A 19-liter mixture consists by volume of 1 part juice to 18 parts water. If x liters of juice and y liters of water are added to this mixture to make a 54-liter mixture consisting by volume of 1 part juice to 2 parts water, what is the value of x?
To solve this problem, we need to set up and solve a system of equations using the given information.
Let's start by setting up the equation for the initial mixture. We are given that the 19-liter mixture consists of 1 part juice to 18 parts water. Since the total number of parts is 1 + 18 = 19, we can determine the amount of juice and water in the initial mixture as follows:
Juice in the initial mixture = (1/19) * 19 liters = 1 liter (equation 1)
Water in the initial mixture = (18/19) * 19 liters = 18 liters (equation 2)
Now, let's consider the final mixture. We are told that the 54-liter mixture consists of 1 part juice to 2 parts water. This means:
Juice in the final mixture = (1/3) * 54 liters (equation 3)
Water in the final mixture = (2/3) * 54 liters (equation 4)
In the final mixture, x liters of juice and y liters of water are added to the initial mixture. Therefore, we can set up the following equations:
Juice in the final mixture = Juice in the initial mixture + x (equation 5)
Water in the final mixture = Water in the initial mixture + y (equation 6)
Now, let's substitute the values from equations 1 and 2 into equations 5 and 6:
1/3 * 54 = 1 + x (equation 7)
2/3 * 54 = 18 + y (equation 8)
Simplifying equations 7 and 8, we have:
18 = 1 + x
36 = 18 + y
So, x = 18 - 1 = 17.
Therefore, the value of x is 17 liters.
Outset:
1 liter juice + 18 liters water = 19 liters
Conclusion:
18 liters juice (54/3)
x = 18 - 1 = 17