Q1. A 20 litre mixture of milk and water contains milk and water in the ratio 3 : 2. 10 litres of the mixture is removed and replaced with pure milk and the operation is repeated once more. At the end of the two removal and replacement, what is the ratio of milk and water in the resultant mixture?
1. 17 : 3
2. 9 : 1
3. 3 : 17
4. 5 : 3
Answer. 9 : 1
Explanation.
The 20 litre mixture contains milk and water in the ratio of 3 : 2. Therefore, the mixture contains 12 litres of milk and 8 litres of water.
Q7. A sample of x litres from a container having a 60 litre mixture of milk and water containing milk and water in the ratio of 2 : 3 is replaced with pure milk so that the container will have milk and water in equal proportions. What is the value of x?
1. 6 litres
2. 10 litres
3. 30 litres
4. None of these
Answer. 10 litres
Explanation.
Let us solve this question by back substituting answer choices.
The rigorous method of solving is outlined in the previous question on removing and replacing mixtures.
The mixture of 60 litres has milk and water in the ratio 2 : 3
i.e., 24 litres of milk and 36 litres of water.When x litres of the mixture is removed, 0.4 x litres of milk and 0.6 x litres of water are removed from it.Take choice (2). According to this choice, x = 10.
So, 10 litres of the mixture is removed.
4 litres of milk and 6 litres of water are removed.
Therefore, there will be 20 litres of milk and 30 litres of water in the container.
Subsequently, when 10 litres of milk is added, the mixture will contain 30 litres of milk and 30 litres of water  i.e. milk and water are in equal proportion.
Q8. A farmer counted the heads of the animals / birds in the farm and found it to be 80. When he counted the legs he found it to be 260. If the farm had either pigeons or horses and nothing else, how many horses were there in the farm? In the farm, each horse had four legs and each pigeon had two legs.
1. 40
2. 30
3. 50
4. 60
Answer. 50
Explanation.
The count of heads gives the total number of animals / birds as each animal / bird has one head.
Let the number of horses be x.
The remaining will be pigeons. The number of pigeons = (80  x).
Each pigeon has 2 legs and each horse has 4 legs.
Therefore, total number of legs = 4x + 2(80  x) = 260
⇒ 4x + 160  2x = 260
⇒ 2x = 100
⇒ x = 50.
x is the number of horses. The number of horses in the farm is 50.
Q9. From a cask of milk containing 30 litres, 6 litres are drawn out and the cask is filled up with water. If the same process is repeated a second, then a third time, what will be the number of litres of milk left in the cask?
1. 0.512 liters
2. 12 liters
3. 14.38 liters
4. 15.36 liters
Answer. 15.36 liters
Explanation.
The problem can be solved by brute force of removing and replacing 6 litres each time and finding the quantity of milk and water after each iteration. Finally, after step 3, we will get to the answer. But it is cumbersome to do that.
The question can be solved more effectively. Let us use that method to solve this question.
Let us find the fraction of milk left after nth operation.
Fraction of milk in the cask after the nth iteration
'x' is initial quantity of milk in the cask 'y' is the quantity of milk drawn out in each iteration and 'n' is the number of iterations.
Fraction of milk left after the 3rd iteration
∴ the quantity of milk in the cask after the 3rd iteration
Q10. A 20 litres mixture of milk and water comprising 60% pure milk is mixed with "x" litres of pure milk. The new mixture comprises 80% milk. What is the value of "x"?
1. 40 litres
2. 20 litres
3. 8 litres
4. 16 litres
Answer. 20 litres
Explanation.
The quantity of the initial mixture of milk and water is 20 litres.
Out of the 20 litres, 60% is pure milk. i.e., 12 litres is milk and the remaining 8 litres is water.
When "x" litres of pure milk is added, we will have (12 + x) litres of milk in the new mixture.
And there will be 20 + x litres of the new mixture.
The question states that (12 + x) litres of milk accounts for 80% of (20 + x) litres of the mixture.
Therefore,
5(12 + x) = 4(20 +x)
60 + 5x = 80 + 4x
x = 20 litres
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1. What is a mixture? 
2. What is an allegation in mixtures? 
3. How can mixtures and allegations be used in reallife situations? 
4. What is the method to solve mixture and allegation problems? 
5. Can mixtures and allegations be used to solve problems other than finding ratios? 
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