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.
2:5
 
3:1
 
6:7
 
3:2

Discuss it
Question 1 Explanation:
Cost Price of 1kg of Type 1 sugar = 6200 p.
Cost Price of 1kg of Type 2 sugar = 7200 p.
Mean Price of 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 = (72006450):(64506200) = 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.
3:2  
1:2  
5:2  
2:1 
Discuss it
Question 2 Explanation:
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 = (128):(80) = 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?
30 Lt  
29.16 Lt  
28.70 Lt  
27.60 Lt 
Discuss it
Question 3 Explanation:
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?
165.4  
170  
169  
175.5 
Discuss it
Question 4 Explanation:
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
x153 153130.5
x153/22.5=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?
8  
6  
10  
9 
Discuss it
Question 5 Explanation:
Let x litrs from 1^{st} drum and 12x 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(12x)
soda from 2^{nd} drum = .5(12x)
total sikanji = .25x+6
total soda = .25x+.5(12x) = 6.25x
ratio = (.25x+6)/(6.25x) = 5/3
.75x+18 = 301.25x
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 the another contains 50% oil. How much vanaspati gee (approx) should one mix from each of the drum so as to get twelve litres of vanaspati gee such that the ratio of gee to oil is 5 : 2?
10  
15  
10.28  
9.57 
Discuss it
Question 6 Explanation:
Let x litrs from 1^{st} drum and 12x litrs from 2^{nd} drum are mixed
gee from 1^{st} drum = .75x
oil from 1^{st} drum = .25x
gee from 2^{nd} drum = .5(12x)
oil from 2^{nd} drum = .5(12x)
total gee = .25x+6
total oil = .25x+.5(12x) = 6.25x
ratio => (.25x+6)/(6.25x) =5/2
0.50x+12 = 301.250x
1.75x= 18
x = 10.28 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 ?
5:21  
21:5  
23:6  
6:23 
Discuss it
Question 7 Explanation:
Say cp of 1 liter whisky is = 1 rs
quantity of whisky in S1 = 2/7
CP of whisky in S1 = 2/7
quantity of whisky in S2 = 6/13
CP of whisky in S2 = 6/13
quantity of whisky in S3 = 7/13+2/7 = 21/91
CP of whisky in S3 = 21/91
By rule of allegation:
S1 S2
CP of 1 ltr whisky CP of 1 ltr whisky
2/7 6/13
21/91
6/1321/91 2/721/91
=21/91 =5/91
whisky from s1: whisky from S2 = 21:5
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 :water left in pot is 8 : 19. How much wine was there in the pot originally?
32  
26  
28  
24 
Discuss it
Question 8 Explanation:
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?
3:2  
5:4  
4:5  
2:3 
Discuss it
Question 9 Explanation:
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,
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.
Concentration of orange pulp in 1st vessel 
Concentration of orange pulp in 2nd vessel 

40% 
19% 

26% 

2619=7 
4026=14 
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?
60  
71  
63  
65 
Discuss it
Question 10 Explanation:
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,
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
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 
Question 11 
In what ratio should a kind of sugar at 8.70 Rs/kg be mixed with another kind of sugar at 9.70 Rs/kg so that the mixture be worth 9 Rs/kg?
3:7  
4:7  
7:4  
7:3 
Discuss it
Question 11 Explanation:
By rule of allegation,
= 0.7 : 0.3 = 7 : 3
Cost of 1 kg sugar of 1st kind 
Cost of 1 kg sugar of 2nd kind 
8.7 
9.70 
9 

9.79 = .7 
9  8.7 = .3 
Question 12 
In what ratio must rice of one kind worth Rs. 50/kg be mixed with another kind of rice of worth Rs. 55/kg such that by selling the mixture at Rs. 57.20/kg, there can be a gain 10%?
3:4  
4:6  
1:3  
3:2 
Discuss it
Question 12 Explanation:
SP of 1 kg mixture = Rs. 57.20
Profit = 10%
CP of 1 kg mixture =100SP/(100+Profit%) =100*57.20/(100+10) =5720/110
=Rs. 52
By rule of allegation
Hence required ratio = 3 : 2
CP of 1 kg rice of 1st kind 
CP of 1 kg rice of 2nd kind 

50 
55 

52 

55  52 = 3 
52  50 = 2 
Question 13 
A drum filled with a mixture of two liquids l1 and l2 in the ratio 5:7. When 9 liters of mixture is taken out and the replaced by l1. Now the ratio of l1 and l2 is 9:7. How many liters of the liquid l1 were there in the drum initially?
15  
20  
18  
25 
Discuss it
Question 13 Explanation:
Let the initial quantity of l1 in the container be 5x.
Let the initial quantity of l2 in the container be 7x.
Now, 9 liters of mixture is drawn off from the container.
Quantity of l1 in 9 liters of the mixture drawn off
=9*5/12=15/4
Quantity of l2 in 9 liters of the mixture drawn off
=9*7/12=21/4
Hence,
Quantity of l1 remaining in the mixture after 9 liters is drawn off
=5x−15/4
Quantity of l2 remaining in the mixture after 9 liters is drawn off
=7x−21/4
Since the container is filled with l1 after 9 liters of mixture is drawn off, quantity of l1 in the mixture
=5x15/4+9=5x+21/4.
Given that the ratio of l1 and l2 becomes 9:7
⇒5x+21/4:7x21/4=9:7
⇒20x+21:28x21=9:7
⇒7(20x+21)=9(28x21)
⇒140x+147=252x189
⇒112x=336
⇒x=3.
Therefore, liters of l1 present in the container initially
=5x=(5*3)=15.
Question 14 
In what ratio a vendor should mix rice at Rs.60 per kg with rice at Rs. 68 per kg so that the final rice mixture must be of worth Rs. 63 per kg?
1:3  
3:1  
2:3  
3:2 
Discuss it
Question 14 Explanation:
By rule of alligation,
Required Ratio = 7.5 : 2.5 = 3 : 1
Cost of 1 kg of 1^{st} kind rice 
Cost of 1 kg of 2^{nd} kind rice 

62 
72 

64.5 

7264.5=7.5 
64.562=2.5 
Question 15 
A vessel is filled with a solution of, 3 parts soda and 5 parts rum. How much of the solution must be taken out and replaced with soda so that the solution contains equal amount of sod and rum?
8/5  
2/5  
1/5  
5/8 
Discuss it
Question 15 Explanation:
Let the total soution = 8 litre.
soda in the solution = 3 litre,
rum in the solution = 5 litre.
Say x Lt of the solution is is taken out and replaced with soda.
soda in the new solution
=3−(3x/8)+x
Quantity of rum in the new mixture
=5−(5x/8)
soda : rum = 1:1
⇒3−(3x/8)+x = 5−(5x/8)
⇒x=8/5
if the quantity of the solution is 8 litre, 8/5 litre of the solution needs to be taken out and replaced with soda so that the solution contains equal amount of soda and rum.
=>1/5^{th} of the solution needs to be taken out and replaced with soda so that the solution contains equal amount of soda and rum.
There are 15 questions to complete.