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Cask I contains wine and water in the respective ratio 6 : 7 and cask II contains wine and water in the respective ratio 9 : 4. In what ratio must the contents of the two casks be mixed to get a mixture of wine and water in the respective ratio 8 : 5?
- a)2 : 5
- b)3 : 5
- c)1 : 5
- d)1 : 2
- e)None of these
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Cask I contains wine and water in the respective ratio 6 : 7 and cask ...
Given information:
- In cask I, the ratio of wine to water is 6:7.
- In cask II, the ratio of wine to water is 9:4.
- The desired ratio of wine to water in the mixture is 8:5.
Approach:
To find the ratio in which the contents of the two casks should be mixed, we can use a weighted average formula. We need to find the weights of the two casks based on their respective wine-water ratios.
Solution:
Step 1: Find the weights of the two casks:
Let the weight of cask I be 'x' and the weight of cask II be 'y'.
The weight ratio of cask I to cask II can be calculated as:
Weight ratio = (ratio of wine in cask I/ratio of wine in cask II) = (6/9) = 2/3
Since the ratio of weights is the inverse of the ratio of wines, the weight ratio of cask I to cask II is 3:2.
Therefore, the weights of cask I and cask II can be represented as:
Weight of cask I = (3/(3+2)) * total weight = (3/5) * total weight
Weight of cask II = (2/(3+2)) * total weight = (2/5) * total weight
Step 2: Find the ratio of wine to water in the mixture:
Let the ratio of wine to water in the mixture be a:b.
Using the weighted average formula, we can write:
(a/b) = [(weight of cask I) * (ratio of wine in cask I) + (weight of cask II) * (ratio of wine in cask II)] / [(weight of cask I) * (ratio of water in cask I) + (weight of cask II) * (ratio of water in cask II)]
Substituting the given values, we get:
(a/b) = [(3/5) * total weight * (6/13) + (2/5) * total weight * (9/13)] / [(3/5) * total weight * (7/13) + (2/5) * total weight * (4/13)]
Simplifying the expression, we get:
(a/b) = (18/65 + 18/65) / (21/65 + 8/65)
(a/b) = (36/65) / (29/65)
(a/b) = 36/29
Step 3: Find the final ratio:
Since the desired ratio of wine to water in the mixture is 8:5, we can equate the two ratios and solve for a:b:
36/29 = 8/5
Cross-multiplying, we get:
5 * 36 = 8 * 29
180 = 232
Since this is not true, the assumption that the two ratios are equal is incorrect.
Therefore, the contents of the two casks cannot be mixed in any ratio to obtain a mixture with a wine-water ratio of 8:5.
Hence, the correct answer is option 'e' (None of these).
- In cask I, the ratio of wine to water is 6:7.
- In cask II, the ratio of wine to water is 9:4.
- The desired ratio of wine to water in the mixture is 8:5.
Approach:
To find the ratio in which the contents of the two casks should be mixed, we can use a weighted average formula. We need to find the weights of the two casks based on their respective wine-water ratios.
Solution:
Step 1: Find the weights of the two casks:
Let the weight of cask I be 'x' and the weight of cask II be 'y'.
The weight ratio of cask I to cask II can be calculated as:
Weight ratio = (ratio of wine in cask I/ratio of wine in cask II) = (6/9) = 2/3
Since the ratio of weights is the inverse of the ratio of wines, the weight ratio of cask I to cask II is 3:2.
Therefore, the weights of cask I and cask II can be represented as:
Weight of cask I = (3/(3+2)) * total weight = (3/5) * total weight
Weight of cask II = (2/(3+2)) * total weight = (2/5) * total weight
Step 2: Find the ratio of wine to water in the mixture:
Let the ratio of wine to water in the mixture be a:b.
Using the weighted average formula, we can write:
(a/b) = [(weight of cask I) * (ratio of wine in cask I) + (weight of cask II) * (ratio of wine in cask II)] / [(weight of cask I) * (ratio of water in cask I) + (weight of cask II) * (ratio of water in cask II)]
Substituting the given values, we get:
(a/b) = [(3/5) * total weight * (6/13) + (2/5) * total weight * (9/13)] / [(3/5) * total weight * (7/13) + (2/5) * total weight * (4/13)]
Simplifying the expression, we get:
(a/b) = (18/65 + 18/65) / (21/65 + 8/65)
(a/b) = (36/65) / (29/65)
(a/b) = 36/29
Step 3: Find the final ratio:
Since the desired ratio of wine to water in the mixture is 8:5, we can equate the two ratios and solve for a:b:
36/29 = 8/5
Cross-multiplying, we get:
5 * 36 = 8 * 29
180 = 232
Since this is not true, the assumption that the two ratios are equal is incorrect.
Therefore, the contents of the two casks cannot be mixed in any ratio to obtain a mixture with a wine-water ratio of 8:5.
Hence, the correct answer is option 'e' (None of these).
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Cask I contains wine and water in the respective ratio 6 : 7 and cask II contains wine and water in the respective ratio 9 : 4. In what ratio must the contents of the two casks be mixed to get a mixture of wine and water in the respective ratio 8 : 5?a)2 : 5b)3 : 5c)1 : 5d)1 : 2e)None of theseCorrect answer is option 'D'. Can you explain this answer?
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