A mixture of water and spirit is placed in two separate vessels denoted by the letters A and B, in the proportions 5:2 and 7:6, respectively. Determine the ratio between which these mixtures should be mixed to generate a new mixture in vessel C that already contains spirit and water in the ratio of 8:5.

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1 point
A. 9: 7
B. 7: 5
C. 5: 7
D. 7: 9

To solve the problem of finding the correct ratio between the mixtures from vessels A and B that should be mixed to generate the new mixture in vessel C with a spirit-to-water ratio of 8:5, we can use the method of **allegation**.

### Step 1: Determine the concentration of spirit in vessels A and B.
- **Vessel A:** The ratio of spirit to water is 5:2.
  - The fraction of spirit in A = \(\frac{5}{5+2} = \frac{5}{7}\).
- **Vessel B:** The ratio of spirit to water is 7:6.
  - The fraction of spirit in B = \(\frac{7}{7+6} = \frac{7}{13}\).

### Step 2: Determine the concentration of spirit in vessel C.
- **Vessel C:** The desired ratio of spirit to water is 8:5.
  - The fraction of spirit in C = \(\frac{8}{8+5} = \frac{8}{13}\).

### Step 3: Apply the allegation rule.
The formula for the allegation is:

\[
\text{Required Ratio} = \frac{\text{Difference of concentration between B and C}}{\text{Difference of concentration between A and C}}
\]

Calculate the differences:
- Difference between the concentration in B and C:
  \[
  \text{Difference}_1 = \frac{7}{13} - \frac{8}{13} = \frac{7-8}{13} = \frac{-1}{13}
  \]
- Difference between the concentration in A and C:
  \[
  \text{Difference}_2 = \frac{5}{7} - \frac{8}{13}
  \]
  First, find a common denominator for these fractions:
  \[
  \text{Difference}_2 = \frac{5 \times 13}{7 \times 13} - \frac{8 \times 7}{13 \times 7} = \frac{65}{91} - \frac{56}{91} = \frac{9}{91}
  \]

### Step 4: Simplify the ratio.
The ratio for mixing A and B should be:
\[
\text{Ratio} = \frac{-1/13}{9/91} = \frac{-1 \times 91}{9 \times 13} = \frac{-91}{117} 
\]
This negative sign is just for the direction, we take the absolute values:
\[
\text{Ratio} = \frac{91}{117} 
\]
Divide by the greatest common divisor:
\[
\frac{91}{117} = \frac{7}{9}
\]

The correct ratio to mix vessels A and B is 7:9.

Thus, the answer is **D. 7:9**.

Would you like to know more details or have any questions?

### Follow-up questions:
1. How does the concept of concentration apply to mixtures?
2. Can you explain the principle of the allegation method in detail?
3. What are some practical applications of the allegation method?
4. How does the ratio influence the final concentration in mixtures?
5. What would happen if the ratios of the two vessels were the same?

**Tip:** Understanding the allegation method is valuable when combining solutions with different concentrations in chemistry, economics, and other fields.

Learn how to determine the correct ratio between mixtures from vessels A and B to achieve the desired spirit-to-water ratio in vessel C. This problem involves the application of ratio and proportion concepts, specifically using the allegation method. Suitable for high school students and above.

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