Reasoning and Quantitative Aptitude Alligation or Mixture
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ALLIGATION OR MIXTURE
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 2
ALLIGATION OR MIXTURE
IMPORTANT FACTS AND FORMULAE
1. 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.
2. Mean Price: The cost price of a unit quantity of the mixture is called the mean price.
3. Rule of Alligation: If two ingredients are mixed, then
(Quantity of cheaper) = (C.P. of dearer) - (Mean price)
(Quantity of dearer) (Mean price) - (C.P. of cheaper)
We present as under:
C.P. of a unit quantity of cheaper C.P. of a unit quantity of dearer
(c) (d)
(m)
(d-m) (m-c)
(Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c).
4. Suppose a container contains x units of liquid from which y units are taken out and replaced
by water. After n operations the quantity of pure liquid=[ x(1-y/x)^n]units.
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 3
SOLVED EXAMPLES
Ex. 1. In what ratio must rice at Rs. 9.30 per kg be mixed with rice at Rs. 10.80 per kg so
that the mixture be worth Rs. 10 per kg ?
Sol. By the rule of alligation, we have:
C.P. of 1 kg rice of 1st kind (in paise) C.P. of 1 kg rice of 2nd kind (in paise)
930 1080 .
.
Mean price
(in paise)
1000
80 70
:. Required ratio = 80 : 70 = 8 : 7.
Ex. 2. How much water must be added to 60 litres of milk at 1 ½ litres for Rs. 20
So as to have a mixture worth Rs.10 2/3 a litre ?
Sol. C.P. of 1 litre of milk = Rs. (20 x 2/3) = Rs. 40/3
c.p of 1 litre of milk c.p of 1 litre of milk
0 Rs.40
3
Mean price
(Rs. 32 )
3
(40/3-32/3)=8/3 (32/3-0)=32/3
Ratio of water and milk =8 : 32 = 8 : 32 = 1 : 4
3 3
Quantity of water to be added to 60 litres of milk = [1/4 X 60 ]litres =15 litre
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 4
Ex. 3. In what ratio must water be mixed with milk to gain 20% by selling the mixture at
cost price?
Sol. Let C.P. of milk be Re. 1 per litre.
Then, S.P. of 1 litre of mixture = Re. 1.
Gain obtained = 20%.
C.P. of 1 litre of mixture = Rs.[(100/120)* 1]=Rs.5/6
By the rule of alligation, we have:
C.P. of 1 litre of water C.P. of1litre ofmilk
0 Re.1
(Re. 5/6)
(1- (5/6))= 1/6 ((5/6)-0)=5/6
Ratio of water and milk = 1/6 : 5/6
Ex. 4. .How many kgs. of wheat costing Rs. 8 per kg must be mixed with 86 kg of rice
costing Rs. 6.40 per kg so that 20% gain may be obtained by Belling the mixture at Rs. 7.20
per kg ?
Sol. S.P. of 1 kg mixture = Rs. 7.20,Gain = 20%.
C.P. of 1 kg mixture = Rs.[(100/120)*7.20]=Rs. 6.
By the rule of alligation, we have:
C_P. of 1 kg wheat of 1st kind C.P. of 1 kg wheat of 2nd kind
(800p) . (540 p)
Mean price
( 600 p)
60 200
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 5
Wheat of 1st kind : Wheat of 2nd kind = 60 : 200 = 3 : 10.
Let x kg of wheat of 1st kind be mixed with 36 kg of wheat of 2nd kind.
Then, 3 : 10 = x : 36 or lOx = 3 * 36 or x = 10.8 kg.
Ex. 5. The milk and water in two vessels A and B are in the ratio 4 : 3 and 2: 3 respectively.
In what ratio, the liquids in both the vessels be mixed to obtain a new mixture in vessel C
containing half milk and half water?
Sol. Let the C.P. of milk be Re. 1 per litre
Milk in 1 litre mixture of A = 4/7litre; Milk in 1 litre mixture of B = 2/5 litre;
Milk in 1 litre mixture of C = ½ litre
C.P. of 1 litre mixture in A = Re .4/7; C.P. of 1 litre mixture in B = Re.2/5
Mean price = Re.1/2
By the rule of alligation, we have:
C.P. of 1 litre mix. in A C.P. of 1 litre mix. in B
(4/7) (2/5)
(1/2)
(1/10) (1/14)
:.Required ratio = 1/10 : 1/14 = 7 : 5
Ex. 6. In what proportion must rice at Rs. 3.10 per kg be mixed with rice at Rs. 3.60 per kg
so that the mixture be worth Rs. 3.25 per kg ?
Sol. By the rule of alligation, we have:
C.P. of 1 kg rice of Cheaper rice (in paise) C.P. of 1 kg rice of dearer rice (in paise)
310 360 .
.
Mean price
(in paise)
325
35 15
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 6
By the allegation rule:
(Quantity of cheaper rice) = 35/15=7/3
(Quantity of dearer rice)
:. They must be mixed in the ratio 7:3.
Ex. 7. How many kilograms of sugar costing Rs.6.10 per kg.must be mixed with 126 kg. of
sugar costing Rs. 2.85 per kg. so that 20% may be gained by selling the mixture at Rs. 4.80
per kg.?
Sol: S.p of 1Kg. of mixture = Rs. 4.80,Gain=20%.
C.P of kg. of mixture = Rs. [(100/120)*4.80] =Rs. 4.
C.P. of 1 kg of Cheaper Sugar (in paise) C.P. of 1 kg rice of dearer sugar (in paise)
285 610
Mean price
(in paise)
400
210 115
(Quantity of cheaper sugar) = 210/115=42/23.
(Quantity of dearer sugar)
If cheaper sugar is 42 kg., dearer one=23 kg.
If cheaper sugar is 126 kg., dearer one=[(23/42)*126]kg.=69 kg.
Ex. 8. In what ratio must a person mix three kinds of wheat costing him Rs. 1.20,Rs. 1.44
and Rs. 1.74 per kg., so that the mixture may be worth Rs. 1.41 per kg?
Sol: step 1. Mix wheats of first and third kind to get a mixture worth Rs. 1.41 per kg.
C.P. of 1kg. Wheat of 1st kind C.P. of I kg. wheat of 3rd kind
120paise 174 paise
Mean price
141 price
33 21
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 7
By allegation rule :
(Quantity of 1st kind of wheat) = 33 = 11
(Quantity of 3rd kind of wheat) 21 7
i.e., they must be mixed in the ratio 11:7.
Step 2. Mix wheats of first and second kind to get a mixture worth Rs. 1.41 per kg.
C.P. of 1kg. Wheat of 1st kind C.P. of I kg. wheat of 2nd kind
120paise 144 paise
Mean price
141 price
3 21
By alligation rule :
(Quantity of 1st kind of wheat) = 3 = 1
(Quantity of 3rd kind of wheat) 21 7
i.e., they must be mixed in the ratio 1:7.
Thus, (Quantity of 2nd kind of wheat)
(Quantity of 1st kind of wheat)
= (Quantity of 2nd kind of wheat) x (Quantity of 1st kind of wheat)
(Quantity of 1st kind of wheat) (Quantity of 3rd kind of wheat)
= [(7/1)*(11/7)]=11/1.
Quantities of wheat of(1st kind : 2nd kind : 3rd kind)=(1:7:7/11) = (11:77:7).
Ex. 9. A butler stole wine from a butt of sherry which contained 40% of spirit and he
replaced what he had stolen by wine containing only 16% spirit. The butt was then of 24%
strength only. How much of the butt did he steal?
Sol:
Wine containing Wine containing
40% spirit 16% spirit
Wine containing
24% spirit
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 8
8 16
By allegation rule:
(Wine with 40% spirit) = 8 = 1
(Wine with 16% spirit) 16 2
i.e., they must be mixed in the ratio (1:2).
Thus 1/3 of the butt of sherry was left and hence the butler drew out 2/3 of the butt.
Ex. 10. The average weekly salary per head of the entire staff of a factory consisting of
supervisors and the labourers is Rs. 60. The average salary per head of the supervisors is
Rs. 400 and that of the labourers is Rs. 56. Given that the number of supervisors is 12, find
the number of labourers in the factory.
Sol: Average salary of labourers Average salary of supervisors
Rs. 56 Rs. 400
Mean
Rs. 60
340 4
By allegation rule:
(Number of labourers) = 340 = 85
(Number of supervisors) 4 1
Thus the number of supervisors is 1, number of labourers = 85.
if the number of supervisors is 12, number of labourers = 85*12 =1020.
Ex. 11. A man possessing Rs. 8400 lent a part of it at 8% simple interest and the remaining
at 6 2/3% simple interest .His total income after 1 ½ years was Rs. 882. Find the sum lent at
different rates.
Sol: Total interest on Rs. 84.. for 1 ½ years is Rs. 882.
Rate of interest = 100 x882x2 = 7%
8400x3
Rate % of 1st sum Rate % of 2nd sum
8% 6 2/3 %
Average Rate
7%
1 1
3
By alligation rule,
Money given at 8% S.I = 1: 1= 1:3
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 9
Money given at 6 2/3% S.I 3
Money lent at 8% = Rs. [8400 *(1/4)] = Rs. 2100.
Money lent at 6 2/3% = Rs. [8400 *(3/4)] = Rs. 6300.
Ex. 12. A man travelled a distance of 80 km. in 7 hours partly on foot at the rate of 8 km.
per hour and partly on bicycle at 16 km. per hour. Find the distance travelled on foot.
Sol. Average distance travelled in one hour =80/7 km.
Dist. Covered in 1 hr. on foot Dist. Covered in 1 hr. on bicycle
8 km. 16 km.
Average in 1 hr.
80 km.
32 7 24
7 7
By alligation rule,
Time taken on foot = 32 = 4:3
Time taken on bicycle 24
Thus out of 7 hours in all, he took 4 hours to travel on foot.
Distance covered on foot in 4 hours=(4x8)km=32 km.
Ex. 13. A sum of Rs. 41 was divided among 50boys and girls. Each boy gets 90 paise and a
girl 65 paise. Find the number of boys and girls.
Sol. Average money received by each =Rs. 41=82 p.
Sum received by each boy Sum received by each girl
90 paise 65 paise
Average
82 paise
17 8
By alligation rule,
Ratio of boys and girls=17:8.
Ex. 14. A lump of two metals weighting 18 gms. Is worth Rs. 87 but if their weights be
interchanged , it would be worth Rs. 78.60. if the price of one metal be Rs. 6.70 per gm.,
find the weight of the other metal in the mixture.
