Soybean meal is 14% protein; cornmeal is 7% protein. How many pounds of each should be mixed together in order to get 280-lb mixture that is 10% protein?
.14s + .07(280-s) = .10*280
.14s + 19.6 - .07s = 28
.07s = 8.4
s = 120
so, use 120 lbs of soybean and 160 lbs of corn
x+y=2
y-x=2
To solve this problem, we can use the method of algebraic equations.
Let's denote the pounds of soybean meal as x and the pounds of cornmeal as y.
According to the given information, we have two equations:
Equation 1: x + y = 280 (since the total weight of the mixture is 280 pounds)
Equation 2: (0.14x + 0.07y) / 280 = 0.10 (since the protein percentage of the mixture is 10%)
Let's solve the equations step-by-step to find the values of x and y.
Step 1: Solve Equation 1 for one variable:
x = 280 - y
Step 2: Substitute the value of x in Equation 2:
(0.14(280 - y) + 0.07y) / 280 = 0.10
Step 3: Simplify the equation:
(39.2 - 0.14y + 0.07y) / 280 = 0.10
Step 4: Combine like terms:
(39.2 - 0.07y) / 280 = 0.10
Step 5: Cross multiply:
39.2 - 0.07y = 0.10 * 280
Step 6: Simplify the equation:
39.2 - 0.07y = 28
Step 7: Rearrange the equation:
-0.07y = 28 - 39.2
Step 8: Simplify further:
-0.07y = -11.2
Step 9: Divide both sides by -0.07 to solve for y:
y = -11.2 / -0.07
Step 10: Calculate y:
y ≈ 160
Now we substitute the value of y into Equation 1 to find x:
x = 280 - y
x = 280 - 160
x = 120
Therefore, to obtain a 280-lb mixture that is 10% protein, we should mix 120 pounds of soybean meal with 160 pounds of cornmeal.
To solve this problem, we can set up a system of equations. Let's denote the number of pounds of soybean meal as "x" and the number of pounds of cornmeal as "y".
Given that soybean meal is 14% protein, we can write the equation for the protein content in soybean meal as:
0.14x
Similarly, for cornmeal, which is 7% protein:
0.07y
Since we want to obtain a 280-pound mixture that is 10% protein, we can set up the equation for the protein content in the mixture as:
0.10(280)
Now, we can set up the equation to solve for the number of pounds of each ingredient:
0.14x + 0.07y = 0.10(280)
Simplifying the equation, we have:
0.14x + 0.07y = 28
To continue, we need an additional equation. One way to obtain this equation is by considering the total weight of the mixture. Since we want a 280-pound mixture, we can write the equation for the total weight as:
x + y = 280
Now, we have a system of equations:
0.14x + 0.07y = 28
x + y = 280
To solve this system, we can use a method such as substitution or elimination. Let's use the elimination method to solve it.
First, we'll multiply both sides of the equation x + y = 280 by 0.07 to eliminate y by multiplying through by the common multiplier:
0.07x + 0.07y = 19.6
Next, we'll subtract the equation 0.14x + 0.07y = 28 from this new equation:
0.07x + 0.07y - (0.14x + 0.07y) = 19.6 - 28
Simplifying, we get:
0.07x - 0.14x = -8.4
Combining like terms:
-0.07x = -8.4
Solving for x:
x = (-8.4) / (-0.07)
Simplifying, we find that x = 120.
Now, we can substitute this value back into the equation x + y = 280:
120 + y = 280
Solving for y:
y = 280 - 120
y = 160
Therefore, to get a 280-pound mixture that is 10% protein, you would need to mix 120 pounds of soybean meal with 160 pounds of cornmeal.