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Mixtures

When two or more than two pure substances/mixtures are mixed in a certain ratio, they create a mixture. Here we shall confine ourselves to mostly homogenous mixtures in view of the questions commonly asked in CAT.

Mixing without Replacement

In this particular type of mixing, two or more than two substances are mixed without any part of any mixture being replaced.
 
Example-1
In a mixture of 420 litres the ratio of milk and water is 6:1. Now, 120 litres of the water is added to the mixture. What is the ratio of milk and water in the final mixture?
Solution
Volume of milk = 360 litres and volume of water = 60 litres.
 
When 120 litres of water is added, volume of water = 180 litres
 
So, the ratio of milk water = 2:1
 
 
Example-2
A milkman mixes 20 litres of water with 80 litres of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he had sold. What is the current proportion of water to milk?
Solution
Ratio of milk and water = 20:80
 
When 1/4th of this mixture is sold, total volume of mixture will be reduced by 25%, so 25% of milk and water both will reduce. So, volume of milk and water after selling out 1/4th of mixture = 60 litres and 15 litres respectively. Addition of 25 litres of water will finally give us the following: volume of milk = 60 litres and volume of water = 40 litres. Hence, the ratio of water and milk = 40:60 = 2:3.
 
 
Example-3
How many litres of fresh water should be mixed with 30 litres of 50% milk solution so that resultant solution is a 10% milk solution?
Solution
Method 1 Using Alligation
 
So, the ratio of fresh water added: milk solution = 4:1
 
Hence, 120 litres of fresh water should be added.
 
Method 2 Principle of constant volume of one component
 
Since we add fresh water, the volume of milk will be constant.
 
Now volume of milk = 15 litres = 10% of the new mixture.
 
So, 100% of the new mixture = 150 litres
 
So, volume of fresh water added = 150 − 30 = 120 litres.
 
Method 3 Principle of inverse proportion
 
We know that concentration is inversely proportional to the volume of solute added.
 
So, in this case 30 × 50% = 10% × (30 + x), where x is the volume of water added.
 
So, x = 120 litres
 
Method 4 Using equation
 
In the final mixture, 2708.png
 
So, x = 120 litres
 

Mixing with replacement

In this particular type of mixing, two or more than two substances are mixed by replacing some part of a mixture. In these types of questions, total volume may or may not be the same and information regarding the same can be obtained from the question.

Case 1
When the quantity withdrawn and the quantity replaced are of the same volume.
 
Initially there are 40 litres of milk, and 4 litres of milk is replaced with 4 litres of water
 
Obviously, there will be 36 litres of milk and 4 litres of water.
 
Now, 4 litres of mixture is replaced with 4 litres of water.
 
The quantity of milk and water being withdrawn here will be in the ratio of 9:1 (36:4). So, quantity of milk withdrawn = 9/10 × 4 = 3.6 l.
 
So, the volume of milk = 32.4
 
And the volume of water = 7.6
 
Now, again 4 litres of mixture is replaced with 4 litres of water
 
The quantity of milk and water being withdrawn here will be in the ratio of 81:19 (32.4:7.6). So, the quantity of milk withdrawn = (81/100) × 4 = 3.24
 
So, the volume of milk = 29.16
 
And the volume of water = 10.84
 
If we summarize the above values, then it looks like
 

 

1st operation

2nd operation

3rd operation

Taken out

Left

Taken out

Left

Taken out

Left

Milk

4

36

3.6

32.4

3.24

29.16

Water

0

4

0.4

7.6

0.76

10.84


It can be seen that the quantity of water or milk withdrawn is 10% of the existing volume of milk or water because only 10% of the total volume of 40 litres taken out.
 
With this we can deduce a standard formula for these kinds of calculations.
 
If V is the initial volume of milk (or any liquid), and x litres of milk is always replaced by water, then quantity of milk left after n such operations = 2717.png
 
This formula is very similar to the standard formula we have seen in the case of Compound Interest 2726.png. The only difference between the two formulae is that while the interest is being added every year (or for the given time-period), volume of milk gets reduced after every operation.
 
Using the values of the above example, quantity of milk left after 3 operations 2735.png
 
The same problem can be solved with straight-line approach of percentage also Since 10% of existing volume is taken out every time, the percentage of milk in the final mixture after the third operation = 72.9%
2744.png
 
Since 100% = 40, so 72.9% = 29.16 litres

Case 2
When the quantity withdrawn and the quantity replaced are of the same volume, but the total volume before replacement does not remain the same.
 
Initially, there are 40 litres of milk, and 4 litres of milk is taken out and 4 litres of water is poured in
 
So, there will be 36 litres of milk and 4 litres of water.
 
Now, 5 litres of mixture is taken out and 5 litres of water is poured in.
 
The quantity of milk and water being withdrawn here will be in the ratio of 36:4. So, the quantity of milk withdrawn = 2753.png
Milk left = 2762.png
 
Again, if now 6 litres of mixture is taken out and 6 litres of water is poured in
Milk left = 2771.png

Case 3
When the quantity withdrawn and the quantity replaced are not of the same volume.
 
Initially there are 40 litres of milk, and 4 litres of milk is taken out and 5 litres of water is poured in

Obviously, there will be 36 litres of milk and 5 litres of water.
 
Now, 5 litres of mixture is taken out and 6 litres of water is poured in then the quantity of milk and water being withdrawn will be in the ratio of 36:5. So, the quantity of milk withdrawn = 2781.png
Milk left = 2790.png
 
Again 6 litres of mixture is taken out and 7 litres of water is poured in.
 
Thus, the volume of milk in the final mixture
2799.png
 
Example
Two vessels A and B of equal capacities contain mixtures of milk and water in the ratio 4:1 and 3:1, respectively. 25% of the mixture from A is taken out and added to B. After mixing it thoroughly, an equal amount is taken out from B and added back to A. The ratio of milk to water in vessel A after the second operation is
  1. 79:21
  2. 83:17
  3. 77:23
  4. 81:19
Solution
Assume there is 20 litres of the mixture in both the vessels.
 
In vessel A, milk = 16 litres and water = 4 litres
 
25% from A to B = milk in B = 15 + 4 = 19 litres
 
= water in B = 5 + 1 = 6 litres
 
ratio = 19:6
 
Equal amount from vessel B to vessel A
= milk in 2808.png
= water in 2817.png
 
Hence, the ratio is 79:21
 




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