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 10
Sol. If one lump is mixed with the quantitiesof metals interchanged then the mixture of the two
lumps would contain 18gm. Of first metal and 18gm. Of second metal and the price of the
mixture would be Rs. (87+78.60) or 165.60.
cost of(18gm. Of 1st metal+18 gm. Of 2nd metal) = Rs. 165.60
So, cost of (1 gm. of 1st metal+1 gm. of 2nd metal) = Rs. 165.60 = Rs. 9.20.
18
(cost of 1 gm. of 1st metal) + (cost of 1 gm. of 2nd metal) = Rs. 9.20.
cost of 1 gm. of 2nd metal = Rs. (Rs. 9.20 – 6.70)= Rs. 2.50.
Now, mean price of lump = Rs. 87 per gm. = Rs. 29
18 6
C.P of 1 gm of 1st metal C.P of 1 gm. of 2nd metal
Rs. 6.70 Rs. 2.50
Mean price
Rs. 29
14 6 56
6 30
By alligation rule,
Quantity of 1st metal = 14 : 56 = 5:4
Quantity of 2nd metal 6 30
In 9 gm. of mix., 2nd metal = 4gm.
In 18 gm. of mix., 2nd metal = 4 x 18 gm. = 8 gm.
9
Ex. 15. A container contains 80 kg. of milk. From this container , 8kg. of milk was taken
out and replaced by water. This process was further repeated two times. How much milk is
now contained by the container?
Remarks. Amount of liquid left after n operations , when the container originally contains x
units of liquids, from which y units is taken out each time is =[ x(1-y/x)^n]units.
Sol. Amount of milk left = 80[(1-(8/80)3)] kg.
= 58.34 kg.
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 11
EXERCISE PROBLEMS:
1. A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the
mixture must be drawn off and replaced with water so that the mixture may be half water and
half syrup?
A. 1
3
B. 1
4
C. 1
5
D. 1
7
Answer: Option C
Explanation:
Suppose the vessel initially contains 8 liters of liquid.
Let x liters of this liquid be replaced with water.
Quantity of water in new mixture = 3 -
3x
8 + x liters
Quantity of syrup in new mixture = 5 -
5x
8 liters
3 -
3x
+ x = 5 -
5x
8 8
5x + 24 = 40 - 5x
10x = 16
x =
8
5
So, part of the mixture replaced =
8 x1
=
1
5 8 5
2. Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety in the ratio 1:1:2.
If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be:
A.Rs. 169.50 B.Rs. 170
C.Rs. 175.50 D.Rs. 180
Answer: Option C
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 12
Explanation:
Since first and second varieties are mixed in equal proportions.
So, their average price = Rs.
126 + 135
= Rs. 130.50
2
So, the mixture is formed by mixing two varieties, one at Rs. 130.50 per kg and the other at say,
Rs. x per kg in the ratio 2 : 2, i.e., 1 : 1. We have to find x.
By the rule of alligation, we have:
Cost of 1 kg of 1st kind Cost of 1 kg tea of 2nd kind
Rs. 130.50
Mean Price
Rs. 153
Rs. x
(x-153)
22.50
x - 153 = 1
22.50
x - 153 = 22.50
x = 175.50
3. A can contains a mixture of two liquids A and B is the ratio 7 : 5. When 9 litres 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.10 B.20
C.21 D.25
Answer: Option C
Explanation:
Suppose the can initially contains 7x and 5x of mixtures A and B respectively.
Quantity of A in mixture left = 7x -
7
x 9 liters = 7x -
21
12 4 liters.
Quantity of B in mixture left = 5x -
5
x 9 liters = 5x -
15
12 4 liters.
7x -
21
4 = 7
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 13
9
5x -
15
4 + 9
28x - 21
=
7
20x + 21 9
252x - 189 = 140x + 147
112x = 336
x = 3.
So, the can contained 21 litres of A.
4. A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second
contains 50% water. How much milk should he mix from each of the containers so as to get 12 liters
of milk such that the ratio of water to milk is 3: 5?
A.4 liters, 8 liters B.6 liters, 6 liters
C.5 liters, 7 liters D.7 liters, 5 liters
Answer: Option B
Explanation:
Let the cost of 1 liter milk be Re. 1
Milk in 1 liter mix. in 1st can =
3
liter, C.P. of 1 liter mix. in 1st can Re.
3
4 4
Milk in 1 liter mix. in 2nd can =
1
liter, C.P. of 1 liter mix. in 2nd can Re.
1
2 2
Milk in 1 liter of final mix. =
5
liter, Mean price = Re.
5
8 8
By the rule of alligation, we have:
C.P. of 1 liter mixture in 1st can C.P. of 1 liter mixture in 2nd can
3
4
Mean Price
5
8
1
2
1
8 1
8
Ratio of two mixtures =
1
:
1
= 1: 1.
8 8
So, quantity of mixture taken from each can =
1
x 12 = 6 litres.
2
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 14
5. 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.3 : 7 B.5 : 7
C.7 : 3 D.7 : 5
Answer: Option C
Explanation:
By the rule of alligation:
Cost of 1 kg pulses of 1st kind Cost of 1 kg pulses of 2nd kind
Rs. 15
Mean Price
Rs. 16.50
Rs. 20
3.50 1.50
Required rate = 3.50 : 1.50 = 7 : 3.
6. A dishonest milkman professes to sell his milk at cost price but he mixes it with water and
thereby gains 25%. The percentage of water in the mixture is:
A.4% B.6 ¼ %
C.20% D.25%
Answer: Option C
Explanation:
Let C.P. of 1 liter milk be Re. 1
Then, S.P. of 1 liter of mixture = Re. 1, Gain = 25%.
C.P. of 1 liter mixture = Re.
100
x 1 =
4
125 5
By the rule of alligation, we have:
C.P. of 1 liter of milk C.P. of 1 liter of water
Re. 1 0
Mean Price
Re.
4
5
4
5
1
5
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 15
Ratio of milk to water =
4
:
1
= 4 : 1.
5 5
Hence, percentage of water in the mixture =[(1/5)*100]%=20% .
7. How many kilogram of sugar costing Rs. 9 per kg must be mixed with 27 kg of sugar costing
Rs. 7 per kg so that there may be a gain of 10% by selling the mixture at Rs. 9.24 per kg?
A.36 kg B.42 kg
C.54 kg D.63 kg
Answer: Option D
Explanation:
S.P. of 1 kg of mixture = Rs. 9.24, Gain 10%.
C.P. of 1 kg of mixture = Rs.
100
x 9.24 = Rs. 8.40
110
By the rule of alligation, we have:
C.P. of 1 kg sugar of 1st kind Cost of 1 kg sugar of 2nd kind
Rs. 9
Mean Price
Rs. 8.40
Rs. 7
1.40 0.60
Ratio of quantities of 1st and 2nd kind = 14 : 6 = 7 : 3.
Let x kg of sugar of 1st be mixed with 27 kg of 2nd kind.
Then, 7 : 3 = x : 27
x =
7 x 27
= 63 kg.
3
8. A container contains 40 litres of milk. From this container 4 litres of milk was taken out and
replaced by water. This process was repeated further two times. How much milk is now
contained by the container?
A.26.34 litres B.27.36 litres
C.28 litres D.29.l6 litres
Answer: Option D
Explanation:
Amount of milk left after 3 operations = [40(1- 4/40 )3] liters
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 16
= 40 x
9
x
9
x
9
= 29.16 liters.
10 10 10
9. A jar full of whisky contains 40% alcohol. A part of this whisky is replaced by another
containing 19% alcohol and now the percentage of alcohol was found to be 26%. The quantity of
whisky replaced is:
A.
1
3
B.
2
3
C.
2
5
D.
3
5
Answer: Option B
Explanation:
By the rule of alligation , we have:
Strength of first jar Strength of 2nd jar
40%
Mean
Strength
26%
19%
7 14
So, ratio of 1st and 2nd quantities = 7 : 14 = 1 : 2
Required quantity replaced =
2
3
10. In what ratio must water be mixed with milk to gain 16 % on selling the mixture at cost
price?
A.1 : 6 B.6 : 1
C.2 : 3 D.4 : 3
Answer: Option A
Explanation:
Let C.P. of 1 litre milk be Re. 1.
S.P. of 1 litre of mixture = Re.1, Gain =
50
%.
3
C.P. of 1 litre of mixture = 100 x
3
x 1 =
6
350 7
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 17
By the rule of alligation, we have:
C.P. of 1 litre of water C.P. of 1 litre of milk
0
Mean Price
Re.
6
7
Re. 1
1
7
6
7
Ratio of water and milk =
1
:
6
= 1 : 6.
7 7
11. Find the ratio in which rice at Rs. 7.20 a kg be mixed with rice at Rs. 5.70 a kg to produce a
mixture worth Rs. 6.30 a kg.
A.1 : 3 B.2 : 3
C.3 : 4 D.4 : 5
Answer: Option B
Explanation:
By the rule of alligation:
Cost of 1 kg of 1st kind Cost of 1 kg of 2nd kind
720 p
Mean Price
630 p
570 p
60 90
Required ratio = 60: 90 = 2: 3.
12. In what ratio must a grocer mix two varieties of tea worth Rs. 60 a kg and Rs. 65 a kg so that
by selling the mixture at Rs. 68.20 a kg he may gain 10%?
A.3 : 2 B.3 : 4
C.3 : 5 D.4 : 5
Answer: Option A
Explanation:
S.P. of 1 kg of the mixture = Rs. 68.20, Gain = 10%.
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 18
C.P. of 1 kg of the mixture = Rs.
100
x 68.20 = Rs. 62.
110
By the rule of alligation, we have:
Cost of 1 kg tea of 1st kind Cost of 1 kg tea of 2nd kind.
Rs. 60
Mean Price
Rs. 62
Rs. 65
3 2
Required ratio = 3 : 2.
13.The cost of Type 1 rice is Rs. 15 per kg and Type 2 rice is Rs. 20 per kg. If both Type 1 and
Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is:
A.Rs. 18 B.Rs. 18.50
C.Rs. 19 D.Rs. 19.50
Answer: Option A
Explanation:
Let the price of the mixed variety be Rs. x per kg.
By rule of alligation, we have:
Cost of 1 kg of Type 1 rice Cost of 1 kg of Type 2 rice
Rs. 15
Mean Price
Rs. x
Rs. 20
(20 - x) (x - 15)
(20 - x)
=
2
(x - 15) 3
60 - 3x = 2x - 30
5x = 90
x = 18.
14.8 liters are drawn from a cask full of wine and is then filled with water. This operation is
performed three more times. The ratio of the quantity of wine now left in cask to that of water is
16 : 81. How much wine did the cask hold originally?
