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Mixture and Alligation Questions

1. The average marks of 10 students of a class is 72 and the average marks of another 12 students of the same class is 75, find the overall average of the class.  



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Correct Ans:73.64
Explanation:
Given
Average marks of 10 students = 72
Average marks = Total marks / No of students
Total marks of 10 students = ( Average marks of 10 students ) * ( No of students )
= 10 x 72 = 720
Given , Average marks of another 12 students = 75
Total marks of another 12 students = 12 x 75 = 900
Total marks of the class = 900 + 720 = 1620
Total number of students in the class = 10 + 12 = 22
Overall average of the class = (Total marks of the class) / (Total number of students in the class)
= 1620 / 22
= 73.64


2. A 20 liters mixture of milk and water comprising 60% pure milk is mixed with "x" liters of pure milk. The new mixture comprises 80% milk. What is the value of "x"? 



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Correct Ans:20
Explanation:
Original mixture comprises 20 liters of milk and water.
Out of the 20 liters, 60% is pure milk.
=> ( 60 / 100 ) x 20 = pure milk
=> 12 liters = pure milk

In 20 liters mixture remaining 8 liters = water

When "x" liters of pure milk is added to 20 liters of mixture
New mixture = ( 20 + x ) liters
Milk in new mixture = ( 12 + x ) liters
Given milk in new mixture = 80% of ( 20 + x )
=> 12 + x = ( 80 / 100 ) * ( 20 + x )
=> 12 + x = ( 4 / 5 ) * ( 20 + x )
=> 5 ( 12 + x ) = 4 ( 20 + x )
=> 60 + 5 x = 80 + 4 x
=> 5 x - 4 x = 80 - 60
=> x = 20 liters


3. A zookeeper counted the heads of the animals in a zoo and found it to be 80. When he counted the legs of the animals he found it to be 260. If the zoo had either pigeons or horses, how many horses were there in the zoo? 



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Correct Ans:50
Explanation:
Solution is:
Given , Zoo had either pigeons or horses
Heads of the animals in Zoo = 80
=> pigeons + horses = 80
Let p = number of pigeons
h = number of horses
=> p = 80 - h
Given , Legs of the animals = 260
Each pigeon has 2 legs and each horse has 4 legs
=> 2p + 4h = 260
Substitute p = 80 - h
=> 2 ( 80 - h ) + 4h = 260
=> 160 - 2h + 4h = 260
=> 2h = 260 - 160
=> 2h = 100
h = 50
So , number of horses in the Zoo = 50


4. How many liters of a 12 litre mixture containing milk and water in the ratio of 2 : 3 be replaced with pure milk so that the resultant mixture contains milk and water in equal proportion? 



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Correct Ans:2 liters
Explanation:
The mixture contains 40% milk and 60% water in it.
Given , Mixture ( milk + water ) = 12 liters
milk ( m ) / water ( w ) = 2 / 3
=> m / w = 2x liters / 3x liters
so, 2x + 3x = 12 liters
5x = 12
x = 12 / 5
x = 2.4
milk , m = 2x liters
= 2 * 2.4 liters
m = 4.8 liters
water , w = 3x liters
= 3 * 2.4 liters
w = 7.2 liters
That is 4.8 liters of milk and 7.2 liters of water.
Now we are replacing the mixture with pure milk so that the amount of milk and water in the mixture is 50% and 50%.
That is wewill end up with 6 litres of milk and 6 litres of water.

=> Water gets reduced by 1.2 litres.
=> To remove 1.2 litres of water from the original mixture containing 60% water, we need to remove (1.2 / 0.6) litres of the mixture = 2 litres.


5. 48 liters of a mixture has 75% alcohol. How much water must be added to it to get 60% alcohol concentration ? (in liters)



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Correct Ans:12
Explanation:
Solution is
Given In 48 litres => Alcohol = 75 %
=> Alcohol (in litres) = ( 75 / 100 ) x 48
= 36000 / 100
= 36 litres
Let x litres of water is added to this mixture to get 60 % alcohol concentration
So Total quantity of new mixture = ( 48 + x ) litres



6. 40 liters of mixture has 65% milk. How much milk should be added to the mixture to make it 85% pure?(in liters)



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Correct Ans:53
Explanation:
Original mixture comprises 40 liters of milk and water.
Out of the 40 liters, 65% is pure milk.
=> ( 65 / 100 ) x 40 = pure milk
=>26 liters = pure milk

