how much low fat milk (1%) fat and how much whole milk (4%) fat should be mixed to make 5 gal of reduced fat mild 2%
Method 1 using Algebra:
Let x=volume of 1% milk in gallons
then (5-x)=volume of 4% milk
5*2%=x*1%+(5-x)*4%
Solve for x:
10=x+20-4x
3x=10
x=3 1/3 gal. (1% milk)
5- 3 1/3 = 1 2/3 gal. (4% milk).
Method 2 (mental calculations)
Mentally put the 1% and 4% on a linear scale as follows:
1%..................................4%
The target percentage is 2%:
1%......2%..............4%
Take the difference:
.....1.............2.........
these are the ratio of the 4% and 1% milk respectively (note the inversion of position), i.e.
4% milk : 1% milk = 1:2
Therefore
volume of 4% = 5*1/(1+2)=1 2/3 gal
volume of 1% = 5*2/(1+2)=3 1/3 gal.
To find out how much low-fat milk (1% fat) and whole milk (4% fat) should be mixed to make 5 gallons of reduced-fat milk (2% fat), we can use the concept of weighted averages.
Let's denote the amount of low-fat milk as L and the amount of whole milk as W.
We know that the desired fat percentage for the mixture is 2%. So, the equation for the fat content in the mixture can be written as:
(1% fat in L gallons of low-fat milk) + (4% fat in W gallons of whole milk) = (2% fat in 5 gallons of reduced-fat milk)
From this, we can create the equation:
(1/100) * L + (4/100) * W = (2/100) * 5
To simplify the equation, we convert the percentages to decimals:
0.01L + 0.04W = 0.02 * 5
0.01L + 0.04W = 0.1
Now, we need another equation to solve for the unknowns L and W. Since we know that the total amount of milk is 5 gallons, we can write:
L + W = 5
Now we have a system of two equations:
0.01L + 0.04W = 0.1
L + W = 5
To solve this system, we can use substitution or elimination method. Let's use the substitution method in this case.
Solving the second equation for L, we get:
L = 5 - W
Now we substitute this expression for L in the first equation:
0.01(5 - W) + 0.04W = 0.1
0.05 - 0.01W + 0.04W = 0.1
0.03W = 0.1 - 0.05
0.03W = 0.05
W = 0.05 / 0.03
W ≈ 1.67
Using this value for W, we can substitute it back into the second equation to find L:
L + 1.67 = 5
L = 5 - 1.67
L ≈ 3.33
Therefore, to make 5 gallons of reduced-fat milk (2% fat), you should mix approximately 3.33 gallons of low-fat milk (1% fat) with approximately 1.67 gallons of whole milk (4% fat).