A.18 liters B.24 liters
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 19
C.32 liters D.42 liters
Answer: Option B
Explanation:
Let the quantity of the wine in the cask originally be x litres.
Then, quantity of wine left in cask after 4 operations = [x(1 – 8/x ) litres.
x(1 - (8/x))4
=
16
x 81
1 -
8
4=
2
4
x 3
x - 8
=
2
x 3
3x - 24 = 2x x = 24.
15. A merchant has 1000 kg of sugar, part of which he sells at 8% profit and the rest at 18%
profit. He gains 14% on the whole. The quantity sold at 18% profit is:
A.400 kg B.560 kg
C.600 kg D.640 kg
Answer: Option C
Explanation:
By the rule of alligation, we have:
Profit on 1st part Profit on 2nd part
8%
Mean Profit
14%
18%
4 6
Ration of 1st and 2nd parts = 4 : 6 = 2 : 3
Quantity of 2nd kind =
3
x 1000
kg
= 600 kg.
5
16. How many liters of water should be added to a 30 litre mixture of milk and water containing
milk and water in the ratio of 7 : 3 such that the resultant mixture has 40% water in it?
A.7 liters B.10 liters
C.5 liters D.None of these
Answer: Option C
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 20
Explanation:
30 liters of the mixture has milk and water in the ratio 7: 3. i.e. the solution has 21 liters of milk
and 9litres of water. When you add more water, the amount of milk in the mixture remains
constant at 21 liters. In the first case, before addition of further water, 21 liters of milk accounts
for 70% by volume. After water is added, the new mixture contains 60% milk and 40% water.
Therefore, the 21litres of milk accounts for 60% by volume.
Hence, 100% volume = 21/0.6= 35 liters.
We started with 30 liters and ended up with 35 liters. Therefore, 5 liters of water was added.
17. A 20 liter mixture of milk and water contains milk and water in the ratio 3: 2. 10 liters of the
mixture is removed and replaced with pure milk and the operation is repeated once more. At the
end of the two removal and replacement, what is the ratio of milk and water in the resultant
mixture?
A.17 : 3 B.9 : 1
C.3 : 17 D.5 : 3
Answer: Option B
Explanation:
The 20 litre mixture contains milk and water in the ratio of 3 : 2. Therefore, there will be 12 litres
of milk in the mixture and 8 litres of water in the mixture.
Step 1. When 10 litres of the mixture is removed, 6 litres of milk is removed and 4 litres of water
is removed. Therefore, there will be 6 litres of milk and 4 litres of water left in the container. It is
then replaced with pure milk of 10 litres. Now the container will have 16 litres of milk and 4
litres of water.
Step 2. When 10 litres of the new mixture is removed, 8 litres of milk and 2 litres of water is
removed. The container will have 8 litres of milk and 2 litres of water in it. Now 10 litres of pure
milk is added. Therefore, the container will have 18 litres of milk and 2 litres of water in it at the
end of the second step.
Therefore, the ratio of milk and water is 18: 2 or 9: 1.
18. How many kgs of Basmati rice costing Rs.42/kg should a shopkeeper mix with 25 kgs of
ordinary rice costing Rs.24 per kg so that he makes a profit of 25% on selling the mixture at
Rs.40/kg?
A.20 kgs B.12.5kgs
C.3 : 16kgs D.200kgs
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 21
Answer: Option A
Explanation:
Let the amount of Basmati rice being mixed be x kgs. As the trader makes 25% profit by selling
the mixture at Rs.40/kg, his cost per kg of the mixture = Rs.32/kg.
i.e. (x * 42) + (25 * 24) = 32 (x + 25)
=> 42x + 600 = 32x + 800
=> 10x = 200 or x = 20 kgs.
19. How many litres of a 12 litre mixture containing milk and water in the ratio of 2 : 3 be
replaced with pure milk so that the resultant mixture contains milk and water in equal
proportion?
A.4 liters B.2 liters
C.1 liter D.1.5 liters
Answer: Option B
Explanation:
The mixture contains 40% milk and 60% water in it. That is 4.8 litres of milk and 7.2 litres of
water.
Now we are replacing the mixture with pure milk so that the amount of milk and water in the
mixture is 50% and 50%.That is we will end up with 6 litres of milk and 6 litres of water.
Water gets reduced by 1.2 litres.
To remove 1.2 litres of water from the original mixture containing 60% water, we need to
remove 1.2 / 0.6 litres of the mixture = 2litres.
20. A zookeeper counted the heads of the animals in a zoo and found it to be 80. When he
counted the legs of the animals he found it to be 260. If the zoo had either pigeons or horses, how
many horses were there in the zoo?
A.40 B.30
C.50 D.60
Answer: Option C
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 22
Explanation:
Let the number of horses = x
Then the number of pigeons = 80 – x.
Each pigeon has 2 legs and each horse has 4 legs.
Therefore, total number of legs = 4x + 2(80-x) = 260
=>4x + 160 – 2x = 260
=>2x = 100
=>x = 50.
21. From a cask of milk containing 30 litres, 6 litres are drawn out and the cask is filled up with
water. If the same process is repeated a second, then a third time, what will be the number of
litres of milk left in the cask?
A.0.512 liters B.12 liters
C.14.38 liters D.15.36 liters
Answer: Option D
Explanation:
The problem can be solved by traditional method but it is cumbersome process to do that. The
problem is simple if its solution is simpler. Hence we will go for a simpler solution for this kind
of problem.
There is a short cut method to find the Quantity of milk left after nth operation.
It is given by [(x – y)/x]n of the whole quantity, where x is initial quantity of milk in the cask y is
the quantity of milk withdrawn in each process and n is the number of process..
Hence from the above rule it can be say that
Quantity of milk left after the 3rd operation = [(30 – 6)/30]3 * 30 = 15.36 liters.
22. In what ratio must wheat at Rs.3.20 pe rkg be mixed with wheat at Rs.2.90 per kg so that the
mixture be worth Rs.3.08 per kg
A.5:7 B.7:9
C.3:2 D.7:5
Answer: Option C
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 23
Explanation:
By the rule of alligation, we have:
C.P of a unit quantity of 1st kind C.P. of a unit quantity of 2nd kind
3.20 2.90
Mean Price (3.80)
0.18 0.12
Required ratio =0.18 : 0.12 = 3:2.
23. Two A and B contain milk and water mixed in the ratio 8:5 and 5:2 respectively. The ratio in
which these two mixtures be mixed to get a new mixture containing milk and a water in the ratio
9:4?
A.5:7 B.2:7
C.3:2 D.7:5
Answer: Option B
Explanation:
Step (i) : Let C.P. of milk be Re.1
Given ratio of mixture in A = 8:5
Milk in 1 liter mixture in A = 8/13 litre
C.P of 1 liter mixture in A = Rs. 8/13
Ratio of Mixture in B = 5:2
milk in 1 liter mixture in B – 5/7 litre
C.P of 1 litre mixture in B = Rs. 5/7
Ratio of new mixture = 9:4
Milk in 1 lit mixture = 9/13
C.P of 1 litre mixture = Rs/ 9/13 (Mean price)
Step (ii) : By the rule of allegation,
i. C.P of 1 liter of mixture in A =8/13
ii. C.P of 1 liter of mixture in B = 5/7
iii. Mean price (p) = 9/13
iv. d – m = 9/13 – 5-7 = 2/91
v. m – c = 9/13 – 8/13 = 1/13
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 24
Required ratio = 2/91: 1/13 = 2:7
24. In what ratio water be mixed with milk costing Rs.12 per liter to obtain a mixture worth of
Rs.8 per litre?
A.1:2 B.2:7
C.3:2 D.7:5
Answer: Option A
Explanation:
By the rule of allegation,
i. C.P of 1 liter of water = 0
ii. C.P of 1 liter of milk = 12
iii. Mean price (p) = 8
iv. d – m = 12-8 = 4
v. m – c = 8-0 = 8
Ratio of water and milk = 4 : 8 = 1 : 2
25.729ml of mixture contains milk and water in the ratio 7:2 how much more water is to be
added to get a new mixture containing milk and water in the ratio 7:3?
A.70ml B.49ml
C.81ml D.96ml
Answer : Option C
Explanation:
Ratio of milk and water in 729 ml = 7:2
Step (i) Milk in 729 ml of mixture = (7/9 x 729 ) ml = 567 ml
water in 729 ml of mixture = 729 – 567 = 162 ml
Step (ii) Let x be the quantity of water added to new mixture, with the ratio 7:3
Quantity of water in the new mixture = (162 + x ) ml
Then 7/3 =
162 x
567
7 (162 + x) = 3 x 567
1134 + 7x = 1701
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 25
7x = 1701 – 1134
X = 567/7 = 81 ml
Quantity of water added to new mixture = 81 ml.
26. A can contains 40kg of milk, from this container 4kg of milk was taken out and replaced by
water. This process was repeated further two times. How much milk is now contained by the
container?
A.29.16kg B.30kg
C.32.49kg D.25.36kg
Answer: Option A
Explanation:
Quantity of milk in the can x = 40 kg
Quantity of milk taken out y = 4 kg
Number of times = 3
Quantity of milk in the can = kg
x
y
x
n
1
= kg
3
40
4
40 1 = 40 (9/10)3 kg = 40 ( 729/1000)
Quantity of milk in the can = 29.16 kg
27. Two vessels A and B contain sprit and water in the ratio 5:2 and 7:6 respectively. Find the
ratio in which these mixture be mixed to obtain a new mixture containing spirit and water in the
ratio 8:5?