In 40 liters mixture, remaining 14 liters = water

When "x" liters of pure milk is added to 40 liters of mixture, to make new mixture 85% pure milk.
New mixture = ( 40 + x ) liters
Milk in new mixture = (26 + x) liters
Given milk in new mixture = 85% of (40 + x)
=> 26 + x = (85/ 100) * (40 + x)
=> 100 * (26 + x) = 85 *(40 + x)
=> 2600 + 100x = 3400 + 85x
=> 100x - 85x = 3400 - 2600
=> 15x = 800
=> x = 800/15
=> x = 53.33
=> x= 53 (approximately)

Hence we need to "add 53 liters of pure milk" to the 40 liters of mixtureto make it 85 % pure.


7. If the average income of a family of ‘x’ members is Rs. 25000 while average income of another family of 6 members is Rs. 32000, then find the value of ‘x’ if the average income of both families is Rs. 28000. 



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Correct Ans:8
Explanation:
Given
Avg income of family one = 25
Avg income of family two = 32
Avg income of both the families = 28

32 - 28 = 4 ; 28 - 25 = 3
Required ratio is 4 : 3
Ratio of the number members in family 1 and 2 is 4 : 3
=>number of members in family 1 = 4x
and, number of members in family 2 = 3x

Given, In 2nd family there are 6 members => 3x = 6
=>x = 2

In family 1, no. of people = 4x= 4 x 2 = 8
Therefore, the value of x (ie., number of members in 1st Family) = 8


8. Two liquids A and B are mixed together in the ratio 2 : 3. The average cost of liquid B is Rs 30 per liter and the average cost of the mixture is Rs. 25 per liter, then find the average cost of liquid A per liter. 



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Correct Ans:17.5
Explanation:
Solution is
Given liquids A and B = 2 : 3
Average cost of mixture = Rs. 25 / liter
Average cost of liquid B = Rs. 30 / liter
Let Average cost of liquid A = x per liter

According to the alligation equation,
( 30 - 25 ) / ( 25 - x ) = 2 / 3
=> 5 / ( 25 - x ) = 2 / 3
=> 5 x 3 / 2 = ( 25 - x )
=> 15 / 2 = 25 - x
=> 7.5 = 25 - x
=> x = 25 - 7.5
x = 17.5
So, Average cost of liquid A =17.5 per liter


9. How many kgs of Basmati rice costing Rs.42/kg should a shopkeeper mix with 25 kgs of ordinary rice costing Rs.24 per kg so that he makes a profit of 25% on selling the mixture at Rs.40/kg?  



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Correct Ans:20
Explanation:
Given, Cost price of Bamati rice = Rs. 42 per kg.
Cost price of Ordinary rice = Rs. 24 per kg.

Selling price of Mixture = Rs. 40 per kg = 25% profit
So, Cost price of mixture = {100 / (100 + profit%)} * Selling price of Mixture
= {100 / (100 + 25)} * 40
= (100/125) * 40
= 32 Rs.
=>Cost price of mixture = 32 Rs.

Given, Quantity of Ordinary rice = 25 kg
Let, the required Quantity of Basmati Rice = "x"



According to alligation equation:
=> x/25 = (32 - 24) / (42 - 32)
=> x/25 = 8/10
=> x = (8/10) * 25
=> x = 20
Therfore,required Quantity of Basmati Rice = "x" = 20 kg


10. Two liquids A and B are mixed together in the ratio 1 : 3. The average cost of liquid B is Rs 30 per liter and the average cost of the mixture is Rs. 25 per liter, then find the average cost of liquid A per liter. 



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Correct Ans:10
Explanation:
Given liquids A and B = 1 : 3
Average cost of mixture = Rs. 25 / liter
Average cost of liquid B = Rs. 30 / liter
Let Average cost of liquid A = x per liter


According to the alligation equation,
( 30 - 25 ) / ( 25 - x ) = 1 / 3
=> 5 / ( 25 - x ) = 1 / 3
=> 5 x 3 = ( 25 - x )
=> 15 = 25 - x
=> x = 25 - 15
=> x = 10
So, Average cost of liquid A =10 per liter


11. Two liquids A and B are mixed together in the ratio 6 : 7. The average cost of liquid A is Rs 65 per liter and the average cost of the mixture is Rs. 60 per liter, then find the average cost of liquid B per liter.



