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 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 Quantity of dearer

=

C.P. of dearer - Mean Price Mean price - C.P. of cheaper

We present as under:

C.P. of a unit quantity of cheaper (c)

C.P. of a unit quantity of dearer (d)

Mean price (m)

(d – m)

(m – c)

(Cheaper quantity) : (Dearer quantity) = (d - m) : (m - c). Suppose a container contains x of liquid from which y units are taken out and replaced by water. y After n operations, the quantity of pure liquid = x 1 - n units. x 4.

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.

C.P of 1 Kg rice of 1st kind 930 p C.P of 1 Kg rice of 2n d kind 1080p Mean Price 1000p 80 70 Required

ratio=80:70

= 8:7

Ex.2. How much water must be added mixture worth I 10 2/3 a Hter? Sol.

to 60 liters of milk at 11/2 liters for Rs 20 so as to have a

C.P of 1 lit of milk = 20*2/3 = 40/3 C.P of 1 lit of water 0 C.P of 1 lit of milk 40/3 Mean Price 32/3 8/3 32/3 Ratio of water and milk =8/3 : 32/3 = 1:4 Quantity of water to be added to 60 lit of milk =1/4*60=15 liters.

Ex.3. How many kg. of salt at 42 P per kg. must a man mix with 25 kg. of salt at 24 P per kg. so that he may , on selling the mixture at 40 P per kg, gain 25% on the outlay? 100 Sol.

Cost price of mixture = 40 x

P = 32 P per kg. 125 …………(By the rule of fraction)

42

24

32

8

10

Ratio = 4 : 5 Thus for every 5 kg. of salt at 24 P, 4 kg . of salt at 42 P is used. 4 :

the required no. of kg = 25 x

= 20.

Ans.

5

Ex.4. A mixture of a certain quantity of milk with 16 litres of water is worth 90 P per litre. If pure milk be worth Rs. 1.08 per litre how much milk is there in the mixture?

Sol.

The main value is 90 P and the price of water is 0 P.

milk

water

108

0

90

90 – 0

108 – 90

By the Alligation Rule , milk and water are in the ratio of 5 : 1. : quantity of milk in the mixture = 5 x 16 = 80 litres.

Ans.

Ex.5. A goldsmith has two qualities of gold - one of 12 carats and another of 16 carats purity . In what proportion should he mix both to make an ornament of 15 carats purity? Sol.

I

II

12

16

15

1

:

3

he should mix both the qualities in the ratio 1 : 3.

Ex.6. A person has a chemical of Rs 25 per litre. In what ratio should water be mixed in that chemical so that after selling the mixture at Rs 20/litre he may get a profit of 25%? Sol.

In this question the allegation method is applicable on prices, so we should get the average price of mixture.

SP of mixture = Rs.20/litre;

profit

= 25%

100 :

Average price = 20 x

= Rs. 16 / litre. 125

Applying the allegation rule :

Chemical

Water

25

0

16

16 :

C : W

9

= 16 : 9 Ans.

Ex.7. A container contained 80 kg of milk. From this container 8 kg 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? Sol. Amount of liquid left after n operations , when the container originally contains x units of liquid from which y units is taken out each time is x–y

n

x

units. x

Thus, in the above case, amount of milk left 80 – 8 = 80

kg

=

58.32 kg.

80

Ex.8. A butler stores wine from a butt of sherry which contained 30% of spirit and he replaced what he had stolen by wine containing only 12% of spirit. The butt was then 18% strong only. How much of the butt did he steal? Sol.

By the allegation rule we find that wine containing 30% of spirit and wine containing 12% of spirit should be mixed in the ratio 1 : 2 to produce a mixture containing 18% of spirit. 30%

12%

18%

6%

12%

Ratio = 6 : 12 = 1 : 2 This means that 1/3 of the butt of sherry was left , i.e. to say , the butler drew out 2/3 of the butt. :

2/3 of the butt was stolen.

Ex.9. 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 allegation , 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 :

Required ratio = 80 : 70

70 = 8 : 7.

Ex.10. 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 per kg? Sol.

By the rule of alligation: Cost of 1 kg pulses of 1st kind

Cost of 1 kg pulses of 2nd kind

Rs.15

Rs.20

Mean price Rs. 16.50

3.50 :

1.50

Required rate = 3.50 : 1.50 = 35 : 15 = 7 : 3.

Ex.11. How many kgs. of wheat costing Rs.8 per kg must be mixed with 36 kg of rice costing Rs.5.40 per kg so that 20% gain may be obtained by selling 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 x 7.20 120 By the rule of allegation, we have :

= Rs. 6.

