Final answer:
To find out how much grape juice is needed for 15/4 quarts of sparkling water based on the original recipe, we set up a proportion and solve for x. The calculation reveals that we would need 15/8 quarts of grape juice for 15/4 quarts of sparkling water.
Step-by-step explanation:
To determine how much grape juice is needed for 15/4 quarts of sparkling water, we will set up a proportion based on the original recipe amounts.
The original recipe calls for 1 1/2 quarts of sparkling water and 3/4 quart of grape juice. This can be written as a ratio: 1 1/2 quarts of sparkling water to 3/4 quart of grape juice, or (3/2):(3/4).
Next, we will set this ratio equal to the new ratio, which has 15/4 quarts of sparkling water to x quarts of grape juice, or (15/4):x. So the proportion will be (3/2)/(3/4) = (15/4)/x.
Now, we cross multiply to solve for x: (3/2) * x = (15/4) * (3/4).
Finally, solve for x by multiplying both sides of the equation: (3/2) * x = (45/16), and then divide both sides by (3/2) to get: x = (45/16) * (2/3), which simplifies to x = 15/8.
Therefore, for 15/4 quarts of sparkling water, we would need 15/8 quarts of grape juice.