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Correct Ans:Rs. 64.29
Explanation:
Given liquids A and B = 6 : 7
Average cost of mixture = Rs. 60/ liter
Average cost of liquid A = Rs. 65 / liter
Let Average cost of liquid B = x per liter


According to the alligation equation,
=> (x - 60) / (65 - 60) = 6 / 7
=> (x - 60) / 5 = 6/7
=> (x - 60) = (6/7) * 5
=> (x - 60) = 30 /7
=> x = (30 /7) + 60
=> x = (30 + 420) /7
=> x = 450 /7
=> x = 64.29

So, Average cost of liquid B = Rs. 64.29per liter.


12. How many liters of water should be added to a 30 liter mixture containing milk and water in the ratio 7:3 such that the resultant mixture has 40 % water in it 



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Correct Ans:5
Explanation:
Solution is
Given
In 30 litres of mixture, milk and water ratio is7 : 3
Let m = milk ; w = water
=> m / w = 7 / 3 = 7x liters / 3x litres
=> 7x + 3x =30
=> 10x =30
=> x = 3
m= 7x litres = 21 litres
w= 3x litres = 9 litres
When we add x litres of water the resulting solution contains 21 litres of milk and (9+x) litres of water
Total quantity = 30 + x
Given , percentage of water in this new mixture =40%
i.e 40% of new mixture =water
=> 40/100(30 + x) =9 + x
=> (2/5) (30 + x) = (9 +x)
=> 2(30 + x) = 5 (9 +x)
=> 60 + 2x = 45 + 5x
=> 5x - 2x = 60 - 45 => 3x= 15 => x=5
Answer is 5


13. How many liters of water should be added to a 30 liter mixture, containing milk and water in the ratio 7:3 such that the resultant mixture has 40 % water in it. 



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Correct Ans:5
Explanation:
Given: 30 Liters of mixture, Milk and water in the ratio 7 : 3 Which means, we have 21 liters of milk and 9 liters of water.

We add water the resulting solution is 21 liters of milk and 9 + x liters of water.
Total quantity = 30 +x.
Water percentage is 40 % = > 40 x (30+x)/100 = 9 + x
=>4(30+x) = 10(9+x)
=> 120 + 4x = 90 + 10x
=> 10x - 4x = 120-90
=>6x = 30
=> x = 5
Thus the quantity of water added = 5 liters


14. 50 liters of mixture has 60% milk. How much milk should be added to the mixture to make it 80% pure?



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Correct Ans:135 liters
Explanation:
Original mixture comprises 50 liters of milk and water.
Out of the 50 liters, 60% is pure milk.
=> ( 60 / 100 ) x 50 = pure milk
=>30 liters = pure milk

In 50 liters mixture, remaining 20 liters = water

When "x" liters of pure milk is added to 50 liters of mixture
New mixture = ( 50 + x ) liters
Milk in new mixture = (30 + x) liters
Given milk in new mixture = 80% of (50 + x)
=> 30 + x = (80 / 100) * (50+ x)
=> 30 + x = (8 / 10) * (50+ x)
=> 10 (30 + x) = 8 (50+ x)
=> 130 + 10x = 400 + 8x
=> 10 x - 8 x = 400 - 130
=> 2x = 270
=> x = 135 liters
Hence we need to add 135 liters of pure milk to the 50 liters of mixtureto make it 80% pure.


15. How many liters of a 12 litre mixture containing milk and water in the ratio of 2 : 3 be replaced with pure milk so that the resultant mixture contains milk and water in equal proportion? 



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Correct Ans:2 liters
Explanation:
The mixture contains 40% milk and 60% water in it.

That is 4.8 liters of milk and 7.2 liters of water.

Now we are replacing the mixture with pure milk so that the amount of milk and water in the mixture is 50% and 50%.

That is we will end up with 6 liters of milk and 6 liters of water. Water gets reduced by 1.2 liters.

To remove 1.2 liters of water from the original mixture containing 60% water, we need to remove 1.2 / 0.6 liters of the mixture = 2 liters.


16. 20 liters of mixture has 70% milk. How much milk should be added to the mixture to make it 90% pure?



