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 20-lb batch of sweets that is 23% nuts and dried fruit?
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amount of brand A needed --- x lb
amount of brand B needed ---- 20-x lb
.25x + .2(20-x) = .23(20)
.25x + 4 - .2x = 4.6
.05x = .6
x = 12
State your conclusion
To determine the amounts of sweet item A and sweet item B to be mixed, we can set up a system of equations based on the given percentages and total weight.
Let's assume x represents the weight (in pounds) of sweet item A and y represents the weight (in pounds) of sweet item B.
Given:
Sweet item A is 25% nuts and dried fruit, which means 0.25x pounds of nuts and dried fruit are in sweet item A.
Sweet item B is 20% nuts and dried fruit, which means 0.20y pounds of nuts and dried fruit are in sweet item B.
We want to form a 20-pound batch of sweets that is 23% nuts and dried fruit, which means 0.23(20) = 4.6 pounds of nuts and dried fruit are in the mixture.
Now we can set up the equations:
Equation 1: x + y = 20 (total weight of the mixture should be 20 pounds)
Equation 2: 0.25x + 0.20y = 4.6 (total weight of nuts and dried fruit in the mixture should be 4.6 pounds)
To solve the system of equations, we can use various methods like substitution or elimination. In this case, let's use substitution.
From Equation 1, we can express x in terms of y:
x = 20 - y
Substitute this value of x into Equation 2:
0.25(20 - y) + 0.20y = 4.6
Simplifying the equation:
5 - 0.25y + 0.20y = 4.6
0.20y - 0.25y = 4.6 - 5
-0.05y = -0.4
y = (-0.4)/(-0.05)
y = 8
Now we have the value of y as 8 pounds. Substitute this value of y back into Equation 1 to find x:
x + 8 = 20
x = 20 - 8
x = 12
Therefore, you should mix 12 pounds of sweet item A and 8 pounds of sweet item B to form a 20-pound batch of sweets that is 23% nuts and dried fruit.
To solve this problem, we can use a mixture problem involving percentages.
Let's assume that you need to mix x pounds of sweet item A and y pounds of sweet item B to form a 20-pound batch of sweets.
Given that brand A granola is 25% nuts and dried fruit, and brand B granola is 20% nuts and dried fruit, we can set up the following equation based on the percentages:
0.25x + 0.20y = 0.23(20)
In this equation, 0.25x represents the amount of nuts and dried fruit in sweet item A, and 0.20y represents the amount of nuts and dried fruit in sweet item B. The right side of the equation represents the desired percentage of nuts and dried fruit in the final 20-pound batch, which is 23%.
Now, we can solve this equation to find the values of x and y.
0.25x + 0.20y = 0.23(20)
0.25x + 0.20y = 4.6
Since we have two variables, we need another equation to solve for x and y. This equation is based on the fact that the total weight of the sweet item A and sweet item B mixture is 20 pounds.
x + y = 20
Now we have a system of two equations:
0.25x + 0.20y = 4.6
x + y = 20
We can solve this system of equations using substitution or elimination to find the values of x and y. Once we have the values of x and y, we will know how much sweet item A (x pounds) and sweet item B (y pounds) should be mixed to form the desired 20-pound batch of sweets.