Math Problem Statement
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.
1 point A. 9: 7 B. 7: 5 C. 5: 7 D. 7: 9
Solution
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 = .
- Vessel B: The ratio of spirit to water is 7:6.
- The fraction of spirit in B = .
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 = .
Step 3: Apply the allegation rule.
The formula for the allegation is:
Calculate the differences:
- Difference between the concentration in B and C:
- Difference between the concentration in A and C: First, find a common denominator for these fractions:
Step 4: Simplify the ratio.
The ratio for mixing A and B should be: This negative sign is just for the direction, we take the absolute values: Divide by the greatest common divisor:
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:
- How does the concept of concentration apply to mixtures?
- Can you explain the principle of the allegation method in detail?
- What are some practical applications of the allegation method?
- How does the ratio influence the final concentration in mixtures?
- 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.
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Math Problem Analysis
Mathematical Concepts
Ratio and Proportion
Allegation Method
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12