Mixtures and Alligation

Question 1

There are two types of sugar. One is priced at Rs 62 per kg and the other is priced at Rs 72 per kg. If the two types are mixed together, the price of new mixture will be Rs 64.50 per kg. Find the ratio of the two types of sugar in this new mixture.

• A. 2:5
• B. 3:1
• C. 6:7
• D. 3:2

Answer: (B)

Cost Price of 1kg of Type 1 sugar = 6200 p. Cost Price of 1kg of Type 2 sugar = 7200 p. Mean Price of 1 kg of mixture = 6450 p. According to the Rule of Alligation, (Quantity of Cheaper):(Quantity of Dearer) = (CP of dearer - Mean Price):(Mean Price - CP of cheaper) Therefore, the required ratio = (7200-6450):(6450-6200) = 750:250 = 3:1.

Question 2

A certain quantity of water is mixed with milk priced at Rs 12 per litre. The price of mixture is Rs 8 per litre. Find out the ratio of water and milk in the new mixture.

• A. 3:2
• B. 1:2
• C. 5:2
• D. 2:1

Answer: (B)

Cost Price of 1 litre of water = Rs 0. Cost Price of 1 litre of milk = Rs 12. Mean Price of Mixture = Rs 8. According to the Rule of Alligation, (Quantity of Cheaper):(Quantity of Dearer) = (CP of dearer - Mean Price):(Mean Price - CP of cheaper) Therefore, Water:Milk = (12-8):(8-0) = 4:8 = 1:2.

Question 3

A drum contains forty liters of whisky. Four liters of whisky is taken out and replaced by soda. This process is carried out twice further. How much whisky is now contained by the container?

• A. 30 Lt
• B. 29.16 Lt
• C. 28.70 Lt
• D. 27.60 Lt

Answer: (B)

Hint: Suppose a solution contains x units of a liquid from which y units are taken out and replaced by water. After n repeated operations, quantity of pure liquid remaining in solution  =x(1−y/x)ⁿ=x(1−y/x)ⁿ units. So, Whisky in the drum now =40(1−4/40)³=40(1−1/10)³ =29.16

Question 4

Rice of rate Rs. 126 per kg and Rs. 135 per kg and 3^rd variety in the ratio 1 : 1 : 2. If the final mixture is worth Rs. 153 per kg, what is the rate of the third variety per kg?

• A. 165.4
• B. 170
• C. 169
• D. 175.5

Answer: (D)

Rice worth Rs. 126 per kg and Rs. 135 per kg are mixed in the ratio 1 : 1 So their average price =(126+135)/2=130.5 Now there are two mixtures one of rate 130.5 /kg and another is of rate say x /kg 130.5               x 153 x-153              153-130.5   x-153/22.5=1/1 x = 175.50

Question 5

A sikanji vendor has two drums of sikanji. The first contains 75% of sikanji. The second contains 50% sikanji. How much sikanji should he mix from each of the drum so as to get twelve litres of sikanji such that the ratio of sikanji to soda is 5 : 3?

• A. 8
• B. 6
• C. 10
• D. 9

Answer: (B)

Let x litrs from 1^st drum and 12-x litrs from 2^nd drum are mixed sikanji from 1^st drum = .75x soda from 1^st drum = .25x sikanji from 2^nd drum = .5(12-x) soda from 2^nd drum = .5(12-x) total sikanji = .25x+6 total soda = .25x+.5(12-x) = 6-.25x ratio = (.25x+6)/(6-.25x) = 5/3 .75x+18 = 30-1.25x 2x =12 x=6 sikanji and 6 soda

Question 6

There are two drums of vanaspati gee. One of them contains 25% of oil (and rest 75% gee) and the another contains 50% oil (and rest 50% gee). How much vanaspati gee (approx) should one mix from each of the drum so as to get 14 litres of vanaspati gee such that the ratio of gee to oil is 5 : 2?

• A. 6, 8
• B. 7, 7
• C. 12, 2
• D. 10, 4

Answer: (C)

Quantity of Ghee in 1st Drum = 75% Quantity of Ghee in 2nd Drum = 50% Total quantity of Ghee required in the final mixture = 14 liters Ratio of Ghee to Oil in the final mixture = 5 : 2 ( i-e 500/7 % Ghee ) By Alligation Rule : So, we have to mix Ghee from 1st and 2nd Drum in the ratio of 6 : 1 Since total quantity of Ghee in the final mixture is 14 liters. So, Ghee to be mixed from 1st Drum = 6/7*14 = 12 liters. And Ghee to be mixed from 2nd Drum = 1/7*14 = 2 liters.

Question 7

Two solutions S1 and S2 contain whisky and soda in the ratio 2 : 5 and 6 : 7 respectively. In what ratio these solutions be mixed to get a new solution S3,  containing whisky and soda in the ratio 5 : 8 ?

• A. 7:9
• B. 21:5
• C. 23:6
• D. 6:23

Answer: (A)

Let the amount taken from S1 be 7x
And amount taken from S2 be 13y

(2x + 6y)/(5x + 7y) = 5/8
16x + 48y = 25x + 35y
9x = 13y
x/y = 13/9

Actual ratios of amounts 
               = 7x/13y
               = (7/13) * (13/9)
               = 7/9

Please refer Mixture and Alligation for direct formula.

Question 8

8 liters of wine is replaced by water from a pot full of wine and repeated this two more times. The ratio of the wine:total left in pot is 8 : 27. 

How much wine was there in the pot originally?

• A. 32
• B. 26
• C. 28
• D. 24

Answer: (D)

Let initial quantity of wine =x litre 

After a total of 1+2=3 operations, 
quantity of wine 

⇒x(1−y/x)n 
⇒ x(1−8/x)3 
⇒x(1−8/x)3 x = 8/27 
⇒(1−8/x)3 = (2/3)3 
⇒(1−8/x)=2/3 
⇒x=24

Question 9

A vessel full of orange juice contains 40% orange pulp. A part of juice is replaced by another juice containing 19% orange pulp and now the percentage of orange pulp is found to be 26%. What quantity of juice is replaced?

• A. 3:2
• B. 5:4
• C. 4:5
• D. 2:3

Answer: (D)

Concentration of orange pulp in 1^st vessel = 40% Concentration of orange pulp in 2^nd vessel = 19% After the mixing, Concentration of orange pulp in the mixture = 26% By rule of alligation,

Concentration of orange pulp in 1st vessel	Concentration of orange pulp in 2nd vessel
40%	19%
26%
26-19=7	40-26=14

Hence ratio of 1^st and 2^nd quantities = 7 : 14 = 1 : 2 i.e., 2/(1+2)=2/3 part of the juice is replaced.

Question 10

How many kg of rice, of cost 9 Rs/kg must be mixed with 27 kg of rice of cost 7 Rs/kg get a gain of 10 % by selling the mixture at 9.24 Rs/kg?

• A. 60
• B. 71
• C. 63
• D. 65

Answer: (C)

Selling Price(SP) of 1 kg mixture= Rs. 9.24 Profit = 10% Cost Price(CP) of 1 kg mixture = 100SP/(100+Profit%) =100*9.24/(100+10) =924/110=8.4 By rule of alligation,

CP of 1 kg rice of 1st kind	CP of 1 kg rice of 2nd kind
Rs. 9	Rs. 7
Rs.8.4
8.4 - 7 = 1.4	9 - 8.4 = 0.6

Ratio = 1.4 : 0.6 = 14 : 6 = 7 : 3 Suppose x kg of kind1 rice is mixed with 27 kg of kind2 rice. then x : 27 = 7 : 3 ⇒3x=27×7 ⇒x=9×7=63