SOLVING SYSTEMS:
A caterer made an ice cream punch by combining fruit juice that cost $2.25 per gallon with ice cream that costs $3.25 per gallon. How many gallons of each were used to make 100 gal of punch costing $2.50 per pound?
-Thank you!
2.25x + 3.25(100-x) = 2.50 * 100
now convert $/gal to $/lb.
To solve this problem, we need to set up a system of equations. Let's denote the number of gallons of fruit juice as x and the number of gallons of ice cream as y.
1. The total number of gallons of punch is 100, so we have the equation:
x + y = 100
2. The cost per gallon of the fruit juice is $2.25 and the cost per gallon of the ice cream is $3.25. The average cost of the punch is $2.50 per gallon, so we can set up the equation:
(2.25x + 3.25y)/100 = 2.50
Now we can solve this system of equations using the substitution method or elimination method.
Let's use the substitution method:
- Solve the first equation for x: x = 100 - y
- Substitute this expression for x in the second equation:
(2.25(100 - y) + 3.25y)/100 = 2.50
Simplifying this equation will give us the value of y. Once we have y, we can substitute it back into the first equation to find x.
To solve this problem, we need to set up a system of equations based on the given information.
Let's assume x represents the number of gallons of fruit juice used and y represents the number of gallons of ice cream used.
Based on the given information:
1. The cost of fruit juice is $2.25 per gallon.
2. The cost of ice cream is $3.25 per gallon.
3. The total cost of the punch is $2.50 per gallon.
From these three pieces of information, we can set up the following equations:
1x + 1y = 100 -- Equation 1 (since we have a total of 100 gallons of punch)
2.25x + 3.25y = 2.50 * 100 -- Equation 2 (since we want the total cost of the punch to equal $2.50 per gallon)
Now, we can solve this system of equations to find the values of x and y.
There are multiple methods to solve this system, such as substitution or elimination. Let's solve it using the substitution method:
From Equation 1:
y = 100 - x
Now substitute this value of y into Equation 2:
2.25x + 3.25(100 - x) = 2.50 * 100
2.25x + 325 - 3.25x = 250
-1x = 250 - 325
-1x = -75
x = 75
Substitute the value of x back into Equation 1:
1(75) + 1y = 100
75 + y = 100
y = 100 - 75
y = 25
Therefore, 75 gallons of fruit juice and 25 gallons of ice cream were used to make 100 gallons of punch costing $2.50 per pound.
To summarize the solution, 75 gallons of fruit juice and 25 gallons of ice cream were used to make 100 gallons of punch costing $2.50 per pound.