A.3:4 B.5:7
C.7:9 D.4:3
Answer: Option C 7:9
Explanation:
Step (i) : Spirit in 1 litre mixture of A = 5/7 litre
Spirit in 1 litre mixture of B = 7/13 litre
Spirit in 1 litre mixture of final mixture = 8/13 litre
Mean quantity = 8/13 litre
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 26
Step (ii) By the rule of allegation,
i. quantity of spirit in A (c) = 5/7
ii. Quantity of spirit in B (d)= 7/13
iii. Mean price (m) = 8/13
iv. d – m = 5/7 – 8/13 = 9/19
v. m – c = 8/13 – 7/13 = 1/13
Required ratio = 1/13 : 9/91 = 7 : 9
28.Two vessels A and B contain milk and water mixed in the ratio 8:5 and 5:2 respectively. The
ratio in which these 2 mixtures be mixed to get a new mixture containing 69 3/13 % milk is :
A.3:4 B.2:7
C.7:9 D.4:3
Answer: Option B
Explanation:
Step (i) : Quantity of milk in 1 lr mixture of A = 8/13 lr
Quantity of milk in 1 lr mixture of B = 5/7 lr
Quantity of milk in 1 lr mixture of final mixture = 69
13
3
%
=
100
1
13
900
x lr
Mean quantity = 9/13 lr
Step (ii) By the rule of allegation,
i. quantity of spirit in A (c) = 8/13
ii. Quantity of spirit in B (d)= 5/7
iii. Mean price (m) = 9/13 lr
iv. d – m = 5/7 – 9/13 = 2/91
v. m – c = 9/13 – 8/13 = 1/13
Required ratio = 2/91 : 1/13 : = 2 : 7
29. The cost of type I rice is Rs.15 p/kg and type II is Rs.20p/kg. Both are mixed in the ratio 2:3,
price P/Kg of the mixed variety is :
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 27
A.Rs.20 B.Rs.30
C.Rs.15 D.Rs.18
Answer: Option D
Explanation:
Step (i) : Let the price of mixed variety be x.Rs
Mean price = Rs.x
Cost of 1 kg of Type I Cost of 1 kg of Type II
Rs. 15 Rs. 20
Mean wt (m)
(20 - x) Rs.x (x - 15)
ratio = 20-x : x - 15
Step (ii) : Mixed variety is in the ratio = 2:3
15
20
x
x
= 2/3
60 – 3x = 2x – 30
X = 90 / 5 = 18
Ratios imply that the price of mixture = Rs. 18 per kg
30. In what ratio must tea at Rs.62 per Kg be mixed with tea at Rs. 72 per Kg so that the mixture
must be worth Rs. 64.50 per Kg?
A. 3 : 1 B. 3 : 2
C. 4 : 3 D. 5 : 3
Answer: Option A
Explanation:
By the rule of alligation:
Cost of 1 kg tea of 1st kind Cost of 1 kg tea of 2nd kind
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 28
Required ratio = 750 : 250 = 3 : 1
31. A bar is creating a new signature drink. They are using two alcoholic ingredients in the drink:
vodka and gin. They are using two non-alcoholic ingredients in the drink: orange juice and
cranberry juice. The alcoholic ingredients contain 40% alcohol. The non-alcoholic ingredients
contain no alcohol. How many liters of non-alcoholic ingredients must be added to 6 liters of
alcoholic ingredients to produce a mixture that is 15% alcohol?
A. 5 B. 10
C. 6 D. 15
Answer: Option B
Explanation:
The alcohol ingredients is 40% in 6 liters of drink
The resultant mixture needs to have 15% alcohol only
Hence we have the following equation :
(6+x) (0.15) = 6(40%) + x(0%)
(6+x) 0.15 = 6(0.4) + 0
Solving for x, we get x = 10.
31. 150 liters of a 20% alcohol solution is mixed with 200 liters of another solution. If the
resulting solution is 18% alcohol, what is the percent of alcohol in the 200-liter solution?
A. 21% B. 16.5%
C. 15.2% D. 18%
Answer: Option B
Explanation:
Total solution is 350 litres .
We have the following equation: 350(0.18) = 200(x) + 150(0.2)
Solving for x , we get x = 0.165 i.e. 16.5%
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 29
32. 8 litres are drawn from a cask full of wine and is then filled with water. This operation is
performed three more times. The ratio of the quantity of wine now left in cask to that of the
water is 16 : 65. How much wine did the cask hold originally?
A. 18 litres B.24 litres
C.32 litres D.42 litres
Answer: Option B
Explanation:
Let the quantity of the wine in the cask originally be x litres.
Then , quantity of wine left in cask after 4 operations = [x(1-(8/x)4] litres.
[x(1-(8/x)4] = 16
x 81
(1-8/x)4 = (2/3)2
x - 8 = 2 3x – 24 = 2x x = 24.
x 3
33. One quality of wheat at Rs. 9.30 per kg is mixed with another quality at certain rate in the
ratio 8 : 7 .If the mixture so formed be worth Rs. 10 per kg,what is the rate per kg of the second
quality of wheat ?
A.Rs. 10.30 B.Rs. 10.60
C.Rs. 10.80 D.Rs.11
Answer: Option C
Explanation:
Let the rate of the second quality be Rs. X per kg.
By the rule of allegation, we have:
C.P of 1 kg. wheat of 1st kind C.P of 1 kg wheat of 2nd kind
930p 100x p
Mean price
1000 P
(100x – 1000)p 70 p
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 30
100x – 1000 = 8
70 7
700x – 7000=560
700x = 7560
x = Rs. 10.80.
QUESTION BANK:
1. In what ratio must rice at Rs.30 per kg mixed with rice at Rs.10.80 per kg so that the
mixture be worth Rs.10per kg .
A. 3:4 B. 5:4
C. 8:7 D. 4:5
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 31
2. In what ratio must wheat at Rs.3.20 pe rkg be mixed with wheat at Rs.2.90 per kg so that
the mixture be worth Rs.3.08 per kg
A. 3:2 B.5:4
C. 2:3 D. 4:5
3. How many kilograms of sugar costing Rs/.9 per kg must be mixed with 27kg of sugar
costing Rs.7 per kg so that there may ba gain of 10% by selling the mixture at Rs.9.24 per
kg ?
A. 50 kg B. 63kg
C. 72kg D. 59kg
4. In what ratio must water be mixed with milk to gain 16 2/3% on selling the mixture at
cost price ?
A. 1:6 B. 5:7
C. 7:5 D. 6:1
5. Two vessels A and B contain milk and water mixed in the ratio 8:5 and 5:2 res. The ratio
in which these two mixtures be mixed to get a new mixture containing milk and a water
in the ratio 9:4?
A. 3:4 B. 5:9
C. 2:7 D. 3:2
6. In what ratio water be mixed with milk costing Rs.12 per liter to obtain a mixture worth
of Rs.8 per litre?
A. 2:4 B. 1:2
C. 5:7 D. 2:1
7. A sum of Rs.4000 is lent out in two parts, one at 8% simple interest and the other at 10%
simple interest. In the annual interest is Rs.352, the sum lent at 8% is ?
A. 1300 B. 1500
C. 2400 D. 3700
8. A merchant has 100kg of sugar, part of which he sells at 8% profit and the rest at 18%
profit. He gains 14% on the whole. The quantity sold at 18% profit is?
A. 120kg B. 500kg
C. 300kg D. 600kg
9. Two vessels A and B contain milk and water mixed in the ratio 4:3 and 2:3 in what ratio
must these mixtures be mixed to form new mixture containing half milk and half water?
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 32
A. 5:6 B. 7:5
C. 6:7 D. 7:8
10. A jar full of whisky contains 40% alcohol. A part of this whisky is replaced by another
containing 19% alcohol and now the percentage of alcohol was found to be 26% the
quantity of whisky replaced is ?
A.1/4 B.1/3
C.2/3 D.2/4
11. 729ml of mixture contains milk and water in the ratio 7:2 how much more wate is to be
added to get a new mixture containing milk and water in the ratio 7:3?
A.70ml B.49ml
C.81ml D.96ml
12. A sum of Rs.312 was divided amng 100boys and girls in such a way that the boy gets
Rs.3.60 and each girl Rs.2.40 the number of girls is ?
A.30 B.40
C.20 D.50
13. A man covered a distance of 2000km in 18 hours partly by bus at 72kmph and partly by
train at 16kmph the distance covered by bus is ?
A.540ml B.960ml
C.720ml D.840ml
14. A sum of rs.36.90 is made up of 180 coins which are either 10 paise coins or 25 p coins.
The number of 10 p coins is ?
A.72 B.54
C.45 D.67
15. A dishonest milk man professes to sell his milk at cost price but he mixed it with water
and thereby gains 25%. The percentage of water in the mixture is ?
A.35% B.25%
C.20% D.40%
16. A mixture of 20kg of spirt and water contains 10 water How much water must be added
to this mixture to raise the percentage of water to 25%?
A.3kg B.4kg
C.5kg D.6kg
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 33
17. A container contains 40kg of milk, from this container 4kg of milk was taken out and
replaced by water. This process was repeated further two times. How much milk is now
contained by the container?
A.43.12 kg B.43.22kg
C.29.16kg D.12.45kg
18. A can contains a mixture of two liquids A and B in the ratio 7:5 when 9 litres 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 liquids A was contained by the can initially?
19. A mixture of milk and water measures 60 gallons. It contains 20% water. How many
gallons of water should be added to it so that water may be 25%?
20. A mixture of spirit and water measure 80 gallons. It contains 20% water. How much
water should be added to it so that water may be 25%?
21. A man lent $2000, part of this at 4% and the rest at 6% per annum simple interest. The
whole annual interest amounted to $92. How much did he lend at 6%?
22. A man invested $2500 into two parts such that if one part be put out at 5% S.I. and other
at 6%, the yearly income may be $140. How much did he invest at 5%?
23. There are two vessels A and B in which the ratio of milk and water are as 5:2 and 8:7
respectively. Two gallons are drawn from vessel A and 3 gallons from vessel B, and are
mixed in another empty vessel. What is the ratio of milk and water in it?
24. Two gallons of mixture in which there is 2/5 of water and the rest spirit is mixed with
five gallons of mixture in which there is 1/3 of water and the rest spirit. What is the ratio
of water and spirit in the new mixture?
A.20 B.21
C.22 D.23
A.6 gallons B.4 gallons
C.8 gallons D.10 gallons
A.8 1/3 gallons B.6 1/3gallons
C.7 1/3 gallons D. 5 1/3 gallons
A.$900 B.$800
C.$600 D.$1000
A.$1250 B.$1500
C.$1000 D.$750
A.106:69 B.103:72
C.89:86 D.101:71
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 34
25. One vessel contains a mixture of 5 parts pure wine and 3 parts soda, whereas the other
vessel contains a mixture of 9 parts pure wine and 5 parts soda. Compare the strength of
the wine.
26. One milk can contains a mixture of milk and water in the ratio 7:5 and the other contains
the mixture of milk and water in which 2/5th is water. Compare their purity.
27. A woman sold 100 oranges at $12.10, some at the rate of 3 for 35 cents and the rest at 7
for 85 cents. How many were sold at the first rate?
28. A merchant has 100 lbs of sugar, part of which he sells at 7% profit and the rest at 17%
profit. He gains 10 % on the whole. Find how much is sold at 7% profit?
A.18:23 B.12:17
C.25:33 D.37:68
A.7:4 B.7:8
C.35:36 D.14:5
A.36:35 B.35:36
C.10:7 D.5:3
A.45 B.21
C.15 D.9
A.70 lbs B.40lbs
C.30 lbs D.50lbs
ALLIGATION OR MIXTURE
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 2
ALLIGATION OR MIXTURE
IMPORTANT FACTS AND FORMULAE
1. 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.
