KOHAPTER

MIXTURE
(ALLIGATIONS)
MIXTURE Solution:
C.P of 1 kg CP. of I kg
Simple Mixture : When two different ingredients are mixed
Cheaper sugar Dearer sugar
together, it is known as a simple mixture. (1340 paise) (1365 paise )
Compound Mixture: Whentwoor more simple mixtures are mixed
together to fonm another mixture, it is known as a compound mixture Average price
Alligation: Aligation is nothing but a faster technique ofsolving (1320 paise)
problems based on the weighted average situation as applied to
the case of two groups being mixed together. 0
The word Aligation' literally means "linking'
Quantity of cheaper sugar 459
Alligation rule: It states that when difierent quantiti dearer sugar 20 4
Quantity of
the same or different ingredients ofdifferent costs a e t
together to produce a mixture of a mean cost, th io of They must be mixed in the ratio 9: 4.
their quantities is inversely proportional to the difense in
EMample 2:
their cost from the mean cost.
Amixture ofa certain quantity of milk with 16 litres of water
Quantity of Cheaper Price of Dearer -Mean Price
is worth 90 P per litre. Ifpure milk be worth Rs. 1.08 per litre,
Quantity of Dearer Mean Price- Price of Cheaper how much milk is there in the mixture?
Graphical representationof Alligation Rule: Solution:
Quantity Quantity The mean value is 90P and the price of water is 0 P.
b
Milk Water
108 0
Mean
Average (d)

b-d d-a
90-0 108- 90
Quantity ofab-d
Quantity of b d-a By the Alligation Rule, milk and water are in the ratio of5:1.
Quantity ofmilkinthe mixture=5 x
16=80 litres.
Applicationsof Alligation Rule
0 To find the mean valueofa mixture when the pricesoftwo or Price ofthe Mixture:
more ingredients, which are mixed together and the proportion When quantities Q, of ingredients M's with the cost C's
in which they are mixed are given. are mixed then cost of the mixture Cm is given by
() To find the proportion in which the ingredients at given
Cm2C
prices must be mixed to produce a mixture at a given price.

Example 1: Example 3:
In what proportion must sugar at Rs. 13.40 per kg be mixed 5 kg of rice of Rs. 6 per kg is mixed with 4 kg of rice to get a
with sugar at Rs. 13.65 per kg, so that the mixture be worth mixture costing Rs. 7 ker kg. Find the price of the costlier
Rs.13.20 akg? rice.
 MATHEMATICAL SKILLS

M-120
same
mixed in the
Solution Thus, the original mixture and liquid Q are
Let the price of the costlier rice be Rs.x. ratio d
out
By direct formula, added. then after taking
I15 litres of liquid Q is should have 15
litres
o
from the jar, there
itres of mixture
7 6x5+4xx mixture lef
9
So, the quantity of mixture in the Jar
63-30=4x4x-33 = 15 15= 30 litres

x8.25 4 and quantity of P in the jar ,4 24 litres

that in
Straight line approach ofAlligation Mixture: Remeimber
Alligation Rule for Compound
Let Q, and Q, be the two quantities, and n, and n, are the number i.e. mixtures ol same
Compound mixture, same mixtures
to make
ngredients are mixed together in different proportion
of elements present in the two
quantities respectively,
a new mixture
Q a:b
Let Mixture I has ingredients A and B in ratio
ind Mixture has ingredients A and B in rati0 X : y.

2 are mixed to
Now, M unit ofmixture I and N unit of mixture
resultant mixture, the
Torni compound miature. Then, in the
where Av is the average of the new group formed then ratio of A and B is
ncorresponds to Q, Avy, n, corresponds to Av - Q, and
(n, tn) coresponds to - 9
M N
Let us consider the previous example.
Quantity of ingredient A-A Mb x
antity of ingredient B B M
Example 4
Skg ofrice at Rs. 6 per kg is mixed with 4 kg ofrice to getamixi
costing Rs. 7 per kg. Find the price of the costlier rice.
Solution: SA
Using straight line method,
Quantity of A in resultant mixture A x(M+N)
A +9B

Quantity ofB in resultant mixture = -x(M +N)

9A +98
4 corresponds to 7 6 and S corresponds to x- 7. ) When q, and qR are known and M and N have to be found
ie. 4 1 out

51.25
Hence, x-7= 1.25
x8.25
Quantity of mixture 1Q_X+y)9 +98
Example 5:
A jar contains a mixture of two liquids P and Q in the ratio
Quantity of mixture 2 Q2 9A
4:1. When 15 litres ofthe mixtureistaken out and 15 litres of 9A +8 a+b
liquid Q is poured into the jar, the ratio becomes 2:3. How
And.
nany litres of liquid P was contained in thejar.
Solution:
Quantity ofmixture

Quantity of resultant mixture

Fraction of Q in original mixture
Q+Q2
Quantityof mixture2
Fraction of Q in resulting mixture = 33
2+3 5 2 Quantity of resultant mixture
Qin mixture Pure Q Q+0

Removal and Replacement

Let a vessel contaigs Q unit of mixture of ingredients A and
B. From this, R unitof mixture is taken out and replaced by
an equal amount of ingredient B only.
MiXTURE (AIligations)
If this
process is repeated n times, then after n M-121
operations The amount of water after
Quantity of Aleft adding x kg of water becomes
Quantity of A originally present
and
Quantity of Bleft
i) Let a vessel contains Q-Quantity of A Left According to question,
Qunit of ingredient A
R unit of only. From this
amount of
ingredient A is taken out and
replaced by an equal
If this
ingredient B.
process is repeated n times, then after
operations.
n
T0x
=2 x =-8
8
Quantity of Aleft Q1-
Quantity of B=1-Quantity of Aleft of the 8 kg mixture is taken out.

