A mixture of 70 litres of wine and water contains 10% water. How much water must be added to make water 12 and 1/2 % of the total mixture ?

Options

A. 2 litres
B. 10 litres
C. 12 litres
D. 4 litres

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and add some clarifications along the way.

### Step 1: Determine the current amount of water in the mixture
The total volume of the mixture is 70 liters, and it contains 10% water.

To find the amount of water:
\[ \text{Amount of water} = 10\% \text{ of } 70 \text{ liters} = \frac{10}{100} \times 70 = 7 \text{ liters} \]

**Hint:** To find the percentage of a quantity, multiply the percentage (as a fraction) by the total amount.

### Step 2: Let \( x \) be the amount of water to be added
We will denote the amount of water we need to add as \( x \) liters.

### Step 3: Write the new total amount of water after adding \( x \)
After adding \( x \) liters of water, the total amount of water in the mixture becomes:
\[ \text{Total water} = 7 + x \text{ liters} \]

### Step 4: Write the new total volume of the mixture
The new total volume of the mixture after adding \( x \) liters of water will be:
\[ \text{Total mixture} = 70 + x \text{ liters} \]

### Step 5: Set up the equation for the new percentage of water
According to the problem, we want the water to make up 12.5% of the new total mixture. Therefore, we can set up the equation:
\[ \frac{7 + x}{70 + x} = \frac{12.5}{100} \]

### Step 6: Simplify the equation
We can rewrite \( 12.5\% \) as a fraction:
\[ \frac{12.5}{100} = \frac{25}{200} = \frac{1}{8} \]
Thus, the equation becomes:
\[ \frac{7 + x}{70 + x} = \frac{1}{8} \]

### Step 7: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 8(7 + x) = 1(70 + x) \]

### Step 8: Expand both sides
Expanding both sides results in:
\[ 56 + 8x = 70 + x \]

### Step 9: Rearrange the equation to isolate \( x \)
Now, we will move all \( x \) terms to one side and constant terms to the other:
\[ 8x - x = 70 - 56 \]
This simplifies to:
\[ 7x = 14 \]

### Step 10: Solve for \( x \)
Dividing both sides by 7 gives:
\[ x = 2 \]

### Conclusion
Thus, the amount of water that must be added to make water 12.5% of the total mixture is **2 liters**.

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