Brand A granola is 25% nuts and dried fruit and brand B granola is 20% nuts and dried fruit. How much of sweet item A and sweet item B should be mixed to form a 10-lb batch of sweets that is 22% nuts and dried fruit?
.25A + .20(10-A) = .22(10)
To determine how much of sweet item A and sweet item B should be mixed, we can use a system of equations:
Let's assume x represents the amount (in pounds) of sweet item A and y represents the amount (in pounds) of sweet item B.
From the problem, we know:
1. Brand A granola is 25% nuts and dried fruit.
2. Brand B granola is 20% nuts and dried fruit.
3. The 10-lb batch of sweets should be 22% nuts and dried fruit.
Based on the above information, we can construct the following equations:
Equation 1: Amount of nuts and dried fruit in sweet item A = 0.25x
Equation 2: Amount of nuts and dried fruit in sweet item B = 0.20y
Equation 3: Amount of nuts and dried fruit in the 10-lb batch = 0.22(10 lb) = 2.2 lb
Since we want to achieve a 10-lb batch with a 22% concentration of nuts and dried fruit, we can set up the equation:
Equation 4: x + y = 10
Now, we have a system of equations:
Equation 1: 0.25x
Equation 2: 0.20y
Equation 3: 2.2
Equation 4: x + y = 10
To solve this system of equations, we can use substitution or elimination.
Method 1: Substitution
Solving Equation 4 for x, we have x = 10 - y
Now we can substitute this value into Equation 1:
0.25(10 - y) + 0.20y = 2.2
Simplifying the equation:
2.5 - 0.25y + 0.20y = 2.2
-0.05y = -0.3
y = -0.3 / -0.05
y = 6
Substituting the value of y back into Equation 4:
x + 6 = 10
x = 10 - 6
x = 4
So, you would need 4 pounds of sweet item A (Brand A granola) and 6 pounds of sweet item B (Brand B granola) to form a 10-lb batch of sweets that is 22% nuts and dried fruit.