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Correct Ans:40 liters
Explanation:
Original mixture comprises 20 liters of milk and water.
Out of the 20 liters, 70 % is pure milk.
=> ( 70 / 100 ) x 20 = pure milk
=>14 liters = pure milk

In 20 liters mixture, remaining 6 liters = water

When "x" liters of pure milk is added to 20 liters of mixture
New mixture = ( 20 + x ) liters
Milk in new mixture = (14 + x) liters
Given milk in new mixture = 90% of (20+ x)
=> 14 + x = (90 / 100) * (20 + x)
=> 14 + x = (9 / 10) * (20+ x)
=> 10 (14 + x) = 9 (20 + x)
=> 140 + 10x = 180 + 9x
=> 10 x - 9 x = 180 - 140
=> x = 40 liters
Hence we need to add 40 liters of pure milk to the 20 liters of mixtureto make it 90% pure.


17. Two liquids A and B are mixed together in the ratio 2 : 3. The average cost of liquid B is Rs 35 per liter and the average cost of the mixture is Rs. 30 per liter, then find the average cost of liquid A per liter.



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Correct Ans:22.5
Explanation:
Given, Ratio of liquids A and B =2 : 3
Average cost of liquid B per liter = Rs. 35
Average cost of mixture per liter = Rs. 30
Let theAverage cost of liquid A per liter = Rs. x


According to alligation equation,
(35-30)/(30-x) = 2/3
=> 5 /(30-x) =2/3
=> 5 * 3 = 2 *(30-x)
=> 15 = 60 - 2x
=> 2x = 60 - 15
=> 2x = 45
=> x = 22.5
Therefore,Average cost of liquid A per liter = Rs. x = Rs. 22.5


18. How many liters of water should be added to a 30 liter mixture of milk and water containing milk and water in the ratio 7:3 such that the resultant mixture has 40 % water in it. 



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Correct Ans:5
Explanation:
Given, Total quantity of mixture = 30 liter
=> Milk + water = 30 liter

Given,Ratio of Milk and water = 7 : 3
=> Milk quantity = 7x liter
and waterquantity = 3x liter

Thus, 7x + 3x = 30
=> 10x = 30
=> x = 3
Hence, Milk =7x= 7 * 3 = 21 liter
Water = 3x = 3 * 3 = 9 liter

If we add "x" liters of water to the 30 liter mixture, to obtain 40% water in it,
the total quantity of resultant mixture = 30 + x liter
water quantity will become = 9 + x liter

thus, 40% of new mixture = water
=> 40% of (30 + x) =9 + x
=> (40/100) * (30 + x) = 9 +x
=> (4/10) * (30 + x) = 9 +x
=> 4 * (30 + x) = 10 * (9 +x)
=> 120 + 4x = 90 + 10x
=> 10x - 4x = 120 - 90
=> 6x = 30
=> x = 5
Thus,to obtain 40% water in new mixture, the quantity of water to be added = 5 liter.


19. How many liters of water should be added to a 30 liter mixture of milk and water containing milk and water in the ratio 7:3 such that the resultant mixture has 40 % water in it. 



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Correct Ans:5
Explanation:

Given, In 30 Liters of mixture, Milk and water in the ratio 7 : 3.
=> milk = 7x liters
water = 3x liters

Now, Milk + water = 30 liters
=> 7x + 3x = 30
=> 10 x = 30
=> x = 3

Milk = 7x liters = 7 * 3 = 21 liters
Water = 3x liters = 3 * 3 = 9 liters

When we add x liters of water, the resulting solution is 21 liters of milk and 9 + x liters of water.
Then,Total quantity = 30+x

Water percentage is 40 %
= > 40 x (30+x)/100 = 9 + x
=>4(30+x) = 10(9+x)
=> 120 + 4x = 90 + 10x
=> 10x – 4x = 120-90
=>6x = 30
=> x = 5
=> Water added = 5 liters



20. Two liquids A and B are mixed together in the ratio 5 : 6. The average cost of liquid B is Rs 15 per liter and the average cost of the mixture is Rs. 10 per liter, then find the average cost of liquid A per liter.



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Correct Ans:Rs. 4
Explanation:
Given, Ratio of liquids A and B = 5 : 6
Average cost of liquid B per liter = Rs. 15
Average cost of mixture per liter = Rs. 10
Let theAverage cost of liquid A per liter = Rs. x



According to alligation equation,
(15 - 10)/(10 - x) = 5/6
=> 5 /(10 - x) = 5/6
=> 5 * 6 = 5 *(10 - x)
=> 30 = 50 - 5x
=> 5x = 50 - 30
=> 5x = 20
=> x = 4
Therefore,Average cost of liquid A per liter = Rs. x = Rs. 4





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