Question: 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?
- 17 : 3
- 9 : 1
- 3 : 17
- 5 : 3
- 11: 2
Correct Answer: B
Solution and Explanation:
Approach Solution 1:
The problem statement informs that:
Given:
- 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.
- The operation is repeated once more.
Find Out:
- The ratio of milk and water in the resultant mixture at the end of the two removals and replacement.
Total 20 Ltrs and ratio of milk and water = 3:2
Therefore, it implies 12 litres of milk and 8 litres of water.
Initially, the ratio of milk to water is 12 : 8
Removing half, we get = 6 : 4
Adding 10 litres of milk means 16 : 4
Removing half, we get = 8 : 2
Again, adding 10 litres of milk means 18 : 2
The ratio of milk and water in the resultant mixture at the end of the two removals and replacement.= 18 : 2 = 9:1
Approach Solution 2:
The problem statement states that:
Given:
- 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.
- The operation is repeated once more.
Find Out:
- The ratio of milk and water in the resultant mixture at the end of the two removals and replacement.
Total solution= 20 litres
Milk : water 3:2, then milk = 12 litres and water = 8 litres
As per the formula, Amount = Concentration * volume
Therefore, the concentration of water in the solution is:
8 = Cw * 20
Cw = 2/5
Step 1:
10 litres from 20 litres solution is removed.
In the leftover solution, the water concentration remains identical. i.e. 2/5.
The initial volume of the solution was 20, the new volume is 10 (after 10 litres is removed)
Step 2:
10 litres of pure milk is added to the solution, therefore the new solution is again 20 litres
As per the formula,
Initial Concentration (Ci) * Initial Volume (Vi) = Final Concentration (Cf) * final Volume (Vf)
(2/5) * 10 = Cf * 20
Cf = 2/5 (1/2)
The water amount will remain identical when initially in a 10-litre solution. However, the water concentration will change when milk is added to the initial 10-litre water
Step 3:
Now again from the Step 2 solution, a 10-litre solution is removed and 10-litre pure milk is added
cf = (2/5) (1/2) (1/2)
cf = 1/10 (this is a new concentration of Water in the solution)
Therefore the concentration of milk is 9/10
Hence, the ratio of milk : water is 9:1.
Approach Solution 3:
The problem statement declares that:
Given:
- 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.
- The operation is repeated once more.
Find Out:
- The ratio of milk and water in the resultant mixture at the end of the two removals and replacement.
Total solution= 20 litres
Milk : water 3:2, then water = 8 litres
Let Wo be the amount of water in the mixture with pure milk = 8 litres
Let Wr be the amount of water in the mixture after the replacement has taken place.
⇒Wr/Wo = (1− R/M)^n
Where R is the amount of mixture replaced by milk in each of the steps, M is the total volume of the mixture and n is the number of times the cycle is repeated.
Hence, Wr/Wo = (1-10/20)^2 = (1/2)^2 = ¼
∴ Wr = Wo/4 = 8/4 = 2 litres
Hence, the mixture will have 18 litres of milk and 2 litres of water.
∴ the ratio of milk and water = 18 : 2 = 9 : 1
“A 20 litre mixture of milk and water contains milk and water”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “Foundations of GMAT Math”. The GMAT Problem Solving questions enhance the candidates’ skills in arithmetic, geometry and algebra. The candidates can go through GMAT Quant practice papers to improve their mathematical proficiency and knowledge.
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