Alligation and Mixture

1

ALLIGATION OR MIXTURE NATIONALISED BANKS & IBPS PO/MT/SO 1.

In a mix tu re of mil k and w ater the proportion of water by weight was 75% . If in the 60 gms mixture 15 gms water was added, what would be the percentage of water in the new mixture ? (1) 75% (2) 88% (3) 90% (4) 100% (5) None of these (Indian Overseas Bank PO Exam. 05.04.2009)

2.

3.

SBI PO EXAMS 1.

2.

3.

In 1kg mixture of sand and iron, 20% is iron. How much sand should be added, so that the proportion of iron becomes 5% ? (1) 3 kg (2) 4 gms (3) 5 gms (4) 6 kg (5) None of these (SBI Associate Banks PO Exam. 14.02.1999) The wheat sold by a grocer contained 10% low quality wheat. What quantity of good quality wheat should be added to 150 kgs of wheat so that the percentage of low quality wheat becomes 5% ? (1) 150 kgs (2) 135 kgs (3) 50 kgs (4) 85 kgs (5) None of these (SBI Associate Banks PO Exam. 21.07.2002) A grocer purchased 2 kg. of rice at the rate of Rs. 15 per kg. and 3 kg. of rice at the rate of Rs. 13 per kg. At what price per kg 1 should he sell the mixture to earn 33 % 3 profit on the cost price ? (1) Rs. 28.00 (2) Rs. 20.00 (3) Rs. 18.40 (4) Rs. 17.40 (5) None of these (SBI PO Exam. 26.11.2006)

(1)

4.

5.

The wheat sold by a grocer contained 10% low quality wheat. What quantity of good quality wheat should be added to 150 kgs of wheat so that the percentage of low quality wheat becomes 5% ? (1) 150 kgs (2) 135 kgs (3) 50 kgs (4) 85 kgs (5) None of these

6.

10m m  10

(2)

2m m  10

(4)

5m m  10

m m  10 (New India Assurance AO Exam. 25.10.2009) When one litre of water is added to a mixture of acid and water, the new mixture contains 20% acid. When one litre of acid is added to the new mixture, then the

(3)

INSURANCE EXAMS 1.

(LIC Assistant Administrative Officer (AAO) Exam. 2006) A man purchased 35 kg of rice at the rate of Rs. 9.50 per kg and 30 kg at the rate of Rs. 10.50 per kg. He mixed the two. Approximately, at what price (in Rupees) per kg should he sell the mixture to make 35 per cent profit in the transaction ? (1) 12 (2) 12.50 (3) 13 (4) 13.50 (LIC Assistant Administrative Officer (AAO) Exam. 07.06.2009) To m litres of a m% solution of acid, x litres of water is mixed to yield (m- 10)% solution of acid. If m > 25, then x equals

1 resulting mixture contains 33 % acid. The 3 percentage of acid in the original mixture was (1) 20% (2) 22% (3) 24% (4) 25% (New India Insurance AAO Exam. 22.05.2011) Two barrels contain a mixture of ethanol and gasoline. The content of the ethanol is 60% in the first barrel and 30% in the second barrel. In what ratio must the mixtures from the first and the second barrel s be take n to form a mix tu re containing 50% ethanol ? (1) 1 : 2 (2) 2 : 1 (3) 2 : 3 (4) 3 : 2 (General Insurance Corporation AAO Exam. 11.12.2011) To x litres of an x% solution of acid, y litres of water is added to get (x -10)% solution of acid. If x > 20, then value of y is

(1)

x2 100

(2)

10 x x  10

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2

7.

10 x 10 x 2 (3) (4) x  10 x  10 (Oriental Insurance Company AAO Exam. 08.04.2012) A and B are two alloys of gold and copper prepared by mixing metals in the ratio 7 : 2 and 7 : 11 respectively. If equal quantities of the alloys are melted to form a third alloy C, the ratio of gold and copper in C will be : (1) 5 : 7 (2) 5 : 9 (3) 7 : 5 (4) 9 : 5 (Oriental Insurance Company AAO Exam. 08.04.2012)

8.