2. Mean Price: The cost price of a unit quantity of the mixture is called the mean price.
3. Rule of Alligation: If two ingredients are mixed, then
(Quantity of cheaper) = (C.P. of dearer) - (Mean price)
(Quantity of dearer) (Mean price) - (C.P. of cheaper)
We present as under:
C.P. of a unit quantity of cheaper C.P. of a unit quantity of dearer
(c) (d)
(m)
(d-m) (m-c)
(Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c).
4. Suppose a container contains x units of liquid from which y units are taken out and replaced
by water. After n operations the quantity of pure liquid=[ x(1-y/x)^n]units.
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 3
SOLVED EXAMPLES
Ex. 1. In what ratio must rice at Rs. 9.30 per kg be mixed with rice at Rs. 10.80 per kg so
that the mixture be worth Rs. 10 per kg ?
Sol. By the rule of alligation, we have:
C.P. of 1 kg rice of 1st kind (in paise) C.P. of 1 kg rice of 2nd kind (in paise)
930 1080 .
.
Mean price
(in paise)
1000
80 70
:. Required ratio = 80 : 70 = 8 : 7.
Ex. 2. How much water must be added to 60 litres of milk at 1 ½ litres for Rs. 20
So as to have a mixture worth Rs.10 2/3 a litre ?
Sol. C.P. of 1 litre of milk = Rs. (20 x 2/3) = Rs. 40/3
c.p of 1 litre of milk c.p of 1 litre of milk
0 Rs.40
3
Mean price
(Rs. 32 )
3
(40/3-32/3)=8/3 (32/3-0)=32/3
Ratio of water and milk =8 : 32 = 8 : 32 = 1 : 4
3 3
Quantity of water to be added to 60 litres of milk = [1/4 X 60 ]litres =15 litre
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 4
Ex. 3. In what ratio must water be mixed with milk to gain 20% by selling the mixture at
cost price?
Sol. Let C.P. of milk be Re. 1 per litre.
Then, S.P. of 1 litre of mixture = Re. 1.
Gain obtained = 20%.
C.P. of 1 litre of mixture = Rs.[(100/120)* 1]=Rs.5/6
By the rule of alligation, we have:
C.P. of 1 litre of water C.P. of1litre ofmilk
0 Re.1
(Re. 5/6)
(1- (5/6))= 1/6 ((5/6)-0)=5/6
Ratio of water and milk = 1/6 : 5/6
Ex. 4. .How many kgs. of wheat costing Rs. 8 per kg must be mixed with 86 kg of rice
costing Rs. 6.40 per kg so that 20% gain may be obtained by Belling the mixture at Rs. 7.20
per kg ?
Sol. S.P. of 1 kg mixture = Rs. 7.20,Gain = 20%.
C.P. of 1 kg mixture = Rs.[(100/120)*7.20]=Rs. 6.
By the rule of alligation, we have:
C_P. of 1 kg wheat of 1st kind C.P. of 1 kg wheat of 2nd kind
(800p) . (540 p)
Mean price
( 600 p)
60 200
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 5
Wheat of 1st kind : Wheat of 2nd kind = 60 : 200 = 3 : 10.
Let x kg of wheat of 1st kind be mixed with 36 kg of wheat of 2nd kind.
Then, 3 : 10 = x : 36 or lOx = 3 * 36 or x = 10.8 kg.
Ex. 5. The milk and water in two vessels A and B are in the ratio 4 : 3 and 2: 3 respectively.
In what ratio, the liquids in both the vessels be mixed to obtain a new mixture in vessel C
containing half milk and half water?
Sol. Let the C.P. of milk be Re. 1 per litre
Milk in 1 litre mixture of A = 4/7litre; Milk in 1 litre mixture of B = 2/5 litre;
Milk in 1 litre mixture of C = ½ litre
C.P. of 1 litre mixture in A = Re .4/7; C.P. of 1 litre mixture in B = Re.2/5
Mean price = Re.1/2
By the rule of alligation, we have:
C.P. of 1 litre mix. in A C.P. of 1 litre mix. in B
(4/7) (2/5)
(1/2)
(1/10) (1/14)
:.Required ratio = 1/10 : 1/14 = 7 : 5
Ex. 6. In what proportion must rice at Rs. 3.10 per kg be mixed with rice at Rs. 3.60 per kg
so that the mixture be worth Rs. 3.25 per kg ?
Sol. By the rule of alligation, we have:
C.P. of 1 kg rice of Cheaper rice (in paise) C.P. of 1 kg rice of dearer rice (in paise)
310 360 .
.
Mean price
(in paise)
325
35 15
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 6
By the allegation rule:
(Quantity of cheaper rice) = 35/15=7/3
(Quantity of dearer rice)
:. They must be mixed in the ratio 7:3.
Ex. 7. How many kilograms of sugar costing Rs.6.10 per kg.must be mixed with 126 kg. of
sugar costing Rs. 2.85 per kg. so that 20% may be gained by selling the mixture at Rs. 4.80
per kg.?
Sol: S.p of 1Kg. of mixture = Rs. 4.80,Gain=20%.
C.P of kg. of mixture = Rs. [(100/120)*4.80] =Rs. 4.
C.P. of 1 kg of Cheaper Sugar (in paise) C.P. of 1 kg rice of dearer sugar (in paise)
285 610
Mean price
(in paise)
400
210 115
(Quantity of cheaper sugar) = 210/115=42/23.
(Quantity of dearer sugar)
If cheaper sugar is 42 kg., dearer one=23 kg.
If cheaper sugar is 126 kg., dearer one=[(23/42)*126]kg.=69 kg.
Ex. 8. In what ratio must a person mix three kinds of wheat costing him Rs. 1.20,Rs. 1.44
and Rs. 1.74 per kg., so that the mixture may be worth Rs. 1.41 per kg?
Sol: step 1. Mix wheats of first and third kind to get a mixture worth Rs. 1.41 per kg.
C.P. of 1kg. Wheat of 1st kind C.P. of I kg. wheat of 3rd kind
120paise 174 paise
Mean price
141 price
33 21
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 7
By allegation rule :
(Quantity of 1st kind of wheat) = 33 = 11
(Quantity of 3rd kind of wheat) 21 7
i.e., they must be mixed in the ratio 11:7.
Step 2. Mix wheats of first and second kind to get a mixture worth Rs. 1.41 per kg.
C.P. of 1kg. Wheat of 1st kind C.P. of I kg. wheat of 2nd kind
120paise 144 paise
Mean price
141 price
3 21
By alligation rule :
(Quantity of 1st kind of wheat) = 3 = 1
(Quantity of 3rd kind of wheat) 21 7
i.e., they must be mixed in the ratio 1:7.
Thus, (Quantity of 2nd kind of wheat)
(Quantity of 1st kind of wheat)
= (Quantity of 2nd kind of wheat) x (Quantity of 1st kind of wheat)
(Quantity of 1st kind of wheat) (Quantity of 3rd kind of wheat)
= [(7/1)*(11/7)]=11/1.
Quantities of wheat of(1st kind : 2nd kind : 3rd kind)=(1:7:7/11) = (11:77:7).
Ex. 9. A butler stole wine from a butt of sherry which contained 40% of spirit and he
replaced what he had stolen by wine containing only 16% spirit. The butt was then of 24%
strength only. How much of the butt did he steal?
Sol:
Wine containing Wine containing
40% spirit 16% spirit
Wine containing
24% spirit
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 8
8 16
By allegation rule:
(Wine with 40% spirit) = 8 = 1
(Wine with 16% spirit) 16 2
i.e., they must be mixed in the ratio (1:2).
Thus 1/3 of the butt of sherry was left and hence the butler drew out 2/3 of the butt.
Ex. 10. The average weekly salary per head of the entire staff of a factory consisting of
supervisors and the labourers is Rs. 60. The average salary per head of the supervisors is
Rs. 400 and that of the labourers is Rs. 56. Given that the number of supervisors is 12, find
the number of labourers in the factory.
Sol: Average salary of labourers Average salary of supervisors
Rs. 56 Rs. 400
Mean
Rs. 60
340 4
By allegation rule:
(Number of labourers) = 340 = 85
(Number of supervisors) 4 1
Thus the number of supervisors is 1, number of labourers = 85.
if the number of supervisors is 12, number of labourers = 85*12 =1020.
Ex. 11. A man possessing Rs. 8400 lent a part of it at 8% simple interest and the remaining
at 6 2/3% simple interest .His total income after 1 ½ years was Rs. 882. Find the sum lent at
different rates.
Sol: Total interest on Rs. 84.. for 1 ½ years is Rs. 882.
Rate of interest = 100 x882x2 = 7%
8400x3
Rate % of 1st sum Rate % of 2nd sum
8% 6 2/3 %
Average Rate
7%
1 1
3
By alligation rule,
Money given at 8% S.I = 1: 1= 1:3
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 9
Money given at 6 2/3% S.I 3
Money lent at 8% = Rs. [8400 *(1/4)] = Rs. 2100.
Money lent at 6 2/3% = Rs. [8400 *(3/4)] = Rs. 6300.
Ex. 12. A man travelled a distance of 80 km. in 7 hours partly on foot at the rate of 8 km.
per hour and partly on bicycle at 16 km. per hour. Find the distance travelled on foot.
Sol. Average distance travelled in one hour =80/7 km.
Dist. Covered in 1 hr. on foot Dist. Covered in 1 hr. on bicycle
8 km. 16 km.
Average in 1 hr.
80 km.
32 7 24
7 7
By alligation rule,
Time taken on foot = 32 = 4:3
Time taken on bicycle 24
Thus out of 7 hours in all, he took 4 hours to travel on foot.
Distance covered on foot in 4 hours=(4x8)km=32 km.
Ex. 13. A sum of Rs. 41 was divided among 50boys and girls. Each boy gets 90 paise and a
girl 65 paise. Find the number of boys and girls.
Sol. Average money received by each =Rs. 41=82 p.
Sum received by each boy Sum received by each girl
90 paise 65 paise
Average
82 paise
17 8
By alligation rule,
Ratio of boys and girls=17:8.
Ex. 14. A lump of two metals weighting 18 gms. Is worth Rs. 87 but if their weights be
interchanged , it would be worth Rs. 78.60. if the price of one metal be Rs. 6.70 per gm.,
find the weight of the other metal in the mixture.
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 10
Sol. If one lump is mixed with the quantitiesof metals interchanged then the mixture of the two
lumps would contain 18gm. Of first metal and 18gm. Of second metal and the price of the
mixture would be Rs. (87+78.60) or 165.60.
cost of(18gm. Of 1st metal+18 gm. Of 2nd metal) = Rs. 165.60
So, cost of (1 gm. of 1st metal+1 gm. of 2nd metal) = Rs. 165.60 = Rs. 9.20.
