Two containers have mixtures of milk and water, Respectively, in the ratios, 3:2 and 6:5. In what ratio should the contents be mixed so that the ratio of milk to water in the final mixture is 4:3?

Options

A. `6:13`
B. `9:14`
C. `10:11`
D. `5:8`

Answer

To solve the problem of mixing two containers of milk and water in the desired ratio, we can follow these steps:

### Step 1: Understand the Ratios
Container A has milk and water in the ratio of 3:2. This means for every 5 parts of the mixture, 3 parts are milk and 2 parts are water.
Container B has milk and water in the ratio of 6:5. This means for every 11 parts of the mixture, 6 parts are milk and 5 parts are water.

### Step 2: Calculate the Fraction of Milk and Water in Each Container
- For Container A:
- Milk = \( \frac{3}{3+2} = \frac{3}{5} \)
- Water = \( \frac{2}{3+2} = \frac{2}{5} \)

- For Container B:
- Milk = \( \frac{6}{6+5} = \frac{6}{11} \)
- Water = \( \frac{5}{6+5} = \frac{5}{11} \)

### Step 3: Set Up the Equation for the Final Mixture
Let the quantities of mixtures from Container A and Container B be \( x \) and \( y \) respectively. We want the final mixture to have a milk to water ratio of 4:3.

The total milk in the final mixture can be expressed as:
\[ \text{Milk from A} + \text{Milk from B} = x \cdot \frac{3}{5} + y \cdot \frac{6}{11} \]

The total water in the final mixture can be expressed as:
\[ \text{Water from A} + \text{Water from B} = x \cdot \frac{2}{5} + y \cdot \frac{5}{11} \]

### Step 4: Set Up the Ratio Equation
We want:
\[ \frac{x \cdot \frac{3}{5} + y \cdot \frac{6}{11}}{x \cdot \frac{2}{5} + y \cdot \frac{5}{11}} = \frac{4}{3} \]

### Step 5: Cross Multiply to Eliminate the Fraction
Cross multiplying gives:
\[ 3 \left( x \cdot \frac{3}{5} + y \cdot \frac{6}{11} \right) = 4 \left( x \cdot \frac{2}{5} + y \cdot \frac{5}{11} \right) \]

### Step 6: Simplify the Equation
Expanding both sides:
\[ \frac{9x}{5} + \frac{18y}{11} = \frac{8x}{5} + \frac{20y}{11} \]

### Step 7: Rearranging the Equation
Bringing like terms together:
\[ \frac{9x}{5} - \frac{8x}{5} = \frac{20y}{11} - \frac{18y}{11} \]
\[ \frac{x}{5} = \frac{2y}{11} \]

### Step 8: Solve for the Ratio of x to y
Cross multiplying gives:
\[ 11x = 10y \implies \frac{x}{y} = \frac{10}{11} \]

### Final Answer
Thus, the contents of the two containers should be mixed in the ratio of **10:11**.

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