A container has 30 litres of water. If 3 litres of water is replaced by 3 litres of spirit and this operation is repeated twice, what will be the quantity of water in the new mixture ? (1) 24 litres (2) 23 litres (3) 24.3 litres (4) 23.3 litres (5) None of these (LIC Assistant Administrative Officer (AAO) Exam. 12.05.2013)

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3

SHORT ANSWERS

2.

2000 = 4000 gms 5  Required answer = 4000 gms - 1000 gms = 3000 gms = 3 kg. (1) Good quality content in 150 kgs of wheat = 90% of 150 = 135 kg. In new mixture, low quality wheat is 5% , so good quality wheat 95%  5% of the new mixture = 15 kg,

3.

15  100 = 300 kg 5  Good quality of wheat added = (300 - 150)kg. = 150 kg. (3) Mixture : 2 kg of rice at Rs. 15/kg + 3 kg of rice at Rs. 13/kg Total weight = 2 + 3 = 5 kg Total cost price = (2 × 15) + (3 × 13) = 30 + 39 = Rs. 69 Cost price per kg of the mixture

 x=

NATIONALISED BANKS & IBPS PO/MT/SO 1.

(5)

l. 3.

(1) (3)

SBI PO EXAMS 2. (1)

 New mixture =

INSURANCE EXAMS 1. 3. 5. 7.

2. (4) 4. (4) 6. (2) 8. (3)

(1) (1) (2) (3)

EXPLANATIONS NATIONALISED BANKS & IBPS PO/MT/SO 1.

(5) In 60 gm of mixture, Quantity of water =

75 = 45gm 100 Quantity of milk = 15 gm After mixing 15 gm of more water, Quantity of water in the new mixture = 45 + 15 = 60 gm  Quantity of water in 75 gm of mixture = 60 gm  100 gm of of mixture will contain

= 60 ×

=

1 Selling price to get 33 % profit 3

10  33

=

=

60 × 100 = 80% of water 75

=

SBI PO EXAMS 1.

(1) Amount of iron in 1 kg mixture = 20% of 1000 gms 20  1000 = gms 100 = 200 gms  Amount of sand in mixture = (1000 - 200) gms = 800 gms Now, let the total mixture is x kg in which iron is 20%  According to the question, 5% of x = 200 gm  5% of x = 200



5 x = 200 100

200 100 gms  x= 5

69 = Rs.13.80 5

100

1 3 × Rs 13.80

400 × Rs 13.80 3  100 4 × Rs 13.80 = Rs 18.40 3

INSURANCE EXAMS 1.

(1) Good quality content in 150 kgs of wheat = 90% of 150 = 135 kg. In new mixture, low quality wheat is 5% , so good quality wheat 95%  5% of the new mixture =15 kg 15  100 = 300 kg 5  Good quality of wheat added = (300 - 150) kg. = 150 kg. (4) CP of 65 kg of the mixture = Rs. (35 × 9.50 + 30 × 10.50) = Rs. (332.5 + 315) = Rs. 647.5

 New mixture =

2.

 647.5   Rate per kg of the mixture = Rs  65   

647.5 135 ×  Required rate = 65 100 LEARN MATHS FROM S.K. RAJU (9811549822, 9811649822)

4 3.

 Rs 13.50/kg (i) Tricky approach

Acid =

Acid =

m m2 m = 100 100

x2 100

Water = x 

m 2 100 New percentage = m  x 100  

x2 100

x2  100 × 100 = x - 10 x y

m2 = m - 10 m x  m2 - 10m + mx - 10x = m2  x (m - 10) = 10m



4.

x2  x  y = x - 10

10m x= m  10 (4) If there be 1 litre of acid in 4 litres of mixture, then in case I. Percentage of acid

 x2 = y2 - 10x - y (x - 10)

7.

