The Royal Fruit Company produces two types of fruit drinks. The first type is
65%
pure fruit juice, and the second type is
90%
pure fruit juice. The company is attempting to produce a fruit drink that contains
75%
pure fruit juice. How many pints of each of the two existing types of drink must be used to make
150
pints of a mixture that is
75%
pure fruit juice?
To solve this problem, we can use a technique called the mixture problem. Let's assume that x pints of the first type (65% pure fruit juice) are used and y pints of the second type (90% pure fruit juice) are used to make 150 pints of a mixture that is 75% pure fruit juice.
The first step is to set up equations based on the given information. We know that the total amount of the mixture is 150 pints, so we have the equation:
x + y = 150 (Equation 1)
We also know that the total amount of pure fruit juice in the mixture should be equal to the sum of the pure fruit juice in each type. The amount of pure fruit juice in the first type can be calculated by multiplying the percentage of pure fruit juice (65%) by the number of pints (x), and the amount of pure fruit juice in the second type can be calculated by multiplying the percentage of pure fruit juice (90%) by the number of pints (y). This gives us:
0.65x + 0.90y = 0.75(150) (Equation 2)
Simplifying Equation 2, we have:
0.65x + 0.90y = 112.50 (Equation 3)
Now we have a system of two equations (Equations 1 and 3) with two variables (x and y) that we can solve to find the values of x and y.
To solve the system of equations, we can use substitution or elimination method. However, in this case, we will use the elimination method to eliminate one variable.
First, we can multiply Equation 1 by -0.65 so that the x-term will have opposite coefficients:
-0.65(x + y) = -0.65(150)
-0.65x - 0.65y = -97.50 (Equation 4)
Now, we can add Equation 4 to Equation 3 to eliminate the x-term:
0.65x + 0.90y + -0.65x - 0.65y = 112.50 + -97.50
0.25y = 15
Dividing both sides of the equation by 0.25, we get:
y = 60
Now we can substitute the value of y into Equation 1 to solve for x:
x + 60 = 150
x = 150 - 60
x = 90
Therefore, we need 90 pints of the first type (65% pure fruit juice) and 60 pints of the second type (90% pure fruit juice) to make 150 pints of a mixture that is 75% pure fruit juice.
To solve this problem, we can use the method of mixture. Let's assume that x pints of the first type of fruit drink (65% pure) will be used, and y pints of the second type of fruit drink (90% pure) will be used to create the mixture.
We need to find the values of x and y that satisfy the given conditions, which are:
1) The total volume of the mixture should be 150 pints, so we have the equation:
x + y = 150
2) The resulting mixture should be 75% pure fruit juice, so we have the equation:
(0.65x + 0.90y) / (x + y) = 0.75
To solve this system of equations, we can use the method of substitution or elimination.
Let's use the substitution method. Solving the first equation for x, we get:
x = 150 - y
Substituting this value of x into the second equation, we have:
(0.65(150 - y) + 0.90y) / (150 - y + y) = 0.75
Simplifying this equation, we get:
(0.65(150 - y) + 0.90y) / 150 = 0.75
Now, let's solve for y:
(0.65(150 - y) + 0.90y) / 150 = 0.75
(97.5 - 0.65y + 0.90y) / 150 = 0.75
(97.5 + 0.25y) / 150 = 0.75
97.5 + 0.25y = 0.75 * 150
97.5 + 0.25y = 112.5
0.25y = 112.5 - 97.5
0.25y = 15
y = 15 / 0.25
y = 60
Now, substitute the value of y back into the first equation to find x:
x = 150 - y
x = 150 - 60
x = 90
Therefore, to make 150 pints of a mixture that is 75% pure fruit juice, you need to use 90 pints of the first type of fruit drink (65% pure) and 60 pints of the second type of fruit drink (90% pure).
add up the juice amounts:
.65x + .90(150-x) = .75(150)
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