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 30 dash l b batch of sweets that is 24% nuts and dried fruit?
Let's assume the amount of sweet item A used in the mixture is x lbs, and the amount of sweet item B used is y lbs.
According to the problem, brand A granola is 25% nuts and dried fruit and brand B granola is 20% nuts and dried fruit. We need to find the amount of sweet item A and sweet item B to mix to create a 30 lb batch of sweets that is 24% nuts and dried fruit.
The total amount of nuts and dried fruit in sweet item A is 25% of x lbs, which is (25/100) * x.
Similarly, the total amount of nuts and dried fruit in sweet item B is 20% of y lbs, which is (20/100) * y.
The total amount of nuts and dried fruit in the 30 lb mixture should be 24% of the total weight, which is (24/100) * 30.
Now we can set up the equation:
(25/100)x + (20/100)y = (24/100) * 30
Simplifying this equation:
0.25x + 0.2y = 0.24 * 30
0.25x + 0.2y = 7.2
To further simplify, we can multiply the equation by 100 to get rid of the decimals:
25x + 20y = 720
Since we have two unknown variables, we need another equation to solve for x and y. We can use the fact that the batch contains 30 lbs of sweets:
x + y = 30
Now we have a system of equations:
25x + 20y = 720
x + y = 30
We can solve this system of equations to find the values of x and y.
To solve this problem, we can use a system of equations to represent the given information. Let's denote the amount of sweet item A as x pounds, and the amount of sweet item B as y pounds.
From the problem, we know that Brand A granola is 25% nuts and dried fruit, and Brand B granola is 20% nuts and dried fruit. We aim to create a 30-pound batch of sweets that is 24% nuts and dried fruit.
The first equation we can form is based on the total weight of the mixture:
x + y = 30
The second equation can be formed using the percentage of nuts and dried fruit in the mixture:
(0.25x + 0.20y) / 30 = 0.24
Let's solve this system of equations to find the values of x and y.
First, we can simplify the second equation:
0.25x + 0.20y = 0.24 * 30
0.25x + 0.20y = 7.2
Next, we can multiply both sides of the first equation by 0.25 to make the coefficients of x the same:
0.25(x + y) = 0.25 * 30
0.25x + 0.25y = 7.5
Now we have a system of equations:
0.25x + 0.20y = 7.2
0.25x + 0.25y = 7.5
We can subtract the second equation from the first equation to eliminate x:
(0.25x + 0.20y) - (0.25x + 0.25y) = 7.2 - 7.5
0.20y - 0.25y = -0.3
-0.05y = -0.3
Simplifying further:
y = (-0.3) / (-0.05)
y = 6
Now we can substitute this value of y into either of the original equations to solve for x. Let's use the first equation:
x + 6 = 30
x = 30 - 6
x = 24
Therefore, we need 24 pounds of sweet item A and 6 pounds of sweet item B to form a 30-pound batch of sweets that is 24% nuts and dried fruit.
why not just write 30-lb batch?
If x is nuts, then the rest (30-x) is fruit. So add up the nuts and fruit:
.25x + .20(30-x) = .24*30