18
(cost of 1 gm. of 1st metal) + (cost of 1 gm. of 2nd metal) = Rs. 9.20.
cost of 1 gm. of 2nd metal = Rs. (Rs. 9.20 – 6.70)= Rs. 2.50.
Now, mean price of lump = Rs. 87 per gm. = Rs. 29
18 6
C.P of 1 gm of 1st metal C.P of 1 gm. of 2nd metal
Rs. 6.70 Rs. 2.50
Mean price
Rs. 29
14 6 56
6 30
By alligation rule,
Quantity of 1st metal = 14 : 56 = 5:4
Quantity of 2nd metal 6 30
In 9 gm. of mix., 2nd metal = 4gm.
In 18 gm. of mix., 2nd metal = 4 x 18 gm. = 8 gm.
9
Ex. 15. A container contains 80 kg. of milk. From this container , 8kg. of milk was taken
out and replaced by water. This process was further repeated two times. How much milk is
now contained by the container?
Remarks. Amount of liquid left after n operations , when the container originally contains x
units of liquids, from which y units is taken out each time is =[ x(1-y/x)^n]units.
Sol. Amount of milk left = 80[(1-(8/80)3)] kg.
= 58.34 kg.
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 11
EXERCISE PROBLEMS:
1. A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the
mixture must be drawn off and replaced with water so that the mixture may be half water and
half syrup?
A. 1
3
B. 1
4
C. 1
5
D. 1
7
Answer: Option C
Explanation:
Suppose the vessel initially contains 8 liters of liquid.
Let x liters of this liquid be replaced with water.
Quantity of water in new mixture = 3 -
3x
8 + x liters
Quantity of syrup in new mixture = 5 -
5x
8 liters
3 -
3x
+ x = 5 -
5x
8 8
5x + 24 = 40 - 5x
10x = 16
x =
8
5
So, part of the mixture replaced =
8 x1
=
1
5 8 5
2. Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety in the ratio 1:1:2.
If the mixture is worth Rs. 153 per kg, the price of the third variety per kg will be:
A.Rs. 169.50 B.Rs. 170
C.Rs. 175.50 D.Rs. 180
Answer: Option C
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 12
Explanation:
Since first and second varieties are mixed in equal proportions.
So, their average price = Rs.
126 + 135
= Rs. 130.50
2
So, the mixture is formed by mixing two varieties, one at Rs. 130.50 per kg and the other at say,
Rs. x per kg in the ratio 2 : 2, i.e., 1 : 1. We have to find x.
By the rule of alligation, we have:
Cost of 1 kg of 1st kind Cost of 1 kg tea of 2nd kind
Rs. 130.50
Mean Price
Rs. 153
Rs. x
(x-153)
22.50
x - 153 = 1
22.50
x - 153 = 22.50
x = 175.50
3. A can contains a mixture of two liquids A and B is the ratio 7 : 5. When 9 litres 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.10 B.20
C.21 D.25
Answer: Option C
Explanation:
Suppose the can initially contains 7x and 5x of mixtures A and B respectively.
Quantity of A in mixture left = 7x -
7
x 9 liters = 7x -
21
12 4 liters.
Quantity of B in mixture left = 5x -
5
x 9 liters = 5x -
15
12 4 liters.
7x -
21
4 = 7
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 13
9
5x -
15
4 + 9
28x - 21
=
7
20x + 21 9
252x - 189 = 140x + 147
112x = 336
x = 3.
So, the can contained 21 litres of A.
4. A milk vendor has 2 cans of milk. The first contains 25% water and the rest milk. The second
contains 50% water. How much milk should he mix from each of the containers so as to get 12 liters
of milk such that the ratio of water to milk is 3: 5?
A.4 liters, 8 liters B.6 liters, 6 liters
C.5 liters, 7 liters D.7 liters, 5 liters
Answer: Option B
Explanation:
Let the cost of 1 liter milk be Re. 1
Milk in 1 liter mix. in 1st can =
3
liter, C.P. of 1 liter mix. in 1st can Re.
3
4 4
Milk in 1 liter mix. in 2nd can =
1
liter, C.P. of 1 liter mix. in 2nd can Re.
1
2 2
Milk in 1 liter of final mix. =
5
liter, Mean price = Re.
5
8 8
By the rule of alligation, we have:
C.P. of 1 liter mixture in 1st can C.P. of 1 liter mixture in 2nd can
3
4
Mean Price
5
8
1
2
1
8 1
8
Ratio of two mixtures =
1
:
1
= 1: 1.
8 8
So, quantity of mixture taken from each can =
1
x 12 = 6 litres.
2
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 14
5. 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.3 : 7 B.5 : 7
C.7 : 3 D.7 : 5
Answer: Option C
Explanation:
By the rule of alligation:
Cost of 1 kg pulses of 1st kind Cost of 1 kg pulses of 2nd kind
Rs. 15
Mean Price
Rs. 16.50
Rs. 20
3.50 1.50
Required rate = 3.50 : 1.50 = 7 : 3.
6. A dishonest milkman professes to sell his milk at cost price but he mixes it with water and
thereby gains 25%. The percentage of water in the mixture is:
A.4% B.6 ¼ %
C.20% D.25%
Answer: Option C
Explanation:
Let C.P. of 1 liter milk be Re. 1
Then, S.P. of 1 liter of mixture = Re. 1, Gain = 25%.
C.P. of 1 liter mixture = Re.
100
x 1 =
4
125 5
By the rule of alligation, we have:
C.P. of 1 liter of milk C.P. of 1 liter of water
Re. 1 0
Mean Price
Re.
4
5
4
5
1
5
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 15
Ratio of milk to water =
4
:
1
= 4 : 1.
5 5
Hence, percentage of water in the mixture =[(1/5)*100]%=20% .
7. How many kilogram of sugar costing Rs. 9 per kg must be mixed with 27 kg of sugar costing
Rs. 7 per kg so that there may be a gain of 10% by selling the mixture at Rs. 9.24 per kg?
A.36 kg B.42 kg
C.54 kg D.63 kg
Answer: Option D
Explanation:
S.P. of 1 kg of mixture = Rs. 9.24, Gain 10%.
C.P. of 1 kg of mixture = Rs.
100
x 9.24 = Rs. 8.40
110
By the rule of alligation, we have:
C.P. of 1 kg sugar of 1st kind Cost of 1 kg sugar of 2nd kind
Rs. 9
Mean Price
Rs. 8.40
Rs. 7
1.40 0.60
Ratio of quantities of 1st and 2nd kind = 14 : 6 = 7 : 3.
Let x kg of sugar of 1st be mixed with 27 kg of 2nd kind.
Then, 7 : 3 = x : 27
x =
7 x 27
= 63 kg.
3
8. A container contains 40 litres of milk. From this container 4 litres of milk was taken out and
replaced by water. This process was repeated further two times. How much milk is now
contained by the container?
A.26.34 litres B.27.36 litres
C.28 litres D.29.l6 litres
Answer: Option D
Explanation:
Amount of milk left after 3 operations = [40(1- 4/40 )3] liters
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 16
= 40 x
9
x
9
x
9
= 29.16 liters.
10 10 10
9. A jar full of whisky contains 40% alcohol. A part of this whisky is replaced by another
containing 19% alcohol and now the percentage of alcohol was found to be 26%. The quantity of
whisky replaced is:
A.
1
3
B.
2
3
C.
2
5
D.
3
5
Answer: Option B
Explanation:
By the rule of alligation , we have:
Strength of first jar Strength of 2nd jar
40%
Mean
Strength
26%
19%
7 14
So, ratio of 1st and 2nd quantities = 7 : 14 = 1 : 2
Required quantity replaced =
2
3
10. In what ratio must water be mixed with milk to gain 16 % on selling the mixture at cost
price?
A.1 : 6 B.6 : 1
C.2 : 3 D.4 : 3
Answer: Option A
Explanation:
Let C.P. of 1 litre milk be Re. 1.
S.P. of 1 litre of mixture = Re.1, Gain =
50
%.
3
C.P. of 1 litre of mixture = 100 x
3
x 1 =
6
350 7
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 17
By the rule of alligation, we have:
C.P. of 1 litre of water C.P. of 1 litre of milk
0
Mean Price
Re.
6
7
Re. 1
1
7
6
7
Ratio of water and milk =
1
:
6
= 1 : 6.
7 7
11. Find the ratio in which rice at Rs. 7.20 a kg be mixed with rice at Rs. 5.70 a kg to produce a
mixture worth Rs. 6.30 a kg.
A.1 : 3 B.2 : 3
C.3 : 4 D.4 : 5
Answer: Option B
Explanation:
By the rule of alligation:
Cost of 1 kg of 1st kind Cost of 1 kg of 2nd kind
720 p
Mean Price
630 p
570 p
60 90
Required ratio = 60: 90 = 2: 3.
12. In what ratio must a grocer mix two varieties of tea worth Rs. 60 a kg and Rs. 65 a kg so that
by selling the mixture at Rs. 68.20 a kg he may gain 10%?
A.3 : 2 B.3 : 4
C.3 : 5 D.4 : 5
Answer: Option A
Explanation:
S.P. of 1 kg of the mixture = Rs. 68.20, Gain = 10%.
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 18
C.P. of 1 kg of the mixture = Rs.
100
x 68.20 = Rs. 62.
110
By the rule of alligation, we have:
Cost of 1 kg tea of 1st kind Cost of 1 kg tea of 2nd kind.
Rs. 60
Mean Price
Rs. 62
Rs. 65
3 2
Required ratio = 3 : 2.
13.The cost of Type 1 rice is Rs. 15 per kg and Type 2 rice is Rs. 20 per kg. If both Type 1 and
Type 2 are mixed in the ratio of 2 : 3, then the price per kg of the mixed variety of rice is:
A.Rs. 18 B.Rs. 18.50
C.Rs. 19 D.Rs. 19.50
Answer: Option A
Explanation:
Let the price of the mixed variety be Rs. x per kg.
By rule of alligation, we have:
Cost of 1 kg of Type 1 rice Cost of 1 kg of Type 2 rice
Rs. 15
Mean Price
Rs. x
Rs. 20
(20 - x) (x - 15)
(20 - x)
=
2
(x - 15) 3
60 - 3x = 2x - 30
5x = 90
x = 18.
14.8 liters are drawn from a cask full of wine and is then filled with water. This operation is
performed three more times. The ratio of the quantity of wine now left in cask to that of water is
16 : 81. How much wine did the cask hold originally?
