The ratio of milk and water in a mixture is `3:7` .We added 10 lt water of the mixture then ratio becomes `1:3` .Find the initial quantity of milk .

Options

A. 10 lt
B. 15 lt
C. 11 lt
D. 13 lt

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the initial ratio of milk and water
The initial ratio of milk to water is given as \(3:7\). This means for every 3 parts of milk, there are 7 parts of water.

### Step 2: Define the quantities
Let the initial quantity of milk be \(3k\) liters and the initial quantity of water be \(7k\) liters, where \(k\) is a common factor.

### Step 3: Calculate the total initial mixture
The total initial mixture is:
\[ 3k + 7k = 10k \text{ liters} \]

### Step 4: Add water to the mixture
According to the problem, we add 10 liters of water to the mixture. Therefore, the new quantity of water becomes:
\[ 7k + 10 \text{ liters} \]

### Step 5: Set up the new ratio
After adding the water, the new ratio of milk to water becomes \(1:3\). This can be expressed as:
\[ \frac{3k}{7k + 10} = \frac{1}{3} \]

### Step 6: Cross-multiply to solve for \(k\)
Cross-multiplying gives us:
\[ 3 \cdot 3k = 1 \cdot (7k + 10) \]
This simplifies to:
\[ 9k = 7k + 10 \]

### Step 7: Rearrange the equation
Subtract \(7k\) from both sides:
\[ 9k - 7k = 10 \]
This simplifies to:
\[ 2k = 10 \]

### Step 8: Solve for \(k\)
Dividing both sides by 2 gives:
\[ k = 5 \]

### Step 9: Find the initial quantity of milk
Now that we have \(k\), we can find the initial quantity of milk:
\[ \text{Initial quantity of milk} = 3k = 3 \times 5 = 15 \text{ liters} \]

### Final Answer
The initial quantity of milk is \(15\) liters.

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