C.P. of 1 kg wheat of 1st kind (800p)

C.P. of 1 kg wheat of 2nd kind (540p)

Mean price (600 p)

60 st

200 nd

Wheat of 1 kind : Wheat of 2 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 10x = 3 x 36 or x = 10.8kg.

Ex.12. 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? Sol.

By the rule of alligation: Cost of 1 kg pulses of 1st kind

Cost of 1 kg pulses of 2nd kind

Rs. 15

Rs. 20

Mean price Rs. 16.50

3.50 :

1.50

Required rate = 3.50 : 1.50 = 7 : 3.

Ex.13. 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?

Sol.

Suppose the vessel initially contains 8 litres of liquid. Let x litres of this liquid be replaced with water. 3x Quantity of water in new mixture =

3-

+ x

litres

8 5x Quantity of syrup in new mixture = 5 –

litres 8

3x :

3–

5x +x

=

5–

8

8

=>

5x + 24 40 - 5x

=>

10x = 16

=>

x =

8 . 5 8 So, part of mixture replaced =

1 x

5

1 =

.

8

5

Ex.14. 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: Sol.

Let C.P. of 1 litre milk be Re. 1 Then, S.P. of 1 litre of mixture = Re. 1, Gain = 25%. 100 C.P. of 1 litre mixture = Re.

4 x 1

=

125

5

By the rule of alligation ,we have: C.P. of 1 litre of milk

C.P. of 1 litre of water

Re.1

0

Mean price Re. 4/5

4/5

:

Ratio of milk to water

1/5

= 4/5 : 1/5 = 4 : 1.

Hence , percentage of water in the mixture = 1/5 x 100 % = 20%

2 Ex.15.

In what ratio must water be mixed with milk to gain 16

on selling the mixture at cost price? 3

Sol.

Let C.P. of 1 litre milk be Re. 1. 50 S.P. of 1 litre of mixture = Re.1, Gain =

%. 3 3

: C.P. of 1 litre of mixture =

6

100 x

x 1

=

350

7

By the rule of allegation , we have: C.P. of 1 litre of water

C.P. of 1 litre of milk

0

Re. 1

Mean price 6 Re. 7

1

6

7

7

:

Ratio of water and milk = 1/7 : 6/7 = 1 : 6.

Ex.16. kg.

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

Sol.

By the rule of alligation: Cost of 1 kg of 1st kind

Cost of 1 kg of 2nd kind

720p

570p

Mean price 630 p

60 : Required ratio: = 60 : 90 = 2 : 3.

90

Ex.17. 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: Sol.

By the rule of alligation, we have: Profit on 1st part

Profit on 2nd part

8%

18%

Mean profit 14%

4

6

Ratio of 1st and 2nd parts = 4 : 6 = 2 : 3

Quantity of 2nd kind =

:

3 x 1000

kg = 600 kg.

5

Ex.18. 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.

Sol.

By the rule of allegation : Cost of 1kg tea of 1st kind

Cost of 1kg tea of 2nd kind

6200p

7200p

Mean price 6450p

750

250

: Required ratio = 750 : 250 = 3 : 1. Ex.19. Milk and water are mixed In a vessel A in the proportion 5 : 2 , and in vessel B in the proportion 8 : 5 . In what proportion should quantities be taken from the two vessels so as to form a mixture in which milk and water will be in the proportion of 9 : 4?

Sol.

In vessel A, milk = 5/7 of the weight of mixture

In vessel B, milk = 8/13 of the weight of mixture . Now we want to form a mixture in which milk will be 9/13 of the weight of this mixture. By alligation rule : 5/7

8/13

9/13

1/13

:

2/91

Required proportion is 1/13 : 2/91 = 7 : 2

Ex.20. A trader has 50 kg of pulses , part of which he sells at 8% profit and the rest at 18% profit . He gain 14% on the whole . What is the quantity sold at 18% profit ? Sol.

By allegation method : I part

II part

8% profit

18% profit

Mean profit 14%

4%

6%

= 4 : 6 = 2 : 3 50 There for the quantity sold at 18% profit =

x 3 = 30 kg. 2 + 3

Ex.21. A trader has 50 kg of rice, a part of which he sells at 10 % profit and the rest at 5% loss. He gains 7% on the whole . What is the quantity sold at 10% gain and 5% loss?

Sol.

I part

II part

10

(-)5

7

12

:

3

Ratio of quantities sold at 10% profit and 5% loss = 12 : 3 = 4 : 1. 50

Therefore , the quantity sold at 10% profit = 10 kg.