A.18 liters B.24 liters
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 19
C.32 liters D.42 liters
Answer: Option B
Explanation:
Let the quantity of the wine in the cask originally be x litres.
Then, quantity of wine left in cask after 4 operations = [x(1 – 8/x ) litres.
x(1 - (8/x))4
=
16
x 81
1 -
8
4=
2
4
x 3
x - 8
=
2
x 3
3x - 24 = 2x x = 24.
15. A merchant has 1000 kg of sugar, part of which he sells at 8% profit and the rest at 18%
profit. He gains 14% on the whole. The quantity sold at 18% profit is:
A.400 kg B.560 kg
C.600 kg D.640 kg
Answer: Option C
Explanation:
By the rule of alligation, we have:
Profit on 1st part Profit on 2nd part
8%
Mean Profit
14%
18%
4 6
Ration of 1st and 2nd parts = 4 : 6 = 2 : 3
Quantity of 2nd kind =
3
x 1000
kg
= 600 kg.
5
16. How many liters of water should be added to a 30 litre mixture of milk and water containing
milk and water in the ratio of 7 : 3 such that the resultant mixture has 40% water in it?
A.7 liters B.10 liters
C.5 liters D.None of these
Answer: Option C
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 20
Explanation:
30 liters of the mixture has milk and water in the ratio 7: 3. i.e. the solution has 21 liters of milk
and 9litres of water. When you add more water, the amount of milk in the mixture remains
constant at 21 liters. In the first case, before addition of further water, 21 liters of milk accounts
for 70% by volume. After water is added, the new mixture contains 60% milk and 40% water.
Therefore, the 21litres of milk accounts for 60% by volume.
Hence, 100% volume = 21/0.6= 35 liters.
We started with 30 liters and ended up with 35 liters. Therefore, 5 liters of water was added.
17. A 20 liter mixture of milk and water contains milk and water in the ratio 3: 2. 10 liters of the
mixture is removed and replaced with pure milk and the operation is repeated once more. At the
end of the two removal and replacement, what is the ratio of milk and water in the resultant
mixture?
A.17 : 3 B.9 : 1
C.3 : 17 D.5 : 3
Answer: Option B
Explanation:
The 20 litre mixture contains milk and water in the ratio of 3 : 2. Therefore, there will be 12 litres
of milk in the mixture and 8 litres of water in the mixture.
Step 1. When 10 litres of the mixture is removed, 6 litres of milk is removed and 4 litres of water
is removed. Therefore, there will be 6 litres of milk and 4 litres of water left in the container. It is
then replaced with pure milk of 10 litres. Now the container will have 16 litres of milk and 4
litres of water.
Step 2. When 10 litres of the new mixture is removed, 8 litres of milk and 2 litres of water is
removed. The container will have 8 litres of milk and 2 litres of water in it. Now 10 litres of pure
milk is added. Therefore, the container will have 18 litres of milk and 2 litres of water in it at the
end of the second step.
Therefore, the ratio of milk and water is 18: 2 or 9: 1.
18. How many kgs of Basmati rice costing Rs.42/kg should a shopkeeper mix with 25 kgs of
ordinary rice costing Rs.24 per kg so that he makes a profit of 25% on selling the mixture at
Rs.40/kg?
A.20 kgs B.12.5kgs
C.3 : 16kgs D.200kgs
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 21
Answer: Option A
Explanation:
Let the amount of Basmati rice being mixed be x kgs. As the trader makes 25% profit by selling
the mixture at Rs.40/kg, his cost per kg of the mixture = Rs.32/kg.
i.e. (x * 42) + (25 * 24) = 32 (x + 25)
=> 42x + 600 = 32x + 800
=> 10x = 200 or x = 20 kgs.
19. How many litres of a 12 litre mixture containing milk and water in the ratio of 2 : 3 be
replaced with pure milk so that the resultant mixture contains milk and water in equal
proportion?
A.4 liters B.2 liters
C.1 liter D.1.5 liters
Answer: Option B
Explanation:
The mixture contains 40% milk and 60% water in it. That is 4.8 litres of milk and 7.2 litres of
water.
Now we are replacing the mixture with pure milk so that the amount of milk and water in the
mixture is 50% and 50%.That is we will end up with 6 litres of milk and 6 litres of water.
Water gets reduced by 1.2 litres.
To remove 1.2 litres of water from the original mixture containing 60% water, we need to
remove 1.2 / 0.6 litres of the mixture = 2litres.
20. A zookeeper counted the heads of the animals in a zoo and found it to be 80. When he
counted the legs of the animals he found it to be 260. If the zoo had either pigeons or horses, how
many horses were there in the zoo?
A.40 B.30
C.50 D.60
Answer: Option C
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 22
Explanation:
Let the number of horses = x
Then the number of pigeons = 80 – x.
Each pigeon has 2 legs and each horse has 4 legs.
Therefore, total number of legs = 4x + 2(80-x) = 260
=>4x + 160 – 2x = 260
=>2x = 100
=>x = 50.
21. From a cask of milk containing 30 litres, 6 litres are drawn out and the cask is filled up with
water. If the same process is repeated a second, then a third time, what will be the number of
litres of milk left in the cask?
A.0.512 liters B.12 liters
C.14.38 liters D.15.36 liters
Answer: Option D
Explanation:
The problem can be solved by traditional method but it is cumbersome process to do that. The
problem is simple if its solution is simpler. Hence we will go for a simpler solution for this kind
of problem.
There is a short cut method to find the Quantity of milk left after nth operation.
It is given by [(x – y)/x]n of the whole quantity, where x is initial quantity of milk in the cask y is
the quantity of milk withdrawn in each process and n is the number of process..
Hence from the above rule it can be say that
Quantity of milk left after the 3rd operation = [(30 – 6)/30]3 * 30 = 15.36 liters.
22. In what ratio must wheat at Rs.3.20 pe rkg be mixed with wheat at Rs.2.90 per kg so that the
mixture be worth Rs.3.08 per kg
A.5:7 B.7:9
C.3:2 D.7:5
Answer: Option C
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 23
Explanation:
By the rule of alligation, we have:
C.P of a unit quantity of 1st kind C.P. of a unit quantity of 2nd kind
3.20 2.90
Mean Price (3.80)
0.18 0.12
Required ratio =0.18 : 0.12 = 3:2.
23. Two A and B contain milk and water mixed in the ratio 8:5 and 5:2 respectively. The ratio in
which these two mixtures be mixed to get a new mixture containing milk and a water in the ratio
9:4?
A.5:7 B.2:7
C.3:2 D.7:5
Answer: Option B
Explanation:
Step (i) : Let C.P. of milk be Re.1
Given ratio of mixture in A = 8:5
Milk in 1 liter mixture in A = 8/13 litre
C.P of 1 liter mixture in A = Rs. 8/13
Ratio of Mixture in B = 5:2
milk in 1 liter mixture in B – 5/7 litre
C.P of 1 litre mixture in B = Rs. 5/7
Ratio of new mixture = 9:4
Milk in 1 lit mixture = 9/13
C.P of 1 litre mixture = Rs/ 9/13 (Mean price)
Step (ii) : By the rule of allegation,
i. C.P of 1 liter of mixture in A =8/13
ii. C.P of 1 liter of mixture in B = 5/7
iii. Mean price (p) = 9/13
iv. d – m = 9/13 – 5-7 = 2/91
v. m – c = 9/13 – 8/13 = 1/13
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 24
Required ratio = 2/91: 1/13 = 2:7
24. In what ratio water be mixed with milk costing Rs.12 per liter to obtain a mixture worth of
Rs.8 per litre?
A.1:2 B.2:7
C.3:2 D.7:5
Answer: Option A
Explanation:
By the rule of allegation,
i. C.P of 1 liter of water = 0
ii. C.P of 1 liter of milk = 12
iii. Mean price (p) = 8
iv. d – m = 12-8 = 4
v. m – c = 8-0 = 8
Ratio of water and milk = 4 : 8 = 1 : 2
25.729ml of mixture contains milk and water in the ratio 7:2 how much more water is to be
added to get a new mixture containing milk and water in the ratio 7:3?
A.70ml B.49ml
C.81ml D.96ml
Answer : Option C
Explanation:
Ratio of milk and water in 729 ml = 7:2
Step (i) Milk in 729 ml of mixture = (7/9 x 729 ) ml = 567 ml
water in 729 ml of mixture = 729 – 567 = 162 ml
Step (ii) Let x be the quantity of water added to new mixture, with the ratio 7:3
Quantity of water in the new mixture = (162 + x ) ml
Then 7/3 =
162 x
567
7 (162 + x) = 3 x 567
1134 + 7x = 1701
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 25
7x = 1701 – 1134
X = 567/7 = 81 ml
Quantity of water added to new mixture = 81 ml.
26. A can contains 40kg of milk, from this container 4kg of milk was taken out and replaced by
water. This process was repeated further two times. How much milk is now contained by the
container?
A.29.16kg B.30kg
C.32.49kg D.25.36kg
Answer: Option A
Explanation:
Quantity of milk in the can x = 40 kg
Quantity of milk taken out y = 4 kg
Number of times = 3
Quantity of milk in the can = kg
x
y
x
n
1
= kg
3
40
4
40 1 = 40 (9/10)3 kg = 40 ( 729/1000)
Quantity of milk in the can = 29.16 kg
27. Two vessels A and B contain sprit and water in the ratio 5:2 and 7:6 respectively. Find the
ratio in which these mixture be mixed to obtain a new mixture containing spirit and water in the
ratio 8:5?
