The Royal Fruit Company produces two types of fruit drinks. The first type is
60%
pure fruit juice, and the second type is
100%
pure fruit juice. The company is attempting to produce a fruit drink that contains
70%
pure fruit juice. How many pints of each of the two existing types of drink must be used to make
120
pints of a mixture that is
70%
pure fruit juice?
To solve this problem, we can use a technique called "weighted average".
Let's assume that x pints of the first type of fruit drink (60% pure) and y pints of the second type of fruit drink (100% pure) are used to make the 120 pints of a mixture that is 70% pure fruit juice.
We know that the total amount of fruit juice in the mixture is equal to the sum of the fruit juice in each type of drink.
For the first type of fruit drink, the amount of fruit juice is 60% of x pints, which can be written as 0.6x.
For the second type of fruit drink, the amount of fruit juice is 100% of y pints, which can be written as 1y or simply y.
The total amount of fruit juice in the mixture is the sum of the fruit juice from both types of drinks, so we have:
0.6x + y = 0.7(120)
Next, we know that the total pints of the mixture is 120, so we have:
x + y = 120
Now we have a system of two equations with two variables:
0.6x + y = 0.7(120)
x + y = 120
We can solve this system of equations using substitution or elimination.
Let's solve it using elimination. Multiply the second equation by -1:
-x - y = -120
Now add this equation to the first equation:
0.6x + y + (-x - y) = 0.7(120) + (-120)
0.6x - x + y - y = 84 - 120
-0.4x = -36
Divide both sides of the equation by -0.4:
x = -36 / -0.4
x = 90
Now substitute the value of x back into the second equation:
90 + y = 120
Subtract 90 from both sides:
y = 120 - 90
y = 30
Therefore, to make 120 pints of a mixture that is 70% pure fruit juice, you would need 90 pints of the first type of fruit drink and 30 pints of the second type of fruit drink.