The ratio of water and alcohol in two different containers is 2 : 3 and 4 : 5. In what ratio we are required to mix the mixtures of two containers in order to get the new mixture in which the ratio of alcohol and water be 7 : 5?

Options

A. `7:3`
B. `5:3`
C. `8:5`
D. `2:7`

Answer

To solve the problem, we need to find the ratio in which the mixtures from two containers should be mixed to achieve a new mixture with a specified ratio of alcohol to water. Let's break down the solution step by step.

### Step 1: Understand the Ratios
- **Container 1** has a ratio of water to alcohol as 2:3.
- **Container 2** has a ratio of water to alcohol as 4:5.
- We want to mix these two containers to achieve a new ratio of alcohol to water as 7:5.

### Step 2: Convert Ratios to Fractions
Convert the ratios of water and alcohol into fractions for easier calculations:
- For **Container 1**:
- Water = 2 parts, Alcohol = 3 parts
- Total parts = 2 + 3 = 5
- Fraction of Water = \( \frac{2}{5} \)
- Fraction of Alcohol = \( \frac{3}{5} \)

- For **Container 2**:
- Water = 4 parts, Alcohol = 5 parts
- Total parts = 4 + 5 = 9
- Fraction of Water = \( \frac{4}{9} \)
- Fraction of Alcohol = \( \frac{5}{9} \)

- For the **New Mixture**:
- Alcohol = 7 parts, Water = 5 parts
- Total parts = 7 + 5 = 12
- Fraction of Water = \( \frac{5}{12} \)
- Fraction of Alcohol = \( \frac{7}{12} \)

### Step 3: Set Up the Equation
Let the quantities of mixtures from Container 1 and Container 2 be \( x \) and \( y \) respectively. The fractions of water and alcohol from the mixtures can be expressed as follows:

- Total Water from both containers:
\[ \text{Water} = \left( \frac{2}{5} \right)x + \left( \frac{4}{9} \right)y \]

- Total Alcohol from both containers:
\[ \text{Alcohol} = \left( \frac{3}{5} \right)x + \left( \frac{5}{9} \right)y \]

### Step 4: Use the New Mixture Ratio
According to the new mixture ratio:
\[ \frac{\text{Alcohol}}{\text{Water}} = \frac{7}{5} \]
This can be rearranged to:
\[ \text{Alcohol} = \frac{7}{5} \times \text{Water} \]

### Step 5: Substitute and Solve
Substituting the expressions for water and alcohol into the equation:
\[ \left( \frac{3}{5} \right)x + \left( \frac{5}{9} \right)y = \frac{7}{5} \left( \left( \frac{2}{5} \right)x + \left( \frac{4}{9} \right)y \right) \]

### Step 6: Clear the Fractions
To eliminate the fractions, multiply through by the least common multiple of the denominators (which is 45):
\[ 45 \left( \left( \frac{3}{5} \right)x + \left( \frac{5}{9} \right)y \right) = 45 \left( \frac{7}{5} \left( \left( \frac{2}{5} \right)x + \left( \frac{4}{9} \right)y \right) \right) \]

This simplifies to:
\[ 27x + 25y = 63 \left( \frac{2}{5}x + \frac{4}{9}y \right) \]

### Step 7: Solve for x and y
After simplifying and solving the equation, we will find the ratio \( x:y \).

### Final Ratio
After solving the equation, we find that the ratio in which the mixtures should be mixed is:
\[ x:y = 5:3 \]