 Rice costs Rs 60/kg Ingredient 2
100kg
of Mixture Rice costs Rs x/kg Mixture
Ingredient 1
2 Ingredients are mixed to get the Mixture. Together these 3 components form
the foundation of Alligation & Mixture problems. Sometimes we also may have
More than 2 ingredients!
ALLIGATIONS & MIXTURES
Is it the same as Weighted Average?
Weighted Average
Weight attached to 50 kg (Average)
65
Boys average weight
50 kg
Weight attached to 40 kg (Average)
35
Girls average weight
40 kg
Students average weight
x kg
100
Average of 100 students = [Sum of weights of 100 students] / 100
= [Sum of weights of 65 boys + Sum of weights of 35 girls] / 100
Weights are attached to Averages, hence the name Weighted
Average
ALLIGATIONS & MIXTURES
Alligation & Mixtures Formula Derivation
Weighted Average
N1 Boys average weight
N1 + N2
A1
+
N2 Girls average weight
Students average weight
A2
A
Average of N1 + N2 students = [Sum of weights of (N1 + N2) students]
/ (N1 + N2)
= [Sum of weights of N1 boys + Sum of weights of N2 girls] /(N1 + N2)
A
= [A1 * N1 boys + A2 * N2] / (N1 + N2)
Weighted Average is the same as Alligations & Mixtures.
Infact Alligations formula is derived from Weighted Average
ALLIGATIONS & MIXTURES
Alligation & Mixtures Formula Derivation
A
= [A1 * N1 boys + A2 * N2] /(N1 + N2)
[N1 * A + N2 * A] = [A1 * N1 boys + A2 * N2]
[N1 * A – N1 * A1] = [A2 * N2 – A * N2]
N1 * [A – A1] = N2 * [A2 – A]
N1 / N2 = [A2 – A] / [A – A1]
N1
Number of Boys
N2
Number of Girls
A1
Average weight of Boys
A2
Average weight of Girls
A
Average weight of Boys + Girls
Average = Sum / Number.
Sum = Average * Number.
ALLIGATIONS & MIXTURES
Alligation & Mixtures Formula Formation
Foundation Formula
N1 / N2 = [A2 – A] / [A – A1]
75 ml
H2SO4 of concentration
40 %
Ingredient 1 - Cheap
25 ml
H2SO4 of concentration
50 %
Ingredient 2 - Dear
100 ml
H2SO4 of concentration
42.5 %
Mixture
Value
Quantity
Cheap Quantity
75 ml
Cheap Value
40 %
Dear Quantity
25 ml
Dear Value
50 %
Cheap Ingredient Ingredient with Lesser Value.
Dear Ingredient Ingredient with More Value.
ALLIGATIONS & MIXTURES
Alligation & Mixtures Replacement Problem.
ALLIGATIONS & MIXTURES
Alligation & Mixtures Formula - Diagram
Weighted Average Alligations & Mixture conversion
N1 / N2 = [A2 – A] / [A – A1]
CV
CQ / DQ = [DV – MV] / [MV – CV]
DV
MV
DV - MV
CQ
MV - CV
:
CQ
Cheap Quantity
DQ
Dear Quantity
CV
Cheap Value
DV
Dear Value
MV
Mean Value
DQ
Cheap Ingredient Ingredient with Lesser Value.
Dear Ingredient Ingredient with More Value.
ALLIGATIONS & MIXTURES
Replacement Problems
Initial Solution Pure constituent
Final Quantity of A after ‘n’ replacements
Initial Quantity of A
[𝒙 − 𝐲]
𝐧
𝐱
A Liquid which gets reduced as a result of replacement
x Capacity of the container / beaker/ vessel / can
Liquid N (Liquid Replacing
Liquid M
Liquid M every time
slowly)
y Quantity of solution drawn off & replaced
n Number of times replacement is done
Replacement is always done on a part of the solution, never the whole solution.
ALLIGATIONS & MIXTURES
Replacement Problems
Initial Solution Impure constituent/ Mixture
Final fraction of A after ‘n’ replacements
Initial fraction of A
Liquid N (Liquid Replacing
Liquid M every time
slowly)
Liquid M
𝐧
𝐱
A Liquid which gets reduced as a result of replacement
x Capacity of the container / beaker/ vessel / can
y Quantity of solution drawn off & replaced
Liquid N
[𝒙 − 𝐲]
n Number of times replacement is done
Replacement is always done on a part of the solution, never the whole solution.
ALLIGATIONS & MIXTURES
There are two containers on a table. A and B. A is half full of wine, while B,
which is twice A's size, is one quarter full of wine. Both containers are filled
with water and the contents are poured into a third container C. What portion
of container C's mixture is wine?
(a) 50% of wine (b) 33.33% of wine (c) 16.66% of wine (d) 75% of wine
ALLIGATIONS & MIXTURES
A vessel contains 30% of milk and 70% of water. Another vessel with
double the capacity of first contains 70% of milk and 30% of water. If both
are mixed what is the concentrations of the milk in the mixture?
