A restaurant has one type of lemonade that is 12% sugar and another that is 7% sugar. How many gallons of each type does the restaurant need to make 20 gallons of a lemonade mixture that is 10% sugar?
I think the answer is 8 gallons of 12% lemonade and 12 gallons of 7% lemonade
if x gallons are 12%, then the rest (20-x) are 7%.
So, adding up all the amounts of sugar, you need to solve
.12x + .07(20-x) = .10(20)
x = 12
You may have worked the problem correctly, but you got your answers switched.
Note also that equal amounts of the two would have produced a final strength of 9.5% sugar. Since 10% is greater than that, you will need more of the 12% than of the 7%.
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To solve this problem, we can use a method called the "mixture" or "alligation" method.
Let's assume we need x gallons of the 12% sugar lemonade and y gallons of the 7% sugar lemonade to make a total of 20 gallons of a 10% sugar lemonade mixture.
To begin, let's express the sugar content of each type of lemonade as a decimal, i.e., the 12% sugar lemonade is 0.12 and the 7% sugar lemonade is 0.07.
Now, let's set up an equation based on the sugar content:
(x * 0.12) + (y * 0.07) = 20 * 0.10
This equation represents the total sugar in the mixture. The left-hand side of the equation calculates the sugar contributed by each type of lemonade, while the right-hand side calculates the total sugar in the desired 10% mixture.
Simplifying the equation, we get:
0.12x + 0.07y = 2
We also know that x + y = 20, since we need a total of 20 gallons.
Now we have a system of equations:
0.12x + 0.07y = 2
x + y = 20
We can solve this system of equations using substitution or elimination.
One way to solve it is by substitution. We can solve the second equation for x and substitute it into the first equation:
x = 20 - y
Substituting this into the first equation, we get:
0.12(20 - y) + 0.07y = 2
Expanding and simplifying:
2.4 - 0.12y + 0.07y = 2
Combining like terms:
0.07y - 0.12y = 2 - 2.4
-0.05y = -0.4
Now, divide both sides by -0.05 to solve for y:
y = (-0.4) / (-0.05) = 8
So, we have found that the restaurant needs 8 gallons of the 7% sugar lemonade.
To find out how much of the 12% sugar lemonade is needed, we can substitute the value of y back into the equation x + y = 20:
x + 8 = 20
Subtracting 8 from both sides:
x = 12
Thus, the restaurant needs 12 gallons of the 12% sugar lemonade and 8 gallons of the 7% sugar lemonade to make 20 gallons of a 10% sugar lemonade mixture.
AMOUNT OF SUGAR=.12*8+.07*12=1.8
BUT 1.8/20=.09 NOT 10 PERCENT