A cask contains a mixture of 49 liters of wine and water in the proportion 5: 2. How much water must be added to it so that the ratio of wine to water maybe 7: 4?
Question
A cask contains a mixture of 49 liters of wine and water in the proportion 5: 2. How much water must be added to it so that the ratio of wine to water maybe 7: 4?
Solution
Step 1: Determine the initial quantity of wine and water
The cask contains a mixture of wine and water in the ratio 5:2. This means for every 5 parts of wine, there are 2 parts of water. The total parts are therefore 5 parts (wine) + 2 parts (water) = 7 parts.
Given that the total volume of the mixture is 49 liters, each part is equal to 49 liters ÷ 7 parts = 7 liters.
Therefore, the initial quantity of wine is 5 parts × 7 liters/part = 35 liters and the initial quantity of water is 2 parts × 7 liters/part = 14 liters.
Step 2: Determine the amount of water to be added
We want the ratio of wine to water to be 7:4. This means for every 7 parts of wine, we want 4 parts of water. Given that the quantity of wine remains constant at 35 liters, each part (for the wine) is equal to 35 liters ÷ 7 parts = 5 liters.
Therefore, for the water to be in the ratio 7:4 with the wine, the quantity of water should be 4 parts × 5 liters/part = 20 liters.
Since we initially had 14 liters of water, we need to add 20 liters - 14 liters = 6 liters of water.
So, 6 liters of water must be added to the cask to make the ratio of wine to water 7:4.
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