Mixture Ratio Calculation: Step-by-Step Solution
This problem requires us to track the quantities of milk and water through several steps to find the final ratio. We will calculate the amounts step by step.
1. Initial Mixture Preparation
A starts with a container of 60 litres of pure milk. The first operation involves removing some milk and adding water.
- Initial quantity of milk = 60 L.
- Quantity of milk removed = 4 L.
- Quantity of milk left after removal = $60 \, \text{L} - 4 \, \text{L} = 56 \, \text{L}$.
- Quantity of water added (replacement) = 4 L.
- The resulting mixture has 56 L of milk and 4 L of water.
- The total volume of the mixture = $56 \, \text{L} + 4 \, \text{L} = 60 \, \text{L}$.
- The proportion (concentration) of milk in this mixture is $\frac{56}{60}$.
- The proportion of water in this mixture is $\frac{4}{60}$.
2. Transaction by B: Selling Mixture
B buys the 60 L mixture prepared by A. B then sells 30 litres of this mixture.
We need to find out how much milk and water are in the 30 L that B sold:
- Amount of milk sold by B = $30 \, \text{L} \times \frac{56}{60} = 30 \, \text{L} \times \frac{14}{15} = 2 \times 14 \, \text{L} = 28 \, \text{L}$.
- Amount of water sold by B = $30 \, \text{L} \times \frac{4}{60} = 30 \, \text{L} \times \frac{1}{15} = 2 \, \text{L}$.
3. Remaining Mixture Calculation
After B sells 30 L, we need to determine the composition of the mixture that remains with B.
- Total volume remaining = Initial Volume - Volume Sold = $60 \, \text{L} - 30 \, \text{L} = 30 \, \text{L}$.
- Amount of milk remaining = Initial Milk - Milk Sold = $56 \, \text{L} - 28 \, \text{L} = 28 \, \text{L}$.
- Amount of water remaining = Initial Water - Water Sold = $4 \, \text{L} - 2 \, \text{L} = 2 \, \text{L}$.
- We can verify the total remaining volume: $28 \, \text{L} (\text{Milk}) + 2 \, \text{L} (\text{Water}) = 30 \, \text{L}$.
4. Final Mixture Composition
B then adds 5 litres of water to this remaining mixture.
- The amount of milk in the mixture remains unchanged: 28 L.
- The amount of water increases: $2 \, \text{L} (\text{existing}) + 5 \, \text{L} (\text{added}) = 7 \, \text{L}$.
- So, the final mixture contains 28 L of milk and 7 L of water.
5. Determining the Final Ratio
The question asks for the ratio of milk to water in the final remaining mixture.
- Final Ratio = Amount of Milk : Amount of Water
- Final Ratio = $28 \, \text{L} : 7 \, \text{L}$
- To simplify the ratio, divide both sides by the greatest common divisor, which is 7.
- Simplified Ratio = $\frac{28}{7} : \frac{7}{7} = 4 : 1$.
The final ratio of milk to water in the remaining mixture is 4:1.
Important Questions from Mixture Problems (Notes)
- 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?
- A $595$ litre of mixture contains milk and water in the ratio $17:18$. How much milk must be added to the mixture so that it contains milk and water in the proportion of $3:2$?
- Alloy A is formed by mixing iron (Fe) and nickel (Ni) in the ratio 3:4, while alloy B is formed by mixing Fe and Ni in the ratio 9:5. If equal quantities of alloys A and B are melted together to form a new alloy C, what will be the ratio of Fe to Ni in the alloy C?
Suppose a tap mixes hot water and cold water in a ratio that depends linearly on the proportion of opening. Water out of the tap has temperature 40°C when the tap is half-open, and 30°C when it is three-fourths open. To get water at 50°C, the tap should be_______________.
- In 80 litres mixture of milk and water, the ratio of amount of milk to that of amount of water is 7 : 3. In order to make this ratio 2 : 1, how many litres of water should be added ?