A.3:4 B.5:7
C.7:9 D.4:3
Answer: Option C 7:9
Explanation:
Step (i) : Spirit in 1 litre mixture of A = 5/7 litre
Spirit in 1 litre mixture of B = 7/13 litre
Spirit in 1 litre mixture of final mixture = 8/13 litre
Mean quantity = 8/13 litre
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 26
Step (ii) By the rule of allegation,
i. quantity of spirit in A (c) = 5/7
ii. Quantity of spirit in B (d)= 7/13
iii. Mean price (m) = 8/13
iv. d – m = 5/7 – 8/13 = 9/19
v. m – c = 8/13 – 7/13 = 1/13
Required ratio = 1/13 : 9/91 = 7 : 9
28.Two vessels A and B contain milk and water mixed in the ratio 8:5 and 5:2 respectively. The
ratio in which these 2 mixtures be mixed to get a new mixture containing 69 3/13 % milk is :
A.3:4 B.2:7
C.7:9 D.4:3
Answer: Option B
Explanation:
Step (i) : Quantity of milk in 1 lr mixture of A = 8/13 lr
Quantity of milk in 1 lr mixture of B = 5/7 lr
Quantity of milk in 1 lr mixture of final mixture = 69
13
3
%
=
100
1
13
900
x lr
Mean quantity = 9/13 lr
Step (ii) By the rule of allegation,
i. quantity of spirit in A (c) = 8/13
ii. Quantity of spirit in B (d)= 5/7
iii. Mean price (m) = 9/13 lr
iv. d – m = 5/7 – 9/13 = 2/91
v. m – c = 9/13 – 8/13 = 1/13
Required ratio = 2/91 : 1/13 : = 2 : 7
29. The cost of type I rice is Rs.15 p/kg and type II is Rs.20p/kg. Both are mixed in the ratio 2:3,
price P/Kg of the mixed variety is :
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 27
A.Rs.20 B.Rs.30
C.Rs.15 D.Rs.18
Answer: Option D
Explanation:
Step (i) : Let the price of mixed variety be x.Rs
Mean price = Rs.x
Cost of 1 kg of Type I Cost of 1 kg of Type II
Rs. 15 Rs. 20
Mean wt (m)
(20 - x) Rs.x (x - 15)
ratio = 20-x : x - 15
Step (ii) : Mixed variety is in the ratio = 2:3
15
20
x
x
= 2/3
60 – 3x = 2x – 30
X = 90 / 5 = 18
Ratios imply that the price of mixture = Rs. 18 per kg
30. In what ratio must tea at Rs.62 per Kg be mixed with tea at Rs. 72 per Kg so that the mixture
must be worth Rs. 64.50 per Kg?
A. 3 : 1 B. 3 : 2
C. 4 : 3 D. 5 : 3
Answer: Option A
Explanation:
By the rule of alligation:
Cost of 1 kg tea of 1st kind Cost of 1 kg tea of 2nd kind
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 28
Required ratio = 750 : 250 = 3 : 1
31. A bar is creating a new signature drink. They are using two alcoholic ingredients in the drink:
vodka and gin. They are using two non-alcoholic ingredients in the drink: orange juice and
cranberry juice. The alcoholic ingredients contain 40% alcohol. The non-alcoholic ingredients
contain no alcohol. How many liters of non-alcoholic ingredients must be added to 6 liters of
alcoholic ingredients to produce a mixture that is 15% alcohol?
A. 5 B. 10
C. 6 D. 15
Answer: Option B
Explanation:
The alcohol ingredients is 40% in 6 liters of drink
The resultant mixture needs to have 15% alcohol only
Hence we have the following equation :
(6+x) (0.15) = 6(40%) + x(0%)
(6+x) 0.15 = 6(0.4) + 0
Solving for x, we get x = 10.
31. 150 liters of a 20% alcohol solution is mixed with 200 liters of another solution. If the
resulting solution is 18% alcohol, what is the percent of alcohol in the 200-liter solution?
A. 21% B. 16.5%
C. 15.2% D. 18%
Answer: Option B
Explanation:
Total solution is 350 litres .
We have the following equation: 350(0.18) = 200(x) + 150(0.2)
Solving for x , we get x = 0.165 i.e. 16.5%
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 29
32. 8 litres are drawn from a cask full of wine and is then filled with water. This operation is
performed three more times. The ratio of the quantity of wine now left in cask to that of the
water is 16 : 65. How much wine did the cask hold originally?
A. 18 litres B.24 litres
C.32 litres D.42 litres
Answer: Option B
Explanation:
Let the quantity of the wine in the cask originally be x litres.
Then , quantity of wine left in cask after 4 operations = [x(1-(8/x)4] litres.
[x(1-(8/x)4] = 16
x 81
(1-8/x)4 = (2/3)2
x - 8 = 2 3x – 24 = 2x x = 24.
x 3
33. One quality of wheat at Rs. 9.30 per kg is mixed with another quality at certain rate in the
ratio 8 : 7 .If the mixture so formed be worth Rs. 10 per kg,what is the rate per kg of the second
quality of wheat ?
A.Rs. 10.30 B.Rs. 10.60
C.Rs. 10.80 D.Rs.11
Answer: Option C
Explanation:
Let the rate of the second quality be Rs. X per kg.
By the rule of allegation, we have:
C.P of 1 kg. wheat of 1st kind C.P of 1 kg wheat of 2nd kind
930p 100x p
Mean price
1000 P
(100x – 1000)p 70 p
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 30
100x – 1000 = 8
70 7
700x – 7000=560
700x = 7560
x = Rs. 10.80.
QUESTION BANK:
1. In what ratio must rice at Rs.30 per kg mixed with rice at Rs.10.80 per kg so that the
mixture be worth Rs.10per kg .
A. 3:4 B. 5:4
C. 8:7 D. 4:5
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 31
2. In what ratio must wheat at Rs.3.20 pe rkg be mixed with wheat at Rs.2.90 per kg so that
the mixture be worth Rs.3.08 per kg
A. 3:2 B.5:4
C. 2:3 D. 4:5
3. How many kilograms of sugar costing Rs/.9 per kg must be mixed with 27kg of sugar
costing Rs.7 per kg so that there may ba gain of 10% by selling the mixture at Rs.9.24 per
kg ?
A. 50 kg B. 63kg
C. 72kg D. 59kg
4. In what ratio must water be mixed with milk to gain 16 2/3% on selling the mixture at
cost price ?
A. 1:6 B. 5:7
C. 7:5 D. 6:1
5. Two vessels A and B contain milk and water mixed in the ratio 8:5 and 5:2 res. The ratio
in which these two mixtures be mixed to get a new mixture containing milk and a water
in the ratio 9:4?
A. 3:4 B. 5:9
C. 2:7 D. 3:2
6. In what ratio water be mixed with milk costing Rs.12 per liter to obtain a mixture worth
of Rs.8 per litre?
A. 2:4 B. 1:2
C. 5:7 D. 2:1
7. A sum of Rs.4000 is lent out in two parts, one at 8% simple interest and the other at 10%
simple interest. In the annual interest is Rs.352, the sum lent at 8% is ?
A. 1300 B. 1500
C. 2400 D. 3700
8. A merchant has 100kg of sugar, part of which he sells at 8% profit and the rest at 18%
profit. He gains 14% on the whole. The quantity sold at 18% profit is?
A. 120kg B. 500kg
C. 300kg D. 600kg
9. Two vessels A and B contain milk and water mixed in the ratio 4:3 and 2:3 in what ratio
must these mixtures be mixed to form new mixture containing half milk and half water?
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 32
A. 5:6 B. 7:5
C. 6:7 D. 7:8
10. A jar full of whisky contains 40% alcohol. A part of this whisky is replaced by another
containing 19% alcohol and now the percentage of alcohol was found to be 26% the
quantity of whisky replaced is ?
A.1/4 B.1/3
C.2/3 D.2/4
11. 729ml of mixture contains milk and water in the ratio 7:2 how much more wate is to be
added to get a new mixture containing milk and water in the ratio 7:3?
A.70ml B.49ml
C.81ml D.96ml
12. A sum of Rs.312 was divided amng 100boys and girls in such a way that the boy gets
Rs.3.60 and each girl Rs.2.40 the number of girls is ?
A.30 B.40
C.20 D.50
13. A man covered a distance of 2000km in 18 hours partly by bus at 72kmph and partly by
train at 16kmph the distance covered by bus is ?
A.540ml B.960ml
C.720ml D.840ml
14. A sum of rs.36.90 is made up of 180 coins which are either 10 paise coins or 25 p coins.
The number of 10 p coins is ?
A.72 B.54
C.45 D.67
15. A dishonest milk man professes to sell his milk at cost price but he mixed it with water
and thereby gains 25%. The percentage of water in the mixture is ?
A.35% B.25%
C.20% D.40%
16. A mixture of 20kg of spirt and water contains 10 water How much water must be added
to this mixture to raise the percentage of water to 25%?
A.3kg B.4kg
C.5kg D.6kg
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 33
17. A container contains 40kg of milk, from this container 4kg of milk was taken out and
replaced by water. This process was repeated further two times. How much milk is now
contained by the container?
A.43.12 kg B.43.22kg
C.29.16kg D.12.45kg
18. A can contains a mixture of two liquids A and B in the ratio 7:5 when 9 litres 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 liquids A was contained by the can initially?
19. A mixture of milk and water measures 60 gallons. It contains 20% water. How many
gallons of water should be added to it so that water may be 25%?
20. A mixture of spirit and water measure 80 gallons. It contains 20% water. How much
water should be added to it so that water may be 25%?
21. A man lent $2000, part of this at 4% and the rest at 6% per annum simple interest. The
whole annual interest amounted to $92. How much did he lend at 6%?
22. A man invested $2500 into two parts such that if one part be put out at 5% S.I. and other
at 6%, the yearly income may be $140. How much did he invest at 5%?
23. There are two vessels A and B in which the ratio of milk and water are as 5:2 and 8:7
respectively. Two gallons are drawn from vessel A and 3 gallons from vessel B, and are
mixed in another empty vessel. What is the ratio of milk and water in it?
24. Two gallons of mixture in which there is 2/5 of water and the rest spirit is mixed with
five gallons of mixture in which there is 1/3 of water and the rest spirit. What is the ratio
of water and spirit in the new mixture?
A.20 B.21
C.22 D.23
A.6 gallons B.4 gallons
C.8 gallons D.10 gallons
A.8 1/3 gallons B.6 1/3gallons
C.7 1/3 gallons D. 5 1/3 gallons
A.$900 B.$800
C.$600 D.$1000
A.$1250 B.$1500
C.$1000 D.$750
A.106:69 B.103:72
C.89:86 D.101:71
Reasoning and Quantitative Aptitude Alligation or Mixture
SITAMS 34
25. One vessel contains a mixture of 5 parts pure wine and 3 parts soda, whereas the other
vessel contains a mixture of 9 parts pure wine and 5 parts soda. Compare the strength of
the wine.
26. One milk can contains a mixture of milk and water in the ratio 7:5 and the other contains
the mixture of milk and water in which 2/5th is water. Compare their purity.
27. A woman sold 100 oranges at $12.10, some at the rate of 3 for 35 cents and the rest at 7
for 85 cents. How many were sold at the first rate?
28. A merchant has 100 lbs of sugar, part of which he sells at 7% profit and the rest at 17%
profit. He gains 10 % on the whole. Find how much is sold at 7% profit?
A.18:23 B.12:17
C.25:33 D.37:68
A.7:4 B.7:8
C.35:36 D.14:5
A.36:35 B.35:36
C.10:7 D.5:3
A.45 B.21
C.15 D.9
A.70 lbs B.40lbs
C.30 lbs D.50lbs
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