Let the volume of liquid in vessels A and B be represented by some unknown quantities $x$ and $y$, respectively.

The ratios of milk and water in vessels A and B are:

• In vessel A, the ratio of milk to water is 5:3.

  • This means that the fraction of milk in vessel A is $\frac{5}{5+3} = \frac{5}{8}$ and the fraction of water is $\frac{3}{8}$.

• In vessel B, the ratio of milk to water is 3:4.

  • This means that the fraction of milk in vessel B is $\frac{3}{3+4} = \frac{3}{7}$ and the fraction of water is $\frac{4}{7}$.

Now, we want to mix the liquids from both vessels to get a new mixture C where the ratio of milk to water is 50:50, i.e., $\frac{1}{2}$ milk and $\frac{1}{2}$ water.

Let the total amount of liquid taken from vessel A be $x$ and from vessel B be $y$.

• The amount of milk from vessel A is $\frac{5}{8}x$, and the amount of water from vessel A is $\frac{3}{8}x$.
• The amount of milk from vessel B is $\frac{3}{7}y$, and the amount of water from vessel B is $\frac{4}{7}y$.

For the mixture C to be 50% milk and 50% water, the amount of milk in the total mixture should be equal to the amount of water in the total mixture. So, we set up the equation:

$\frac{5}{8}x + \frac{3}{7}y = \frac{3}{8}x + \frac{4}{7}y$

Simplifying this:

$\frac{5}{8}x - \frac{3}{8}x = \frac{4}{7}y - \frac{3}{7}y$

$\frac{2}{8}x = \frac{1}{7}y$

$\frac{x}{4} = \frac{y}{7}$

Multiplying both sides by 28 to eliminate the fractions:

$7x = 4y$

Thus, the ratio of $x$ to $y$ is:

$\frac{x}{y} = \frac{4}{7}$

Conclusion:

To get a mixture with 50% milk and 50% water, the liquids from vessels A and B should be mixed in the ratio 4:7.

Would you like further details or have any questions?

Here are 5 related questions to extend this topic:

1. How would the ratio change if the percentage of milk in vessel C were 60% instead of 50%?
2. How do you solve for the total volume of the mixture if you know the volumes of vessels A and B?
3. What if vessel A had a different milk-to-water ratio, say 2:1? How would that affect the solution?
4. How would you find the exact volume of milk and water if the volumes of vessels A and B were given?
5. How can you verify that the final mixture meets the desired milk-to-water ratio?

Tip: When dealing with ratios and mixtures, it’s helpful to express all parts of the mixture as fractions to set up balanced equations easily.