1 × 100 = 20% 4 1 Case II, Percentage of acid

Gold =

=

2 9 In 1 kg of alloy B,

2 100 1 × 100 = = 33 % 6 3 3 Percentage of acid in original mixture  = 25% (2) By alligation, Mixture-1 Mixture -II

Ethanol-

3 5

7 18

Copper =

=

5.

10 x 10 x = x  10 x  10 because x > 20 (3) In 1 kg of alloy A,

y=

Ethanol-

3 10

7 11 Copper = 18 18  Ratio of gold and copper in alloy C

Gold =

7 7 2 11 + : + = 21 : 15 = 7 : 5 9 18 9 18 (3) 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

= 8.

n

2 1 1 3 = = 2 10 10 5  Required ratio

1 1 : =2:1 5 10 (2) In litres of solution,

= 6.

3 1 1 - = 5 2 10

y  = x 1   units x  Remaining water 3   = 30  1    30 

=

2

30  9  9 = 24.3 litres 100

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5 MODEL EXERCISES 1.

2.

3.

4.

5.

6.

7.

several litres of acid were drawn off a 54 L vessel full of acid and an equal amount of water added, Again, the same volume of the mixture was drawn off and replaced by water. As a result the vessel contained 24 L of pure acid. How much of the acid was drawn off initially ? (1) 12 L (2) 16 L (3) 18 L (4) 24 L (5) None of these 37.85% and 92% alcoholic solutions are mixed to get 35 L of an 89% alcoholic solution. How many litres of each solution are there in the new mixture ? (1) 10 of the first and 25 of the second (2) 20 of the first and 15 of the second (3) 15 of the first and 20 of the second (4) None of the above (5) None of these Three containers, A, B and C are having mixtures of milk and water in the ratio 1 : 5, 3 : 5 and 5 : 7 respectively. If the capacities of the containers are in the ratio 5 : 4 : 5, then find the ratio of the milk to the water, if the mixtures of all the three containers are mixed together. (1) 51 : 115 (2) 52 : 115 (3) 53 : 115 (4) 54 : 115 (5) None of these A chemist has 10 L of a solution that is 10% nitric acid by volume. He wants to dilute the solution to 40% strength by adding water. How many litres of water must be add ? (1) 15 (2) 20 (3) 18 (4) 25 (5) None of these 5 L of water is added to a certain quantity of pure milk costing Rs 3/L. If by selling the mixture at the same price as before, a profit of 20% is made, then what is the amount of pure milk in the mixture ? (1) 20 L (2) 30 L (3) 25 L (4) 35 L (5) None of these How many kg of sugar costing Rs 5.75 per kg should be mixed with 75 kg of cheaper sugar costing Rs 4.50 per kg so that the mixture is worth Rs 5.50 per kg ? (1) 350 kg (2) 300 kg (3) 250 kg (4) 325 kg (5) None of these The amount of water (in ml.) that should be added to reduce 9 mL lotion, containing

8.

9.

10.

50% alcohol, to a lotion containing 30% alcohol is : (1) 5 mL (2) 4 mL (3) 3 mL (4) 6 mL (5) None of these 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 per litre he may get a profit of 25% ? (1) 13 : 16 (2) 12 : 15 (3) 16 : 9 (4) 19 : 22 (5) None of these 50 g of an alloy of gold and silver contains 80% gold (by weight). The quantity of gold, that is to be mixed up with this alloy, so that it may contain 95% gold, is (1) 200 g (2) 50 g (3) 150 g (4) 10 g (5) None of these A bottle contains 3/4 of milk and the rest water. How much of the mixture must be taken away and replaced by an equal quantity of water so that the mixture has half milk and half water ? (1) 25%

11.

12.

13.

