Question

A container has a mixture of milk and water in the ratio 5 : 2. If 14 liters of this mixture is removed and replaced with water, the ratio becomes 5 : 4. What is the initial quantity of the mixture?

• A. 63 L
• B. 35 L
• C. 49 L
• D. 56 L

Hint:

In mixture replacement problems, focus on the substance whose quantity is not directly being replenished. The final amount of that substance is equal to its initial amount in the reduced volume. For this problem: Final Milk = (Initial Milk Proportion) \(\times\) (Initial Volume - Volume Removed). This can often lead to a faster solution.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We start with a mixture of milk and water in a 5:2 ratio. A certain amount of the mixture (14 L) is removed, and the same amount of pure water is added. This changes the ratio. We need to find the total volume of the mixture before this operation.
Step 2: Key Formula or Approach:
We can solve this algebraically. Let the initial quantities of milk and water be \(5x\) and \(2x\) liters, respectively. The total initial volume is \(7x\). The key principle is that when the mixture is removed, the quantities of milk and water decrease in the same ratio (5:2). We can then track the changes to find the value of \(x\). An alternative and often quicker method is to focus on the component that is not being added (in this case, milk).
Step 3: Detailed Explanation:
Let's use the algebraic method.
Initial quantity of milk = \(5x\) L.
Initial quantity of water = \(2x\) L.
Initial total mixture = \(7x\) L.
Operation: 14 liters of mixture are removed.
The milk and water are removed in the ratio 5:2.
- Amount of milk removed = \(14 \times \frac{5}{5+2} = 14 \times \frac{5}{7} = 10\) L.
- Amount of water removed = \(14 \times \frac{2}{5+2} = 14 \times \frac{2}{7} = 4\) L.
Quantities after removal:
- Milk remaining = \(5x - 10\) L.
- Water remaining = \(2x - 4\) L.
Operation: 14 liters of water are added.
- Final quantity of milk = \(5x - 10\) L.
- Final quantity of water = \((2x - 4) + 14 = 2x + 10\) L.
The new ratio of milk to water is given as 5 : 4.
\[ \frac{\text{Final Milk}}{\text{Final Water}} = \frac{5x - 10}{2x + 10} = \frac{5}{4} \] Now, we solve for \(x\) by cross-multiplying:
\[ 4(5x - 10) = 5(2x + 10) \] \[ 20x - 40 = 10x + 50 \] \[ 20x - 10x = 50 + 40 \] \[ 10x = 90 \] \[ x = 9 \] The initial quantity of the mixture was \(7x\).
\[ \text{Initial Quantity} = 7 \times 9 = 63 \text{ L} \] Step 4: Final Answer
The calculation consistently shows that the initial quantity of the mixture is 63 L.