Two solutions of acid and water containing acid and water in the ratio 2 : 7 and 4 : 5, respectively are mixed in the ratio 2 : 5. What is the ratio of acid and water in the resulting solution?

Options

A. 8 : 13
B. 3 : 7
C. 8 : 17
D. 2 : 1

Answer

To solve the problem, we need to find the ratio of acid and water in the resulting solution when two solutions are mixed. Let's break it down step by step.

### Step 1: Understand the Ratios of the Two Solutions
The first solution has acid and water in the ratio of 2:7. This means:
- Acid in Solution 1 = 2 parts
- Water in Solution 1 = 7 parts

The second solution has acid and water in the ratio of 4:5. This means:
- Acid in Solution 2 = 4 parts
- Water in Solution 2 = 5 parts

### Step 2: Determine the Total Parts in Each Solution
For Solution 1:
- Total parts = 2 (acid) + 7 (water) = 9 parts

For Solution 2:
- Total parts = 4 (acid) + 5 (water) = 9 parts

### Step 3: Calculate the Amount of Acid and Water in Each Solution
Let's assume we take 2 parts of Solution 1 and 5 parts of Solution 2 (as given in the problem).

For Solution 1 (2 parts):
- Acid = (2/9) * 2 = 4/9 parts
- Water = (7/9) * 2 = 14/9 parts

For Solution 2 (5 parts):
- Acid = (4/9) * 5 = 20/9 parts
- Water = (5/9) * 5 = 25/9 parts

### Step 4: Combine the Amounts of Acid and Water
Now, we combine the acid and water from both solutions:
- Total Acid = (4/9) + (20/9) = 24/9 parts
- Total Water = (14/9) + (25/9) = 39/9 parts

### Step 5: Find the Ratio of Acid to Water in the Resulting Solution
Now we need to find the ratio of acid to water:
- Ratio of Acid to Water = Total Acid : Total Water = (24/9) : (39/9)

This simplifies to:
- Ratio of Acid to Water = 24 : 39

### Step 6: Simplify the Ratio
To simplify the ratio 24:39, we can divide both numbers by their greatest common divisor (GCD), which is 3:
- 24 ÷ 3 = 8
- 39 ÷ 3 = 13

Thus, the final ratio of acid to water in the resulting solution is:
**8 : 13**