1 (2) 33 % 3 (4) 50%

(3) 45% (5) None of these A mixture of 45 L of spirit and water contains 20% of water in it. How much water must be added to it make the water 25% in the new mixture ? (1) 5 L (2) 3 L (3) 4 L (4) 6 L (5) None of these In a mixture of 60 L, the ratio of milk and water is 2 : 1. If this ratio is to be 1 : 2, then the quantity of water to be further added is (1) 30 L (2) 20 L (3) 40 L (4) 60 L (5) None of these One test tube contains some acid and another test tube contains an equal quantity of water. To prepare a solution, 20 g of the acid is poured into the second test tube. Then, two-thirds of the so formed solution is poured from the second tube into the first. If the fluid in the first test tube is four times that in the second, what quantity of water was taken initially ? (1) 80 g (2) 60 g (3) 40 g (4) None of these (5) All of these

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6 14.

15.

16.

One type of liquid contains 25% of milk, the other contains 30% of milk. A container is filled with 6 parts of the first liquid and 4 parts of the second liquid. The percentage of milk in the mixture is— (1) 27% (2) 31% (3) 29% (4) 33% (5) None of these A solution of sugar syrup has 15% sugar. Another solution has 5% sugar. How many litres of the second solution must be added to 20 L of the first solution to make a solution of 20% sugar. (1) 10 L (2) 5 L (3) 15 L (4) 20 L (5) None of these Mixture of milk and water has been kept in two separate containers. Ratio of milk to water in one of the containers is 5 : 1 and that in the other container is 7 : 2. In that ratio the mixture of these two containes should be added together, so that the quantity of milk in the new mixture may become 80% ? (1) 2 : 3 (2) 3 : 2 (3) 4 : 5 (4) 3 : 4 (5) None of these

17.

18.

Vijay purchased two different kinds of alcohol. In the first mixture the ratio of alcohol to water is 3 : 4 and in the second mixture it is 5 : 6. If he mixes the two given mixtures and makes a third mixture of 18 litres in which the ratio of alcohol to water is 4 : 5, the quantity of the first mixture (whose ratio is 3 : 4) that is required to make 18 litres of the third kind of mixture is (1) 6 (2) 7 (3) 8 (4) 9 (5) None of these Two casks of 48 litres and 42 litres are filled with mixtures of milk and water; the proportions in the two casks being respectively, 13 : 7 and 18 : 17. If the contents of the two casks be mixed, and 20 litres of water be added to the whole, what will be the proportion of milk and water in the resulting mixture ? (1) 5 : 12 (2) 7 : 13 (3) 12 : 13 (4) 8 : 15 (5) None of these

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7

SHORT ANSWERS l. 3. 5. 7. 9. 11. 13. 15. 17.

(3) (3) (3) (4) (3) (2) (4) (4) (2)

2. (4) 4. (1) 6. (2) 8. (3) 10. (2) 12. (4) 14. (1) 16. (1) 18. (3)

1.

(3) According to question, if a container contains y units of liquid and x units of liquid is taken out. If this operation is repeated n times. The final quantity of the luquid in the container is

Then, (10 + x) × 5.

EXPLANATIONS

 x 1 x  y 

n

 x = 15 L (3) Let the quantity of pure milk be x L. If 5 L of water is added to it, then cost of (5 + x)L = Rs (3x + 5)  Rs Profit = Rs 15 Given, 20% of 3x =15 3x = 15 5  3x = 75  x = 25 L  The amount of pure milk in the mixture was 25 L. (2) Using Alligation Method, Sugar I Sugar II 5.75 4.50



6.

x    24 = 54  1  54   

4 =1 100

2

2

x  24 4  =  1   = 54 9  54  2

x  2   1   = 54 3  

x 1 = 54 3  x = 18 L (4) Let x L of 37.85% of alcoholic solution and (35 - x)L of 92% alcoholic solution are required to get 35 L of 89% solution.  37.85% of x + 92% of (35 - x) = 89% of 35  x = 1.94  1.94 L of 1st solution and 33.06 L of Ilnd solution are there in the new mixture. (3) By question ratio of milk and water

5.50 - 4.50 5.75 - 5.50 1.00 0.25 i.e., 4 : 1 Hence, the required quantity of Sugar I



2.

