__Question No. 01__**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?**

__Solution:__
Let the C.P. of milk be Rs. 1 per litre

Milk in 1 litre mixture of A = 4/7 litre; 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 = Rs. 4/7

C.P. of
1 litre mixture in B = Rs. 2/5

Mean price = Rs. 1/2

By the rule of alligation, we have:

Alligation and Mixture Solution-01 |

Therefore, required ratio = (1/10): (1/14) = 7: 5

__Question No. 02__**In what ratio must water be mixed with milk to gain 20 % by selling the mixture at cost price?**

__Solution:__
Let C.P. of milk be Rs. 1 per litre.

Then, S.P. of 1 litre of mixture = Rs. 1.

Gain obtained = 20%.

Therefore, C.P. of 1 litre of mixture = Rs. [(100/120)
× 1] = 5/6

By the rule of alligation, we have:

Alligation and Mixture Solution-02 |

Therefore, Ratio of water and milk = (1/6): (5/6) = 1:
5.

__Question No. 03__**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:**

__ution:__Alligation and Mixture Solution-03 |

(Cheaper quantity): (Dearer quantity) =
(d - m): (m - c).

=> (1080 - 1000): (1000 - 930) => 80: 70 =>8: 7

Therefore, required ratio =8: 7.

__Question No. 04__**A vessel is filled with liquid, 3 parts of which are water and 5 parts acid. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half acid?**

__Solution:__
Suppose the vessel initially contains 8
litres of liquid.

Let, ‘x’ litres of this liquid be
replaced with water.

Quantity of water in new mixture = [3 -
(3

*x*/8) +*x*] litres
Quantity of acid in new mixture = [5 -
(5

*x*/8)] litres
Therefore, [3 - (3

*x*/8) +*x*] = [5 - (5*x*/8)]
=> 5

*x*+ 24 = 40 - 5*x*=> 10*x*= 16 =>*x*= 8/5
So, parts of mixture replaced = (8/5) ×
(1/8) = 1/5.

__Question No. 05__**Coffee wroth 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:**

__Solution:__
Since 1st and 2nd variety is mixed in
equal proportions.

So, their average price = Rs. [126 +
135)/2] = Rs. 130.50

So, the mixture is formed by mixing two verities,
one at Rs. 135.50 per kg and the other at say, Rs.

*x*per kg in the ratio of 2 : 2, i.e. the price of 3rd kind is Rs.*x*.
By the rule of alligation, we have:

Alligation and Mixture Solution-05 |

Therefore, (

*x*- 153) = 22.50 =>*x*= 175.50
Hence, the price of 3rd variety per kg is
Rs. 175.50

__Question No. 06__**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?**

__Solution:__
By the rule of alligation, we have:

Therefore, the required rate = 3.50: 1.50
= 7: 3

__Question No. 07__
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:

__Solution:__
By the rule of alligation, we have:

Alligation and Mixture Solution-07 |

So, the ratio of 1st and 2nd parts = 4: 6
= 2: 3

Therefore, the quantity of 2nd kind =
(3/5) × 1000 = 600 kg.

__Question No. 08__**A dishonest 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:**

__Solution:__
Let, C.P of 1 litre milk be Rs. 1

Then, S.P of 1 litre of mixture = Rs. 1,
Gain = 25%

Therefore, C.P of 1 litre mixture = Rs.
(100/125) × 1 = 4/5

By the rule of alligation, we have:

Alligation and Mixture Solution-08 |

Therefore, ratio of milk to water = (4/5):
(1/5)

Hence, percentage of water in the mixture
= [(1/5) × 100] % = 20%

__Question No. 09__**A jar full of water contains 40% sugar dissolved in it. A part of the water is replaced by another containing 19% of sugar and now the percentage of sugar dissolved was found to be 26%. The quantity of sugar replaced is:**

__Solution:__
By the rule of alligation, we have:

Alligation and Mixture Solution-09 |

So, the ratio of 1st and 2nd quantities =
7: 14 = 1: 2

Therefore, required quantity replaced =
2/3

__Question No. 10__**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:**

__Solution:__
Let the price of the mixed variety be Rs.

*x*per kg.
By the rule of alligation, we have:

Alligation and Mixture Solution-10 |

Therefore, (20 -

*x*)/ (*x*- 15) = 2/3
=> (60 - 3

*x*) = (2*x*- 30)
=> 5

*x*= 90
=>

*x*= 18
Therefore, the price of the mixed variety
is Rs. 18 per kg.

**Alligation and Mixtures:**

**Formula: Alligation and Mixtures Formulas**

**Solved Examples: Solved Examples: Set 01 Solved Examples: Set 02**

**Practice Test: Practice Test: 01**

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