Example 6: I f in x litres mixture of A and the

Acontainercontains 40 litres of milk. From this B, ratio of A a B isa:b,
the quantity of B to be added in order to make the ratio c
litres of milk was taken out container, 4 :d
and replaced by water. This
process was repeated further two times. I is X(ad- bc)
low much milk is
now contained
by the container? c(a+b)
(a) 26.34 itres (6) 27.36 litres
(c) 28 litres (d) 29.16 1itres Example8:
Solution: The ratio of water and milk in a 30 litres mixture
is 7 :3. Find
(d) the quantity of water to be added to the
Milk Water
mixture in order to
To start with make this ratio 6: 1.
40 litres
After Ist Solution
operation 36 litres 4 litres In this example the ratio of water: milk is given and water is
further added. But in the above formula ratio of :
After 2nd operation 36-x36 4-x
4
4+4 A B is
40 40 given and quantity B is added. So the formula in this
changed
32.4 litres 4-0.4 +4 Scenario becomes
x(bc - ad)
7.6litres
Quantity of B added =

d(atb
After 3rd operation 324324 76 404
40
Required quantity 30(3x6-7x)_30(18-7)
32.4-3.24 -76-0.76SALENN l(7+3) Ix10
29.16 10.84
. The quantity of
milk in the container is 29.16 litres.
10 30x-33litres.
A mixture contains A and B in the ratio
Examples 7: a:b. lfx litres
added to the mixture, A and B become in the of B
is
A dishonest hair dresser uses a
mixture having 5 parts pure ratio
After shave lotion and 3 parts of a:c.Then the quantity of Ain the mixture is given by ax
pure water. After out taking
some portion of the mixture, he adds equal amount of pure c-b
bx
water to the remaining portion of the mixture such that the
amount of Aftershave lotion and water become
andthat of Bis given
by-h
part of the mixture taken out is
equal. The Example 9:
(a) 13 A mixture contains beer and soda in the ratio
(6) 1/5 of 8: 3. On
(c) /4 adding 3 litres of soda, the ratio of beer to soda becomes 2
(d) 16
Solution: 1(i.e., 8:4). Find the quantity of beer and soda in the mixture.
6) Let quantity of pure After shave lotion Solution:
=

Skg
and quantity of pure water 3 =

kg Quantity of beer in the mixture= 24 litres

Total quantity of the mixture 8 kg =

Again letx kg of mixture is taken out of 8kg of mixture. 3x3

and the quantity of soda in the mixture = 9 litres.
4-3
Now, the amount of Aftershave lotion left
Example 10:
Mira's expenditure and savings in the ratio 3:2.Her income
and the amount of water lef= increase by 10%. Her expenditure also increases by 12%. By
how many %does her saving increase?
 M-122
MATHEMATICAL SKILLS
Solution:
Expenditure Example 13
12 Saving KS500 n invested in two such part that if one invested a
16%, and the other at 5% the total interest in one year iron
(% increase in
exp) %increase in saving) both investments is Rs 85. How much invested at 507
Solution:
t the whole money is invested at 6% the annual income iS

10 6% ofRs. I,500 Rs. 90. Ifthe whole money is invested at

(% increase in income) 5%, the annual income is s% ofRs 1,500 Rs 75. But real
income Rs85
Applying the alligation rule, we have
6% 5%
Rs 90 Rs 75
We get two values
(given)
of x, 7 and 13 But to get a viable answer,
we must keep in mind the
central value (10) must lie between
x and 12. Thus the
value of xshould be 7 and not 13.
required % increase = 7% Rs 85
Example 1l:
A vessel of
80 litre is filled with milk and water. 70% of milk
and 30% of water is taken out
of the vessel. It is found that Rs 10 Rs 5
the vessel is vacated by 55%. Find the initial
and water.
quantity of milk
Solution: Money investedat 5%= Rs 1,500 Rs500
Here theo values of milk and water that is
taken from the
vessel should be taken into consideration. Example 14
milk water Three vessels containing mixtures of milk and water are of
70% 30% capacities which are in the ratio 1:2:3. The ratios of milk and
water in the three vessels are 4:1,3:2 and 2:3 respectively.
one fodrth the contents of the first vessel, one-third of
that of the second vessel and half of that of the third vessel
55 % are mixe6, ýhat is the ratio of milk and water in the new mix-

Solution
25% Park of milk in the resultant solution
15%
5:3
Ratio of milk to water =5:3 .2...2.
80 Part of water in the resultant solution
quantity of milk= 5+3
x5=50 litres
80
andquantity of water= x3= 30 litres
Example 12 73
Nine litres are from drawn from a case Ratio of milk to water 672:73.
full of water and it is 360
then filled with milk .Nine litres of Example 15
mixture are drawn and the
cask is again filled with milk.The quantity Sea water contains 5 % salt
of water now left in
the cask is to that of the milk in it as 16:9. How by weight. How many kg of fresh
much does the water must be added to 60
caskhold? kg of sea water for the content of
salt in solution to be made 3%.
Solution: Solution:
Let there be x litres in the cask From Let
the above formula we
x kg of fresh water is added to sea water
have, after n operations

Water left in vessel after n salt 5%of60 3

operations
-

Whole quàntity of milk in vessel saltwater 60+x 100

(given 3% salt in solution)
3 B
Thus in this case,1 ) . 60+x 100 X=40 kg.
x45 litr 40 kg of fresh waer must be added
to sea water.