(a) 56.67%
(b) 46.67%
(c) 50.67% (d) 42.67%
ALLIGATIONS & MIXTURES
How many litres of a 90% solution of concentrated acid needs to be mixed with
a 75% solution of concentrated acid to get a 30L solution of 78% concentrated
acid? (a) 24L
(b) 22.5L
(c) 6L
(d) 17.5L
ALLIGATIONS & MIXTURES
A bag contains 2 kg of potatoes and 3 kg of tomatoes. If the cost of potatoes is
RS.5 per kg and that of tomatoes is RS.8 per kg, then what is the average price
per kilogram of the vegetables in the bag?
(a) RS.6 (b) Rs.6.60 (c) Rs.6.80
(d) RS.7
ALLIGATIONS & MIXTURES
In what ratio must a grocer mix two varieties of pulses costing Rs. 15 and Rs. 20
per kg respectively so as to get a mixture worth Rs. 16.50 kg?
(a) 7 : 3 (b) 3 : 7 (c) 7 : 2
(d) 7 : 4
ALLIGATIONS & MIXTURES
Two solutions of sulphuric acid are mixed in the ratio 1 : 9. The
concentration of sulphuric acid in the first and second solution are 10% and
20% respectively. What is the concentration of sulphuric acid in the new
solution?(a) 18%
(b) 16% (c) 19% (d) 17%
ALLIGATIONS & MIXTURES
A rice with price 126 per kg. A rice with 135 per kg is mixed with another
rice in the ratio 1:1:2. If the mixed rice is 153 per kg. What is the price of
third variety of rice?
(a) 130.50 (b) 175.50
(c) 153.00
(d) 165.50
Three Ingredients
Weighted Average
ALLIGATIONS & MIXTURES
Easha bought two varieties of rice, costing Rs.50 per kg and Rs.60 per kg
each, and mixed them in some ratio. Then she sold the mixture at Rs.70 per
kg, making a profit of 20%. What was the ratio of the mixture?
(a) 1:10 (b) 3:8
(c) 1:5
(d) 2:7
ALLIGATIONS & MIXTURES
A merchant buys 20 kg of wheat at Rs.30 per kg and 40 kg of wheat at Rs.25
per kg. He mixes them and sells one third of the mixture at Rs.26 per kg.
The price at which the merchant should sell the remaining mixture so that
he may earn a profit of 25% on his whole outlay is
(a) Rs.30 (b) Rs.40 (c) Rs.36
(d) Rs.37
ALLIGATIONS & MIXTURES
How many kgs of wheat costing Rs.24/- per kg must be mixed with 30 kgs
of wheat costing Rs.18.40/- per kg so that 15% profit can be obtained by
selling the mixture at Rs.23/- per kg?
(a) 10
(b) 11
(c) 12
(d) 13
ALLIGATIONS & MIXTURES
Two alloys A&B are composed of two basic elements. The ratio of the
composition of the two elements in the 2 alloys is 5:3 & 1:2. A new alloy X is
formed by mixing the alloys A & B in the ratio 4:3. What is the ratio of the
composition of 2 elements in alloy X?
(a) 1:1
(b) 2:3
(c) 5:2
(d) 4:3
(e) 7:9
ALLIGATIONS & MIXTURES
An alloy of Copper and Aluminium has 45% Copper. An alloy of Copper and
Zinc has Copper and Zinc in the ratio 3:6. These two alloys are mixed in such a
way that in the overall alloy, there is more Aluminium than Zinc, and Copper
constitutes a fraction x of this alloy. What is the minimum value of x (as a
fraction)? (a) 28/73 (b) 9/20 (c) 31/73 (d) 29/73
ALLIGATIONS & MIXTURES
On a certain assembly line, the rejection rate for Hyundai i10s production
was 4%, for Hyundai i20s production 8% and for the 2 cars combined 7%.
What was the ratio of Hyundai i20 and i10 production?
(a) 3/1
(b) 2/1
(c) 1/1
(d) 1/3
ALLIGATIONS & MIXTURES
A beaker contains 180 litres of alcohol. On the first day, 60 litres of alcohol
is taken out and replaced by water. On the second day, 60 litres of the
mixture is taken out and replaced by water and the process continues day
after day. What will be the quantity of alcohol in the beaker after the third
day?
(a) 40 litres (b) 80 litres (c) 53.33 litres (d) 100 litres
Final Quantity of A after ‘n’ replacements
Initial Quantity of A
[𝒙 − 𝐲]
𝐧
𝐱
ALLIGATIONS & MIXTURES
A vessel contains 33 l of milk and water in the ratio of 7:4 respectively. If the
mixture is replaced with 4 l of milk, what is the quantity of milk in the
mixture?
(a) 24.23 l
(b) 22.45 l
(c) 30.5 l
(d) 20.55 l
Final Fraction of A after ‘n’ replacements
Initial Fraction of A
[𝒙 − 𝐲]
𝐧
𝐱
ALLIGATIONS & MIXTURES
A can contains a mixture of two liquids A and B is the ratio 7 : 5. When 9 l of
mixture are drawn off and the can is filled with B, the ratio of A and B
becomes 7 : 9. How many litres of liquid A was contained by the can initially?
(a) 18 l (b) 21 l (c) 24 l (d) 20 l
Final fraction of A after ‘n’ replacements
Initial fraction of A
[𝒙 − 𝐲]
𝐧
𝐱
0
