Ordinary corn syrup contains dextrose sugar. A process called enzyme conversion changes it into a syrup that is 42% fructose. This new syrup can be further treated with fractionation columns to make a syrup that is 95% fructose (high fructose corn syrup). Manufacturers of soft drinks, ice cream, and frozen desserts blend 95% fructose syrup and 42% fructose syrup to produce a 55% fructose syrup.† Use a system of equations to find the number of gallons of 42% fructose syrup and the number of gallons of 95% fructose syrup needed to produce 3000 gallons of 55% fructose syrup. Round to the nearest whole number.

Question

Ordinary corn syrup contains dextrose sugar. A process called enzyme conversion changes it into a syrup that is 42% fructose. This new syrup can be further treated with fractionation columns to make a syrup that is 95% fructose (high fructose corn syrup). Manufacturers of soft drinks, ice cream, and frozen desserts blend 95% fructose syrup and 42% fructose syrup to produce a 55% fructose syrup.† Use a system of equations to find the number of gallons of 42% fructose syrup and the number of gallons of 95% fructose syrup needed to produce 3000 gallons of 55% fructose syrup. Round to the nearest whole number.

Answer 1

If there are x gallons of 42% syrup, and y gallons of 95% syrup, then

x+y = 3000
.42x + .95y = .55(3000)

now just solve for x and y

Answer 2

To solve this problem using a system of equations, let's represent the unknown quantities as variables. Let's assume x represents the number of gallons of 42% fructose syrup, and y represents the number of gallons of 95% fructose syrup.

We have the following information from the problem statement:
1) The total number of gallons of syrup produced is 3000.
2) The resulting syrup is 55% fructose.

Using this information, we can set up two equations:

Equation 1: x + y = 3000 (representing the total number of gallons of syrup)

Equation 2: (0.42x + 0.95y) / 3000 = 0.55 (representing the fructose content in the resulting syrup)

Let's solve this system of equations:

Equation 1: x + y = 3000

To isolate one of the variables, let's solve for x in terms of y:
x = 3000 - y

Substituting this expression for x in Equation 2:
(0.42(3000 - y) + 0.95y) / 3000 = 0.55

Now, let's simplify and solve for y:

(1260 - 0.42y + 0.95y) / 3000 = 0.55
(0.53y + 1260) / 3000 = 0.55
0.53y + 1260 = 1650

Subtract 1260 from both sides:
0.53y = 390

Divide both sides by 0.53:
y = 390 / 0.53
y ≈ 735.85

Since y represents the number of gallons of 95% fructose syrup, and it needs to be a whole number, we round it to the nearest whole number:
y ≈ 736

Substituting this value back into Equation 1 to solve for x:
x + 736 = 3000
x = 3000 - 736
x ≈ 2264

Hence, approximately 2264 gallons of 42% fructose syrup and 736 gallons of 95% fructose syrup are needed to produce 3000 gallons of 55% fructose syrup.