The milk and water in vessels A and B are in the ratio 5:3 and 3:4 respectively and what resear the liquids in both vessels be mixed to get a new mixture whether C consisting 50% milk and 50% water

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.

The milk and water in vessels A and B are in the ratio 5:3 and 3:4 respectively. In what ratio should the liquids in both vessels be mixed to get a new mixture consisting of 50% milk and 50% water?

Solve a problem involving the mixing of liquids from two vessels with different milk-to-water ratios to achieve a new mixture with 50% milk and 50% water. This solution uses ratio and proportion concepts along with linear equations, making it suitable for students in Grades 9-12.

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