Alligation and Mixtures Solved Examples - Set 02

Question No. 01
A cup contains milk and water in the ratio 3:1. How much mixture should be taken out and water added to make the ratio 1:1?
(A) 1/3 part of mixture
(B) 1/2 part of mixture
(C) 1/4 part of mixture
(D) 1/5 part of mixture
Answer: Option A
Explanation:
Let y part of mixture is replaced by water
Therefore, (4x - y)/4x = 2x/3x
=> 12x² - 3xy = 8x²
=> y = 4x/3
I.e. 1/3 part of mixture.

Question No. 02
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 Rs. 1
Milk in 1 liter mix in 1st can = 3/4 liter, C.P of 1 liter mix in 1st can Rs. 3/4
Milk in 1 liter mix in 2nd can = 1/2 liter, C.P of 1 liter mix in 2nd can Rs. 1/2
Milk in 1 liter of final mix = 5/8 liter, Mean price = Rs. 5/8
By the rule of alligation, we have:
Alligation and Mixtures question number 02
∴ Ratio of two mixtures = (1/8) : (1/8) = 1 : 1
So, quantity of mixture taken from each can = (½ × 12) = 6 liters

Question No. 03
In 2 gallons mixture of spirit and water, the percentage of water is 12. In another 3 gallons mixture of spirit and water, the percentage of water is 5. These two mixtures are poured in a third pot and ½ gallon more water is added to it. Find the percentage of water in this new mixture.
(A) 17 1/11 %
(B) 17 3/11 %
(C) 16 2/11 %
(D) 16 13/27 %
Answer: Option C
Explanation:
W1 = 12% of 2 = 24/100 = 0.24
∴ W2 = 5% of 3 = 0.15
    W3 = 0.5
∴ W1 + W2 + W3 = 0.24 + 0.15 + 0.5 = 0.89
∴ New W% = (0.89/5.5) × 100 = 16 2/11 %

Question No. 04
How many kg 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. [(100/110) × 9.24] = Rs. 8.40
By the rule of alligation, we have:
Alligation and Mixtures question number 04
∴ Ratio of quantities of 1st and 2nd kind = 14 : 6 = 7 : 3
Let, x kg of sugar of 1st kind be mixed with 27 kg of 2nd kind
Then, 7 : 3 = x : 27
=> x = (7 × 27)/3 = 63 kg.

Question No. 05
Three identical vessels contain the mixture of spirit and water. The ratio of spirit and water in each glass is 2:3, 3:4 and 4:5 respectively. The mixtures of all the three vessels are poured into a big pot. The ratio of spirit and water in the new mixture is
(A) 401/544
(B) 27/37
(C) 19/37
(D) 13/37
Answer: Option A
Explanation:
Net spirit = (2/5) + (3/7) + (4/9) = 401/315
∴ Net water = 3 - (401/315) = 544/325
Hence required ratio = 401/544

Question No. 06
A container contains 40 liters of milk. From this container 4 liters of milk was taken out and replaced with water. This process was repeated further 2 times. How much milk is now contained by the container?
(A) 26.34 liters
(B) 27.36 liters
(C) 28 liters
(D) 29.16 liters
Answer: Option D
Explanation:
Amount of milk left after 3 operations = [40 × {1 - (4/40)}3] liters
                                                                     = [40 × (9/10) × (9/10) × (9/10)] = 29.16 liters

Question No. 07
In what ratio must water be mixed with milk to gain 16⅔ % 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 liter milk be Rs.1
S.P of 1 liter of mixture = Rs. 1, gain = (50/3) %
∴ C.P of 1 liter of mixture = [100 × (3/350) × 1] = 6/7
By the rule of alligation, we have:
Alligation and Mixtures question number 07
∴ Ratio of water and milk = (1/7) : (6/7) = 1 : 6

Question No. 08
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, we have:
Alligation and Mixtures question number 08
∴ Required ratio = 60 : 90 = 2 : 3

Question No. 09
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 mixture = 68.20, Gain = 10%
C.P of 1 kg of the mixture = Rs [(100/110) × 68.20] = Rs. 62
By the rule of alligation, we have:
Alligation and Mixtures question number 09
∴ Required ratio = 3 : 2

Question No. 10
A can contains a mixture of two liquids ‘A’ and ‘B’ in the ratio 7 : 5. When 9 liters of mixture are drawn off and the can is filled with ‘B’, the ratio of ‘A’ and ‘B’ becomes 7 : 9. how many liters 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 - (712 × 9)] liters = [7x - (21/4)] liters
Quantity of B in the mixture left = [5x - (512 × 9)] liters = [5x - (15/4)] liters
∴ [7x - (21/4)]/ [5x - (15/4)] = 7/9
=> (28x - 21)/(20x + 21) = 7/9
=> 252x - 189 = 140x + 147
=> 112x = 336
=> x = 3
So, the can contained 21 liters of A.

Question No. 11
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 16:65. How much wine did the cask hold originally?
(A) 18 liters
(B) 24 liters
(C) 32 liters
(D) 42 liters
Answer: Option B
Explanation:
Let the quantity of the wine in the cask originally be x liters
Then, quantity of wine left in the cask after 4 operations = [x {1 - (8/x)}4] liters
[x {1 - (8/x)}4]/x = 16/81
=> {1 - (8/x)}4 = (2/3)4
=> [(x - 8)/x] = 2/3
=> 3x - 24 = 2x
=> x = 24

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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