Problem Overview: Mixing Container Contents

This problem involves determining the precise ratio at which mixtures from two identical containers, labeled P and Q, should be combined to achieve a specific final milk-to-water ratio. Container P holds a mixture with a milk-to-water ratio of 1:1, while container Q contains a mixture with a ratio of 3:1. The goal is to find the mixing ratio (from P to Q) that results in a final mixture with a milk-to-water ratio of 5:4.

Analyzing Container Compositions

First, let's determine the proportion (fraction) of milk and water in each container:

Setting Up the Mixing Ratio Equation

Let's assume we need to take '$x$' units of the mixture from container P and '$y$' units of the mixture from container Q. The total volume of the final mixture will be '$x + y$' units.

We can calculate the total amount of milk in the final mixture by summing the milk contributions from both containers:

• Amount of milk from P = $x \times \text{Milk fraction in P} = x \times \frac{1}{2}$
• Amount of milk from Q = $y \times \text{Milk fraction in Q} = y \times \frac{3}{4}$

Total milk in the final mixture = $\frac{x}{2} + \frac{3y}{4}$.

The problem states that the final mixture should have a milk-to-water ratio of 5:4. This means the fraction of milk in the final mixture should be $\frac{5}{9}$.

We can set up an equation based on the milk concentration:

$$ \frac{\text{Total milk in final mixture}}{\text{Total volume of final mixture}} = \text{Target milk fraction} $$

$$ \frac{\frac{x}{2} + \frac{3y}{4}}{x+y} = \frac{5}{9} $$

Solving for the Required Ratio (x:y)

Now, we solve this equation to find the ratio '$x:y$'.

1. Simplify the numerator of the fraction: To combine $\frac{x}{2}$ and $\frac{3y}{4}$, find a common denominator, which is 4. $$ \frac{x}{2} + \frac{3y}{4} = \frac{2x}{4} + \frac{3y}{4} = \frac{2x + 3y}{4} $$
2. Substitute this back into the equation: $$ \frac{\frac{2x + 3y}{4}}{x+y} = \frac{5}{9} $$
3. Simplify the left side: $$ \frac{2x + 3y}{4(x+y)} = \frac{5}{9} $$
4. Cross-multiply to eliminate the denominators: $$ 9(2x + 3y) = 5 \times 4(x+y) $$
5. Expand both sides of the equation: $$ 18x + 27y = 20(x+y) $$ $$ 18x + 27y = 20x + 20y $$
6. Rearrange the terms to group '$x$' terms on one side and '$y$' terms on the other: $$ 27y - 20y = 20x - 18x $$
7. Simplify the equation: $$ 7y = 2x $$
8. Express the ratio '$x:y$'. Divide both sides by '$y$' and then by 2: $$ \frac{x}{y} = \frac{7}{2} $$

Thus, the mixture should be taken in the ratio 7:2 from container P to container Q.

Verifying the Solution Using Water Concentration

We can verify this result by considering the water concentration. The total amount of water in the final mixture is:

• Amount of water from P = $x \times \text{Water fraction in P} = x \times \frac{1}{2}$
• Amount of water from Q = $y \times \text{Water fraction in Q} = y \times \frac{1}{4}$

Total water in the final mixture = $\frac{x}{2} + \frac{y}{4}$.

The target fraction of water in the final mixture is $\frac{4}{9}$.

The equation for water concentration is:

$$ \frac{\frac{x}{2} + \frac{y}{4}}{x+y} = \frac{4}{9} $$

Simplifying the numerator gives $\frac{2x+y}{4}$. So the equation becomes:

$$ \frac{2x + y}{4(x+y)} = \frac{4}{9} $$

Cross-multiplying:

$$ 9(2x + y) = 16(x+y) $$

$$ 18x + 9y = 16x + 16y $$

Rearranging terms:

$$ 18x - 16x = 16y - 9y $$

$$ 2x = 7y $$

$$ \frac{x}{y} = \frac{7}{2} $$

This confirms that the required ratio is indeed 7:2.

Final Answer: Required Ratio

The mixtures should be taken out from containers P and Q in the ratio 7:2 to achieve the desired milk-to-water ratio of 5:4 in the final mixture.