Solving Mixture Ratio Problems: Container A & B
This problem involves calculating the ratio of milk and water in a second container (Container B) based on how its contents are mixed with another container (Container A) to achieve a specific final mixture ratio.
1. Understanding the Initial Ratios
We are given the ratio of milk to water in two containers:
- Container A: Milk : Water = 1:3. This means for every 1 part milk, there are 3 parts water. The total parts are \(1 + 3 = 4\). So, milk constitutes \(\frac{1}{4}\) of the mixture and water constitutes \(\frac{3}{4}\).
- Container B: Milk : Water = \(m:n\). This means milk constitutes \(\frac{m}{m+n}\) and water constitutes \(\frac{n}{m+n}\) of the mixture. Our goal is to find the value of \(\frac{m}{n}\).
2. Calculating Mixture Volumes
The mixtures from Container A and Container B are mixed in the ratio 2:3. The total final volume is 20 litres.
- Volume from A = \(20 \text{ litres} \times \frac{2}{2+3} = 20 \times \frac{2}{5} = 8 \text{ litres}\).
- Volume from B = \(20 \text{ litres} \times \frac{3}{2+3} = 20 \times \frac{3}{5} = 12 \text{ litres}\).
3. Calculating Final Mixture Composition
The final 20-litre mixture has a milk to water ratio of 3:7.
- Total parts = \(3 + 7 = 10\).
- Amount of Milk in Final Mixture = \(20 \text{ litres} \times \frac{3}{10} = 6 \text{ litres}\).
- Amount of Water in Final Mixture = \(20 \text{ litres} \times \frac{7}{10} = 14 \text{ litres}\).
4. Determining Milk Content from Container B
We can find the amount of milk contributed by Container B by subtracting the milk from Container A from the total milk in the final mixture.
- Milk from Container A = \(8 \text{ litres (volume of A)} \times \frac{1}{4} \text{ (milk fraction in A)} = 2 \text{ litres}\).
- Milk from Container B = Total Milk - Milk from A
- Milk from Container B = \(6 \text{ litres} - 2 \text{ litres} = 4 \text{ litres}\).
5. Finding the Ratio \(\frac{m}{n}\)
We know that 12 litres of mixture were taken from Container B, and this contained 4 litres of milk. The fraction of milk in Container B is \(\frac{m}{m+n}\).
- Milk fraction in B = \(\frac{\text{Milk from B}}{\text{Volume from B}} = \frac{4 \text{ litres}}{12 \text{ litres}} = \frac{1}{3}\).
- So, we have the equation: \(\frac{m}{m+n} = \frac{1}{3}\).
Now, we solve for \(\frac{m}{n}\):
- Cross-multiply: \(3 \times m = 1 \times (m+n)\)
- Simplify: \(3m = m + n\)
- Rearrange to find the relationship between m and n: \(3m - m = n\)
- \(2m = n\)
- To find the ratio \(\frac{m}{n}\), divide both sides by \(n\) and then by 2: \(\frac{2m}{n} = 1 \implies \frac{m}{n} = \frac{1}{2}\).
Alternatively, we could calculate using water:
- Water from Container A = \(8 \text{ litres} \times \frac{3}{4} = 6 \text{ litres}\).
- Water from Container B = Total Water - Water from A = \(14 \text{ litres} - 6 \text{ litres} = 8 \text{ litres}\).
- Water fraction in B = \(\frac{\text{Water from B}}{\text{Volume from B}} = \frac{8 \text{ litres}}{12 \text{ litres}} = \frac{2}{3}\).
- So, \(\frac{n}{m+n} = \frac{2}{3}\).
- Cross-multiply: \(3n = 2(m+n)\)
- \(3n = 2m + 2n\)
- \(n = 2m\)
- \(\frac{m}{n} = \frac{1}{2}\).
Both methods confirm the result.
Similar Questions
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Important Questions from Mixture and Alligation
A bottle contains 20 litres of liquid A. 4 litres of liquid A is taken out of it d replaced by same quantity of liquid B. Again 4 litre of the mixture is taken out and replaced by same quantity of liquid B. What is the ratio of quantity of liquid A to that of liquid B in the final mixture?
The average score of a batsman after his 50th innings was 46.4. After 60th innings, his average Score increases by 2.6. What was his average score in the last ten innings?
If 1 litre of water weighs 1 kg, then how many cubic millimeters of water will weigh 0.1 gm?
A vessel full of water weighs 4o kg. If it is one-third filled, its weight becomes 20 kg. What is the weight of the empty vessel?
Two equal glasses of same type are respectively 1/3 and 1/4 full of milk. They are then filled up with water and the contents are mixed in a pot. What is the ratio of milk and water in the pot?