Note :

x 4 = 40 kg and the quantity sold at 5% loss = 50 – 40 = 4 + 1

Whenever there is loss, take the negative value. Here , difference between 7 and (-5) = 7 -(-5) = 7 + 5 = 12. Never take the difference that counts negative value.

Ex.22. 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 : Sol.

Let the price of the mixed variety be Rs. x per kg. By the rule of allegation , we have : Cost of 1 kg of Type 1 rice

Cost of 1 kg of Type 1 rice

Rs.15

Rs.20

Mean price Rs. x

(20 – x)

(x – 15)

(20 – x) :

2 =

(x – 15)

= > 60 - 3x

=

2x – 30

= > 5x

= 90 = > x = 18.

3

So, price of the mixture is Rs.18 per kg.

Ex.23. 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? Sol.

Amount of milk left after 3 operations

3

4 =

40

9

1–

litres

=

40 x

40

9 x

10

9 x

10

= 29.16 litres. 10

Ex.24. Two vessels A and B contain spirit and water mixed in the ratio 5 : 2 and 7 : 6 respectively . Find the ratio in which these mixture be mixed to obtain a new mixture in vessel C containing spirit and water in the ratio 8 : 5? Sol.

Let the C.P. of sprit be Re. 1 per litre. Spirit in 1 litre mix. of A = 5/7 litre; C.P. of 1 litre mix. in A = Re. 5/7. Spirit in 1 litre mix. of B = 7/13 litre; C.P. of 1 litre mix. in B = Re. 7/13. SpiriT in 1 litre mix. of C

= 8/13 litre; Mean price = Re. 8/13.

By the rule of allegation , we have : C.P. of 1 litre mixture in A

C.P. of 1 litre mixture in B

(5/7)

(7/13)

Mean price (8/13)

(1/13)

:

Required ratio = 1/13

(9/91)

: 9/91

= 7 : 9.

Ex.25. 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%? Sol.

S.P. of 1 kg of the mixture = Rs. 68.20, Gain = 10%. 100 C.P. of 1 kg of the mixture = Rs.

x 68.20 110

= Rs. 62.

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

Rs. 65

Mean price Rs.62

3

:

2

Required ratio

= 3 : 2.

1 Ex.26. a litre?

How much water must be added to 60 litres of milk at 1

Sol.

C.P. of 1 litre of milk = Rs.

2 litres for Rs.20 so as to have a mixture worth Rs. 10

2 2 20 x

3

40 = Rs.

.

3

3

C.P. of 1 litre of water

C.P. of 1 litre of milk 40 Rs. 3

0

Mean price 32 Rs. 3

40

32

3

8

32

= 3

32

- 0 3

3

8 : Ratio of water and milk =

32 :

3

= 8 : 32 = 1 : 4. 3 1

: Quantity of water to be added to 60 litres of milk =

x 60 litres 4

= 15 litres.

= 3

Ex.27. 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 syrup ? Sol.

Suppose the vessel initially contains 8 litres of liquid . Let x litres of this liquid be replaced with water . 3x Quantity of water in new mixture =

3–

+ x litres. 8

5x Quantity of syrup in new mixture =

5-

litres. 8

3x :

3–

5x +x

=

8

5–

8

=>

5x + 24 = 40 – 5x

=>

8

10x = 16 = > x =

. 5

8 So, part of the mixture replaced =

1 x

5

1 =

8

. 5

Ex.28.

In what ratio must water to be mixed with milk to gain 20% by selling the mixture at cost price?

Sol.

Let the C.P of milk be Re 1 per liter Then S.P of 1 liter of mixture = Re.1 Gain obtained =20%. Therefore C.P of 1 liter mixture = Rs(100/120*1) =5/6 C.P of 1 liter of water 0 C.P of 1 liter of milk1 Mean Price 5/6 1/6 5/6 Ratio of water and milk =1/6 : 5/6 = 1:5.

Ex.29. 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 per Kg? Sol.

Cost of 1 Kg pulses of 1 kind 15 Cost of 1 Kg pulses of 2nd kind 20 Mean Price Rs 16.50 3.50 , 1.50 Required ratio =3.50 : 1.50 = 35:15 = 7:3.

Ex.30. 4Kg s of rice at Rs 5 per Kg is mixed with 8 Kg of rice at Rs 6 per Kg .Find the average of the mixture? Sol.

rice of 5 Rs per Kg rice of 6 Rs per Kg Average price Aw 6-Aw Aw-5 (6-Aw)/(Aw-5) = 4/8 =1/2 12-2Aw = Aw-5 3Aw = 17 Aw = 5.66 per Kg.

price