3.

4.

7.

8.

9   50  30 

9  20 = = 6 mL 30 30 (3) Selling price of mixture = Rs 20 Cost price of mixture

=

100 × 20 = Rs16 125 By the rule of alligation,

=

3 5 5 7 1  5  =   5   4   5 :   5   4   5  8 12   6 8 12  6 = 106 : 230 = 53 : 115 (1) According to question, quantity of nitric acid 1 =1L 10 Water = 10 - 1 = 9 L Let x L of water be added.

75 × 4 = 300 kg 1 (4) Required quantity of water

=

= 10 ×

9.

So, required ratio = 16 : 9 (3) According to question, quantity of gold 80 = 40 g 100 Let x g of gold is mixed with this alloy.

in alloy = 50 ×

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8 Quantity of water in mixture

40  x × 100 = 95 50  x

1 × 60 = 20 L 3 Let quantity of water to be further added be x.

=

40  x 95 = 50  x 100  4000 + 100x = 4750 + 95x



10.

750 = 150 g x= 5 (2) According to question, Ratio of milk and water = 3 : 1

13.

1 Water = × 100 = 25 4 Let x L of mixture is taken away, then

3x   quantity of milk left =  3   4  

First test tube = (x - 20) + (x + 20) ×

 x and water left =  1   + x  4

Second test tube = (x + 20) ×

Given, 3 -

3- 1=

11.

2=

6x 4

x=

4 3

(x - 20) +

3x x - +x 4 4

14.

20 × 45 = 9 L 100 Let x L of water must be added to make the water 25% in the

2 1 (x + 20) = 4× (x + 20) 3 3

2 (x + 20) 3  3x - 60 = 2x - 40  x = 100 g (1) According to question, milk in mixture

6 ×

9 x 25 = 45  x 100  36 + 4x = 45 + x  3x = 9 x = 3L (4) By question quantity of milk in mixture

2 × 60 = 40 L 3

25 30 270 + 4 × = 100 100 100

270  10 = 27% 100 (4) Let x L of second solution must be added.

For 10 parts = 15.

15  20  5  x = 10 20  x  300 + 5x = 200 + 10x

Then,

9 x × 100 = 25 45  x



=

1 3

 x - 20 =

4 1 Required percentage = × 100 = 33 % 3 4 3 (2) According to the question, Quantity of water in the mixture



2 3

A/q,

3x x =1- +x 4 4

=

12.

40 1 = 20  x 2  20 + x = 80  x = 60 L (4) Initially, let x g of water and Acid was taken. Initially 1st process First test tube = (x - 20) g Second test tube = (x + 20) g 2nd process

Then,

100 = 20L 5 (1) Percentage of milk in first mixture

x =

16.

5 250 ×100 = % 6 3 Percentage of milk in second mixture

=

7 700 = × 100 = % 9 9 Using Alligation method,

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9 18.

(3) In mixture A of 48 litres, Milk=

20 10 : =2:3 9 3 (2) Using Alligation method,

So, required ratio = 17.

13 156 × 48 = litres 20 5

7 84 × 48 = litres 20 5 In mixture B of 42 litres,

Water =

Milk =  

18 108 × 42 = litres 35 5

17 102 × 42 = litres 35 5 In the final mixture,

Water =

5 4 1 - = 11 9 99

4 3 1 - = 9 7 63

1 1 1 1  Ratio = 99 : 63 = : 7 11 = 7:11  Required quantity

=

7 × 18 = 7 litres 18

Milk =

156 108 264 + = litres 5 5 5

84 102 186 + = litres 5 5 5  Required ratio

Water =

264 186 : + 20 5 5 264 : 286 = 12 : 13

=

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